id sid tid token lemma pos cana-1656 1 1 communications communication NOUN cana-1656 1 2 on on ADP cana-1656 1 3 applied apply VERB cana-1656 1 4 nonlinear nonlinear ADJ cana-1656 1 5 analysis analysis NOUN cana-1656 1 6 issn issn NOUN cana-1656 1 7 : : PUNCT cana-1656 1 8 1074 1074 NUM cana-1656 1 9 - - PUNCT cana-1656 1 10 133x 133x NUM cana-1656 1 11 vol vol NOUN cana-1656 1 12 32 32 NUM cana-1656 1 13 no no NOUN cana-1656 1 14 . . NOUN cana-1656 1 15 1 1 NUM cana-1656 1 16 ( ( PUNCT cana-1656 1 17 2025 2025 NUM cana-1656 1 18 ) ) PUNCT cana-1656 1 19 352 352 NUM cana-1656 1 20 https://internationalpubls.com https://internationalpubls.com X cana-1656 1 21 an an DET cana-1656 1 22 approach approach NOUN cana-1656 1 23 for for ADP cana-1656 1 24 cipher cipher ADJ cana-1656 1 25 communication communication NOUN cana-1656 1 26 utilizing utilize VERB cana-1656 1 27 the the DET cana-1656 1 28 concepts concept NOUN cana-1656 1 29 of of ADP cana-1656 1 30 elliptic elliptic ADJ cana-1656 1 31 curve curve NOUN cana-1656 1 32 cryptography cryptography NOUN cana-1656 1 33 s. s. PROPN cana-1656 1 34 sarvalakshmi1 sarvalakshmi1 PROPN cana-1656 1 35 * * PROPN cana-1656 1 36 , , PUNCT cana-1656 1 37 ch ch PROPN cana-1656 1 38 . . PUNCT cana-1656 1 39 suneetha2 suneetha2 PROPN cana-1656 2 1 1 1 NUM cana-1656 2 2 * * NOUN cana-1656 2 3 . . PUNCT cana-1656 3 1 research research NOUN cana-1656 3 2 scholar scholar NOUN cana-1656 3 3 in in ADP cana-1656 3 4 mathematics mathematic NOUN cana-1656 3 5 , , PUNCT cana-1656 3 6 gitam gitam NOUN cana-1656 3 7 ( ( PUNCT cana-1656 3 8 deemed deem VERB cana-1656 3 9 to to PART cana-1656 3 10 be be AUX cana-1656 3 11 university university NOUN cana-1656 3 12 ) ) PUNCT cana-1656 3 13 , , PUNCT cana-1656 3 14 visakhapatnam visakhapatnam PROPN cana-1656 3 15 , , PUNCT cana-1656 3 16 india india PROPN cana-1656 3 17 2 2 NUM cana-1656 3 18 . . PUNCT cana-1656 3 19 associate associate ADJ cana-1656 3 20 professor professor NOUN cana-1656 3 21 in in ADP cana-1656 3 22 mathematics mathematics PROPN cana-1656 3 23 , , PUNCT cana-1656 3 24 gitam gitam NOUN cana-1656 3 25 ( ( PUNCT cana-1656 3 26 deemed deem VERB cana-1656 3 27 to to PART cana-1656 3 28 be be AUX cana-1656 3 29 university university NOUN cana-1656 3 30 ) ) PUNCT cana-1656 3 31 , , PUNCT cana-1656 3 32 visakhapatnam visakhapatnam PROPN cana-1656 3 33 , , PUNCT cana-1656 3 34 india india PROPN cana-1656 3 35 1*corresponding 1*corresponding PROPN cana-1656 3 36 author author NOUN cana-1656 3 37 mail mail NOUN cana-1656 3 38 : : PUNCT cana-1656 4 1 radhika1483@gmail.com radhika1483@gmail.com X cana-1656 4 2 article article NOUN cana-1656 4 3 history history NOUN cana-1656 4 4 : : PