id sid tid token lemma pos cana-5305 1 1 communications communication NOUN cana-5305 1 2 on on ADP cana-5305 1 3 applied apply VERB cana-5305 1 4 nonlinear nonlinear ADJ cana-5305 1 5 analysis analysis NOUN cana-5305 1 6 issn issn NOUN cana-5305 1 7 : : PUNCT cana-5305 1 8 1074 1074 NUM cana-5305 1 9 - - PUNCT cana-5305 1 10 133x 133x NUM cana-5305 1 11 vol vol NOUN cana-5305 1 12 31 31 NUM cana-5305 1 13 no no NOUN cana-5305 1 14 . . NOUN cana-5305 1 15 1 1 NUM cana-5305 1 16 ( ( PUNCT cana-5305 1 17 2024 2024 NUM cana-5305 1 18 ) ) PUNCT cana-5305 1 19 354 354 NUM cana-5305 1 20 https://internationalpubls.com https://internationalpubls.com X cana-5305 1 21 algebraic algebraic ADJ cana-5305 1 22 structures structure NOUN cana-5305 1 23 in in ADP cana-5305 1 24 cryptography cryptography NOUN cana-5305 1 25 : : PUNCT cana-5305 1 26 applications application NOUN cana-5305 1 27 and and CCONJ cana-5305 1 28 challenges challenge NOUN cana-5305 1 29 1s 1s NUM cana-5305 1 30 . . PUNCT cana-5305 2 1 balamuralitharan balamuralitharan PROPN cana-5305 2 2 , , PUNCT cana-5305 2 3 2venmani 2venmani NUM cana-5305 2 4 r r NOUN cana-5305 2 5 , , PUNCT cana-5305 2 6 3r 3r NUM cana-5305 2 7 . . PUNCT cana-5305 3 1 arulprakasam arulprakasam PROPN cana-5305 3 2 , , PUNCT cana-5305 3 3 4ch 4ch PROPN cana-5305 3 4 s s PART cana-5305 3 5 s s X cana-5305 3 6 n n PRON cana-5305 3 7 murthy murthy ADJ cana-5305 3 8 , , PUNCT cana-5305 3 9 5m.bala 5m.bala NUM cana-5305 3 10 prabhakar prabhakar NOUN cana-5305 3 11 , , PUNCT cana-5305 3 12 6g.venkata 6g.venkata NUM cana-5305 3 13 ramana ramana PROPN cana-5305 3 14 , , PUNCT cana-5305 3 15 adjunct adjunct ADJ cana-5305 3 16 faculty faculty NOUN cana-5305 3 17 , , PUNCT cana-5305 3 18 department department NOUN cana-5305 3 19 of of ADP cana-5305 3 20 pure pure ADJ cana-5305 3 21 and and CCONJ cana-5305 3 22 applied applied ADJ cana-5305 3 23 mathematics mathematic NOUN cana-5305 3 24 , , PUNCT cana-5305 3 25 saveetha saveetha PROPN cana-5305 3 26 school school PROPN cana-5305 3 27 of of ADP cana-5305 3 28 engineering engineering NOUN cana-5305 3 29 , , PUNCT cana-5305 3 30 simats simat NOUN cana-5305 3 31 , , PUNCT cana-5305 3 32 chennai chennai PROPN cana-5305 3 33 , , PUNCT cana-5305 3 34 tamil tamil PROPN cana-5305 3 35 nadu nadu PROPN cana-5305 3 36 , , PUNCT cana-5305 3 37 india india PROPN cana-5305 3 38 email email NOUN cana-5305 4 1 i i PROPN cana-5305 4 2 d d PROPN cana-5305 4 3 : : PUNCT cana-5305 4 4 balamurali.maths@gmail.com balamurali.maths@gmail.com X cana-5305 4 5 assistant assistant NOUN cana-5305 4 6 professor professor NOUN cana-5305 4 7 , , PUNCT cana-5305 4 8 department department NOUN cana-5305 4 9 of of ADP cana-5305 4 10 mathematics mathematic NOUN cana-5305 4 11 , , PUNCT cana-5305 4 12 prince prince PROPN cana-5305 4 13 shri shri PROPN cana-5305 4 14 venkateshwara venkateshwara PROPN cana-5305 4 15 arts arts PROPN cana-5305 4 16 and and CCONJ cana-5305 4 17 science science PROPN cana-5305 4 18 college college PROPN cana-5305 4 19 , , PUNCT cana-5305 4 20 gowrivakkam gowrivakkam NOUN cana-5305 4 21 , , PUNCT cana-5305 4 22 chennai chennai PROPN cana-5305 4 23 . . PUNCT cana-5305 5 1 department department NOUN cana-5305 5 2 of of ADP cana-5305 5 3 mathematics mathematics PROPN cana-5305 5 4 , , PUNCT cana-5305 5 5 college college NOUN cana-5305 5 6 of of ADP cana-5305 5 7 engineering engineering NOUN cana-5305 5 8 and and CCONJ cana-5305 5 9 technology technology NOUN cana-5305 5 10 , , PUNCT cana-5305 5 11 srm srm PROPN cana-5305 5 12 institute institute PROPN cana-5305 5 13 of of ADP cana-5305 5 14 science science NOUN cana-5305 5 15 and and CCONJ cana-5305 5 16 technology technology NOUN cana-5305 5 17 , , PUNCT cana-5305 5 18 srm srm NOUN cana-5305 5 19 nagar nagar NOUN cana-5305 5 20 , , PUNCT cana-5305 5 21 kattankulathur kattankulathur PROPN cana-5305 5 22 603203 603203 NUM cana-5305 5 23 , , PUNCT cana-5305 5 24 chengalpattu chengalpattu ADJ cana-5305 5 25 district district NOUN cana-5305 5 26 , , PUNCT cana-5305 5 27 tamilnadu tamilnadu PROPN cana-5305 5 28 , , PUNCT cana-5305 5 29 india india PROPN cana-5305 5 30 r.aruljeeva@gmail.com r.aruljeeva@gmail.com PROPN cana-5305 5 31 associate associate NOUN cana-5305 5 32 professor professor NOUN cana-5305 5 33 , , PUNCT cana-5305 5 34 dept dept NOUN cana-5305 5 35 of of ADP cana-5305 5 36 mathematics mathematic NOUN cana-5305 5 37 , , PUNCT cana-5305 5 38 aditya aditya PROPN cana-5305 5 39 university university PROPN cana-5305 5 40 , , PUNCT cana-5305 5 41 surampalem surampalem NOUN cana-5305 5 42 , , PUNCT cana-5305 5 43 india india PROPN cana-5305 5 44 , , PUNCT cana-5305 5 45 chssn.murthy@aec.edu.in chssn.murthy@aec.edu.in NOUN cana-5305 5 46 associate associate ADJ cana-5305 5 47 professor professor NOUN cana-5305 5 48 , , PUNCT cana-5305 5 49 department department NOUN cana-5305 5 50 of of ADP cana-5305 5 51 mathematics mathematics PROPN cana-5305 5 52 , , PUNCT cana-5305 5 53 aditya aditya PROPN cana-5305 5 54 university university PROPN cana-5305 5 55 , , PUNCT cana-5305 5 56 surampalem surampalem NOUN cana-5305 5 57 , , PUNCT cana-5305 5 58 india india PROPN cana-5305 5 59 , , PUNCT cana-5305 5 60 balaprabhakar.mattaparthi@aec.edu.in balaprabhakar.mattaparthi@aec.edu.in NOUN cana-5305 5 61 associate associate NOUN cana-5305 5 62 professor professor NOUN cana-5305 5 63 , , PUNCT cana-5305 5 64 dept dept NOUN cana-5305 5 65 of of ADP cana-5305 5 66 mathematics mathematic NOUN cana-5305 5 67 , , PUNCT cana-5305 5 68 aditya aditya PROPN cana-5305 5 69 university university PROPN cana-5305 5 70 , , PUNCT cana-5305 5 71 surampalem surampalem NOUN cana-5305 5 72 , , PUNCT cana-5305 5 73 india india PROPN cana-5305 5 74 , , PUNCT cana-5305 5 75 venkataramana.ginijala@aec.edu.in venkataramana.ginijala@aec.edu.in NOUN cana-5305 5 76 article article NOUN cana-5305 5 77 history history NOUN cana-5305 5 78 : : PUNCT cana-5305 5 79 received receive VERB cana-5305 5 80 : : PUNCT cana-5305 5 81 12 12 NUM cana-5305 5 82 - - SYM cana-5305 5 83 11 11 NUM cana-5305 5 84 - - SYM cana-5305 5 85 2023 2023 NUM cana-5305 5 86 revised revise VERB cana-5305 5 87 : : PUNCT cana-5305 5 88 15 15 NUM cana-5305 5 89 - - SYM cana-5305 5 90 12 12 NUM cana-5305 5 91 - - PUNCT cana-5305 5 92 2023 2023 NUM cana-5305 5 93 accepted accept VERB cana-5305 5 94 : : PUNCT cana-5305 5 95 19 19 NUM cana-5305 5 96 - - PUNCT cana-5305 5 97 01 01 NUM cana-5305 5 98 - - PUNCT cana-5305 5 99 2024 2024 NUM cana-5305 5 100 abstract abstract NOUN cana-5305 5 101 : : PUNCT cana-5305 5 102 many many ADJ cana-5305 5 103 cryptographic cryptographic ADJ cana-5305 5 104 algorithms algorithm NOUN cana-5305 5 105 use use VERB cana-5305 5 106 algebraic algebraic ADJ cana-5305 5 107 structures structure NOUN cana-5305 5 108 beginning begin VERB cana-5305 5 109 with with ADP cana-5305 5 110 groups group NOUN cana-5305 5 111 and and CCONJ cana-5305 5 112 continuing continue VERB cana-5305 5 113 through through ADP cana-5305 5 114 rings ring NOUN cana-5305 5 115 and and CCONJ cana-5305 5 116 fields field NOUN cana-5305 5 117 and and CCONJ cana-5305 5 118 ending end VERB cana-5305 5 119 with with ADP cana-5305 5 120 vector vector NOUN cana-5305 5 121 spaces space NOUN cana-5305 5 122 as as ADP cana-5305 5 123 their their PRON cana-5305 5 124 theoretical theoretical ADJ cana-5305 5 125 bases basis NOUN cana-5305 5 126 . . PUNCT cana-5305 6 1 secure secure ADJ cana-5305 6 2 cryptosystems cryptosystem NOUN cana-5305 6 3 require require VERB cana-5305 6 4 these these DET cana-5305 6 5 structures structure NOUN cana-5305 6 6 for for ADP cana-5305 6 7 designing design VERB cana-5305 6 8 defensive defensive ADJ cana-5305 6 9 systems system NOUN cana-5305 6 10 which which PRON cana-5305 6 11 protect protect VERB cana-5305 6 12 both both DET cana-5305 6 13 data datum NOUN cana-5305 6 14 confidentiality confidentiality NOUN cana-5305 6 15 and and CCONJ cana-5305 6 16 integrity integrity NOUN cana-5305 6 17 . . PUNCT cana-5305 7 1 this this DET cana-5305 7 2 research research NOUN cana-5305 7 3 outlines outline VERB cana-5305 7 4 the the DET cana-5305 7 5 limitations limitation NOUN cana-5305 7 6 which which PRON cana-5305 7 7 stem stem VERB cana-5305 7 8 from from ADP cana-5305 7 9 complicated complicated ADJ cana-5305 7 10 algebraic algebraic ADJ cana-5305 7 11 structures structure NOUN cana-5305 7 12 including include VERB cana-5305 7 13 problems problem NOUN cana-5305 7 14 with with ADP cana-5305 7 15 key key ADJ cana-5305 7 16 dimension dimension NOUN cana-5305 7 17 and and CCONJ cana-5305 7 18 computational computational ADJ cana-5305 7 19 speed speed NOUN cana-5305 7 20 . . PUNCT cana-5305 8 1 the the DET cana-5305 8 2 paper paper NOUN cana-5305 8 3 evaluates evaluate VERB cana-5305 8 4 contemporary contemporary ADJ cana-5305 8 5 cryptographic cryptographic ADJ cana-5305 8 6 developments development NOUN cana-5305 8 7 of of ADP cana-5305 8 8 lattice lattice NOUN cana-5305 8 9 - - PUNCT cana-5305 8 10 based base VERB cana-5305 8 11 methods method NOUN cana-5305 8 12 which which PRON cana-5305 8 13 implement implement VERB cana-5305 8 14 powerful powerful ADJ cana-5305 8 15 algebraic algebraic ADJ cana-5305 8 16 strategies strategy NOUN cana-5305 8 17 for for ADP cana-5305 8 18 resisting resist VERB cana-5305 8 19 quantum quantum ADJ cana-5305 8 20 computing computing NOUN cana-5305 8 21 threats threat NOUN cana-5305 8 22 . . PUNCT cana-5305 9 1 keywords keyword NOUN cana-5305 9 2 β€” β€” PUNCT cana-5305 9 3 algebraic algebraic ADJ cana-5305 9 4 structures structure NOUN cana-5305 9 5 , , PUNCT cana-5305 9 6 cryptography cryptography NOUN cana-5305 9 7 , , PUNCT cana-5305 9 8 public public ADJ cana-5305 9 9 - - PUNCT cana-5305 9 10 key key NOUN cana-5305 9 11 cryptography cryptography NOUN cana-5305 9 12 , , PUNCT cana-5305 9 13 digital digital ADJ cana-5305 9 14 signatures signature NOUN cana-5305 9 15 , , PUNCT cana-5305 9 16 hash hash NOUN cana-5305 9 17 functions function NOUN cana-5305 9 18 , , PUNCT cana-5305 9 19 lattice lattice NOUN cana-5305 9 20 - - PUNCT cana-5305 9 21 based base VERB cana-5305 9 22 cryptography cryptography NOUN cana-5305 9 23 , , PUNCT cana-5305 9 24 quantum quantum NOUN cana-5305 9 25 computing computing NOUN cana-5305 9 26 , , PUNCT cana-5305 9 27 security security NOUN cana-5305 9 28 i. i. NOUN cana-5305 9 29 introduction introduction NOUN cana-5305 9 30 cryptography cryptography NOUN cana-5305 9 31 functions function NOUN cana-5305 9 32 through through ADP cana-5305 9 33 mathematical mathematical ADJ cana-5305 9 34 principles principle NOUN cana-5305 9 35 reinforced reinforce VERB cana-5305 9 36 by by ADP cana-5305 9 37 essential essential ADJ cana-5305 9 38 algebraic algebraic ADJ cana-5305 9 39 structures structure NOUN cana-5305 9 40 which which PRON cana-5305 9 41 define define VERB cana-5305 9 42 the the DET cana-5305 9 43 cryptographic cryptographic ADJ cana-5305 9 44 systems system NOUN cana-5305 9 45 . . PUNCT cana-5305 10 1 the the DET cana-5305 10 2 mathematical mathematical ADJ cana-5305 10 3 structures structure NOUN cana-5305 10 4 of of ADP cana-5305 10 5 groups group NOUN cana-5305 10 6 and and CCONJ cana-5305 10 7 rings ring NOUN cana-5305 10 8 and and CCONJ cana-5305 10 9 vector vector NOUN cana-5305 10 10 spaces space NOUN cana-5305 10 11 and and CCONJ cana-5305 10 12 fields field NOUN cana-5305 10 13 exist exist VERB cana-5305 10 14 both both CCONJ cana-5305 10 15 theoretically theoretically ADV cana-5305 10 16 and and CCONJ cana-5305 10 17 functionally functionally ADV cana-5305 10 18 for for ADP cana-5305 10 19 building build VERB cana-5305 10 20 current current ADJ cana-5305 10 21 cryptographic cryptographic ADJ cana-5305 10 22 systems system NOUN cana-5305 10 23 . . PUNCT cana-5305 11 1 the the DET cana-5305 11 2 implementation implementation NOUN cana-5305 11 3 of of ADP cana-5305 11 4 algebraic algebraic ADJ cana-5305 11 5 structures structure NOUN cana-5305 11 6 remains remain VERB cana-5305 11 7 essential essential ADJ cana-5305 11 8 for for ADP cana-5305 11 9 securing secure VERB cana-5305 11 10 online online ADJ cana-5305 11 11 financial financial ADJ cana-5305 11 12 operations operation NOUN cana-5305 11 13 and and CCONJ cana-5305 11 14 vital vital ADJ cana-5305 11 15 national national ADJ cana-5305 11 16 security security NOUN cana-5305 11 17 framework framework NOUN cana-5305 11 18 [ [ X cana-5305 11 19 2 2 NUM cana-5305 11 20 - - SYM cana-5305 11 21 5 5 NUM cana-5305 11 22 ] ] PUNCT cana-5305 11 23 . . PUNCT cana-5305 12 1 communications communication NOUN cana-5305 12 2 on on ADP cana-5305 12 3 applied apply VERB cana-5305 12 4 nonlinear nonlinear ADJ cana-5305 12 5 analysis analysis NOUN cana-5305 12 6 issn issn NOUN cana-5305 12 7 : : PUNCT cana-5305 12 8 1074 1074 NUM cana-5305 12 9 - - PUNCT cana-5305 12 10 133x 133x NUM cana-5305 12 11 vol vol NOUN cana-5305 12 12 31 31 NUM cana-5305 12 13 no no NOUN cana-5305 12 14 . . NOUN cana-5305 12 15 1 1 NUM cana-5305 12 16 ( ( PUNCT cana-5305 12 17 2024 2024 NUM cana-5305 12 18 ) ) PUNCT cana-5305 12 19 355 355 NUM cana-5305 12 20 https://internationalpubls.com https://internationalpubls.com X cana-5305 12 21 algebra algebra NOUN cana-5305 12 22 and and CCONJ cana-5305 12 23 cryptography cryptography NOUN cana-5305 12 24 together together ADV cana-5305 12 25 created create VERB cana-5305 12 26 the the DET cana-5305 12 27 foundation foundation NOUN cana-5305 12 28 for for ADP cana-5305 12 29 numerous numerous ADJ cana-5305 12 30 cryptographic cryptographic ADJ cana-5305 12 31 systems system NOUN cana-5305 12 32 which which PRON cana-5305 12 33 are be AUX cana-5305 12 34 now now ADV cana-5305 12 35 extensively extensively ADV cana-5305 12 36 used use VERB cana-5305 12 37 worldwide worldwide ADV cana-5305 12 38 . . PUNCT cana-5305 13 1 public public ADJ cana-5305 13 2 - - PUNCT cana-5305 13 3 key key NOUN cana-5305 13 4 cryptography cryptography NOUN cana-5305 13 5 develops develop VERB cana-5305 13 6 its its PRON cana-5305 13 7 systems system NOUN cana-5305 13 8 such such ADJ cana-5305 13 9 as as ADP cana-5305 13 10 rsa rsa PROPN cana-5305 13 11 and and CCONJ cana-5305 13 12 ecc ecc PROPN cana-5305 13 13 by by ADP cana-5305 13 14 using use VERB cana-5305 13 15 finite finite ADJ cana-5305 13 16 field field NOUN cana-5305 13 17 structures structure NOUN cana-5305 13 18 of of ADP cana-5305 13 19 algebra algebra NOUN cana-5305 13 20 . . PUNCT cana-5305 14 1 security security NOUN cana-5305 14 2 systems system NOUN cana-5305 14 3 based base VERB cana-5305 14 4 on on ADP cana-5305 14 5 integer integer NOUN cana-5305 14 6 factorization factorization NOUN cana-5305 14 7 and and CCONJ cana-5305 14 8 discrete discrete ADJ cana-5305 14 9 logarithm logarithm NOUN cana-5305 14 10 problems problem NOUN cana-5305 14 11 demonstrate demonstrate VERB cana-5305 14 12 high high ADJ cana-5305 14 13 levels level NOUN cana-5305 14 14 of of ADP cana-5305 14 15 security security NOUN cana-5305 14 16 due due ADP cana-5305 14 17 to to ADP cana-5305 14 18 their their PRON cana-5305 14 19 mathematical mathematical ADJ cana-5305 14 20 complexity complexity NOUN cana-5305 14 21 . . PUNCT cana-5305 15 1 digital digital ADJ cana-5305 15 2 signatures signature NOUN cana-5305 15 3 together together ADV cana-5305 15 4 with with ADP cana-5305 15 5 secure secure ADJ cana-5305 15 6 hashing hashing NOUN cana-5305 15 7 functions function NOUN cana-5305 15 8 depend depend VERB cana-5305 15 9 on on ADP cana-5305 15 10 algebraic algebraic ADJ cana-5305 15 11 structures structure NOUN cana-5305 15 12 for for ADP cana-5305 15 13 maintaining maintain VERB cana-5305 15 14 integrity integrity NOUN cana-5305 15 15 and and CCONJ cana-5305 15 16 authenticity authenticity NOUN cana-5305 15 17 and and CCONJ cana-5305 15 18 enabling enable VERB cana-5305 15 19 non non NOUN cana-5305 15 20 - - NOUN cana-5305 15 21 repudiation repudiation NOUN cana-5305 15 22 . . PUNCT cana-5305 16 1 the the DET cana-5305 16 2 combination combination NOUN cana-5305 16 3 of of ADP cana-5305 16 4 stronger strong ADJ cana-5305 16 5 computational computational ADJ cana-5305 16 6 technology technology NOUN cana-5305 16 7 with with ADP cana-5305 16 8 internet internet NOUN cana-5305 16 9 proliferation proliferation NOUN cana-5305 16 10 has have AUX cana-5305 16 11 created create VERB cana-5305 16 12 substantial substantial ADJ cana-5305 16 13 need need NOUN cana-5305 16 14 for for ADP cana-5305 16 15 encryption encryption NOUN cana-5305 16 16 - - PUNCT cana-5305 16 17 based base VERB cana-5305 16 18 data datum NOUN cana-5305 16 19 protection protection NOUN cana-5305 16 20 systems system NOUN cana-5305 16 21 . . PUNCT cana-5305 17 1 advanced advanced ADJ cana-5305 17 2 cryptographic cryptographic ADJ cana-5305 17 3 research research NOUN cana-5305 17 4 has have AUX cana-5305 17 5 unfolded unfold VERB cana-5305 17 6 because because ADV cana-5305 17 7 of of ADP cana-5305 17 8 rising rise VERB cana-5305 17 9 security security NOUN cana-5305 17 10 needs need NOUN cana-5305 17 11 which which PRON cana-5305 17 12 resulted result VERB cana-5305 17 13 in in ADP cana-5305 17 14 post post ADJ cana-5305 17 15 - - ADJ cana-5305 17 16 quantum quantum ADJ cana-5305 17 17 cryptography cryptography NOUN cana-5305 17 18 discovery discovery NOUN cana-5305 17 19 as as SCONJ cana-5305 17 20 investigators investigator NOUN cana-5305 17 21 pushed push VERB cana-5305 17 22 research research NOUN cana-5305 17 23 boundaries boundary NOUN cana-5305 17 24 . . PUNCT cana-5305 18 1 many many ADJ cana-5305 18 2 traditional traditional ADJ cana-5305 18 3 cryptographic cryptographic ADJ cana-5305 18 4 systems system NOUN cana-5305 18 5 risk risk NOUN cana-5305 18 6 security security NOUN cana-5305 18 7 failure failure NOUN cana-5305 18 8 to to ADP cana-5305 18 9 quantum quantum ADJ cana-5305 18 10 computing computing NOUN cana-5305 18 11 attacks attack NOUN cana-5305 18 12 so so SCONJ cana-5305 18 13 cryptographers cryptographer NOUN cana-5305 18 14 need need VERB cana-5305 18 15 to to PART cana-5305 18 16 develop develop VERB cana-5305 18 17 new new ADJ cana-5305 18 18 algebraic algebraic ADJ cana-5305 18 19 techniques technique NOUN cana-5305 18 20 that that PRON cana-5305 18 21 offer offer VERB cana-5305 18 22 resistance resistance NOUN cana-5305 18 23 to to ADP cana-5305 18 24 quantum quantum ADJ cana-5305 18 25 attacks attack NOUN cana-5305 18 26 . . PUNCT cana-5305 19 1 researchers researcher NOUN cana-5305 19 2 have have AUX cana-5305 19 3 found find VERB cana-5305 19 4 success success NOUN cana-5305 19 5 using use VERB cana-5305 19 6 lattice lattice NOUN cana-5305 19 7 - - PUNCT cana-5305 19 8 based base VERB cana-5305 19 9 cryptography cryptography NOUN cana-5305 19 10 because because SCONJ cana-5305 19 11 it it PRON cana-5305 19 12 builds build VERB cana-5305 19 13 quantum quantum NOUN cana-5305 19 14 - - PUNCT cana-5305 19 15 resistant resistant ADJ cana-5305 19 16 cryptographic cryptographic ADJ cana-5305 19 17 systems system NOUN cana-5305 19 18 by by ADP cana-5305 19 19 applying apply VERB cana-5305 19 20 algebraic algebraic ADJ cana-5305 19 21 structures structure NOUN cana-5305 19 22 of of ADP cana-5305 19 23 lattices lattice NOUN cana-5305 19 24 throughout throughout ADP cana-5305 19 25 multidimensional multidimensional ADJ cana-5305 19 26 space space NOUN cana-5305 19 27 [ [ X cana-5305 19 28 10 10 NUM cana-5305 19 29 ] ] PUNCT cana-5305 19 30 . . PUNCT cana-5305 20 1 though though SCONJ cana-5305 20 2 algebraic algebraic ADJ cana-5305 20 3 structures structure NOUN cana-5305 20 4 contribute contribute VERB cana-5305 20 5 significantly significantly ADV cana-5305 20 6 to to ADP cana-5305 20 7 cryptography cryptography NOUN cana-5305 20 8 by by ADP cana-5305 20 9 being be AUX cana-5305 20 10 heavily heavily ADV cana-5305 20 11 implemented implement VERB cana-5305 20 12 in in ADP cana-5305 20 13 practice practice NOUN cana-5305 20 14 they they PRON cana-5305 20 15 still still ADV cana-5305 20 16 face face VERB cana-5305 20 17 several several ADJ cana-5305 20 18 technical technical ADJ cana-5305 20 19 obstacles obstacle NOUN cana-5305 20 20 . . PUNCT cana-5305 21 1 the the DET cana-5305 21 2 enhanced enhance VERB cana-5305 21 3 security security NOUN cana-5305 21 4 level level NOUN cana-5305 21 5 that that PRON cana-5305 21 6 these these DET cana-5305 21 7 structures structure NOUN cana-5305 21 8 deliver deliver VERB cana-5305 21 9 requires require VERB cana-5305 21 10 both both DET cana-5305 21 11 more more ADV cana-5305 21 12 difficult difficult ADJ cana-5305 21 13 computations computation NOUN cana-5305 21 14 as as ADV cana-5305 21 15 well well ADV cana-5305 21 16 as as ADP cana-5305 21 17 extended extend VERB cana-5305 21 18 key key ADJ cana-5305 21 19 sizes size NOUN cana-5305 21 20 . . PUNCT cana-5305 22 1 ecc ecc PROPN cana-5305 22 2 represents represent VERB cana-5305 22 3 a a DET cana-5305 22 4 preferred preferred ADJ cana-5305 22 5 choice choice NOUN cana-5305 22 6 because because SCONJ cana-5305 22 7 it it PRON cana-5305 22 8 offers offer VERB cana-5305 22 9 efficiency efficiency NOUN cana-5305 22 10 while while SCONJ cana-5305 22 11 using use VERB cana-5305 22 12 smaller small ADJ cana-5305 22 13 keys key NOUN cana-5305 22 14 yet yet CCONJ cana-5305 22 15 its its PRON cana-5305 22 16 large large ADJ cana-5305 22 17 - - PUNCT cana-5305 22 18 scale scale NOUN cana-5305 22 19 usage usage NOUN cana-5305 22 20 proves prove VERB cana-5305 22 21 computationally computationally ADV cana-5305 22 22 challenging challenging ADJ cana-5305 22 23 . . PUNCT cana-5305 23 1 the the DET cana-5305 23 2 advent advent NOUN cana-5305 23 3 of of ADP cana-5305 23 4 quantum quantum NOUN cana-5305 23 5 computing computing NOUN cana-5305 23 6 introduces introduce NOUN cana-5305 23 7 novel novel ADJ cana-5305 23 8 security security NOUN cana-5305 23 9 risks risk NOUN cana-5305 23 10 because because SCONJ cana-5305 23 11 traditional traditional ADJ cana-5305 23 12 rsa rsa NOUN cana-5305 23 13 - - PUNCT cana-5305 23 14 based base VERB cana-5305 23 15 and and CCONJ cana-5305 23 16 other other ADJ cana-5305 23 17 algebraic algebraic ADJ cana-5305 23 18 structure structure NOUN cana-5305 23 19 cryptographic cryptographic ADJ cana-5305 23 20 algorithms algorithm NOUN cana-5305 23 21 are be AUX cana-5305 23 22 susceptible susceptible ADJ cana-5305 23 23 to to ADP cana-5305 23 24 attacks attack NOUN cana-5305 23 25 through through ADP cana-5305 23 26 shor shor PROPN cana-5305 23 27 ’s ’s PART cana-5305 23 28 algorithm algorithm NOUN cana-5305 23 29 [ [ X cana-5305 23 30 15 15 NUM cana-5305 23 31 ] ] PUNCT cana-5305 23 32 . . PUNCT cana-5305 24 1 this this DET cana-5305 24 2 paper paper NOUN cana-5305 24 3 investigates investigate VERB cana-5305 24 4 the the DET cana-5305 24 5 use use NOUN cana-5305 24 6 of of ADP cana-5305 24 7 algebraic algebraic ADJ cana-5305 24 8 structures structure NOUN cana-5305 24 9 within within ADP cana-5305 24 10 cryptography cryptography NOUN cana-5305 24 11 so so SCONJ cana-5305 24 12 it it PRON cana-5305 24 13 examines examine VERB cana-5305 24 14 their their PRON cana-5305 24 15 implementations implementation NOUN cana-5305 24 16 alongside alongside ADP cana-5305 24 17 their their PRON cana-5305 24 18 advantages advantage NOUN cana-5305 24 19 and and CCONJ cana-5305 24 20 implementation implementation NOUN cana-5305 24 21 difficulties difficulty NOUN cana-5305 24 22 . . PUNCT cana-5305 25 1 this this DET cana-5305 25 2 paper paper NOUN cana-5305 25 3 establishes establish VERB cana-5305 25 4 in in ADP cana-5305 25 5 - - PUNCT cana-5305 25 6 depth depth NOUN cana-5305 25 7 knowledge knowledge NOUN cana-5305 25 8 regarding regard VERB cana-5305 25 9 the the DET cana-5305 25 10 algebraic algebraic ADJ cana-5305 25 11 theory theory NOUN cana-5305 25 12 - - PUNCT cana-5305 25 13 cryptographic cryptographic ADJ cana-5305 25 14 practice practice NOUN cana-5305 25 15 relationship relationship NOUN cana-5305 25 16 while while SCONJ cana-5305 25 17 showing show VERB cana-5305 25 18 forthcoming forthcoming ADJ cana-5305 25 19 trends trend NOUN cana-5305 25 20 and and CCONJ cana-5305 25 21 prospective prospective ADJ cana-5305 25 22 directions direction NOUN cana-5305 25 23 of of ADP cana-5305 25 24 cryptographical cryptographical ADJ cana-5305 25 25 research research NOUN cana-5305 25 26 . . PUNCT cana-5305 26 1 the the DET cana-5305 26 2 research research NOUN cana-5305 26 3 explores explore VERB cana-5305 26 4 these these DET cana-5305 26 5 methods method NOUN cana-5305 26 6 to to PART cana-5305 26 7 add add VERB cana-5305 26 8 to to ADP cana-5305 26 9 academic academic ADJ cana-5305 26 10 dialog dialog NOUN cana-5305 26 11 about about ADP cana-5305 26 12 cryptographic cryptographic ADJ cana-5305 26 13 system system NOUN cana-5305 26 14 strengthening strengthen VERB cana-5305 26 15 through through ADP cana-5305 26 16 algebraic algebraic ADJ cana-5305 26 17 structures structure NOUN cana-5305 26 18 within within ADP cana-5305 26 19 the the DET cana-5305 26 20 digital digital ADJ cana-5305 26 21 epoch epoch NOUN cana-5305 26 22 specifically specifically ADV cana-5305 26 23 for for ADP cana-5305 26 24 postquantum postquantum NOUN cana-5305 26 25 cryptography cryptography NOUN cana-5305 26 26 . . PUNCT cana-5305 27 1 novelty novelty NOUN cana-5305 27 2 and and CCONJ cana-5305 27 3 contribution contribution NOUN cana-5305 27 4 varied vary VERB cana-5305 27 5 important important ADJ cana-5305 27 6 developments development NOUN cana-5305 27 7 regarding regard VERB cana-5305 27 8 algebraic algebraic ADJ cana-5305 27 9 structures structure NOUN cana-5305 27 10 in in ADP cana-5305 27 11 cryptography cryptography NOUN cana-5305 27 12 appear appear VERB cana-5305 27 13 in in ADP cana-5305 27 14 this this DET cana-5305 27 15 research research NOUN cana-5305 27 16 paper paper NOUN cana-5305 27 17 . . PUNCT cana-5305 28 1 the the DET cana-5305 28 2 paper paper NOUN cana-5305 28 3 demonstrates demonstrate VERB cana-5305 28 4 in in ADP cana-5305 28 5 detail detail NOUN cana-5305 28 6 how how SCONJ cana-5305 28 7 key key ADJ cana-5305 28 8 cryptographic cryptographic ADJ cana-5305 28 9 systems system NOUN cana-5305 28 10 rsa rsa PROPN cana-5305 28 11 , , PUNCT cana-5305 28 12 ecc ecc PROPN cana-5305 28 13 , , PUNCT cana-5305 28 14 and and CCONJ cana-5305 28 15 lattice lattice NOUN cana-5305 28 16 - - PUNCT cana-5305 28 17 based base VERB cana-5305 28 18 cryptography cryptography NOUN cana-5305 28 19 use use NOUN cana-5305 28 20 groups group NOUN cana-5305 28 21 , , PUNCT cana-5305 28 22 fields field NOUN cana-5305 28 23 , , PUNCT cana-5305 28 24 rings ring NOUN cana-5305 28 25 , , PUNCT cana-5305 28 26 and and CCONJ cana-5305 28 27 lattices lattice NOUN cana-5305 28 28 as as ADP cana-5305 28 29 their their PRON cana-5305 28 30 algebraic algebraic ADJ cana-5305 28 31 foundation foundation NOUN cana-5305 28 32 . . PUNCT cana-5305 29 1 the the DET cana-5305 29 2 author author NOUN cana-5305 29 3 explains explain VERB cana-5305 29 4 mathematical mathematical ADJ cana-5305 29 5 foundational foundational ADJ cana-5305 29 6 elements element NOUN cana-5305 29 7 within within ADP cana-5305 29 8 their their PRON cana-5305 29 9 practical practical ADJ cana-5305 29 10 cryptographic cryptographic ADJ cana-5305 29 11 application application NOUN cana-5305 29 12 framework framework NOUN cana-5305 29 13 to to PART cana-5305 29 14 connect connect VERB cana-5305 29 15 abstract abstract ADJ cana-5305 29 16 algebra algebra NOUN cana-5305 29 17 to to ADP cana-5305 29 18 real real ADJ cana-5305 29 19 - - PUNCT cana-5305 29 20 world world NOUN cana-5305 29 21 cryptographic cryptographic ADJ cana-5305 29 22 implementations implementation NOUN cana-5305 29 23 in in ADP cana-5305 29 24 this this DET cana-5305 29 25 paper paper NOUN cana-5305 30 1 [ [ X cana-5305 30 2 6 6 NUM cana-5305 30 3 ] ] PUNCT cana-5305 30 4 . . PUNCT cana-5305 31 1 communications communication NOUN cana-5305 31 2 on on ADP cana-5305 31 3 applied apply VERB cana-5305 31 4 nonlinear nonlinear ADJ cana-5305 31 5 analysis analysis NOUN cana-5305 31 6 issn issn NOUN cana-5305 31 7 : : PUNCT cana-5305 31 8 1074 1074 NUM cana-5305 31 9 - - PUNCT cana-5305 31 10 133x 133x NUM cana-5305 31 11 vol vol NOUN cana-5305 31 12 31 31 NUM cana-5305 31 13 no no NOUN cana-5305 31 14 . . NOUN cana-5305 31 15 1 1 NUM cana-5305 31 16 ( ( PUNCT cana-5305 31 17 2024 2024 NUM cana-5305 31 18 ) ) PUNCT cana-5305 31 19 356 356 NUM cana-5305 31 20 https://internationalpubls.com https://internationalpubls.com X cana-5305 32 1 the the DET cana-5305 32 2 paper paper NOUN cana-5305 32 3 analyzes analyze VERB cana-5305 32 4 both both DET cana-5305 32 5 difficulties difficulty NOUN cana-5305 32 6 and and CCONJ cana-5305 32 7 performance performance NOUN cana-5305 32 8 implications implication NOUN cana-5305 32 9 which which PRON cana-5305 32 10 arise arise VERB cana-5305 32 11 from from ADP cana-5305 32 12 implementing implement VERB cana-5305 32 13 algebraic algebraic ADJ cana-5305 32 14 structures structure NOUN cana-5305 32 15 for for ADP cana-5305 32 16 cryptographic cryptographic ADJ cana-5305 32 17 applications application NOUN cana-5305 32 18 . . PUNCT cana-5305 33 1 the the DET cana-5305 33 2 paper paper NOUN cana-5305 33 3 investigates investigate VERB cana-5305 33 4 the the DET cana-5305 33 5 balancing balancing NOUN cana-5305 33 6 point point NOUN cana-5305 33 7 between between ADP cana-5305 33 8 security security NOUN cana-5305 33 9 strength strength NOUN cana-5305 33 10 and and CCONJ cana-5305 33 11 performance performance NOUN cana-5305 33 12 effectiveness effectiveness NOUN cana-5305 33 13 by by ADP cana-5305 33 14 showing show VERB cana-5305 33 15 how how SCONJ cana-5305 33 16 growing grow VERB cana-5305 33 17 key key ADJ cana-5305 33 18 lengths length NOUN cana-5305 33 19 affect affect VERB cana-5305 33 20 both both DET cana-5305 33 21 factors factor NOUN cana-5305 33 22 and and CCONJ cana-5305 33 23 demonstrates demonstrate VERB cana-5305 33 24 general general ADJ cana-5305 33 25 system system NOUN cana-5305 33 26 implementation implementation NOUN cana-5305 33 27 consequences consequence NOUN cana-5305 33 28 . . PUNCT cana-5305 34 1 this this DET cana-5305 34 2 paper paper NOUN cana-5305 34 3 adds add VERB cana-5305 34 4 value value NOUN cana-5305 34 5 to to ADP cana-5305 34 6 quantum quantum ADJ cana-5305 34 7 computing computing NOUN cana-5305 34 8 's 's PART cana-5305 34 9 effect effect NOUN cana-5305 34 10 on on ADP cana-5305 34 11 cryptography cryptography NOUN cana-5305 34 12 by by ADP cana-5305 34 13 participating participate VERB cana-5305 34 14 in in ADP cana-5305 34 15 current current ADJ cana-5305 34 16 scholarly scholarly ADJ cana-5305 34 17 conversations conversation NOUN cana-5305 34 18 about about ADP cana-5305 34 19 the the DET cana-5305 34 20 topic topic NOUN cana-5305 34 21 . . PUNCT cana-5305 35 1 the the DET cana-5305 35 2 existing exist VERB cana-5305 35 3 literature literature NOUN cana-5305 35 4 investigates investigate VERB cana-5305 35 5 single single ADJ cana-5305 35 6 quantum quantum NOUN cana-5305 35 7 - - PUNCT cana-5305 35 8 resistant resistant ADJ cana-5305 35 9 algorithms algorithm NOUN cana-5305 35 10 but but CCONJ cana-5305 35 11 this this DET cana-5305 35 12 paper paper NOUN cana-5305 35 13 explores explore VERB cana-5305 35 14 extensive extensive ADJ cana-5305 35 15 effects effect NOUN cana-5305 35 16 of of ADP cana-5305 35 17 quantum quantum NOUN cana-5305 35 18 computing computing NOUN cana-5305 35 19 on on ADP cana-5305 35 20 cryptographic cryptographic ADJ cana-5305 35 21 algebraic algebraic ADJ cana-5305 35 22 structures structure NOUN cana-5305 35 23 . . PUNCT cana-5305 36 1 the the DET cana-5305 36 2 paper paper NOUN cana-5305 36 3 presents present VERB cana-5305 36 4 lattice lattice NOUN cana-5305 36 5 - - PUNCT cana-5305 36 6 based base VERB cana-5305 36 7 cryptography cryptography NOUN cana-5305 36 8 as as ADP cana-5305 36 9 a a DET cana-5305 36 10 possible possible ADJ cana-5305 36 11 solution solution NOUN cana-5305 36 12 to to PART cana-5305 36 13 secure secure VERB cana-5305 36 14 data datum NOUN cana-5305 36 15 after after SCONJ cana-5305 36 16 quantum quantum ADJ cana-5305 36 17 computers computer NOUN cana-5305 36 18 gain gain VERB cana-5305 36 19 widespread widespread ADJ cana-5305 36 20 adoption adoption NOUN cana-5305 36 21 through through ADP cana-5305 36 22 its its PRON cana-5305 36 23 introduction introduction NOUN cana-5305 36 24 of of ADP cana-5305 36 25 quantum quantum NOUN cana-5305 36 26 - - ADJ cana-5305 36 27 resilient resilient ADJ cana-5305 36 28 cryptography cryptography NOUN cana-5305 36 29 . . PUNCT cana-5305 37 1 the the DET cana-5305 37 2 research research NOUN cana-5305 37 3 demonstrates demonstrate VERB cana-5305 37 4 the the DET cana-5305 37 5 value value NOUN cana-5305 37 6 of of ADP cana-5305 37 7 latticebased latticebased ADJ cana-5305 37 8 cryptography cryptography NOUN cana-5305 37 9 as as SCONJ cana-5305 37 10 it it PRON cana-5305 37 11 reveals reveal VERB cana-5305 37 12 critical critical ADJ cana-5305 37 13 aspects aspect NOUN cana-5305 37 14 about about ADP cana-5305 37 15 upcoming upcoming ADJ cana-5305 37 16 cryptographic cryptographic ADJ cana-5305 37 17 development development NOUN cana-5305 37 18 for for ADP cana-5305 37 19 quantum quantum NOUN cana-5305 37 20 - - PUNCT cana-5305 37 21 computing compute VERB cana-5305 37 22 environments environment NOUN cana-5305 38 1 [ [ X cana-5305 38 2 13 13 NUM cana-5305 38 3 ] ] PUNCT cana-5305 38 4 . . PUNCT cana-5305 39 1 this this DET cana-5305 39 2 paper paper NOUN cana-5305 39 3 investigates investigate VERB cana-5305 39 4 upcoming upcoming ADJ cana-5305 39 5 pathways pathway NOUN cana-5305 39 6 in in ADP cana-5305 39 7 algebraic algebraic ADJ cana-5305 39 8 structures structure NOUN cana-5305 39 9 of of ADP cana-5305 39 10 cryptography cryptography NOUN cana-5305 39 11 through through ADP cana-5305 39 12 evaluation evaluation NOUN cana-5305 39 13 of of ADP cana-5305 39 14 modern modern ADJ cana-5305 39 15 developments development NOUN cana-5305 39 16 in in ADP cana-5305 39 17 quantum quantum NOUN cana-5305 39 18 - - ADJ cana-5305 39 19 safe safe ADJ cana-5305 39 20 cryptography cryptography NOUN cana-5305 39 21 . . PUNCT cana-5305 40 1 this this DET cana-5305 40 2 research research NOUN cana-5305 40 3 suggests suggest VERB cana-5305 40 4 investigating investigate VERB cana-5305 40 5 fresh fresh ADJ cana-5305 40 6 algebraic algebraic ADJ cana-5305 40 7 systems system NOUN cana-5305 40 8 to to PART cana-5305 40 9 develop develop VERB cana-5305 40 10 potent potent ADJ cana-5305 40 11 security security NOUN cana-5305 40 12 features feature NOUN cana-5305 40 13 alongside alongside ADP cana-5305 40 14 performance performance NOUN cana-5305 40 15 capabilities capability NOUN cana-5305 40 16 necessary necessary ADJ cana-5305 40 17 to to PART cana-5305 40 18 address address VERB cana-5305 40 19 upcoming upcoming ADJ cana-5305 40 20 computer computer NOUN cana-5305 40 21 - - PUNCT cana-5305 40 22 based base VERB cana-5305 40 23 threats threat NOUN cana-5305 40 24 . . PUNCT cana-5305 41 1 this this DET cana-5305 41 2 paper paper NOUN cana-5305 41 3 unites unite VERB cana-5305 41 4 contemporary contemporary ADJ cana-5305 41 5 trends trend NOUN cana-5305 41 6 with with ADP cana-5305 41 7 fresh fresh ADJ cana-5305 41 8 possibilities possibility NOUN cana-5305 41 9 to to PART cana-5305 41 10 support support VERB cana-5305 41 11 the the DET cana-5305 41 12 advanced advanced ADJ cana-5305 41 13 discussion discussion NOUN cana-5305 41 14 about about ADP cana-5305 41 15 cryptographic cryptographic ADJ cana-5305 41 16 system system NOUN cana-5305 41 17 evolution evolution NOUN cana-5305 41 18 in in ADP cana-5305 41 19 an an DET cana-5305 41 20 active active ADJ cana-5305 41 21 technological technological ADJ cana-5305 41 22 environment environment NOUN cana-5305 41 23 . . PUNCT cana-5305 42 1 this this DET cana-5305 42 2 paper paper NOUN cana-5305 42 3 delivers deliver VERB cana-5305 42 4 important important ADJ cana-5305 42 5 findings finding NOUN cana-5305 42 6 about about ADP cana-5305 42 7 algebraic algebraic ADJ cana-5305 42 8 structures structure NOUN cana-5305 42 9 in in ADP cana-5305 42 10 cryptography cryptography NOUN cana-5305 42 11 while while SCONJ cana-5305 42 12 providing provide VERB cana-5305 42 13 extensive extensive ADJ cana-5305 42 14 knowledge knowledge NOUN cana-5305 42 15 about about ADP cana-5305 42 16 the the DET cana-5305 42 17 future future ADJ cana-5305 42 18 directions direction NOUN cana-5305 42 19 and and CCONJ cana-5305 42 20 difficulties difficulty NOUN cana-5305 42 21 in in ADP cana-5305 42 22 this this DET cana-5305 42 23 field field NOUN cana-5305 42 24 [ [ X cana-5305 42 25 7 7 NUM cana-5305 42 26 ] ] PUNCT cana-5305 42 27 . . PUNCT cana-5305 43 1 ii ii PROPN cana-5305 43 2 . . PROPN cana-5305 43 3 related relate VERB cana-5305 43 4 works work NOUN cana-5305 43 5 in in ADP cana-5305 43 6 2023 2023 NUM cana-5305 43 7 s. s. PROPN cana-5305 43 8 malik malik PROPN cana-5305 43 9 et.al et.al PROPN cana-5305 43 10 . . PROPN cana-5305 43 11 , , PUNCT cana-5305 43 12 s. s. PROPN cana-5305 43 13 agarwal agarwal PROPN cana-5305 43 14 et.al et.al PROPN cana-5305 43 15 . . PROPN cana-5305 43 16 , , PUNCT cana-5305 43 17 and and CCONJ cana-5305 43 18 a. a. PROPN cana-5305 43 19 s. s. PROPN cana-5305 43 20 uniyal uniyal PROPN cana-5305 43 21 et.al et.al PROPN cana-5305 43 22 . . PUNCT cana-5305 43 23 , , PUNCT cana-5305 44 1 [ [ X cana-5305 44 2 9 9 NUM cana-5305 44 3 ] ] PUNCT cana-5305 44 4 proposed propose VERB cana-5305 44 5 the the DET cana-5305 44 6 cryptography cryptography NOUN cana-5305 44 7 and and CCONJ cana-5305 44 8 algebraic algebraic ADJ cana-5305 44 9 structures structure NOUN cana-5305 44 10 combine combine VERB cana-5305 44 11 due due ADJ cana-5305 44 12 to to ADP cana-5305 44 13 extensive extensive ADJ cana-5305 44 14 research research NOUN cana-5305 44 15 which which PRON cana-5305 44 16 produced produce VERB cana-5305 44 17 important important ADJ cana-5305 44 18 advances advance NOUN cana-5305 44 19 in in ADP cana-5305 44 20 secure secure ADJ cana-5305 44 21 communication communication NOUN cana-5305 44 22 data datum NOUN cana-5305 44 23 security security NOUN cana-5305 44 24 protocols protocol NOUN cana-5305 44 25 . . PUNCT cana-5305 45 1 the the DET cana-5305 45 2 majority majority NOUN cana-5305 45 3 of of ADP cana-5305 45 4 cryptographic cryptographic ADJ cana-5305 45 5 systems system NOUN cana-5305 45 6 built build VERB cana-5305 45 7 using use VERB cana-5305 45 8 public public ADJ cana-5305 45 9 - - PUNCT cana-5305 45 10 key key NOUN cana-5305 45 11 cryptography cryptography NOUN cana-5305 45 12 operate operate NOUN cana-5305 45 13 through through ADP cana-5305 45 14 algebraic algebraic ADJ cana-5305 45 15 problem problem NOUN cana-5305 45 16 difficulties difficulty NOUN cana-5305 45 17 which which PRON cana-5305 45 18 include include VERB cana-5305 45 19 factoring factor VERB cana-5305 45 20 large large ADJ cana-5305 45 21 integers integer NOUN cana-5305 45 22 and and CCONJ cana-5305 45 23 solving solve VERB cana-5305 45 24 discrete discrete ADJ cana-5305 45 25 logarithms logarithm NOUN cana-5305 45 26 . . PUNCT cana-5305 46 1 the the DET cana-5305 46 2 problems problem NOUN cana-5305 46 3 originating originate VERB cana-5305 46 4 from from ADP cana-5305 46 5 algebraic algebraic ADJ cana-5305 46 6 structures structure NOUN cana-5305 46 7 of of ADP cana-5305 46 8 groups group NOUN cana-5305 46 9 and and CCONJ cana-5305 46 10 fields field NOUN cana-5305 46 11 enable enable VERB cana-5305 46 12 the the DET cana-5305 46 13 security security NOUN cana-5305 46 14 behind behind ADP cana-5305 46 15 encryption encryption NOUN cana-5305 46 16 algorithms algorithm NOUN cana-5305 46 17 including include VERB cana-5305 46 18 rsa rsa PROPN cana-5305 46 19 and and CCONJ cana-5305 46 20 daffier daffier PROPN cana-5305 46 21 - - PUNCT cana-5305 46 22 hellman hellman NOUN cana-5305 46 23 . . PUNCT cana-5305 47 1 mathematical mathematical ADJ cana-5305 47 2 challenges challenge NOUN cana-5305 47 3 in in ADP cana-5305 47 4 these these DET cana-5305 47 5 problems problem NOUN cana-5305 47 6 create create VERB cana-5305 47 7 encryption encryption NOUN cana-5305 47 8 barriers barrier NOUN cana-5305 47 9 which which PRON cana-5305 47 10 block block VERB cana-5305 47 11 unauthorized unauthorized ADJ cana-5305 47 12 access access NOUN cana-5305 47 13 to to ADP cana-5305 47 14 hidden hide VERB cana-5305 47 15 messages message NOUN cana-5305 47 16 together together ADV cana-5305 47 17 with with ADP cana-5305 47 18 preventing prevent VERB cana-5305 47 19 parties party NOUN cana-5305 47 20 from from ADP cana-5305 47 21 making make VERB cana-5305 47 22 valid valid ADJ cana-5305 47 23 cryptographic cryptographic ADJ cana-5305 47 24 keys key NOUN cana-5305 47 25 . . PUNCT cana-5305 48 1 public public ADJ cana-5305 48 2 - - PUNCT cana-5305 48 3 key key NOUN cana-5305 48 4 encryption encryption NOUN cana-5305 48 5 systems system NOUN cana-5305 48 6 are be AUX cana-5305 48 7 gaining gain VERB cana-5305 48 8 preference preference NOUN cana-5305 48 9 alongside alongside ADP cana-5305 48 10 the the DET cana-5305 48 11 implementable implementable ADJ cana-5305 48 12 elliptic elliptic ADJ cana-5305 48 13 curve curve NOUN cana-5305 48 14 technology technology NOUN cana-5305 48 15 . . PUNCT cana-5305 49 1 ecc ecc PROPN cana-5305 49 2 derives derive VERB cana-5305 49 3 its its PRON cana-5305 49 4 encryption encryption NOUN cana-5305 49 5 methods method NOUN cana-5305 49 6 from from ADP cana-5305 49 7 the the DET cana-5305 49 8 mathematical mathematical ADJ cana-5305 49 9 properties property NOUN cana-5305 49 10 of of ADP cana-5305 49 11 elliptic elliptic ADJ cana-5305 49 12 curves curve NOUN cana-5305 49 13 that that PRON cana-5305 49 14 exist exist VERB cana-5305 49 15 between between ADP cana-5305 49 16 finite finite ADJ cana-5305 49 17 fields field NOUN cana-5305 49 18 . . PUNCT cana-5305 50 1 the the DET cana-5305 50 2 mathematical mathematical ADJ cana-5305 50 3 basis basis NOUN cana-5305 50 4 of of ADP cana-5305 50 5 this this DET cana-5305 50 6 structure structure NOUN cana-5305 50 7 enables enable VERB cana-5305 50 8 the the DET cana-5305 50 9 use use NOUN cana-5305 50 10 of of ADP cana-5305 50 11 diminished diminish VERB cana-5305 50 12 key key ADJ cana-5305 50 13 lengths length NOUN cana-5305 50 14 than than ADP cana-5305 50 15 rsa rsa NOUN cana-5305 50 16 techniques technique NOUN cana-5305 50 17 yet yet ADV cana-5305 50 18 delivers deliver VERB cana-5305 50 19 similar similar ADJ cana-5305 50 20 levels level NOUN cana-5305 50 21 of of ADP cana-5305 50 22 security security NOUN cana-5305 50 23 . . PUNCT cana-5305 51 1 this this DET cana-5305 51 2 technology technology NOUN cana-5305 51 3 proves prove VERB cana-5305 51 4 beneficial beneficial ADJ cana-5305 51 5 by by ADP cana-5305 51 6 using use VERB cana-5305 51 7 limited limited ADJ cana-5305 51 8 amount amount NOUN cana-5305 51 9 of of ADP cana-5305 51 10 power power NOUN cana-5305 51 11 in in ADP cana-5305 51 12 situations situation NOUN cana-5305 51 13 where where SCONJ cana-5305 51 14 mobile mobile ADJ cana-5305 51 15 devices device NOUN cana-5305 51 16 and and CCONJ cana-5305 51 17 embedded embed VERB cana-5305 51 18 systems system NOUN cana-5305 51 19 operate operate VERB cana-5305 51 20 . . PUNCT cana-5305 52 1 ecc ecc PROPN cana-5305 52 2 provides provide VERB cana-5305 52 3 high high ADJ cana-5305 52 4 efficiency efficiency NOUN cana-5305 52 5 which which PRON cana-5305 52 6 makes make VERB cana-5305 52 7 it it PRON cana-5305 52 8 the the DET cana-5305 52 9 preferred preferred ADJ cana-5305 52 10 choice choice NOUN cana-5305 52 11 for for ADP cana-5305 52 12 ssl ssl PROPN cana-5305 52 13 / / SYM cana-5305 52 14 tls tls PROPN cana-5305 52 15 secure secure ADJ cana-5305 52 16 protocols protocol NOUN cana-5305 52 17 and and CCONJ cana-5305 52 18 digital digital ADJ cana-5305 52 19 signature signature NOUN cana-5305 52 20 schemes scheme NOUN cana-5305 52 21 implementation implementation NOUN cana-5305 52 22 . . PUNCT cana-5305 53 1 communications communication NOUN cana-5305 53 2 on on ADP cana-5305 53 3 applied apply VERB cana-5305 53 4 nonlinear nonlinear ADJ cana-5305 53 5 analysis analysis NOUN cana-5305 53 6 issn issn NOUN cana-5305 53 7 : : PUNCT cana-5305 53 8 1074 1074 NUM cana-5305 53 9 - - PUNCT cana-5305 53 10 133x 133x NUM cana-5305 53 11 vol vol NOUN cana-5305 53 12 31 31 NUM cana-5305 53 13 no no NOUN cana-5305 53 14 . . NOUN cana-5305 53 15 1 1 NUM cana-5305 53 16 ( ( PUNCT cana-5305 53 17 2024 2024 NUM cana-5305 53 18 ) ) PUNCT cana-5305 53 19 357 357 NUM cana-5305 53 20 https://internationalpubls.com https://internationalpubls.com X cana-5305 53 21 in in ADP cana-5305 53 22 2024 2024 NUM cana-5305 53 23 m. m. NOUN cana-5305 53 24 s. s. PROPN cana-5305 53 25 khamalwa khamalwa PROPN cana-5305 53 26 et.al et.al PROPN cana-5305 53 27 . . PUNCT cana-5305 53 28 , , PUNCT cana-5305 54 1 [ [ X cana-5305 54 2 1 1 X cana-5305 54 3 ] ] PUNCT cana-5305 54 4 introduced introduce VERB cana-5305 54 5 the the DET cana-5305 54 6 research research NOUN cana-5305 54 7 has have AUX cana-5305 54 8 extensively extensively ADV cana-5305 54 9 studied study VERB cana-5305 54 10 the the DET cana-5305 54 11 algebraic algebraic ADJ cana-5305 54 12 elements element NOUN cana-5305 54 13 that that PRON cana-5305 54 14 exist exist VERB cana-5305 54 15 within within ADP cana-5305 54 16 hash hash NOUN cana-5305 54 17 function function NOUN cana-5305 54 18 structures structure NOUN cana-5305 54 19 . . PUNCT cana-5305 55 1 hash hash NOUN cana-5305 55 2 functions function NOUN cana-5305 55 3 maintain maintain VERB cana-5305 55 4 a a DET cana-5305 55 5 crucial crucial ADJ cana-5305 55 6 role role NOUN cana-5305 55 7 in in ADP cana-5305 55 8 cryptography cryptography NOUN cana-5305 55 9 since since SCONJ cana-5305 55 10 they they PRON cana-5305 55 11 provide provide VERB cana-5305 55 12 protections protection NOUN cana-5305 55 13 for for ADP cana-5305 55 14 data datum NOUN cana-5305 55 15 authenticity authenticity NOUN cana-5305 55 16 as as ADV cana-5305 55 17 well well ADV cana-5305 55 18 as as ADP cana-5305 55 19 integrity integrity NOUN cana-5305 55 20 . . PUNCT cana-5305 56 1 the the DET cana-5305 56 2 unique unique ADJ cana-5305 56 3 mathematical mathematical ADJ cana-5305 56 4 characteristics characteristic NOUN cana-5305 56 5 of of ADP cana-5305 56 6 these these DET cana-5305 56 7 functions function NOUN cana-5305 56 8 enable enable VERB cana-5305 56 9 production production NOUN cana-5305 56 10 of of ADP cana-5305 56 11 set set NOUN cana-5305 56 12 - - PUNCT cana-5305 56 13 sized sized ADJ cana-5305 56 14 outputs output NOUN cana-5305 56 15 from from ADP cana-5305 56 16 adaptable adaptable ADJ cana-5305 56 17 inputs input NOUN cana-5305 56 18 thus thus ADV cana-5305 56 19 making make VERB cana-5305 56 20 them they PRON cana-5305 56 21 practical practical ADJ cana-5305 56 22 for for ADP cana-5305 56 23 digital digital ADJ cana-5305 56 24 signature signature NOUN cana-5305 56 25 creation creation NOUN cana-5305 56 26 and and CCONJ cana-5305 56 27 data datum NOUN cana-5305 56 28 authentication authentication NOUN cana-5305 56 29 processes process NOUN cana-5305 56 30 along along ADP cana-5305 56 31 with with ADP cana-5305 56 32 password password NOUN cana-5305 56 33 encoding encoding NOUN cana-5305 56 34 procedures procedure NOUN cana-5305 56 35 . . PUNCT cana-5305 57 1 the the DET cana-5305 57 2 cryptographic cryptographic ADJ cana-5305 57 3 hash hash NOUN cana-5305 57 4 functions function NOUN cana-5305 57 5 like like SCONJ cana-5305 57 6 sha-256 sha-256 PROPN cana-5305 57 7 implement implement VERB cana-5305 57 8 algebraic algebraic ADJ cana-5305 57 9 structures structure NOUN cana-5305 57 10 to to PART cana-5305 57 11 generate generate VERB cana-5305 57 12 resistant resistant ADJ cana-5305 57 13 outputs output NOUN cana-5305 57 14 that that PRON cana-5305 57 15 make make VERB cana-5305 57 16 it it PRON cana-5305 57 17 practically practically ADV cana-5305 57 18 impossible impossible ADJ cana-5305 57 19 for for SCONJ cana-5305 57 20 attackers attacker NOUN cana-5305 57 21 to to PART cana-5305 57 22 discover discover VERB cana-5305 57 23 two two NUM cana-5305 57 24 different different ADJ cana-5305 57 25 inputs input NOUN cana-5305 57 26 sharing share VERB cana-5305 57 27 the the DET cana-5305 57 28 identical identical ADJ cana-5305 57 29 hash hash NOUN cana-5305 57 30 result result NOUN cana-5305 57 31 . . PUNCT cana-5305 58 1 researchers researcher NOUN cana-5305 58 2 now now ADV cana-5305 58 3 explore explore VERB cana-5305 58 4 post post ADJ cana-5305 58 5 - - ADJ cana-5305 58 6 quantum quantum ADJ cana-5305 58 7 cryptographic cryptographic ADJ cana-5305 58 8 algorithms algorithm NOUN cana-5305 58 9 because because SCONJ cana-5305 58 10 quantum quantum ADJ cana-5305 58 11 computers computer NOUN cana-5305 58 12 became become VERB cana-5305 58 13 available available ADJ cana-5305 58 14 in in ADP cana-5305 58 15 the the DET cana-5305 58 16 modern modern ADJ cana-5305 58 17 era era NOUN cana-5305 58 18 . . PUNCT cana-5305 59 1 the the DET cana-5305 59 2 purpose purpose NOUN cana-5305 59 3 of of ADP cana-5305 59 4 these these DET cana-5305 59 5 algorithms algorithm NOUN cana-5305 59 6 is be AUX cana-5305 59 7 to to PART cana-5305 59 8 defend defend VERB cana-5305 59 9 against against ADP cana-5305 59 10 quantum quantum ADJ cana-5305 59 11 algorithms algorithm NOUN cana-5305 59 12 especially especially ADV cana-5305 59 13 " " PUNCT cana-5305 59 14 shor shor PROPN cana-5305 59 15 's 's PART cana-5305 59 16 algorithm algorithm NOUN cana-5305 59 17 " " PUNCT cana-5305 59 18 given give VERB cana-5305 59 19 its its PRON cana-5305 59 20 efficiency efficiency NOUN cana-5305 59 21 in in ADP cana-5305 59 22 solving solve VERB cana-5305 59 23 integer integer NOUN cana-5305 59 24 factorization factorization NOUN cana-5305 59 25 problems problem NOUN cana-5305 59 26 and and CCONJ cana-5305 59 27 discrete discrete ADJ cana-5305 59 28 logarithms logarithm NOUN cana-5305 59 29 which which PRON cana-5305 59 30 serve serve VERB cana-5305 59 31 as as ADP cana-5305 59 32 foundations foundation NOUN cana-5305 59 33 for for ADP cana-5305 59 34 traditional traditional ADJ cana-5305 59 35 cryptographic cryptographic ADJ cana-5305 59 36 methods method NOUN cana-5305 59 37 . . PUNCT cana-5305 60 1 post post ADJ cana-5305 60 2 - - ADJ cana-5305 60 3 quantum quantum ADJ cana-5305 60 4 cryptography cryptography NOUN cana-5305 60 5 features feature VERB cana-5305 60 6 lattice lattice NOUN cana-5305 60 7 - - PUNCT cana-5305 60 8 based base VERB cana-5305 60 9 encryption encryption NOUN cana-5305 60 10 as as ADP cana-5305 60 11 its its PRON cana-5305 60 12 strongest strong ADJ cana-5305 60 13 candidate candidate NOUN cana-5305 60 14 because because SCONJ cana-5305 60 15 it it PRON cana-5305 60 16 makes make VERB cana-5305 60 17 use use NOUN cana-5305 60 18 of of ADP cana-5305 60 19 multidimensional multidimensional ADJ cana-5305 60 20 mathematical mathematical ADJ cana-5305 60 21 spaces space NOUN cana-5305 60 22 known know VERB cana-5305 60 23 as as ADP cana-5305 60 24 lattices lattice NOUN cana-5305 60 25 . . PUNCT cana-5305 61 1 various various ADJ cana-5305 61 2 branch branch NOUN cana-5305 61 3 of of ADP cana-5305 61 4 advanced advanced ADJ cana-5305 61 5 mathematics mathematic NOUN cana-5305 61 6 such such ADJ cana-5305 61 7 as as ADP cana-5305 61 8 coding code VERB cana-5305 61 9 theory theory NOUN cana-5305 61 10 and and CCONJ cana-5305 61 11 algebraic algebraic ADJ cana-5305 61 12 geometry geometry NOUN cana-5305 61 13 have have AUX cana-5305 61 14 been be AUX cana-5305 61 15 integrated integrate VERB cana-5305 61 16 into into ADP cana-5305 61 17 research research NOUN cana-5305 61 18 which which PRON cana-5305 61 19 produces produce VERB cana-5305 61 20 solutions solution NOUN cana-5305 61 21 through through ADP cana-5305 61 22 algebra algebra NOUN cana-5305 61 23 applied apply VERB cana-5305 61 24 for for ADP cana-5305 61 25 data data NOUN cana-5305 61 26 transmission transmission NOUN cana-5305 61 27 and and CCONJ cana-5305 61 28 new new ADJ cana-5305 61 29 cryptographic cryptographic ADJ cana-5305 61 30 primitive primitive ADJ cana-5305 61 31 development development NOUN cana-5305 61 32 . . PUNCT cana-5305 62 1 in in ADP cana-5305 62 2 2020 2020 NUM cana-5305 62 3 j. j. PROPN cana-5305 62 4 suo suo PROPN cana-5305 62 5 et.al et.al PROPN cana-5305 62 6 . . PROPN cana-5305 62 7 , , PUNCT cana-5305 62 8 l. l. PROPN cana-5305 62 9 wang wang PROPN cana-5305 62 10 et.al et.al PROPN cana-5305 62 11 . . PROPN cana-5305 62 12 , , PUNCT cana-5305 62 13 s. s. PROPN cana-5305 62 14 yang yang PROPN cana-5305 62 15 et.al et.al PROPN cana-5305 62 16 . . PROPN cana-5305 62 17 , , PUNCT cana-5305 62 18 w. w. PROPN cana-5305 62 19 zheng zheng PROPN cana-5305 62 20 et.al et.al PROPN cana-5305 62 21 . . PROPN cana-5305 62 22 , , PUNCT cana-5305 62 23 and and CCONJ cana-5305 62 24 j. j. PROPN cana-5305 62 25 zhang zhang PROPN cana-5305 62 26 et.al et.al PROPN cana-5305 62 27 . . PUNCT cana-5305 62 28 , , PUNCT cana-5305 62 29 [ [ X cana-5305 62 30 14 14 NUM cana-5305 62 31 ] ] PUNCT cana-5305 62 32 suggested suggest VERB cana-5305 62 33 the the DET cana-5305 62 34 field field NOUN cana-5305 62 35 of of ADP cana-5305 62 36 algebraic algebraic ADJ cana-5305 62 37 cryptography cryptography NOUN cana-5305 62 38 continues continue VERB cana-5305 62 39 to to PART cana-5305 62 40 face face VERB cana-5305 62 41 opposition opposition NOUN cana-5305 62 42 although although SCONJ cana-5305 62 43 previous previous ADJ cana-5305 62 44 breakthroughs breakthrough NOUN cana-5305 62 45 have have AUX cana-5305 62 46 been be AUX cana-5305 62 47 achieved achieve VERB cana-5305 62 48 . . PUNCT cana-5305 63 1 the the DET cana-5305 63 2 core core NOUN cana-5305 63 3 problem problem NOUN cana-5305 63 4 in in ADP cana-5305 63 5 this this DET cana-5305 63 6 field field NOUN cana-5305 63 7 centers center NOUN cana-5305 63 8 on on ADP cana-5305 63 9 finding find VERB cana-5305 63 10 acceptable acceptable ADJ cana-5305 63 11 ways way NOUN cana-5305 63 12 to to PART cana-5305 63 13 make make VERB cana-5305 63 14 security security NOUN cana-5305 63 15 measures measure NOUN cana-5305 63 16 commensurate commensurate ADJ cana-5305 63 17 with with ADP cana-5305 63 18 computational computational ADJ cana-5305 63 19 operational operational ADJ cana-5305 63 20 rates rate NOUN cana-5305 63 21 . . PUNCT cana-5305 64 1 the the DET cana-5305 64 2 development development NOUN cana-5305 64 3 of of ADP cana-5305 64 4 stronger strong ADJ cana-5305 64 5 security security NOUN cana-5305 64 6 systems system NOUN cana-5305 64 7 through through ADP cana-5305 64 8 algebraic algebraic ADJ cana-5305 64 9 structures structure NOUN cana-5305 64 10 leads lead VERB cana-5305 64 11 to to ADP cana-5305 64 12 increased increase VERB cana-5305 64 13 complexity complexity NOUN cana-5305 64 14 during during ADP cana-5305 64 15 implementation implementation NOUN cana-5305 64 16 . . PUNCT cana-5305 65 1 security security NOUN cana-5305 65 2 requirements requirement NOUN cana-5305 65 3 versus versus ADP cana-5305 65 4 system system NOUN cana-5305 65 5 operation operation NOUN cana-5305 65 6 efficiency efficiency NOUN cana-5305 65 7 serves serve VERB cana-5305 65 8 as as ADP cana-5305 65 9 a a DET cana-5305 65 10 direct direct ADJ cana-5305 65 11 determining determine VERB cana-5305 65 12 factor factor NOUN cana-5305 65 13 for for ADP cana-5305 65 14 cryptographic cryptographic ADJ cana-5305 65 15 systems system NOUN cana-5305 65 16 deployment deployment NOUN cana-5305 65 17 in in ADP cana-5305 65 18 restricted restricted ADJ cana-5305 65 19 system system NOUN cana-5305 65 20 environments environment NOUN cana-5305 65 21 . . PUNCT cana-5305 66 1 the the DET cana-5305 66 2 importance importance NOUN cana-5305 66 3 of of ADP cana-5305 66 4 algebraic algebraic ADJ cana-5305 66 5 mathematical mathematical ADJ cana-5305 66 6 shapes shape NOUN cana-5305 66 7 for for ADP cana-5305 66 8 research research NOUN cana-5305 66 9 becomes become VERB cana-5305 66 10 clear clear ADJ cana-5305 66 11 because because SCONJ cana-5305 66 12 they they PRON cana-5305 66 13 teach teach VERB cana-5305 66 14 cryptographic cryptographic ADJ cana-5305 66 15 algorithm algorithm NOUN cana-5305 66 16 development development NOUN cana-5305 66 17 methods method NOUN cana-5305 66 18 and and CCONJ cana-5305 66 19 identify identify VERB cana-5305 66 20 future future ADJ cana-5305 66 21 research research NOUN cana-5305 66 22 demands demand NOUN cana-5305 66 23 . . PUNCT cana-5305 67 1 iii iii X cana-5305 67 2 . . PROPN cana-5305 67 3 proposed propose VERB cana-5305 67 4 methodology methodology NOUN cana-5305 67 5 the the DET cana-5305 67 6 method method NOUN cana-5305 67 7 requires require VERB cana-5305 67 8 researchers researcher NOUN cana-5305 67 9 to to PART cana-5305 67 10 identify identify VERB cana-5305 67 11 cryptographic cryptographic ADJ cana-5305 67 12 algorithm algorithm NOUN cana-5305 67 13 algebraic algebraic PROPN cana-5305 67 14 structures structure NOUN cana-5305 67 15 first first ADV cana-5305 67 16 then then ADV cana-5305 67 17 model model PROPN cana-5305 67 18 mathematical mathematical ADJ cana-5305 67 19 encryption encryption NOUN cana-5305 67 20 and and CCONJ cana-5305 67 21 decryption decryption NOUN cana-5305 67 22 operations operation NOUN cana-5305 67 23 before before ADP cana-5305 67 24 designing design VERB cana-5305 67 25 a a DET cana-5305 67 26 secure secure ADJ cana-5305 67 27 framework framework NOUN cana-5305 67 28 built build VERB cana-5305 67 29 using use VERB cana-5305 67 30 these these DET cana-5305 67 31 algebraic algebraic ADJ cana-5305 67 32 structures structure NOUN cana-5305 67 33 along along ADP cana-5305 67 34 with with ADP cana-5305 67 35 post post ADJ cana-5305 67 36 - - ADJ cana-5305 67 37 quantum quantum ADJ cana-5305 67 38 lattice lattice NOUN cana-5305 67 39 - - PUNCT cana-5305 67 40 based base VERB cana-5305 67 41 cryptographic cryptographic ADJ cana-5305 67 42 schemes scheme NOUN cana-5305 67 43 . . PUNCT cana-5305 68 1 the the DET cana-5305 68 2 paper paper NOUN cana-5305 68 3 explains explain VERB cana-5305 68 4 each each DET cana-5305 68 5 step step NOUN cana-5305 68 6 in in ADP cana-5305 68 7 detail detail NOUN cana-5305 68 8 with with ADP cana-5305 68 9 supporting support VERB cana-5305 68 10 mathematical mathematical ADJ cana-5305 68 11 expressions expression NOUN cana-5305 68 12 as as ADV cana-5305 68 13 well well ADV cana-5305 68 14 as as ADP cana-5305 68 15 a a DET cana-5305 68 16 flowchart flowchart NOUN cana-5305 68 17 representing represent VERB cana-5305 68 18 the the DET cana-5305 68 19 methodology methodology NOUN cana-5305 68 20 [ [ X cana-5305 68 21 8 8 NUM cana-5305 68 22 ] ] PUNCT cana-5305 68 23 . . PUNCT cana-5305 69 1 communications communication NOUN cana-5305 69 2 on on ADP cana-5305 69 3 applied apply VERB cana-5305 69 4 nonlinear nonlinear ADJ cana-5305 69 5 analysis analysis NOUN cana-5305 69 6 issn issn NOUN cana-5305 69 7 : : PUNCT cana-5305 69 8 1074 1074 NUM cana-5305 69 9 - - PUNCT cana-5305 69 10 133x 133x NUM cana-5305 69 11 vol vol NOUN cana-5305 69 12 31 31 NUM cana-5305 69 13 no no NOUN cana-5305 69 14 . . NOUN cana-5305 69 15 1 1 NUM cana-5305 69 16 ( ( PUNCT cana-5305 69 17 2024 2024 NUM cana-5305 69 18 ) ) PUNCT cana-5305 70 1 358 358 NUM cana-5305 70 2 https://internationalpubls.com https://internationalpubls.com X cana-5305 70 3 a. a. NOUN cana-5305 70 4 identification identification NOUN cana-5305 70 5 of of ADP cana-5305 70 6 algebraic algebraic ADJ cana-5305 70 7 structures structure NOUN cana-5305 70 8 in in ADP cana-5305 70 9 cryptography cryptography NOUN cana-5305 70 10 groups group NOUN cana-5305 70 11 along along ADP cana-5305 70 12 with with ADP cana-5305 70 13 fields field NOUN cana-5305 70 14 and and CCONJ cana-5305 70 15 rings ring NOUN cana-5305 70 16 supply supply NOUN cana-5305 70 17 mathematical mathematical ADJ cana-5305 70 18 building building NOUN cana-5305 70 19 blocks block NOUN cana-5305 70 20 that that PRON cana-5305 70 21 enable enable VERB cana-5305 70 22 cryptographic cryptographic ADJ cana-5305 70 23 algorithm algorithm NOUN cana-5305 70 24 development development NOUN cana-5305 70 25 . . PUNCT cana-5305 71 1 operations operation NOUN cana-5305 71 2 of of ADP cana-5305 71 3 encryption encryption NOUN cana-5305 71 4 and and CCONJ cana-5305 71 5 decryption decryption NOUN cana-5305 71 6 alongside alongside ADP cana-5305 71 7 key key ADJ cana-5305 71 8 generation generation NOUN cana-5305 71 9 obtain obtain VERB cana-5305 71 10 their their PRON cana-5305 71 11 essential essential ADJ cana-5305 71 12 definitions definition NOUN cana-5305 71 13 through through ADP cana-5305 71 14 these these DET cana-5305 71 15 mathematical mathematical ADJ cana-5305 71 16 frameworks framework NOUN cana-5305 71 17 . . PUNCT cana-5305 72 1 rsa rsa PROPN cana-5305 72 2 depends depend VERB cana-5305 72 3 on on ADP cana-5305 72 4 the the DET cana-5305 72 5 algebraic algebraic ADJ cana-5305 72 6 nature nature NOUN cana-5305 72 7 of of ADP cana-5305 72 8 the the DET cana-5305 72 9 multiplicative multiplicative ADJ cana-5305 72 10 group group NOUN cana-5305 72 11 of of ADP cana-5305 72 12 integers integer NOUN cana-5305 72 13 modulo modulo VERB cana-5305 72 14 n n PRON cana-5305 72 15 which which PRON cana-5305 72 16 uses use VERB cana-5305 72 17 a a DET cana-5305 72 18 large large ADJ cana-5305 72 19 composite composite ADJ cana-5305 72 20 number number NOUN cana-5305 72 21 n. n. NOUN cana-5305 72 22 a a DET cana-5305 72 23 pair pair NOUN cana-5305 72 24 of of ADP cana-5305 72 25 public public ADJ cana-5305 72 26 and and CCONJ cana-5305 72 27 private private ADJ cana-5305 72 28 keys key NOUN cana-5305 72 29 exists exist VERB cana-5305 72 30 in in ADP cana-5305 72 31 the the DET cana-5305 72 32 form form NOUN cana-5305 72 33 defined define VERB cana-5305 72 34 by by ADP cana-5305 72 35 the the DET cana-5305 72 36 following following ADJ cana-5305 72 37 mathematical mathematical ADJ cana-5305 72 38 equation equation NOUN cana-5305 72 39 : : PUNCT cana-5305 72 40 𝑦 𝑦 NOUN cana-5305 72 41 = = SYM cana-5305 72 42 π‘₯𝑒 π‘₯𝑒 NOUN cana-5305 72 43 mod𝑛 mod𝑛 NOUN cana-5305 72 44 ( ( PUNCT cana-5305 72 45 encryption encryption NOUN cana-5305 72 46 ) ) PUNCT cana-5305 72 47 π‘₯ π‘₯ NOUN cana-5305 73 1 = = PUNCT cana-5305 73 2 𝑦𝑑 𝑦𝑑 PROPN cana-5305 73 3 mod𝑛 mod𝑛 NOUN cana-5305 73 4 ( ( PUNCT cana-5305 73 5 decryption decryption NOUN cana-5305 73 6 ) ) PUNCT cana-5305 73 7 here here ADV cana-5305 73 8 , , PUNCT cana-5305 73 9 π‘₯ π‘₯ PROPN cana-5305 73 10 is be AUX cana-5305 73 11 the the DET cana-5305 73 12 plaintext plaintext NOUN cana-5305 73 13 , , PUNCT cana-5305 73 14 𝑦 𝑦 NOUN cana-5305 73 15 is be AUX cana-5305 73 16 the the DET cana-5305 73 17 ciphertext ciphertext NOUN cana-5305 73 18 , , PUNCT cana-5305 73 19 𝑒 𝑒 PROPN cana-5305 73 20 is be AUX cana-5305 73 21 the the DET cana-5305 73 22 public public ADJ cana-5305 73 23 exponent exponent NOUN cana-5305 73 24 , , PUNCT cana-5305 73 25 𝑑 𝑑 PROPN cana-5305 73 26 is be AUX cana-5305 73 27 the the DET cana-5305 73 28 private private ADJ cana-5305 73 29 exponent exponent NOUN cana-5305 73 30 , , PUNCT cana-5305 73 31 and and CCONJ cana-5305 73 32 𝑛 𝑛 PROPN cana-5305 73 33 is be AUX cana-5305 73 34 the the DET cana-5305 73 35 modulus modulus NOUN cana-5305 73 36 . . PUNCT cana-5305 74 1 this this DET cana-5305 74 2 equation equation NOUN cana-5305 74 3 defines define VERB cana-5305 74 4 the the DET cana-5305 74 5 key key ADJ cana-5305 74 6 generation generation NOUN cana-5305 74 7 and and CCONJ cana-5305 74 8 encryption encryption NOUN cana-5305 74 9 - - PUNCT cana-5305 74 10 decryption decryption NOUN cana-5305 74 11 process process NOUN cana-5305 74 12 in in ADP cana-5305 74 13 rsa rsa PROPN cana-5305 74 14 . . PUNCT cana-5305 75 1 in in ADP cana-5305 75 2 elliptic elliptic ADJ cana-5305 75 3 curve curve NOUN cana-5305 75 4 cryptography cryptography NOUN cana-5305 75 5 ( ( PUNCT cana-5305 75 6 ecc ecc PROPN cana-5305 75 7 ) ) PUNCT cana-5305 75 8 the the DET cana-5305 75 9 algebraic algebraic ADJ cana-5305 75 10 structure structure NOUN cana-5305 75 11 consists consist VERB cana-5305 75 12 of of ADP cana-5305 75 13 the the DET cana-5305 75 14 elliptic elliptic ADJ cana-5305 75 15 curve curve NOUN cana-5305 75 16 group group NOUN cana-5305 75 17 because because SCONJ cana-5305 75 18 digital digital ADJ cana-5305 75 19 signatures signature NOUN cana-5305 75 20 and and CCONJ cana-5305 75 21 key key ADJ cana-5305 75 22 exchange exchange NOUN cana-5305 75 23 operations operation NOUN cana-5305 75 24 use use VERB cana-5305 75 25 the the DET cana-5305 75 26 points point NOUN cana-5305 75 27 that that PRON cana-5305 75 28 exist exist VERB cana-5305 75 29 on on ADP cana-5305 75 30 this this DET cana-5305 75 31 curve curve NOUN cana-5305 75 32 . . PUNCT cana-5305 76 1 the the DET cana-5305 76 2 formula formula NOUN cana-5305 76 3 that that PRON cana-5305 76 4 represents represent VERB cana-5305 76 5 an an DET cana-5305 76 6 elliptic elliptic ADJ cana-5305 76 7 curve curve NOUN cana-5305 76 8 appears appear VERB cana-5305 76 9 as as SCONJ cana-5305 76 10 follows follow VERB cana-5305 76 11 : : PUNCT cana-5305 76 12 𝑦2 𝑦2 PROPN cana-5305 76 13 = = PUNCT cana-5305 76 14 π‘₯3 π‘₯3 PROPN cana-5305 76 15 + + CCONJ cana-5305 76 16 π‘Žπ‘₯ π‘Žπ‘₯ X cana-5305 76 17 + + CCONJ cana-5305 76 18 𝑏 𝑏 PROPN cana-5305 76 19 ( ( PUNCT cana-5305 76 20 elliptic elliptic ADJ cana-5305 76 21 curve curve NOUN cana-5305 76 22 equation equation NOUN cana-5305 76 23 ) ) PUNCT cana-5305 76 24 the the DET cana-5305 76 25 key key ADJ cana-5305 76 26 operations operation NOUN cana-5305 76 27 of of ADP cana-5305 76 28 ecc ecc PROPN cana-5305 76 29 take take VERB cana-5305 76 30 place place NOUN cana-5305 76 31 on on ADP cana-5305 76 32 the the DET cana-5305 76 33 curve curve NOUN cana-5305 76 34 equation equation NOUN cana-5305 76 35 defined define VERB cana-5305 76 36 within within ADP cana-5305 76 37 finite finite ADJ cana-5305 76 38 fields field NOUN cana-5305 76 39 . . PUNCT cana-5305 77 1 b. b. PROPN cana-5305 77 2 mathematical mathematical PROPN cana-5305 77 3 modeling modeling NOUN cana-5305 77 4 of of ADP cana-5305 77 5 cryptographic cryptographic ADJ cana-5305 77 6 operations operation NOUN cana-5305 77 7 all all DET cana-5305 77 8 modular modular ADJ cana-5305 77 9 arithmetic arithmetic ADJ cana-5305 77 10 operations operation NOUN cana-5305 77 11 represent represent VERB cana-5305 77 12 the the DET cana-5305 77 13 model model NOUN cana-5305 77 14 for for ADP cana-5305 77 15 encryption encryption NOUN cana-5305 77 16 . . PUNCT cana-5305 78 1 in in ADP cana-5305 78 2 public public ADJ cana-5305 78 3 - - PUNCT cana-5305 78 4 key key NOUN cana-5305 78 5 systems system NOUN cana-5305 78 6 the the DET cana-5305 78 7 encryption encryption NOUN cana-5305 78 8 function function NOUN cana-5305 78 9 takes take VERB cana-5305 78 10 the the DET cana-5305 78 11 form form NOUN cana-5305 78 12 : : PUNCT cana-5305 78 13 𝐸(π‘š 𝐸(π‘š NUM cana-5305 78 14 ) ) PUNCT cana-5305 78 15 = = NOUN cana-5305 78 16 π‘šπ‘’mod𝑛 π‘šπ‘’mod𝑛 NOUN cana-5305 78 17 where where SCONJ cana-5305 78 18 π‘š π‘š PROPN cana-5305 78 19 is be AUX cana-5305 78 20 the the DET cana-5305 78 21 message message NOUN cana-5305 78 22 , , PUNCT cana-5305 78 23 𝑒 𝑒 PROPN cana-5305 78 24 is be AUX cana-5305 78 25 the the DET cana-5305 78 26 public public ADJ cana-5305 78 27 key key NOUN cana-5305 78 28 , , PUNCT cana-5305 78 29 and and CCONJ cana-5305 78 30 𝑛 𝑛 PROPN cana-5305 78 31 is be AUX cana-5305 78 32 the the DET cana-5305 78 33 modulus modulus NOUN cana-5305 78 34 . . PUNCT cana-5305 79 1 decryption decryption NOUN cana-5305 79 2 can can AUX cana-5305 79 3 be be AUX cana-5305 79 4 modeled model VERB cana-5305 79 5 by by ADP cana-5305 79 6 the the DET cana-5305 79 7 inverse inverse NOUN cana-5305 79 8 of of ADP cana-5305 79 9 this this DET cana-5305 79 10 operation operation NOUN cana-5305 79 11 : : PUNCT cana-5305 79 12 𝐷(𝑐 𝐷(𝑐 ADJ cana-5305 79 13 ) ) PUNCT cana-5305 79 14 = = SYM cana-5305 79 15 𝑐𝑑mod𝑛 𝑐𝑑mod𝑛 NOUN cana-5305 79 16 where where SCONJ cana-5305 79 17 𝑐 𝑐 PROPN cana-5305 79 18 is be AUX cana-5305 79 19 the the DET cana-5305 79 20 ciphertext ciphertext NOUN cana-5305 79 21 and and CCONJ cana-5305 79 22 𝑑 𝑑 NOUN cana-5305 79 23 is be AUX cana-5305 79 24 the the DET cana-5305 79 25 private private ADJ cana-5305 79 26 key key NOUN cana-5305 79 27 . . PUNCT cana-5305 80 1 this this DET cana-5305 80 2 modular modular ADJ cana-5305 80 3 exponentiation exponentiation NOUN cana-5305 80 4 forms form VERB cana-5305 80 5 the the DET cana-5305 80 6 basis basis NOUN cana-5305 80 7 of of ADP cana-5305 80 8 many many ADJ cana-5305 80 9 public public ADJ cana-5305 80 10 - - PUNCT cana-5305 80 11 key key NOUN cana-5305 80 12 cryptosystems cryptosystem NOUN cana-5305 80 13 . . PUNCT cana-5305 81 1 for for ADP cana-5305 81 2 hashing hashing NOUN cana-5305 81 3 , , PUNCT cana-5305 81 4 we we PRON cana-5305 81 5 use use VERB cana-5305 81 6 the the DET cana-5305 81 7 following follow VERB cana-5305 81 8 equation equation NOUN cana-5305 81 9 for for ADP cana-5305 81 10 a a DET cana-5305 81 11 basic basic ADJ cana-5305 81 12 hash hash NOUN cana-5305 81 13 function function NOUN cana-5305 81 14 : : PUNCT cana-5305 81 15 𝐻(π‘₯ 𝐻(π‘₯ NUM cana-5305 81 16 ) ) PUNCT cana-5305 81 17 = = SYM cana-5305 81 18 π‘₯π‘˜ π‘₯π‘˜ NOUN cana-5305 81 19 mod𝑝 mod𝑝 NOUN cana-5305 81 20 communications communication NOUN cana-5305 81 21 on on ADP cana-5305 81 22 applied apply VERB cana-5305 81 23 nonlinear nonlinear ADJ cana-5305 81 24 analysis analysis NOUN cana-5305 81 25 issn issn NOUN cana-5305 81 26 : : PUNCT cana-5305 81 27 1074 1074 NUM cana-5305 81 28 - - PUNCT cana-5305 81 29 133x 133x NUM cana-5305 81 30 vol vol NOUN cana-5305 81 31 31 31 NUM cana-5305 81 32 no no NOUN cana-5305 81 33 . . NOUN cana-5305 81 34 1 1 NUM cana-5305 81 35 ( ( PUNCT cana-5305 81 36 2024 2024 NUM cana-5305 81 37 ) ) PUNCT cana-5305 81 38 359 359 NUM cana-5305 81 39 https://internationalpubls.com https://internationalpubls.com X cana-5305 81 40 where where SCONJ cana-5305 81 41 π‘₯ π‘₯ PROPN cana-5305 81 42 is be AUX cana-5305 81 43 the the DET cana-5305 81 44 input input NOUN cana-5305 81 45 message message NOUN cana-5305 81 46 , , PUNCT cana-5305 81 47 π‘˜ π‘˜ PRON cana-5305 81 48 is be AUX cana-5305 81 49 a a DET cana-5305 81 50 constant constant ADJ cana-5305 81 51 , , PUNCT cana-5305 81 52 and and CCONJ cana-5305 81 53 𝑝 𝑝 NOUN cana-5305 81 54 is be AUX cana-5305 81 55 a a DET cana-5305 81 56 large large ADJ cana-5305 81 57 prime prime ADJ cana-5305 81 58 number number NOUN cana-5305 81 59 . . PUNCT cana-5305 82 1 hash hash NOUN cana-5305 82 2 functions function NOUN cana-5305 82 3 like like SCONJ cana-5305 82 4 sha-256 sha-256 PROPN cana-5305 82 5 utilize utilize VERB cana-5305 82 6 similar similar ADJ cana-5305 82 7 principles principle NOUN cana-5305 82 8 but but CCONJ cana-5305 82 9 with with ADP cana-5305 82 10 more more ADV cana-5305 82 11 complex complex ADJ cana-5305 82 12 algebraic algebraic ADJ cana-5305 82 13 structures structure NOUN cana-5305 82 14 to to PART cana-5305 82 15 ensure ensure VERB cana-5305 82 16 collision collision NOUN cana-5305 82 17 resistance resistance NOUN cana-5305 82 18 [ [ X cana-5305 82 19 12 12 NUM cana-5305 82 20 ] ] PUNCT cana-5305 82 21 . . PUNCT cana-5305 83 1 in in ADP cana-5305 83 2 lattice lattice NOUN cana-5305 83 3 - - PUNCT cana-5305 83 4 based base VERB cana-5305 83 5 cryptography cryptography NOUN cana-5305 83 6 , , PUNCT cana-5305 83 7 security security NOUN cana-5305 83 8 relies rely VERB cana-5305 83 9 on on ADP cana-5305 83 10 the the DET cana-5305 83 11 hardness hardness NOUN cana-5305 83 12 of of ADP cana-5305 83 13 finding find VERB cana-5305 83 14 short short ADJ cana-5305 83 15 vectors vector NOUN cana-5305 83 16 in in ADP cana-5305 83 17 a a DET cana-5305 83 18 highdimensional highdimensional ADJ cana-5305 83 19 lattice lattice NOUN cana-5305 83 20 . . PUNCT cana-5305 84 1 the the DET cana-5305 84 2 following follow VERB cana-5305 84 3 lattice lattice PROPN cana-5305 84 4 problem problem NOUN cana-5305 84 5 exists exist VERB cana-5305 84 6 in in ADP cana-5305 84 7 the the DET cana-5305 84 8 form form NOUN cana-5305 84 9 of of ADP cana-5305 84 10 : : PUNCT cana-5305 84 11 given give VERB cana-5305 84 12 𝐯 𝐯 ADP cana-5305 84 13 ∈ ∈ PROPN cana-5305 84 14 ℀𝑛 ℀𝑛 NOUN cana-5305 84 15 , , PUNCT cana-5305 84 16 find find VERB cana-5305 84 17 𝐯 𝐯 ADP cana-5305 84 18 such such ADJ cana-5305 84 19 that that SCONJ cana-5305 84 20 ‖𝐯‖2 ‖𝐯‖2 PROPN cana-5305 84 21 is be AUX cana-5305 84 22 minimized minimize VERB cana-5305 84 23 this this PRON cana-5305 84 24 represents represent VERB cana-5305 84 25 the the DET cana-5305 84 26 shortest short ADJ cana-5305 84 27 vector vector NOUN cana-5305 84 28 problem problem NOUN cana-5305 84 29 ( ( PUNCT cana-5305 84 30 svp svp NOUN cana-5305 84 31 ) ) PUNCT cana-5305 84 32 , , PUNCT cana-5305 84 33 which which PRON cana-5305 84 34 forms form VERB cana-5305 84 35 the the DET cana-5305 84 36 security security NOUN cana-5305 84 37 basis basis NOUN cana-5305 84 38 for for ADP cana-5305 84 39 latticebased latticebased ADJ cana-5305 84 40 encryption encryption NOUN cana-5305 84 41 schemes scheme NOUN cana-5305 84 42 . . PUNCT cana-5305 85 1 c. c. NOUN cana-5305 85 2 development development NOUN cana-5305 85 3 of of ADP cana-5305 85 4 secure secure ADJ cana-5305 85 5 cryptographic cryptographic ADJ cana-5305 85 6 framework framework NOUN cana-5305 85 7 the the DET cana-5305 85 8 development development NOUN cana-5305 85 9 of of ADP cana-5305 85 10 the the DET cana-5305 85 11 cryptographic cryptographic ADJ cana-5305 85 12 framework framework NOUN cana-5305 85 13 consists consist VERB cana-5305 85 14 of of ADP cana-5305 85 15 merging merge VERB cana-5305 85 16 the the DET cana-5305 85 17 identified identify VERB cana-5305 85 18 algebraic algebraic ADJ cana-5305 85 19 structures structure NOUN cana-5305 85 20 . . PUNCT cana-5305 86 1 a a DET cana-5305 86 2 secure secure ADJ cana-5305 86 3 key key ADJ cana-5305 86 4 exchange exchange NOUN cana-5305 86 5 uses use VERB cana-5305 86 6 rsa rsa PROPN cana-5305 86 7 in in ADP cana-5305 86 8 conjunction conjunction NOUN cana-5305 86 9 with with ADP cana-5305 86 10 ecc ecc PROPN cana-5305 86 11 through through ADP cana-5305 86 12 the the DET cana-5305 86 13 hybrid hybrid ADJ cana-5305 86 14 approach approach NOUN cana-5305 86 15 to to PART cana-5305 86 16 protect protect VERB cana-5305 86 17 against against ADP cana-5305 86 18 quantum quantum NOUN cana-5305 86 19 attacks attack NOUN cana-5305 86 20 by by ADP cana-5305 86 21 employing employ VERB cana-5305 86 22 lattice lattice NOUN cana-5305 86 23 - - PUNCT cana-5305 86 24 based base VERB cana-5305 86 25 encryption encryption NOUN cana-5305 86 26 . . PUNCT cana-5305 87 1 the the DET cana-5305 87 2 hybrid hybrid ADJ cana-5305 87 3 encryption encryption NOUN cana-5305 87 4 methodology methodology NOUN cana-5305 87 5 has have VERB cana-5305 87 6 a a DET cana-5305 87 7 system system NOUN cana-5305 87 8 model model NOUN cana-5305 87 9 which which PRON cana-5305 87 10 represents represent VERB cana-5305 87 11 the the DET cana-5305 87 12 following follow VERB cana-5305 87 13 sequence sequence NOUN cana-5305 87 14 : : PUNCT cana-5305 87 15 ciphertext ciphertext PROPN cana-5305 87 16 = = PUNCT cana-5305 88 1 𝐸rsa(π‘š)‖𝐸lattice 𝐸rsa(π‘š)‖𝐸lattice X cana-5305 88 2 ( ( PUNCT cana-5305 88 3 π‘š π‘š X cana-5305 88 4 ) ) PUNCT cana-5305 88 5 the the DET cana-5305 88 6 implementation implementation NOUN cana-5305 88 7 uses use VERB cana-5305 88 8 rsa rsa PROPN cana-5305 88 9 for for ADP cana-5305 88 10 classical classical ADJ cana-5305 88 11 security security NOUN cana-5305 88 12 together together ADV cana-5305 88 13 with with ADP cana-5305 88 14 lattice lattice NOUN cana-5305 88 15 - - PUNCT cana-5305 88 16 based base VERB cana-5305 88 17 schemes scheme NOUN cana-5305 88 18 for for ADP cana-5305 88 19 quantum quantum ADJ cana-5305 88 20 resistance resistance NOUN cana-5305 88 21 . . PUNCT cana-5305 89 1 the the DET cana-5305 89 2 hybrid hybrid ADJ cana-5305 89 3 framework framework NOUN cana-5305 89 4 maintains maintain VERB cana-5305 89 5 backward backward ADJ cana-5305 89 6 compatibility compatibility NOUN cana-5305 89 7 during during ADP cana-5305 89 8 this this DET cana-5305 89 9 time time NOUN cana-5305 89 10 to to PART cana-5305 89 11 create create VERB cana-5305 89 12 quantum quantum NOUN cana-5305 89 13 computing computing NOUN cana-5305 89 14 readiness readiness NOUN cana-5305 89 15 for for ADP cana-5305 89 16 future future ADJ cana-5305 89 17 operations operation NOUN cana-5305 89 18 . . PUNCT cana-5305 90 1 d. d. PROPN cana-5305 90 2 quantum quantum PROPN cana-5305 90 3 - - PUNCT cana-5305 90 4 safe safe ADJ cana-5305 90 5 cryptography cryptography NOUN cana-5305 90 6 via via ADP cana-5305 90 7 lattice lattice NOUN cana-5305 90 8 - - PUNCT cana-5305 90 9 based base VERB cana-5305 90 10 structures structure NOUN cana-5305 90 11 the the DET cana-5305 90 12 security security NOUN cana-5305 90 13 system system NOUN cana-5305 90 14 of of ADP cana-5305 90 15 lattice lattice NOUN cana-5305 90 16 - - PUNCT cana-5305 90 17 based base VERB cana-5305 90 18 cryptography cryptography NOUN cana-5305 90 19 depends depend VERB cana-5305 90 20 on on ADP cana-5305 90 21 the the DET cana-5305 90 22 difficulty difficulty NOUN cana-5305 90 23 of of ADP cana-5305 90 24 lattice lattice NOUN cana-5305 90 25 problems problem NOUN cana-5305 90 26 to to PART cana-5305 90 27 defend defend VERB cana-5305 90 28 against against ADP cana-5305 90 29 quantum quantum NOUN cana-5305 90 30 attacks attack NOUN cana-5305 90 31 . . PUNCT cana-5305 91 1 the the DET cana-5305 91 2 basic basic ADJ cana-5305 91 3 lattice lattice NOUN cana-5305 91 4 problem problem NOUN cana-5305 91 5 establishes establish VERB cana-5305 91 6 as as ADP cana-5305 91 7 : : PUNCT cana-5305 91 8 𝐀 𝐀 PROPN cana-5305 91 9 β‹… β‹… NOUN cana-5305 91 10 𝐯 𝐯 X cana-5305 91 11 = = SYM cana-5305 91 12 𝐛 𝐛 X cana-5305 91 13 where where SCONJ cana-5305 91 14 𝐀 𝐀 PROPN cana-5305 91 15 is be AUX cana-5305 91 16 a a DET cana-5305 91 17 matrix matrix NOUN cana-5305 91 18 defining define VERB cana-5305 91 19 the the DET cana-5305 91 20 lattice lattice NOUN cana-5305 91 21 , , PUNCT cana-5305 91 22 𝐯 𝐯 PROPN cana-5305 91 23 is be AUX cana-5305 91 24 the the DET cana-5305 91 25 vector vector NOUN cana-5305 91 26 to to PART cana-5305 91 27 be be AUX cana-5305 91 28 solved solve VERB cana-5305 91 29 , , PUNCT cana-5305 91 30 and and CCONJ cana-5305 91 31 𝐛 𝐛 PROPN cana-5305 91 32 is be AUX cana-5305 91 33 the the DET cana-5305 91 34 target target NOUN cana-5305 91 35 vector vector NOUN cana-5305 91 36 . . PUNCT cana-5305 92 1 nesplet nesplet PROPN cana-5305 92 2 solves solve VERB cana-5305 92 3 this this DET cana-5305 92 4 computational computational ADJ cana-5305 92 5 equation equation NOUN cana-5305 92 6 at at ADP cana-5305 92 7 such such DET cana-5305 92 8 a a DET cana-5305 92 9 high high ADJ cana-5305 92 10 level level NOUN cana-5305 92 11 that that SCONJ cana-5305 92 12 even even ADV cana-5305 92 13 quantum quantum ADJ cana-5305 92 14 computers computer NOUN cana-5305 92 15 can can AUX cana-5305 92 16 not not PART cana-5305 92 17 handle handle VERB cana-5305 92 18 it it PRON cana-5305 92 19 . . PUNCT cana-5305 93 1 the the DET cana-5305 93 2 basis basis NOUN cana-5305 93 3 for for ADP cana-5305 93 4 lattice lattice NOUN cana-5305 93 5 - - PUNCT cana-5305 93 6 based base VERB cana-5305 93 7 encryption encryption NOUN cana-5305 93 8 algorithms algorithm NOUN cana-5305 93 9 including include VERB cana-5305 93 10 learning learn VERB cana-5305 93 11 with with ADP cana-5305 93 12 errors error NOUN cana-5305 93 13 ( ( PUNCT cana-5305 93 14 lwe lwe ADV cana-5305 93 15 ) ) PUNCT cana-5305 93 16 represents represent VERB cana-5305 93 17 the the DET cana-5305 93 18 solution solution NOUN cana-5305 93 19 developed develop VERB cana-5305 93 20 from from ADP cana-5305 93 21 this this DET cana-5305 93 22 problem problem NOUN cana-5305 93 23 which which PRON cana-5305 93 24 quantum quantum NOUN cana-5305 93 25 computers computer NOUN cana-5305 93 26 view view VERB cana-5305 93 27 as as ADP cana-5305 93 28 quantum quantum NOUN cana-5305 93 29 - - PUNCT cana-5305 93 30 resistant resistant ADJ cana-5305 93 31 . . PUNCT cana-5305 94 1 e. e. PROPN cana-5305 94 2 cryptographic cryptographic ADJ cana-5305 94 3 operations operation NOUN cana-5305 94 4 flowchart flowchart NOUN cana-5305 94 5 the the DET cana-5305 94 6 proposed propose VERB cana-5305 94 7 framework framework NOUN cana-5305 94 8 involves involve VERB cana-5305 94 9 steps step NOUN cana-5305 94 10 that that PRON cana-5305 94 11 use use VERB cana-5305 94 12 cryptographic cryptographic ADJ cana-5305 94 13 operations operation NOUN cana-5305 94 14 as as SCONJ cana-5305 94 15 shown show VERB cana-5305 94 16 through through ADP cana-5305 94 17 this this DET cana-5305 94 18 flowchart flowchart NOUN cana-5305 94 19 . . PUNCT cana-5305 95 1 the the DET cana-5305 95 2 flowchart flowchart NOUN cana-5305 95 3 displays display VERB cana-5305 95 4 both both CCONJ cana-5305 95 5 classical classical ADJ cana-5305 95 6 and and CCONJ cana-5305 95 7 quantum quantum NOUN cana-5305 95 8 - - ADJ cana-5305 95 9 safe safe ADJ cana-5305 95 10 cryptographic cryptographic ADJ cana-5305 95 11 system system NOUN cana-5305 95 12 operations operation NOUN cana-5305 95 13 which which PRON cana-5305 95 14 depict depict VERB cana-5305 95 15 the the DET cana-5305 95 16 handling handling NOUN cana-5305 95 17 of of ADP cana-5305 95 18 encryption encryption NOUN cana-5305 95 19 and and CCONJ cana-5305 95 20 decryption decryption NOUN cana-5305 95 21 tasks task NOUN cana-5305 95 22 alongside alongside ADP cana-5305 95 23 key key ADJ cana-5305 95 24 generation generation NOUN cana-5305 95 25 implementation implementation NOUN cana-5305 95 26 based base VERB cana-5305 95 27 on on ADP cana-5305 95 28 algebraic algebraic ADJ cana-5305 95 29 structures structure NOUN cana-5305 95 30 . . PUNCT cana-5305 96 1 communications communication NOUN cana-5305 96 2 on on ADP cana-5305 96 3 applied apply VERB cana-5305 96 4 nonlinear nonlinear ADJ cana-5305 96 5 analysis analysis NOUN cana-5305 96 6 issn issn NOUN cana-5305 96 7 : : PUNCT cana-5305 96 8 1074 1074 NUM cana-5305 96 9 - - PUNCT cana-5305 96 10 133x 133x NUM cana-5305 96 11 vol vol NOUN cana-5305 96 12 31 31 NUM cana-5305 96 13 no no NOUN cana-5305 96 14 . . NOUN cana-5305 96 15 1 1 NUM cana-5305 96 16 ( ( PUNCT cana-5305 96 17 2024 2024 NUM cana-5305 96 18 ) ) PUNCT cana-5305 96 19 360 360 NUM cana-5305 96 20 https://internationalpubls.com https://internationalpubls.com X cana-5305 96 21 figure figure NOUN cana-5305 96 22 1 1 NUM cana-5305 96 23 : : PUNCT cana-5305 96 24 workflow workflow NOUN cana-5305 96 25 of of ADP cana-5305 96 26 the the DET cana-5305 96 27 proposed propose VERB cana-5305 96 28 hybrid hybrid ADJ cana-5305 96 29 cryptographic cryptographic ADJ cana-5305 96 30 framework framework NOUN cana-5305 96 31 f. f. PROPN cana-5305 96 32 implementation implementation PROPN cana-5305 96 33 and and CCONJ cana-5305 96 34 evaluation evaluation NOUN cana-5305 96 35 performance performance NOUN cana-5305 96 36 evaluation evaluation NOUN cana-5305 96 37 of of ADP cana-5305 96 38 the the DET cana-5305 96 39 cryptographic cryptographic ADJ cana-5305 96 40 framework framework NOUN cana-5305 96 41 happens happen VERB cana-5305 96 42 after after ADP cana-5305 96 43 its its PRON cana-5305 96 44 implementation implementation NOUN cana-5305 96 45 . . PUNCT cana-5305 97 1 the the DET cana-5305 97 2 evaluation evaluation NOUN cana-5305 97 3 tests test NOUN cana-5305 97 4 will will AUX cana-5305 97 5 run run VERB cana-5305 97 6 on on ADP cana-5305 97 7 various various ADJ cana-5305 97 8 platforms platform NOUN cana-5305 97 9 to to PART cana-5305 97 10 examine examine VERB cana-5305 97 11 the the DET cana-5305 97 12 performance performance NOUN cana-5305 97 13 metrics metric NOUN cana-5305 97 14 such such ADJ cana-5305 97 15 as as ADP cana-5305 97 16 computational computational ADJ cana-5305 97 17 overhead overhead NOUN cana-5305 97 18 and and CCONJ cana-5305 97 19 key key ADJ cana-5305 97 20 size size NOUN cana-5305 97 21 and and CCONJ cana-5305 97 22 security security NOUN cana-5305 97 23 strength strength NOUN cana-5305 97 24 . . PUNCT cana-5305 98 1 time time NOUN cana-5305 98 2 required require VERB cana-5305 98 3 for for ADP cana-5305 98 4 encryption encryption NOUN cana-5305 98 5 and and CCONJ cana-5305 98 6 decryption decryption NOUN cana-5305 98 7 runs run VERB cana-5305 98 8 alongside alongside ADP cana-5305 98 9 key key ADJ cana-5305 98 10 size size NOUN cana-5305 98 11 measurements measurement NOUN cana-5305 98 12 and and CCONJ cana-5305 98 13 quantum quantum NOUN cana-5305 98 14 - - PUNCT cana-5305 98 15 resistant resistant ADJ cana-5305 98 16 system system NOUN cana-5305 98 17 assessment assessment NOUN cana-5305 98 18 as as ADP cana-5305 98 19 key key ADJ cana-5305 98 20 performance performance NOUN cana-5305 98 21 factors factor NOUN cana-5305 98 22 . . PUNCT cana-5305 99 1 iv iv X cana-5305 99 2 . . PROPN cana-5305 99 3 result result PROPN cana-5305 99 4 & & CCONJ cana-5305 99 5 discussions discussion NOUN cana-5305 99 6 the the DET cana-5305 99 7 cryptographic cryptographic ADJ cana-5305 99 8 system system NOUN cana-5305 99 9 was be AUX cana-5305 99 10 applied apply VERB cana-5305 99 11 to to ADP cana-5305 99 12 classical classical ADJ cana-5305 99 13 and and CCONJ cana-5305 99 14 quantum quantum NOUN cana-5305 99 15 - - ADJ cana-5305 99 16 safe safe ADJ cana-5305 99 17 systems system NOUN cana-5305 99 18 during during ADP cana-5305 99 19 testing testing NOUN cana-5305 99 20 which which PRON cana-5305 99 21 led lead VERB cana-5305 99 22 researchers researcher NOUN cana-5305 99 23 to to PART cana-5305 99 24 compare compare VERB cana-5305 99 25 them they PRON cana-5305 99 26 with with ADP cana-5305 99 27 rsa rsa PROPN cana-5305 99 28 and and CCONJ cana-5305 99 29 ecc ecc PROPN cana-5305 99 30 - - PUNCT cana-5305 99 31 based base VERB cana-5305 99 32 systems system NOUN cana-5305 99 33 . . PUNCT cana-5305 100 1 the the DET cana-5305 100 2 research research NOUN cana-5305 100 3 team team NOUN cana-5305 100 4 analyzed analyze VERB cana-5305 100 5 performance performance NOUN cana-5305 100 6 metrics metric NOUN cana-5305 100 7 using use VERB cana-5305 100 8 three three NUM cana-5305 100 9 criteria criterion NOUN cana-5305 100 10 including include VERB cana-5305 100 11 operational operational ADJ cana-5305 100 12 speed speed NOUN cana-5305 100 13 together together ADV cana-5305 100 14 with with ADP cana-5305 100 15 cryptographic cryptographic ADJ cana-5305 100 16 strength strength NOUN cana-5305 100 17 level level NOUN cana-5305 100 18 and and CCONJ cana-5305 100 19 programming programming NOUN cana-5305 100 20 platform platform NOUN cana-5305 100 21 adaptability adaptability NOUN cana-5305 100 22 [ [ X cana-5305 100 23 11 11 NUM cana-5305 100 24 ] ] PUNCT cana-5305 100 25 . . PUNCT cana-5305 101 1 a a DET cana-5305 101 2 random random ADJ cana-5305 101 3 collection collection NOUN cana-5305 101 4 of of ADP cana-5305 101 5 messages message NOUN cana-5305 101 6 with with ADP cana-5305 101 7 different different ADJ cana-5305 101 8 lengths length NOUN cana-5305 101 9 underwent undergo VERB cana-5305 101 10 experimental experimental ADJ cana-5305 101 11 evaluation evaluation NOUN cana-5305 101 12 according accord VERB cana-5305 101 13 to to ADP cana-5305 101 14 the the DET cana-5305 101 15 setup setup NOUN cana-5305 101 16 . . PUNCT cana-5305 102 1 rsa rsa PROPN cana-5305 102 2 algorithm algorithm PROPN cana-5305 102 3 and and CCONJ cana-5305 102 4 ecc ecc PROPN cana-5305 102 5 together together ADV cana-5305 102 6 with with ADP cana-5305 102 7 their their PRON cana-5305 102 8 hybrid hybrid ADJ cana-5305 102 9 encryption encryption NOUN cana-5305 102 10 system system NOUN cana-5305 102 11 ( ( PUNCT cana-5305 102 12 joining join VERB cana-5305 102 13 rsa rsa PROPN cana-5305 102 14 with with ADP cana-5305 102 15 ecc ecc PROPN cana-5305 102 16 and and CCONJ cana-5305 102 17 lattice lattice NOUN cana-5305 102 18 - - PUNCT cana-5305 102 19 based base VERB cana-5305 102 20 encryption encryption NOUN cana-5305 102 21 ) ) PUNCT cana-5305 102 22 underwent underwent NOUN cana-5305 102 23 evaluation evaluation NOUN cana-5305 102 24 regarding regard VERB cana-5305 102 25 encryption encryption NOUN cana-5305 102 26 speed speed NOUN cana-5305 102 27 , , PUNCT cana-5305 102 28 decryption decryption NOUN cana-5305 102 29 speed speed NOUN cana-5305 102 30 , , PUNCT cana-5305 102 31 key key ADJ cana-5305 102 32 length length NOUN cana-5305 102 33 strength strength NOUN cana-5305 102 34 and and CCONJ cana-5305 102 35 security security NOUN cana-5305 102 36 level level NOUN cana-5305 102 37 . . PUNCT cana-5305 103 1 maintenance maintenance NOUN cana-5305 103 2 of of ADP cana-5305 103 3 security security NOUN cana-5305 103 4 in in ADP cana-5305 103 5 the the DET cana-5305 103 6 post post ADJ cana-5305 103 7 - - ADJ cana-5305 103 8 quantum quantum ADJ cana-5305 103 9 period period NOUN cana-5305 103 10 became become VERB cana-5305 103 11 the the DET cana-5305 103 12 essential essential ADJ cana-5305 103 13 factor factor NOUN cana-5305 103 14 which which PRON cana-5305 103 15 contributed contribute VERB cana-5305 103 16 to to ADP cana-5305 103 17 the the DET cana-5305 103 18 communications communication NOUN cana-5305 103 19 on on ADP cana-5305 103 20 applied apply VERB cana-5305 103 21 nonlinear nonlinear ADJ cana-5305 103 22 analysis analysis NOUN cana-5305 103 23 issn issn NOUN cana-5305 103 24 : : PUNCT cana-5305 103 25 1074 1074 NUM cana-5305 103 26 - - PUNCT cana-5305 103 27 133x 133x NUM cana-5305 103 28 vol vol NOUN cana-5305 103 29 31 31 NUM cana-5305 103 30 no no NOUN cana-5305 103 31 . . NOUN cana-5305 103 32 1 1 NUM cana-5305 103 33 ( ( PUNCT cana-5305 103 34 2024 2024 NUM cana-5305 103 35 ) ) PUNCT cana-5305 103 36 361 361 NUM cana-5305 103 37 https://internationalpubls.com https://internationalpubls.com NUM cana-5305 103 38 system system NOUN cana-5305 103 39 's 's PART cana-5305 103 40 evaluation evaluation NOUN cana-5305 103 41 . . PUNCT cana-5305 104 1 the the DET cana-5305 104 2 rsa rsa PROPN cana-5305 104 3 and and CCONJ cana-5305 104 4 ecc ecc PROPN cana-5305 104 5 systems system NOUN cana-5305 104 6 showed show VERB cana-5305 104 7 their their PRON cana-5305 104 8 key key ADJ cana-5305 104 9 generation generation NOUN cana-5305 104 10 times time NOUN cana-5305 104 11 to to PART cana-5305 104 12 be be AUX cana-5305 104 13 the the DET cana-5305 104 14 most most ADV cana-5305 104 15 critical critical ADJ cana-5305 104 16 factor factor NOUN cana-5305 104 17 which which PRON cana-5305 104 18 created create VERB cana-5305 104 19 bottlenecks bottleneck NOUN cana-5305 104 20 because because SCONJ cana-5305 104 21 rsa rsa PROPN cana-5305 104 22 needed need VERB cana-5305 104 23 larger large ADJ cana-5305 104 24 key key ADJ cana-5305 104 25 sizes size NOUN cana-5305 104 26 compared compare VERB cana-5305 104 27 to to ADP cana-5305 104 28 ecc ecc PROPN cana-5305 104 29 systems system NOUN cana-5305 104 30 for for ADP cana-5305 104 31 matching match VERB cana-5305 104 32 security security NOUN cana-5305 104 33 levels level NOUN cana-5305 104 34 . . PUNCT cana-5305 105 1 the the DET cana-5305 105 2 hybrid hybrid ADJ cana-5305 105 3 system system NOUN cana-5305 105 4 could could AUX cana-5305 105 5 reduce reduce VERB cana-5305 105 6 key key ADJ cana-5305 105 7 lengths length NOUN cana-5305 105 8 effectively effectively ADV cana-5305 105 9 because because SCONJ cana-5305 105 10 it it PRON cana-5305 105 11 implements implement VERB cana-5305 105 12 the the DET cana-5305 105 13 lattice lattice NOUN cana-5305 105 14 - - PUNCT cana-5305 105 15 based base VERB cana-5305 105 16 cryptographic cryptographic ADJ cana-5305 105 17 method method NOUN cana-5305 105 18 which which PRON cana-5305 105 19 preserves preserve VERB cana-5305 105 20 high high ADJ cana-5305 105 21 security security NOUN cana-5305 105 22 levels level NOUN cana-5305 105 23 . . PUNCT cana-5305 106 1 table table NOUN cana-5305 106 2 1 1 NUM cana-5305 106 3 shows show VERB cana-5305 106 4 the the DET cana-5305 106 5 key key ADJ cana-5305 106 6 generation generation NOUN cana-5305 106 7 duration duration NOUN cana-5305 106 8 and and CCONJ cana-5305 106 9 key key ADJ cana-5305 106 10 length length NOUN cana-5305 106 11 information information NOUN cana-5305 106 12 of of ADP cana-5305 106 13 all all DET cana-5305 106 14 three three NUM cana-5305 106 15 systems system NOUN cana-5305 106 16 according accord VERB cana-5305 106 17 to to ADP cana-5305 106 18 the the DET cana-5305 106 19 evaluation evaluation NOUN cana-5305 106 20 results result NOUN cana-5305 106 21 . . PUNCT cana-5305 107 1 rsa rsa PROPN cana-5305 107 2 proved prove VERB cana-5305 107 3 to to PART cana-5305 107 4 have have VERB cana-5305 107 5 the the DET cana-5305 107 6 biggest big ADJ cana-5305 107 7 key key ADJ cana-5305 107 8 dimensions dimension NOUN cana-5305 107 9 however however ADV cana-5305 107 10 ecc ecc PROPN cana-5305 107 11 functioned function VERB cana-5305 107 12 as as ADP cana-5305 107 13 a a DET cana-5305 107 14 better well ADJ cana-5305 107 15 efficiency efficiency NOUN cana-5305 107 16 solution solution NOUN cana-5305 107 17 . . PUNCT cana-5305 108 1 table table NOUN cana-5305 108 2 1 1 NUM cana-5305 108 3 : : PUNCT cana-5305 108 4 comparison comparison NOUN cana-5305 108 5 of of ADP cana-5305 108 6 key key ADJ cana-5305 108 7 generation generation NOUN cana-5305 108 8 time time NOUN cana-5305 108 9 and and CCONJ cana-5305 108 10 key key ADJ cana-5305 108 11 sizes size NOUN cana-5305 108 12 cryptographic cryptographic ADJ cana-5305 108 13 system system NOUN cana-5305 108 14 key key ADJ cana-5305 108 15 size size NOUN cana-5305 108 16 ( ( PUNCT cana-5305 108 17 bits bits NOUN cana-5305 108 18 ) ) PUNCT cana-5305 108 19 key key ADJ cana-5305 108 20 generation generation NOUN cana-5305 108 21 time time NOUN cana-5305 108 22 ( ( PUNCT cana-5305 108 23 ms ms PROPN cana-5305 108 24 ) ) PUNCT cana-5305 108 25 rsa rsa PROPN cana-5305 108 26 2048 2048 NUM cana-5305 108 27 1200 1200 NUM cana-5305 109 1 ecc ecc PROPN cana-5305 109 2 256 256 NUM cana-5305 109 3 800 800 NUM cana-5305 109 4 hybrid hybrid NOUN cana-5305 109 5 ( ( PUNCT cana-5305 109 6 rsa rsa NOUN cana-5305 109 7 + + CCONJ cana-5305 109 8 ecc ecc PROPN cana-5305 109 9 + + CCONJ cana-5305 109 10 lattice lattice NOUN cana-5305 109 11 ) ) PUNCT cana-5305 109 12 512 512 NUM cana-5305 109 13 1100 1100 NUM cana-5305 109 14 a a DET cana-5305 109 15 test test NOUN cana-5305 109 16 of of ADP cana-5305 109 17 message message NOUN cana-5305 109 18 processing processing NOUN cana-5305 109 19 time time NOUN cana-5305 109 20 was be AUX cana-5305 109 21 conducted conduct VERB cana-5305 109 22 with with ADP cana-5305 109 23 encryption encryption NOUN cana-5305 109 24 and and CCONJ cana-5305 109 25 decryption decryption NOUN cana-5305 109 26 operations operation NOUN cana-5305 109 27 performed perform VERB cana-5305 109 28 on on ADP cana-5305 109 29 plaintexts plaintexts NOUN cana-5305 109 30 with with ADP cana-5305 109 31 sizes size NOUN cana-5305 109 32 spanning span VERB cana-5305 109 33 from from ADP cana-5305 109 34 512 512 NUM cana-5305 109 35 bits bit NOUN cana-5305 109 36 to to ADP cana-5305 109 37 4096 4096 NUM cana-5305 109 38 bits bit NOUN cana-5305 109 39 . . PUNCT cana-5305 110 1 both both DET cana-5305 110 2 encryption encryption NOUN cana-5305 110 3 and and CCONJ cana-5305 110 4 decryption decryption NOUN cana-5305 110 5 operations operation NOUN cana-5305 110 6 with with ADP cana-5305 110 7 the the DET cana-5305 110 8 hybrid hybrid ADJ cana-5305 110 9 system system NOUN cana-5305 110 10 proceeded proceed VERB cana-5305 110 11 faster fast ADV cana-5305 110 12 than than ADP cana-5305 110 13 rsa rsa NOUN cana-5305 110 14 procedures procedure NOUN cana-5305 110 15 because because SCONJ cana-5305 110 16 elliptic elliptic ADJ cana-5305 110 17 curve curve NOUN cana-5305 110 18 and and CCONJ cana-5305 110 19 lattice lattice NOUN cana-5305 110 20 - - PUNCT cana-5305 110 21 based base VERB cana-5305 110 22 procedures procedure NOUN cana-5305 110 23 demonstrated demonstrate VERB cana-5305 110 24 improved improved ADJ cana-5305 110 25 efficiency efficiency NOUN cana-5305 110 26 . . PUNCT cana-5305 111 1 the the DET cana-5305 111 2 hybrid hybrid ADJ cana-5305 111 3 system system NOUN cana-5305 111 4 achieved achieve VERB cana-5305 111 5 slower slow ADJ cana-5305 111 6 speeds speed NOUN cana-5305 111 7 than than ADP cana-5305 111 8 ecc ecc PROPN cana-5305 111 9 for for ADP cana-5305 111 10 smaller small ADJ cana-5305 111 11 data datum NOUN cana-5305 111 12 because because SCONJ cana-5305 111 13 lattice lattice NOUN cana-5305 111 14 encryption encryption NOUN cana-5305 111 15 introduced introduce VERB cana-5305 111 16 extra extra ADJ cana-5305 111 17 processing processing NOUN cana-5305 111 18 requirements requirement NOUN cana-5305 111 19 . . PUNCT cana-5305 112 1 security security NOUN cana-5305 112 2 enhancements enhancement NOUN cana-5305 112 3 of of ADP cana-5305 112 4 hybrid hybrid NOUN cana-5305 112 5 systems system NOUN cana-5305 112 6 became become VERB cana-5305 112 7 more more ADV cana-5305 112 8 apparent apparent ADJ cana-5305 112 9 while while SCONJ cana-5305 112 10 processing process VERB cana-5305 112 11 bigger big ADJ cana-5305 112 12 messages message NOUN cana-5305 112 13 because because SCONJ cana-5305 112 14 lattice lattice NOUN cana-5305 112 15 - - PUNCT cana-5305 112 16 based base VERB cana-5305 112 17 encryption encryption NOUN cana-5305 112 18 methods method NOUN cana-5305 112 19 deliver deliver VERB cana-5305 112 20 faster fast ADJ cana-5305 112 21 processing processing NOUN cana-5305 112 22 for for ADP cana-5305 112 23 larger large ADJ cana-5305 112 24 data datum NOUN cana-5305 112 25 sets set NOUN cana-5305 112 26 . . PUNCT cana-5305 113 1 figure figure NOUN cana-5305 113 2 1 1 NUM cana-5305 113 3 : : PUNCT cana-5305 113 4 encryption encryption NOUN cana-5305 113 5 time time NOUN cana-5305 113 6 ( ( PUNCT cana-5305 113 7 ms ms NOUN cana-5305 113 8 ) ) PUNCT cana-5305 113 9 vs vs ADP cana-5305 113 10 message message NOUN cana-5305 113 11 size size NOUN cana-5305 113 12 ( ( PUNCT cana-5305 113 13 bits bits PROPN cana-5305 113 14 ) ) PUNCT cana-5305 113 15 the the DET cana-5305 113 16 hybrid hybrid ADJ cana-5305 113 17 system system NOUN cana-5305 113 18 provides provide VERB cana-5305 113 19 improved improved ADJ cana-5305 113 20 scalability scalability NOUN cana-5305 113 21 with with ADP cana-5305 113 22 large large ADJ cana-5305 113 23 messages message NOUN cana-5305 113 24 but but CCONJ cana-5305 113 25 ecc ecc NOUN cana-5305 113 26 proves prove VERB cana-5305 113 27 more more ADV cana-5305 113 28 efficient efficient ADJ cana-5305 113 29 with with ADP cana-5305 113 30 small small ADJ cana-5305 113 31 messages message NOUN cana-5305 113 32 according accord VERB cana-5305 113 33 to to ADP cana-5305 113 34 the the DET cana-5305 113 35 data datum NOUN cana-5305 113 36 . . PUNCT cana-5305 114 1 designing design VERB cana-5305 114 2 cryptographic cryptographic ADJ cana-5305 114 3 systems system NOUN cana-5305 114 4 needs need VERB cana-5305 114 5 to to PART cana-5305 114 6 account account VERB cana-5305 114 7 for for ADP cana-5305 114 8 this this DET cana-5305 114 9 consideration consideration NOUN cana-5305 114 10 in in ADP cana-5305 114 11 the the DET cana-5305 114 12 planning planning NOUN cana-5305 114 13 of of ADP cana-5305 114 14 mobile mobile ADJ cana-5305 114 15 communication communication NOUN cana-5305 114 16 systems system NOUN cana-5305 114 17 as as ADV cana-5305 114 18 well well ADV cana-5305 114 19 as as ADP cana-5305 114 20 1200 1200 NUM cana-5305 114 21 1500 1500 NUM cana-5305 114 22 1800 1800 NUM cana-5305 115 1 600 600 NUM cana-5305 115 2 800 800 NUM cana-5305 115 3 1000 1000 NUM cana-5305 115 4 1100 1100 NUM cana-5305 115 5 1300 1300 NUM cana-5305 115 6 1500 1500 NUM cana-5305 115 7 0 0 NUM cana-5305 116 1 200 200 NUM cana-5305 116 2 400 400 NUM cana-5305 116 3 600 600 NUM cana-5305 116 4 800 800 NUM cana-5305 116 5 1000 1000 NUM cana-5305 116 6 1200 1200 NUM cana-5305 116 7 1400 1400 NUM cana-5305 116 8 1600 1600 NUM cana-5305 116 9 1800 1800 NUM cana-5305 116 10 2000 2000 NUM cana-5305 116 11 512 512 NUM cana-5305 116 12 2048 2048 NUM cana-5305 116 13 4096 4096 NUM cana-5305 116 14 encryption encryption NOUN cana-5305 116 15 time time NOUN cana-5305 116 16 ( ( PUNCT cana-5305 116 17 ms ms NOUN cana-5305 116 18 ) ) PUNCT cana-5305 116 19 vs vs ADP cana-5305 116 20 message message NOUN cana-5305 116 21 size size NOUN cana-5305 116 22 ( ( PUNCT cana-5305 116 23 bits bits NOUN cana-5305 116 24 ) ) PUNCT cana-5305 116 25 hybrid hybrid NOUN cana-5305 116 26 system system NOUN cana-5305 116 27 encryption encryption NOUN cana-5305 116 28 time time NOUN cana-5305 116 29 ( ( PUNCT cana-5305 116 30 ms ms PROPN cana-5305 116 31 ) ) PUNCT cana-5305 116 32 ecc ecc PROPN cana-5305 116 33 encryption encryption NOUN cana-5305 116 34 time time NOUN cana-5305 116 35 ( ( PUNCT cana-5305 116 36 ms ms PROPN cana-5305 116 37 ) ) PUNCT cana-5305 116 38 rsa rsa PROPN cana-5305 116 39 encryption encryption NOUN cana-5305 116 40 time time NOUN cana-5305 116 41 ( ( PUNCT cana-5305 116 42 ms ms NOUN cana-5305 116 43 ) ) PUNCT cana-5305 116 44 communications communication NOUN cana-5305 116 45 on on ADP cana-5305 116 46 applied apply VERB cana-5305 116 47 nonlinear nonlinear ADJ cana-5305 116 48 analysis analysis NOUN cana-5305 116 49 issn issn NOUN cana-5305 116 50 : : PUNCT cana-5305 116 51 1074 1074 NUM cana-5305 116 52 - - PUNCT cana-5305 116 53 133x 133x NUM cana-5305 116 54 vol vol NOUN cana-5305 116 55 31 31 NUM cana-5305 116 56 no no NOUN cana-5305 116 57 . . NOUN cana-5305 116 58 1 1 NUM cana-5305 116 59 ( ( PUNCT cana-5305 116 60 2024 2024 NUM cana-5305 116 61 ) ) PUNCT cana-5305 116 62 362 362 NUM cana-5305 116 63 https://internationalpubls.com https://internationalpubls.com X cana-5305 116 64 cloud cloud NOUN cana-5305 116 65 storage storage NOUN cana-5305 116 66 solutions solution NOUN cana-5305 116 67 and and CCONJ cana-5305 116 68 iot iot ADJ cana-5305 116 69 device device NOUN cana-5305 116 70 applications application NOUN cana-5305 116 71 . . PUNCT cana-5305 117 1 hybrid hybrid ADJ cana-5305 117 2 systems system NOUN cana-5305 117 3 feature feature VERB cana-5305 117 4 quantum quantum NOUN cana-5305 117 5 - - PUNCT cana-5305 117 6 resistant resistant ADJ cana-5305 117 7 security security NOUN cana-5305 117 8 that that PRON cana-5305 117 9 necessitates necessitate VERB cana-5305 117 10 their their PRON cana-5305 117 11 increased increase VERB cana-5305 117 12 cost cost NOUN cana-5305 117 13 because because SCONJ cana-5305 117 14 they they PRON cana-5305 117 15 secure secure VERB cana-5305 117 16 upcoming upcoming ADJ cana-5305 117 17 communication communication NOUN cana-5305 117 18 from from ADP cana-5305 117 19 upcoming upcoming ADJ cana-5305 117 20 quantum quantum ADJ cana-5305 117 21 computing computing NOUN cana-5305 117 22 threats threat NOUN cana-5305 117 23 . . PUNCT cana-5305 118 1 a a DET cana-5305 118 2 security security NOUN cana-5305 118 3 assessment assessment NOUN cana-5305 118 4 was be AUX cana-5305 118 5 conducted conduct VERB cana-5305 118 6 by by ADP cana-5305 118 7 applying apply VERB cana-5305 118 8 tests test NOUN cana-5305 118 9 on on ADP cana-5305 118 10 the the DET cana-5305 118 11 systems system NOUN cana-5305 118 12 to to PART cana-5305 118 13 identify identify VERB cana-5305 118 14 vulnerability vulnerability NOUN cana-5305 118 15 to to ADP cana-5305 118 16 standard standard ADJ cana-5305 118 17 attacks attack NOUN cana-5305 118 18 such such ADJ cana-5305 118 19 as as ADP cana-5305 118 20 brute brute ADJ cana-5305 118 21 - - PUNCT cana-5305 118 22 force force NOUN cana-5305 118 23 methods method NOUN cana-5305 118 24 and and CCONJ cana-5305 118 25 chosen choose VERB cana-5305 118 26 - - PUNCT cana-5305 118 27 plaintext plaintext NOUN cana-5305 118 28 operations operation NOUN cana-5305 118 29 and and CCONJ cana-5305 118 30 quantum quantum NOUN cana-5305 118 31 - - PUNCT cana-5305 118 32 based base VERB cana-5305 118 33 threats threat NOUN cana-5305 118 34 . . PUNCT cana-5305 119 1 integers integer NOUN cana-5305 119 2 in in ADP cana-5305 119 3 rsa rsa PROPN cana-5305 119 4 and and CCONJ cana-5305 119 5 ecc ecc PROPN cana-5305 119 6 systems system NOUN cana-5305 119 7 suffer suffer VERB cana-5305 119 8 from from ADP cana-5305 119 9 quantum quantum ADJ cana-5305 119 10 attack attack NOUN cana-5305 119 11 vulnerabilities vulnerability NOUN cana-5305 119 12 that that PRON cana-5305 119 13 shor shor PROPN cana-5305 119 14 's 's PART cana-5305 119 15 algorithm algorithm NOUN cana-5305 119 16 exploits exploit VERB cana-5305 119 17 to to PART cana-5305 119 18 break break VERB cana-5305 119 19 the the DET cana-5305 119 20 discrete discrete ADJ cana-5305 119 21 logarithm logarithm NOUN cana-5305 119 22 and and CCONJ cana-5305 119 23 integer integer NOUN cana-5305 119 24 factorization factorization NOUN cana-5305 119 25 problems problem NOUN cana-5305 119 26 easily easily ADV cana-5305 119 27 . . PUNCT cana-5305 120 1 analysts analyst NOUN cana-5305 120 2 evaluated evaluate VERB cana-5305 120 3 the the DET cana-5305 120 4 proposed propose VERB cana-5305 120 5 hybrid hybrid NOUN cana-5305 120 6 system system NOUN cana-5305 120 7 through through ADP cana-5305 120 8 analysis analysis NOUN cana-5305 120 9 of of ADP cana-5305 120 10 the the DET cana-5305 120 11 learning learning NOUN cana-5305 120 12 with with ADP cana-5305 120 13 errors error NOUN cana-5305 120 14 ( ( PUNCT cana-5305 120 15 lwe lwe PROPN cana-5305 120 16 ) ) PUNCT cana-5305 120 17 problem problem NOUN cana-5305 120 18 whose whose DET cana-5305 120 19 status status NOUN cana-5305 120 20 as as ADP cana-5305 120 21 a a DET cana-5305 120 22 key key ADJ cana-5305 120 23 challenge challenge NOUN cana-5305 120 24 exists exist VERB cana-5305 120 25 in in ADP cana-5305 120 26 latticebased latticebased ADJ cana-5305 120 27 cryptography cryptography NOUN cana-5305 120 28 . . PUNCT cana-5305 121 1 the the DET cana-5305 121 2 lwe lwe PROPN cana-5305 121 3 problem problem NOUN cana-5305 121 4 maintained maintain VERB cana-5305 121 5 an an DET cana-5305 121 6 uncrack uncrack NOUN cana-5305 121 7 able able ADJ cana-5305 121 8 status status NOUN cana-5305 121 9 despite despite SCONJ cana-5305 121 10 quantum quantum NOUN cana-5305 121 11 attacks attack NOUN cana-5305 121 12 because because SCONJ cana-5305 121 13 there there PRON cana-5305 121 14 is be VERB cana-5305 121 15 no no DET cana-5305 121 16 known know VERB cana-5305 121 17 method method NOUN cana-5305 121 18 for for ADP cana-5305 121 19 quantum quantum ADJ cana-5305 121 20 computers computer NOUN cana-5305 121 21 to to PART cana-5305 121 22 easily easily ADV cana-5305 121 23 solve solve VERB cana-5305 121 24 it it PRON cana-5305 121 25 . . PUNCT cana-5305 122 1 figure figure VERB cana-5305 122 2 2 2 NUM cana-5305 122 3 : : PUNCT cana-5305 122 4 decryption decryption NOUN cana-5305 122 5 time time NOUN cana-5305 122 6 ( ( PUNCT cana-5305 122 7 ms ms NOUN cana-5305 122 8 ) ) PUNCT cana-5305 122 9 vs vs ADP cana-5305 122 10 message message NOUN cana-5305 122 11 size size NOUN cana-5305 122 12 ( ( PUNCT cana-5305 122 13 bits bits NOUN cana-5305 122 14 ) ) PUNCT cana-5305 122 15 decryption decryption NOUN cana-5305 122 16 times time NOUN cana-5305 122 17 demonstrated demonstrate VERB cana-5305 122 18 parallel parallel ADJ cana-5305 122 19 behavior behavior NOUN cana-5305 122 20 when when SCONJ cana-5305 122 21 the the DET cana-5305 122 22 hybrid hybrid ADJ cana-5305 122 23 system system NOUN cana-5305 122 24 processed process VERB cana-5305 122 25 larger large ADJ cana-5305 122 26 messages message NOUN cana-5305 122 27 because because SCONJ cana-5305 122 28 it it PRON cana-5305 122 29 achieved achieve VERB cana-5305 122 30 enhanced enhanced ADJ cana-5305 122 31 scalability scalability NOUN cana-5305 122 32 . . PUNCT cana-5305 123 1 lattice lattice NOUN cana-5305 123 2 - - PUNCT cana-5305 123 3 based base VERB cana-5305 123 4 encryption encryption NOUN cana-5305 123 5 added add VERB cana-5305 123 6 some some DET cana-5305 123 7 duration duration NOUN cana-5305 123 8 to to ADP cana-5305 123 9 decryption decryption NOUN cana-5305 123 10 procedures procedure NOUN cana-5305 123 11 in in ADP cana-5305 123 12 comparison comparison NOUN cana-5305 123 13 to to ADP cana-5305 123 14 ecc ecc PROPN cana-5305 123 15 . . PROPN cana-5305 123 16 message message NOUN cana-5305 123 17 encryption encryption NOUN cana-5305 123 18 using use VERB cana-5305 123 19 latticebased latticebased ADJ cana-5305 123 20 cryptography cryptography NOUN cana-5305 123 21 required require VERB cana-5305 123 22 longer long ADJ cana-5305 123 23 decryption decryption NOUN cana-5305 123 24 times time NOUN cana-5305 123 25 particularly particularly ADV cana-5305 123 26 with with ADP cana-5305 123 27 short short ADJ cana-5305 123 28 messages message NOUN cana-5305 123 29 because because SCONJ cana-5305 123 30 of of ADP cana-5305 123 31 its its PRON cana-5305 123 32 added add VERB cana-5305 123 33 cryptographic cryptographic ADJ cana-5305 123 34 complexity complexity NOUN cana-5305 123 35 . . PUNCT cana-5305 124 1 the the DET cana-5305 124 2 fundamental fundamental ADJ cana-5305 124 3 challenge challenge NOUN cana-5305 124 4 in in ADP cana-5305 124 5 designing design VERB cana-5305 124 6 nextgeneration nextgeneration NOUN cana-5305 124 7 cryptographic cryptographic ADJ cana-5305 124 8 environments environment NOUN cana-5305 124 9 exists exist VERB cana-5305 124 10 because because SCONJ cana-5305 124 11 security security NOUN cana-5305 124 12 will will AUX cana-5305 124 13 strengthen strengthen VERB cana-5305 124 14 significantly significantly ADV cana-5305 124 15 following follow VERB cana-5305 124 16 the the DET cana-5305 124 17 emergence emergence NOUN cana-5305 124 18 of of ADP cana-5305 124 19 quantum quantum ADJ cana-5305 124 20 computation computation NOUN cana-5305 124 21 . . PUNCT cana-5305 125 1 deciding decide VERB cana-5305 125 2 between between ADP cana-5305 125 3 reduced reduce VERB cana-5305 125 4 key key ADJ cana-5305 125 5 sizes size NOUN cana-5305 125 6 or or CCONJ cana-5305 125 7 additional additional ADJ cana-5305 125 8 workload workload NOUN cana-5305 125 9 necessary necessary ADJ cana-5305 125 10 for for ADP cana-5305 125 11 evaluation evaluation NOUN cana-5305 125 12 involved involve VERB cana-5305 125 13 multiple multiple ADJ cana-5305 125 14 factors factor NOUN cana-5305 125 15 when when SCONJ cana-5305 125 16 operating operating NOUN cana-5305 125 17 system system NOUN cana-5305 125 18 and and CCONJ cana-5305 125 19 iot iot NOUN cana-5305 125 20 devices device NOUN cana-5305 125 21 faced face VERB cana-5305 125 22 limited limited ADJ cana-5305 125 23 conditions condition NOUN cana-5305 125 24 . . PUNCT cana-5305 126 1 the the DET cana-5305 126 2 hybrid hybrid ADJ cana-5305 126 3 system system NOUN cana-5305 126 4 functions function NOUN cana-5305 126 5 at at ADP cana-5305 126 6 a a DET cana-5305 126 7 decreased decreased ADJ cana-5305 126 8 speed speed NOUN cana-5305 126 9 but but CCONJ cana-5305 126 10 enables enable VERB cana-5305 126 11 improved improved ADJ cana-5305 126 12 security security NOUN cana-5305 126 13 features feature VERB cana-5305 126 14 appropriate appropriate ADJ cana-5305 126 15 for for ADP cana-5305 126 16 device device NOUN cana-5305 126 17 protection protection NOUN cana-5305 126 18 until until ADP cana-5305 126 19 the the DET cana-5305 126 20 rise rise NOUN cana-5305 126 21 of of ADP cana-5305 126 22 quantum quantum NOUN cana-5305 126 23 computing computing NOUN cana-5305 126 24 technology technology NOUN cana-5305 126 25 . . PUNCT cana-5305 127 1 the the DET cana-5305 127 2 system system NOUN cana-5305 127 3 provides provide VERB cana-5305 127 4 capability capability NOUN cana-5305 127 5 to to PART cana-5305 127 6 create create VERB cana-5305 127 7 brief brief ADJ cana-5305 127 8 keys key NOUN cana-5305 127 9 using use VERB cana-5305 127 10 decreased decrease VERB cana-5305 127 11 system system NOUN cana-5305 127 12 resources resource NOUN cana-5305 127 13 essential essential ADJ cana-5305 127 14 for for ADP cana-5305 127 15 resourcelimited resourcelimited ADJ cana-5305 127 16 devices device NOUN cana-5305 127 17 . . PUNCT cana-5305 128 1 due due ADP cana-5305 128 2 to to ADP cana-5305 128 3 its its PRON cana-5305 128 4 quantum quantum ADJ cana-5305 128 5 resistance resistance NOUN cana-5305 128 6 capabilities capability NOUN cana-5305 128 7 this this DET cana-5305 128 8 hybrid hybrid ADJ cana-5305 128 9 system system NOUN cana-5305 128 10 demonstrates demonstrate VERB cana-5305 128 11 1600 1600 NUM cana-5305 128 12 1800 1800 NUM cana-5305 128 13 2100 2100 NUM cana-5305 129 1 650 650 NUM cana-5305 129 2 700 700 NUM cana-5305 129 3 900 900 NUM cana-5305 129 4 1200 1200 NUM cana-5305 129 5 1400 1400 NUM cana-5305 129 6 1600 1600 NUM cana-5305 129 7 0 0 NUM cana-5305 129 8 500 500 NUM cana-5305 129 9 1000 1000 NUM cana-5305 129 10 1500 1500 NUM cana-5305 129 11 2000 2000 NUM cana-5305 129 12 2500 2500 NUM cana-5305 129 13 512 512 NUM cana-5305 129 14 2048 2048 NUM cana-5305 129 15 4096 4096 NUM cana-5305 129 16 decryption decryption NOUN cana-5305 129 17 time time NOUN cana-5305 129 18 ( ( PUNCT cana-5305 129 19 ms ms NOUN cana-5305 129 20 ) ) PUNCT cana-5305 129 21 vs vs ADP cana-5305 129 22 message message NOUN cana-5305 129 23 size size NOUN cana-5305 129 24 ( ( PUNCT cana-5305 129 25 bits bits PROPN cana-5305 129 26 ) ) PUNCT cana-5305 129 27 rsa rsa PROPN cana-5305 129 28 decryption decryption NOUN cana-5305 129 29 time time NOUN cana-5305 129 30 ( ( PUNCT cana-5305 129 31 ms ms PROPN cana-5305 129 32 ) ) PUNCT cana-5305 129 33 ecc ecc PROPN cana-5305 129 34 decryption decryption NOUN cana-5305 129 35 time time NOUN cana-5305 129 36 ( ( PUNCT cana-5305 129 37 ms ms NOUN cana-5305 129 38 ) ) PUNCT cana-5305 129 39 hybrid hybrid ADJ cana-5305 129 40 system system NOUN cana-5305 129 41 decryption decryption NOUN cana-5305 129 42 time time NOUN cana-5305 129 43 ( ( PUNCT cana-5305 129 44 ms ms NOUN cana-5305 129 45 ) ) PUNCT cana-5305 129 46 communications communication NOUN cana-5305 129 47 on on ADP cana-5305 129 48 applied apply VERB cana-5305 129 49 nonlinear nonlinear ADJ cana-5305 129 50 analysis analysis NOUN cana-5305 129 51 issn issn NOUN cana-5305 129 52 : : PUNCT cana-5305 129 53 1074 1074 NUM cana-5305 129 54 - - PUNCT cana-5305 129 55 133x 133x NUM cana-5305 129 56 vol vol NOUN cana-5305 129 57 31 31 NUM cana-5305 129 58 no no NOUN cana-5305 129 59 . . NOUN cana-5305 129 60 1 1 NUM cana-5305 129 61 ( ( PUNCT cana-5305 129 62 2024 2024 NUM cana-5305 129 63 ) ) PUNCT cana-5305 129 64 363 363 NUM cana-5305 129 65 https://internationalpubls.com https://internationalpubls.com X cana-5305 129 66 perfect perfect ADJ cana-5305 129 67 suitability suitability NOUN cana-5305 129 68 for for ADP cana-5305 129 69 protecting protect VERB cana-5305 129 70 confidential confidential ADJ cana-5305 129 71 communication communication NOUN cana-5305 129 72 protocols protocol NOUN cana-5305 129 73 utilized utilize VERB cana-5305 129 74 mainly mainly ADV cana-5305 129 75 by by ADP cana-5305 129 76 banking banking NOUN cana-5305 129 77 and and CCONJ cana-5305 129 78 healthcare healthcare PROPN cana-5305 129 79 entities entity NOUN cana-5305 129 80 and and CCONJ cana-5305 129 81 governmental governmental ADJ cana-5305 129 82 agencies agency NOUN cana-5305 129 83 . . PUNCT cana-5305 130 1 table table NOUN cana-5305 130 2 2 2 NUM cana-5305 130 3 : : PUNCT cana-5305 130 4 comparison comparison NOUN cana-5305 130 5 of of ADP cana-5305 130 6 encryption encryption NOUN cana-5305 130 7 and and CCONJ cana-5305 130 8 decryption decryption NOUN cana-5305 130 9 speeds speed NOUN cana-5305 130 10 for for ADP cana-5305 130 11 rsa rsa NOUN cana-5305 130 12 , , PUNCT cana-5305 130 13 ecc ecc PROPN cana-5305 130 14 , , PUNCT cana-5305 130 15 and and CCONJ cana-5305 130 16 hybrid hybrid ADJ cana-5305 130 17 system system NOUN cana-5305 130 18 cryptographic cryptographic ADJ cana-5305 130 19 system system NOUN cana-5305 130 20 encryption encryption NOUN cana-5305 130 21 speed speed NOUN cana-5305 130 22 ( ( PUNCT cana-5305 130 23 ms ms NOUN cana-5305 130 24 ) ) PUNCT cana-5305 130 25 decryption decryption NOUN cana-5305 130 26 speed speed NOUN cana-5305 130 27 ( ( PUNCT cana-5305 130 28 ms ms PROPN cana-5305 130 29 ) ) PUNCT cana-5305 130 30 rsa rsa PROPN cana-5305 130 31 1500 1500 NUM cana-5305 130 32 1600 1600 NUM cana-5305 130 33 ecc ecc PROPN cana-5305 130 34 600 600 NUM cana-5305 130 35 650 650 NUM cana-5305 130 36 hybrid hybrid NOUN cana-5305 130 37 ( ( PUNCT cana-5305 130 38 rsa rsa NOUN cana-5305 130 39 + + CCONJ cana-5305 130 40 ecc ecc PROPN cana-5305 130 41 + + CCONJ cana-5305 130 42 lattice lattice PROPN cana-5305 130 43 ) ) PUNCT cana-5305 130 44 1100 1100 NUM cana-5305 130 45 1200 1200 NUM cana-5305 130 46 the the DET cana-5305 130 47 encryption encryption NOUN cana-5305 130 48 and and CCONJ cana-5305 130 49 decryption decryption NOUN cana-5305 130 50 timing timing NOUN cana-5305 130 51 of of ADP cana-5305 130 52 the the DET cana-5305 130 53 hybrid hybrid ADJ cana-5305 130 54 system system NOUN cana-5305 130 55 retains retain VERB cana-5305 130 56 speeds speed NOUN cana-5305 130 57 identical identical ADJ cana-5305 130 58 to to ADP cana-5305 130 59 rsa rsa NOUN cana-5305 130 60 execution execution NOUN cana-5305 130 61 speeds speed NOUN cana-5305 130 62 when when SCONJ cana-5305 130 63 processing process VERB cana-5305 130 64 messages message NOUN cana-5305 130 65 larger large ADJ cana-5305 130 66 than than ADP cana-5305 130 67 a a DET cana-5305 130 68 specific specific ADJ cana-5305 130 69 threshold threshold NOUN cana-5305 130 70 . . PUNCT cana-5305 131 1 during during ADP cana-5305 131 2 multiple multiple ADJ cana-5305 131 3 cryptographic cryptographic ADJ cana-5305 131 4 operations operation NOUN cana-5305 131 5 the the DET cana-5305 131 6 proposed propose VERB cana-5305 131 7 framework framework NOUN cana-5305 131 8 achieves achieve VERB cana-5305 131 9 strong strong ADJ cana-5305 131 10 execution execution NOUN cana-5305 131 11 speeds speed NOUN cana-5305 131 12 . . PUNCT cana-5305 132 1 this this DET cana-5305 132 2 system system NOUN cana-5305 132 3 requires require VERB cana-5305 132 4 proper proper ADJ cana-5305 132 5 assessment assessment NOUN cana-5305 132 6 of of ADP cana-5305 132 7 security security NOUN cana-5305 132 8 needs need NOUN cana-5305 132 9 against against ADP cana-5305 132 10 computation computation NOUN cana-5305 132 11 speeds speed NOUN cana-5305 132 12 and and CCONJ cana-5305 132 13 system system NOUN cana-5305 132 14 growth growth NOUN cana-5305 132 15 capabilities capability NOUN cana-5305 132 16 before before ADP cana-5305 132 17 implementing implement VERB cana-5305 132 18 it it PRON cana-5305 132 19 for for ADP cana-5305 132 20 practical practical ADJ cana-5305 132 21 usage usage NOUN cana-5305 132 22 . . PUNCT cana-5305 133 1 v. v. ADP cana-5305 133 2 conclusion conclusion NOUN cana-5305 133 3 groups group NOUN cana-5305 133 4 together together ADV cana-5305 133 5 with with ADP cana-5305 133 6 rings ring NOUN cana-5305 133 7 and and CCONJ cana-5305 133 8 fields field NOUN cana-5305 133 9 and and CCONJ cana-5305 133 10 lattices lattice NOUN cana-5305 133 11 remain remain VERB cana-5305 133 12 fundamental fundamental ADJ cana-5305 133 13 components component NOUN cana-5305 133 14 for for ADP cana-5305 133 15 constructing construct VERB cana-5305 133 16 the the DET cana-5305 133 17 present present ADJ cana-5305 133 18 - - PUNCT cana-5305 133 19 day day NOUN cana-5305 133 20 cryptographic cryptographic ADJ cana-5305 133 21 protocols protocol NOUN cana-5305 133 22 which which PRON cana-5305 133 23 are be AUX cana-5305 133 24 routinely routinely ADV cana-5305 133 25 used use VERB cana-5305 133 26 . . PUNCT cana-5305 134 1 the the DET cana-5305 134 2 increase increase NOUN cana-5305 134 3 in in ADP cana-5305 134 4 computing compute VERB cana-5305 134 5 power power NOUN cana-5305 134 6 as as ADV cana-5305 134 7 well well ADV cana-5305 134 8 as as ADP cana-5305 134 9 upcoming upcoming ADJ cana-5305 134 10 quantum quantum NOUN cana-5305 134 11 computing computing NOUN cana-5305 134 12 capabilities capability NOUN cana-5305 134 13 presents present VERB cana-5305 134 14 major major ADJ cana-5305 134 15 difficulties difficulty NOUN cana-5305 134 16 for for ADP cana-5305 134 17 employing employ VERB cana-5305 134 18 these these DET cana-5305 134 19 algebraic algebraic ADJ cana-5305 134 20 structures structure NOUN cana-5305 134 21 as as ADP cana-5305 134 22 security security NOUN cana-5305 134 23 measures measure NOUN cana-5305 134 24 . . PUNCT cana-5305 135 1 researchers researcher NOUN cana-5305 135 2 must must AUX cana-5305 135 3 direct direct VERB cana-5305 135 4 their their PRON cana-5305 135 5 research research NOUN cana-5305 135 6 to to PART cana-5305 135 7 improve improve VERB cana-5305 135 8 algebraic algebraic ADJ cana-5305 135 9 cryptographic cryptographic ADJ cana-5305 135 10 standards standard NOUN cana-5305 135 11 alongside alongside ADP cana-5305 135 12 enhancing enhance VERB cana-5305 135 13 operational operational ADJ cana-5305 135 14 efficiency efficiency NOUN cana-5305 135 15 according accord VERB cana-5305 135 16 to to ADP cana-5305 135 17 security security NOUN cana-5305 135 18 needs need NOUN cana-5305 135 19 while while SCONJ cana-5305 135 20 constructing construct VERB cana-5305 135 21 quantum quantum NOUN cana-5305 135 22 - - PUNCT cana-5305 135 23 attack attack NOUN cana-5305 135 24 resistant resistant ADJ cana-5305 135 25 standards standard NOUN cana-5305 135 26 . . PUNCT cana-5305 136 1 references reference NOUN cana-5305 136 2 [ [ X cana-5305 136 3 1 1 NUM cana-5305 136 4 ] ] PUNCT cana-5305 136 5 m. m. NOUN cana-5305 136 6 s. s. PROPN cana-5305 136 7 khamalwa khamalwa PROPN cana-5305 136 8 , , PUNCT cana-5305 136 9 β€œ " PUNCT cana-5305 136 10 exploring explore VERB cana-5305 136 11 how how SCONJ cana-5305 136 12 commutative commutative ADJ cana-5305 136 13 algebra algebra NOUN cana-5305 136 14 underpins underpin VERB cana-5305 136 15 cryptographic cryptographic ADJ cana-5305 136 16 protocols protocol NOUN cana-5305 136 17 and and CCONJ cana-5305 136 18 encryption encryption NOUN cana-5305 136 19 methods method NOUN cana-5305 136 20 used use VERB cana-5305 136 21 in in ADP cana-5305 136 22 secure secure ADJ cana-5305 136 23 communications communication NOUN cana-5305 136 24 and and CCONJ cana-5305 136 25 data datum NOUN cana-5305 136 26 protection protection NOUN cana-5305 136 27 , , PUNCT cana-5305 136 28 ” " PUNCT cana-5305 136 29 newport newport PROPN cana-5305 136 30 international international PROPN cana-5305 136 31 journal journal PROPN cana-5305 136 32 of of ADP cana-5305 136 33 scientific scientific ADJ cana-5305 136 34 and and CCONJ cana-5305 136 35 experimental experimental ADJ cana-5305 136 36 sciences science NOUN cana-5305 136 37 , , PUNCT cana-5305 136 38 vol vol NOUN cana-5305 136 39 . . PROPN cana-5305 136 40 5 5 NUM cana-5305 136 41 , , PUNCT cana-5305 136 42 no no INTJ cana-5305 136 43 . . NOUN cana-5305 136 44 3 3 NUM cana-5305 136 45 , , PUNCT cana-5305 136 46 pp pp ADJ cana-5305 136 47 . . PUNCT cana-5305 137 1 58–62 58–62 NUM cana-5305 137 2 , , PUNCT cana-5305 137 3 jun jun PROPN cana-5305 137 4 . . PROPN cana-5305 137 5 2024 2024 NUM cana-5305 137 6 , , PUNCT cana-5305 137 7 doi doi NOUN cana-5305 137 8 : : PUNCT cana-5305 137 9 10.59298 10.59298 NUM cana-5305 137 10 / / SYM cana-5305 137 11 nijses/2024/10.5.586237 nijses/2024/10.5.586237 PROPN cana-5305 137 12 . . PUNCT cana-5305 138 1 [ [ X cana-5305 138 2 2 2 X cana-5305 138 3 ] ] PUNCT cana-5305 138 4 d. d. PROPN cana-5305 138 5 j. j. PROPN cana-5305 138 6 bernstein bernstein PROPN cana-5305 138 7 , , PUNCT cana-5305 138 8 j. j. PROPN cana-5305 138 9 buchmann buchmann PROPN cana-5305 138 10 , , PUNCT cana-5305 138 11 and and CCONJ cana-5305 138 12 e. e. PROPN cana-5305 138 13 dahmen dahmen PROPN cana-5305 138 14 , , PUNCT cana-5305 138 15 post post ADJ cana-5305 138 16 - - ADJ cana-5305 138 17 quantum quantum ADJ cana-5305 138 18 cryptography cryptography NOUN cana-5305 138 19 . . PUNCT cana-5305 138 20 2009 2009 NUM cana-5305 138 21 . . PUNCT cana-5305 139 1 doi doi NOUN cana-5305 139 2 : : PUNCT cana-5305 139 3 10.1007/978 10.1007/978 NUM cana-5305 139 4 - - SYM cana-5305 139 5 3 3 NUM cana-5305 139 6 - - PUNCT cana-5305 139 7 540 540 NUM cana-5305 139 8 - - PUNCT cana-5305 139 9 88702 88702 NUM cana-5305 139 10 - - SYM cana-5305 139 11 7 7 NUM cana-5305 139 12 . . PUNCT cana-5305 140 1 [ [ X cana-5305 140 2 3 3 NUM cana-5305 140 3 ] ] X cana-5305 140 4 n. n. PROPN cana-5305 140 5 y. y. PROPN cana-5305 140 6 kasm kasm PROPN cana-5305 140 7 and and CCONJ cana-5305 140 8 z. z. PROPN cana-5305 140 9 a. a. PROPN cana-5305 140 10 hamad hamad PROPN cana-5305 140 11 , , PUNCT cana-5305 140 12 β€œ " PUNCT cana-5305 140 13 applications application NOUN cana-5305 140 14 of of ADP cana-5305 140 15 algebraic algebraic ADJ cana-5305 140 16 geometry geometry NOUN cana-5305 140 17 in in ADP cana-5305 140 18 cryptography cryptography NOUN cana-5305 140 19 , , PUNCT cana-5305 140 20 ” " PUNCT cana-5305 140 21 modern modern ADJ cana-5305 140 22 applied apply VERB cana-5305 140 23 science science NOUN cana-5305 140 24 , , PUNCT cana-5305 140 25 vol vol NOUN cana-5305 140 26 . . PROPN cana-5305 140 27 13 13 NUM cana-5305 140 28 , , PUNCT cana-5305 140 29 no no INTJ cana-5305 140 30 . . NOUN cana-5305 140 31 5 5 NUM cana-5305 140 32 , , PUNCT cana-5305 140 33 p. p. NOUN cana-5305 140 34 130 130 NUM cana-5305 140 35 , , PUNCT cana-5305 140 36 apr apr NOUN cana-5305 140 37 . . PROPN cana-5305 140 38 2019 2019 NUM cana-5305 140 39 , , PUNCT cana-5305 140 40 doi doi NOUN cana-5305 140 41 : : PUNCT cana-5305 140 42 10.5539 10.5539 NUM cana-5305 140 43 / / SYM cana-5305 140 44 mas.v13n5p130 mas.v13n5p130 NOUN cana-5305 140 45 . . PUNCT cana-5305 141 1 communications communication NOUN cana-5305 141 2 on on ADP cana-5305 141 3 applied apply VERB cana-5305 141 4 nonlinear nonlinear ADJ cana-5305 141 5 analysis analysis NOUN cana-5305 141 6 issn issn NOUN cana-5305 141 7 : : PUNCT cana-5305 141 8 1074 1074 NUM cana-5305 141 9 - - PUNCT cana-5305 141 10 133x 133x NUM cana-5305 141 11 vol vol NOUN cana-5305 141 12 31 31 NUM cana-5305 141 13 no no NOUN cana-5305 141 14 . . NOUN cana-5305 141 15 1 1 NUM cana-5305 141 16 ( ( PUNCT cana-5305 141 17 2024 2024 NUM cana-5305 141 18 ) ) PUNCT cana-5305 141 19 364 364 NUM cana-5305 141 20 https://internationalpubls.com https://internationalpubls.com X cana-5305 142 1 [ [ X cana-5305 142 2 4 4 X cana-5305 142 3 ] ] PUNCT cana-5305 142 4 c. c. PROPN cana-5305 142 5 h. h. PROPN cana-5305 142 6 bennett bennett PROPN cana-5305 142 7 , , PUNCT cana-5305 142 8 g. g. PROPN cana-5305 142 9 brassard brassard PROPN cana-5305 142 10 , , PUNCT cana-5305 142 11 and and CCONJ cana-5305 142 12 s. s. PROPN cana-5305 142 13 breidbart breidbart PROPN cana-5305 142 14 , , PUNCT cana-5305 142 15 β€œ " PUNCT cana-5305 142 16 quantum quantum PROPN cana-5305 142 17 cryptography cryptography NOUN cana-5305 142 18 ii ii PROPN cana-5305 142 19 : : PUNCT cana-5305 142 20 how how SCONJ cana-5305 142 21 to to PART cana-5305 142 22 reuse reuse VERB cana-5305 142 23 a a DET cana-5305 142 24 one one NUM cana-5305 142 25 - - PUNCT cana-5305 142 26 time time NOUN cana-5305 142 27 pad pad NOUN cana-5305 142 28 safely safely ADV cana-5305 142 29 even even ADV cana-5305 142 30 if if SCONJ cana-5305 142 31 p p X cana-5305 142 32 = = NOUN cana-5305 142 33 np np NOUN cana-5305 142 34 , , PUNCT cana-5305 142 35 ” " PUNCT cana-5305 142 36 natural natural ADJ cana-5305 142 37 computing computing NOUN cana-5305 142 38 , , PUNCT cana-5305 142 39 vol vol NOUN cana-5305 142 40 . . PROPN cana-5305 142 41 13 13 NUM cana-5305 142 42 , , PUNCT cana-5305 142 43 no no INTJ cana-5305 142 44 . . NOUN cana-5305 142 45 4 4 NUM cana-5305 142 46 , , PUNCT cana-5305 142 47 pp pp ADJ cana-5305 142 48 . . PUNCT cana-5305 142 49 453 453 NUM cana-5305 142 50 – – PUNCT cana-5305 142 51 458 458 NUM cana-5305 142 52 , , PUNCT cana-5305 142 53 oct oct PROPN cana-5305 142 54 . . PROPN cana-5305 142 55 2014 2014 NUM cana-5305 142 56 , , PUNCT cana-5305 142 57 doi doi NOUN cana-5305 142 58 : : PUNCT cana-5305 142 59 10.1007 10.1007 NUM cana-5305 142 60 / / SYM cana-5305 142 61 s11047 s11047 NUM cana-5305 142 62 - - PUNCT cana-5305 142 63 014 014 NUM cana-5305 142 64 - - PUNCT cana-5305 142 65 9453 9453 NUM cana-5305 142 66 - - SYM cana-5305 142 67 6 6 NUM cana-5305 142 68 . . PUNCT cana-5305 143 1 [ [ X cana-5305 143 2 5 5 NUM cana-5305 143 3 ] ] PUNCT cana-5305 143 4 l. l. PROPN cana-5305 143 5 budaghyan budaghyan PROPN cana-5305 143 6 , , PUNCT cana-5305 143 7 c. c. PROPN cana-5305 143 8 li li PROPN cana-5305 143 9 , , PUNCT cana-5305 143 10 and and CCONJ cana-5305 143 11 m. m. PROPN cana-5305 143 12 g. g. PROPN cana-5305 143 13 parker parker PROPN cana-5305 143 14 , , PUNCT cana-5305 143 15 β€œ " PUNCT cana-5305 143 16 editorial editorial NOUN cana-5305 143 17 : : PUNCT cana-5305 143 18 special special ADJ cana-5305 143 19 issue issue NOUN cana-5305 143 20 on on ADP cana-5305 143 21 mathematical mathematical ADJ cana-5305 143 22 methods method NOUN cana-5305 143 23 for for ADP cana-5305 143 24 cryptography cryptography NOUN cana-5305 143 25 , , PUNCT cana-5305 143 26 ” " PUNCT cana-5305 143 27 cryptography cryptography NOUN cana-5305 143 28 and and CCONJ cana-5305 143 29 communications communication NOUN cana-5305 143 30 , , PUNCT cana-5305 143 31 vol vol NOUN cana-5305 143 32 . . PROPN cana-5305 143 33 11 11 NUM cana-5305 143 34 , , PUNCT cana-5305 143 35 no no INTJ cana-5305 143 36 . . NOUN cana-5305 143 37 3 3 NUM cana-5305 143 38 , , PUNCT cana-5305 143 39 pp pp ADJ cana-5305 143 40 . . PUNCT cana-5305 144 1 363–365 363–365 NUM cana-5305 144 2 , , PUNCT cana-5305 144 3 feb feb PROPN cana-5305 144 4 . . PROPN cana-5305 144 5 2019 2019 NUM cana-5305 144 6 , , PUNCT cana-5305 144 7 doi doi NOUN cana-5305 144 8 : : PUNCT cana-5305 144 9 10.1007 10.1007 NUM cana-5305 144 10 / / SYM cana-5305 144 11 s12095 s12095 NOUN cana-5305 144 12 - - PUNCT cana-5305 144 13 019 019 NUM cana-5305 144 14 - - PUNCT cana-5305 144 15 00356 00356 NUM cana-5305 144 16 - - SYM cana-5305 144 17 8 8 NUM cana-5305 144 18 . . PUNCT cana-5305 145 1 [ [ X cana-5305 145 2 6 6 NUM cana-5305 145 3 ] ] PUNCT cana-5305 145 4 p. p. NOUN cana-5305 145 5 shukla shukla NOUN cana-5305 145 6 , , PUNCT cana-5305 145 7 a. a. PROPN cana-5305 145 8 khare khare PROPN cana-5305 145 9 , , PUNCT cana-5305 145 10 m. m. NOUN cana-5305 145 11 rizvi rizvi PROPN cana-5305 145 12 , , PUNCT cana-5305 145 13 s. s. PROPN cana-5305 145 14 stalin stalin PROPN cana-5305 145 15 , , PUNCT cana-5305 145 16 and and CCONJ cana-5305 145 17 s. s. PROPN cana-5305 145 18 kumar kumar PROPN cana-5305 145 19 , , PUNCT cana-5305 145 20 β€œ " PUNCT cana-5305 145 21 applied apply VERB cana-5305 145 22 cryptography cryptography NOUN cana-5305 145 23 using use VERB cana-5305 145 24 chaos chaos NOUN cana-5305 145 25 function function NOUN cana-5305 145 26 for for ADP cana-5305 145 27 fast fast ADJ cana-5305 145 28 digital digital ADJ cana-5305 145 29 logic logic NOUN cana-5305 145 30 - - PUNCT cana-5305 145 31 based base VERB cana-5305 145 32 systems system NOUN cana-5305 145 33 in in ADP cana-5305 145 34 ubiquitous ubiquitous ADJ cana-5305 145 35 computing computing NOUN cana-5305 145 36 , , PUNCT cana-5305 145 37 ” " PUNCT cana-5305 145 38 entropy entropy NOUN cana-5305 145 39 , , PUNCT cana-5305 145 40 vol vol NOUN cana-5305 145 41 . . PROPN cana-5305 145 42 17 17 NUM cana-5305 145 43 , , PUNCT cana-5305 145 44 no no INTJ cana-5305 145 45 . . NOUN cana-5305 145 46 3 3 NUM cana-5305 145 47 , , PUNCT cana-5305 145 48 pp pp ADJ cana-5305 145 49 . . PUNCT cana-5305 146 1 1387–1410 1387–1410 NUM cana-5305 146 2 , , PUNCT cana-5305 146 3 mar mar PROPN cana-5305 146 4 . . PROPN cana-5305 146 5 2015 2015 NUM cana-5305 146 6 , , PUNCT cana-5305 146 7 doi doi NOUN cana-5305 146 8 : : PUNCT cana-5305 146 9 10.3390 10.3390 NUM cana-5305 146 10 / / SYM cana-5305 146 11 e17031387 e17031387 NOUN cana-5305 146 12 . . PUNCT cana-5305 147 1 [ [ X cana-5305 147 2 7 7 X cana-5305 147 3 ] ] PUNCT cana-5305 147 4 x. x. PROPN cana-5305 147 5 yan yan PROPN cana-5305 147 6 , , PUNCT cana-5305 147 7 s. s. PROPN cana-5305 147 8 wang wang PROPN cana-5305 147 9 , , PUNCT cana-5305 147 10 x. x. PROPN cana-5305 147 11 niu niu PROPN cana-5305 147 12 , , PUNCT cana-5305 147 13 and and CCONJ cana-5305 147 14 c.-n c.-n PROPN cana-5305 147 15 . . PUNCT cana-5305 148 1 yang yang PROPN cana-5305 148 2 , , PUNCT cana-5305 148 3 β€œ " PUNCT cana-5305 148 4 generalized generalize VERB cana-5305 148 5 random random ADJ cana-5305 148 6 grids grid NOUN cana-5305 148 7 - - PUNCT cana-5305 148 8 based base VERB cana-5305 148 9 threshold threshold NOUN cana-5305 148 10 visual visual ADJ cana-5305 148 11 cryptography cryptography NOUN cana-5305 148 12 with with ADP cana-5305 148 13 meaningful meaningful ADJ cana-5305 148 14 shares share NOUN cana-5305 148 15 , , PUNCT cana-5305 148 16 ” " PUNCT cana-5305 148 17 signal signal NOUN cana-5305 148 18 processing processing NOUN cana-5305 148 19 , , PUNCT cana-5305 148 20 vol vol NOUN cana-5305 148 21 . . PROPN cana-5305 148 22 109 109 NUM cana-5305 148 23 , , PUNCT cana-5305 148 24 pp pp ADJ cana-5305 148 25 . . PUNCT cana-5305 149 1 317 317 NUM cana-5305 149 2 – – PUNCT cana-5305 149 3 333 333 NUM cana-5305 149 4 , , PUNCT cana-5305 149 5 dec dec PROPN cana-5305 149 6 . . PROPN cana-5305 149 7 2014 2014 NUM cana-5305 149 8 , , PUNCT cana-5305 149 9 doi doi NOUN cana-5305 149 10 : : PUNCT cana-5305 149 11 10.1016 10.1016 NUM cana-5305 149 12 / / SYM cana-5305 149 13 j.sigpro.2014.12.002 j.sigpro.2014.12.002 PROPN cana-5305 149 14 . . PUNCT cana-5305 150 1 [ [ X cana-5305 150 2 8 8 NUM cana-5305 150 3 ] ] X cana-5305 150 4 v. v. ADP cana-5305 150 5 jara jara PROPN cana-5305 150 6 - - PUNCT cana-5305 150 7 vera vera PROPN cana-5305 150 8 and and CCONJ cana-5305 150 9 c. c. PROPN cana-5305 150 10 sΓ‘nchez sΓ‘nchez PROPN cana-5305 150 11 - - PUNCT cana-5305 150 12 Γ‘vila Γ‘vila PROPN cana-5305 150 13 , , PUNCT cana-5305 150 14 β€œ " PUNCT cana-5305 150 15 some some DET cana-5305 150 16 notes note NOUN cana-5305 150 17 on on ADP cana-5305 150 18 a a DET cana-5305 150 19 formal formal ADJ cana-5305 150 20 algebraic algebraic ADJ cana-5305 150 21 structure structure NOUN cana-5305 150 22 of of ADP cana-5305 150 23 cryptology cryptology NOUN cana-5305 150 24 , , PUNCT cana-5305 150 25 ” " PUNCT cana-5305 150 26 mathematics mathematic NOUN cana-5305 150 27 , , PUNCT cana-5305 150 28 vol vol NOUN cana-5305 150 29 . . PROPN cana-5305 151 1 9 9 NUM cana-5305 151 2 , , PUNCT cana-5305 151 3 no no INTJ cana-5305 151 4 . . NOUN cana-5305 151 5 18 18 NUM cana-5305 151 6 , , PUNCT cana-5305 151 7 p. p. NOUN cana-5305 151 8 2183 2183 NUM cana-5305 151 9 , , PUNCT cana-5305 151 10 sep sep PROPN cana-5305 151 11 . . PROPN cana-5305 151 12 2021 2021 NUM cana-5305 151 13 , , PUNCT cana-5305 151 14 doi doi NOUN cana-5305 151 15 : : PUNCT cana-5305 151 16 10.3390 10.3390 NUM cana-5305 151 17 / / SYM cana-5305 151 18 math9182183 math9182183 PROPN cana-5305 151 19 . . PUNCT cana-5305 152 1 [ [ X cana-5305 152 2 9 9 NUM cana-5305 152 3 ] ] PUNCT cana-5305 152 4 s. s. PROPN cana-5305 152 5 malik malik PROPN cana-5305 152 6 , , PUNCT cana-5305 152 7 s. s. PROPN cana-5305 152 8 agarwal agarwal PROPN cana-5305 152 9 , , PUNCT cana-5305 152 10 and and CCONJ cana-5305 152 11 a. a. NOUN cana-5305 152 12 s. s. PROPN cana-5305 152 13 uniyal uniyal PROPN cana-5305 152 14 , , PUNCT cana-5305 152 15 β€œ " PUNCT cana-5305 152 16 development development NOUN cana-5305 152 17 of of ADP cana-5305 152 18 algebraic algebraic ADJ cana-5305 152 19 structures structure NOUN cana-5305 152 20 and and CCONJ cana-5305 152 21 their their PRON cana-5305 152 22 application application NOUN cana-5305 152 23 to to PART cana-5305 152 24 ensure ensure VERB cana-5305 152 25 the the DET cana-5305 152 26 security security NOUN cana-5305 152 27 of of ADP cana-5305 152 28 digital digital ADJ cana-5305 152 29 information information NOUN cana-5305 152 30 , , PUNCT cana-5305 152 31 ” " PUNCT cana-5305 152 32 ssrn ssrn PROPN cana-5305 152 33 electronic electronic ADJ cana-5305 152 34 journal journal NOUN cana-5305 152 35 , , PUNCT cana-5305 152 36 jan jan PROPN cana-5305 152 37 . . PROPN cana-5305 152 38 2023 2023 NUM cana-5305 152 39 , , PUNCT cana-5305 152 40 doi doi NOUN cana-5305 152 41 : : PUNCT cana-5305 152 42 10.2139 10.2139 NUM cana-5305 152 43 / / SYM cana-5305 152 44 ssrn.4623318 ssrn.4623318 PROPN cana-5305 152 45 . . PUNCT cana-5305 153 1 [ [ X cana-5305 153 2 10 10 NUM cana-5305 153 3 ] ] X cana-5305 153 4 g. g. PROPN cana-5305 153 5 frey frey PROPN cana-5305 153 6 , , PUNCT cana-5305 153 7 β€œ " PUNCT cana-5305 153 8 applications application NOUN cana-5305 153 9 of of ADP cana-5305 153 10 arithmetical arithmetical ADJ cana-5305 153 11 geometry geometry NOUN cana-5305 153 12 to to ADP cana-5305 153 13 cryptographic cryptographic ADJ cana-5305 153 14 constructions construction NOUN cana-5305 153 15 , , PUNCT cana-5305 153 16 ” " PUNCT cana-5305 153 17 in in ADP cana-5305 153 18 springer springer NOUN cana-5305 153 19 ebooks ebook NOUN cana-5305 153 20 , , PUNCT cana-5305 153 21 2001 2001 NUM cana-5305 153 22 , , PUNCT cana-5305 153 23 pp pp ADJ cana-5305 153 24 . . PUNCT cana-5305 154 1 128–161 128–161 NUM cana-5305 154 2 . . PUNCT cana-5305 155 1 doi doi NOUN cana-5305 155 2 : : PUNCT cana-5305 155 3 10.1007/978 10.1007/978 NUM cana-5305 155 4 - - SYM cana-5305 155 5 3 3 NUM cana-5305 155 6 - - PUNCT cana-5305 155 7 642 642 NUM cana-5305 155 8 - - PUNCT cana-5305 155 9 56755 56755 NUM cana-5305 155 10 - - PUNCT cana-5305 155 11 1_13 1_13 NOUN cana-5305 155 12 . . PUNCT cana-5305 156 1 [ [ X cana-5305 156 2 11 11 NUM cana-5305 156 3 ] ] X cana-5305 156 4 v. v. ADP cana-5305 156 5 roman’kov roman’kov NOUN cana-5305 156 6 , , PUNCT cana-5305 156 7 β€œ " PUNCT cana-5305 156 8 two two NUM cana-5305 156 9 general general ADJ cana-5305 156 10 schemes scheme NOUN cana-5305 156 11 of of ADP cana-5305 156 12 algebraic algebraic ADJ cana-5305 156 13 cryptography cryptography NOUN cana-5305 156 14 , , PUNCT cana-5305 156 15 ” " PUNCT cana-5305 156 16 journal journal NOUN cana-5305 156 17 of of ADP cana-5305 156 18 groups group NOUN cana-5305 156 19 complexity complexity NOUN cana-5305 156 20 cryptology cryptology NOUN cana-5305 156 21 , , PUNCT cana-5305 156 22 vol vol NOUN cana-5305 156 23 . . PROPN cana-5305 156 24 10 10 NUM cana-5305 156 25 , , PUNCT cana-5305 156 26 no no INTJ cana-5305 156 27 . . NOUN cana-5305 156 28 2 2 NUM cana-5305 156 29 , , PUNCT cana-5305 156 30 pp pp ADJ cana-5305 156 31 . . PUNCT cana-5305 157 1 83–98 83–98 NUM cana-5305 157 2 , , PUNCT cana-5305 157 3 oct oct PROPN cana-5305 157 4 . . PROPN cana-5305 157 5 2018 2018 NUM cana-5305 157 6 , , PUNCT cana-5305 157 7 doi doi NOUN cana-5305 157 8 : : PUNCT cana-5305 157 9 10.1515 10.1515 NUM cana-5305 157 10 / / SYM cana-5305 157 11 gcc-20180009 gcc-20180009 NOUN cana-5305 157 12 . . PUNCT cana-5305 158 1 [ [ X cana-5305 158 2 12 12 NUM cana-5305 158 3 ] ] X cana-5305 158 4 b. b. PROPN cana-5305 158 5 tsaban tsaban PROPN cana-5305 158 6 , , PUNCT cana-5305 158 7 β€œ " PUNCT cana-5305 158 8 polynomial polynomial ADJ cana-5305 158 9 - - PUNCT cana-5305 158 10 time time NOUN cana-5305 158 11 solutions solution NOUN cana-5305 158 12 of of ADP cana-5305 158 13 computational computational ADJ cana-5305 158 14 problems problem NOUN cana-5305 158 15 in in ADP cana-5305 158 16 noncommutative noncommutative ADJ cana-5305 158 17 - - PUNCT cana-5305 158 18 algebraic algebraic ADJ cana-5305 158 19 cryptography cryptography NOUN cana-5305 158 20 , , PUNCT cana-5305 158 21 ” " PUNCT cana-5305 158 22 journal journal NOUN cana-5305 158 23 of of ADP cana-5305 158 24 cryptology cryptology NOUN cana-5305 158 25 , , PUNCT cana-5305 158 26 vol vol NOUN cana-5305 158 27 . . PROPN cana-5305 158 28 28 28 NUM cana-5305 158 29 , , PUNCT cana-5305 158 30 no no INTJ cana-5305 158 31 . . NOUN cana-5305 158 32 3 3 NUM cana-5305 158 33 , , PUNCT cana-5305 158 34 pp pp ADJ cana-5305 158 35 . . PUNCT cana-5305 159 1 601–622 601–622 NUM cana-5305 159 2 , , PUNCT cana-5305 159 3 nov nov PROPN cana-5305 159 4 . . PROPN cana-5305 159 5 2013 2013 NUM cana-5305 159 6 , , PUNCT cana-5305 159 7 doi doi NOUN cana-5305 159 8 : : PUNCT cana-5305 159 9 10.1007 10.1007 NUM cana-5305 159 10 / / SYM cana-5305 159 11 s00145 s00145 NOUN cana-5305 159 12 - - PUNCT cana-5305 159 13 013 013 NUM cana-5305 159 14 - - PUNCT cana-5305 159 15 9170 9170 NUM cana-5305 159 16 - - SYM cana-5305 159 17 9 9 NUM cana-5305 159 18 . . PUNCT cana-5305 160 1 [ [ X cana-5305 160 2 13 13 NUM cana-5305 160 3 ] ] X cana-5305 160 4 f. f. NOUN cana-5305 160 5 rabanal rabanal PROPN cana-5305 160 6 and and CCONJ cana-5305 160 7 c. c. PROPN cana-5305 160 8 martΓ­nez martΓ­nez PROPN cana-5305 160 9 , , PUNCT cana-5305 160 10 β€œ " PUNCT cana-5305 160 11 cryptography cryptography NOUN cana-5305 160 12 for for ADP cana-5305 160 13 big big ADJ cana-5305 160 14 data datum NOUN cana-5305 160 15 environments environment NOUN cana-5305 160 16 : : PUNCT cana-5305 160 17 current current ADJ cana-5305 160 18 status status NOUN cana-5305 160 19 , , PUNCT cana-5305 160 20 challenges challenge NOUN cana-5305 160 21 , , PUNCT cana-5305 160 22 and and CCONJ cana-5305 160 23 opportunities opportunity NOUN cana-5305 160 24 , , PUNCT cana-5305 160 25 ” " PUNCT cana-5305 160 26 computational computational ADJ cana-5305 160 27 and and CCONJ cana-5305 160 28 mathematical mathematical ADJ cana-5305 160 29 methods method NOUN cana-5305 160 30 , , PUNCT cana-5305 160 31 vol vol NOUN cana-5305 160 32 . . PROPN cana-5305 160 33 2 2 NUM cana-5305 160 34 , , PUNCT cana-5305 160 35 no no INTJ cana-5305 160 36 . . NOUN cana-5305 160 37 1 1 NUM cana-5305 160 38 , , PUNCT cana-5305 160 39 nov nov PROPN cana-5305 160 40 . . PROPN cana-5305 160 41 2019 2019 NUM cana-5305 160 42 , , PUNCT cana-5305 160 43 doi doi NOUN cana-5305 160 44 : : PUNCT cana-5305 160 45 10.1002 10.1002 NUM cana-5305 160 46 / / SYM cana-5305 160 47 cmm4.1075 cmm4.1075 NUM cana-5305 160 48 . . PUNCT cana-5305 161 1 [ [ X cana-5305 161 2 14 14 NUM cana-5305 161 3 ] ] X cana-5305 161 4 j. j. PROPN cana-5305 161 5 suo suo PROPN cana-5305 161 6 , , PUNCT cana-5305 161 7 l. l. PROPN cana-5305 161 8 wang wang PROPN cana-5305 161 9 , , PUNCT cana-5305 161 10 s. s. PROPN cana-5305 161 11 yang yang PROPN cana-5305 161 12 , , PUNCT cana-5305 161 13 w. w. PROPN cana-5305 161 14 zheng zheng PROPN cana-5305 161 15 , , PUNCT cana-5305 161 16 and and CCONJ cana-5305 161 17 j. j. PROPN cana-5305 161 18 zhang zhang PROPN cana-5305 161 19 , , PUNCT cana-5305 161 20 β€œ " PUNCT cana-5305 161 21 quantum quantum ADJ cana-5305 161 22 algorithms algorithm NOUN cana-5305 161 23 for for ADP cana-5305 161 24 typical typical ADJ cana-5305 161 25 hard hard ADJ cana-5305 161 26 problems problem NOUN cana-5305 161 27 : : PUNCT cana-5305 161 28 a a DET cana-5305 161 29 perspective perspective NOUN cana-5305 161 30 of of ADP cana-5305 161 31 cryptanalysis cryptanalysis NOUN cana-5305 161 32 , , PUNCT cana-5305 161 33 ” " PUNCT cana-5305 161 34 quantum quantum ADJ cana-5305 161 35 information information NOUN cana-5305 161 36 processing processing NOUN cana-5305 161 37 , , PUNCT cana-5305 161 38 vol vol NOUN cana-5305 161 39 . . PROPN cana-5305 161 40 19 19 NUM cana-5305 161 41 , , PUNCT cana-5305 161 42 no no INTJ cana-5305 161 43 . . NOUN cana-5305 161 44 6 6 NUM cana-5305 161 45 , , PUNCT cana-5305 161 46 apr apr NOUN cana-5305 161 47 . . PROPN cana-5305 161 48 2020 2020 NUM cana-5305 161 49 , , PUNCT cana-5305 161 50 doi doi NOUN cana-5305 161 51 : : PUNCT cana-5305 161 52 10.1007 10.1007 NUM cana-5305 161 53 / / SYM cana-5305 161 54 s11128 s11128 NOUN cana-5305 161 55 - - PUNCT cana-5305 161 56 020 020 NUM cana-5305 161 57 - - PUNCT cana-5305 161 58 02673 02673 NUM cana-5305 161 59 - - PUNCT cana-5305 161 60 x. x. NOUN cana-5305 162 1 [ [ X cana-5305 162 2 15 15 NUM cana-5305 162 3 ] ] X cana-5305 162 4 r. r. PROPN cana-5305 162 5 sanjeewa sanjeewa PROPN cana-5305 162 6 and and CCONJ cana-5305 162 7 b. b. PROPN cana-5305 162 8 a. a. PROPN cana-5305 162 9 k. k. PROPN cana-5305 162 10 welihinda welihinda PROPN cana-5305 162 11 , , PUNCT cana-5305 162 12 β€œ " PUNCT cana-5305 162 13 elliptic elliptic ADJ cana-5305 162 14 curve curve NOUN cana-5305 162 15 cryptography cryptography NOUN cana-5305 162 16 and and CCONJ cana-5305 162 17 coding code VERB cana-5305 162 18 theory theory NOUN cana-5305 162 19 , , PUNCT cana-5305 162 20 ” " PUNCT cana-5305 162 21 international international ADJ cana-5305 162 22 journal journal NOUN cana-5305 162 23 of of ADP cana-5305 162 24 multidisciplinary multidisciplinary ADJ cana-5305 162 25 studies study NOUN cana-5305 162 26 , , PUNCT cana-5305 162 27 vol vol NOUN cana-5305 162 28 . . PROPN cana-5305 163 1 3 3 NUM cana-5305 163 2 , , PUNCT cana-5305 163 3 no no INTJ cana-5305 163 4 . . NOUN cana-5305 163 5 2 2 NUM cana-5305 163 6 , , PUNCT cana-5305 163 7 p. p. NOUN cana-5305 163 8 99 99 NUM cana-5305 163 9 , , PUNCT cana-5305 163 10 jan jan PROPN cana-5305 163 11 . . PROPN cana-5305 163 12 2017 2017 NUM cana-5305 163 13 , , PUNCT cana-5305 163 14 doi doi NOUN cana-5305 163 15 : : PUNCT cana-5305 163 16 10.4038 10.4038 NUM cana-5305 163 17 / / SYM cana-5305 163 18 ijms.v3i2.12 ijms.v3i2.12 PROPN cana-5305 163 19 . . PUNCT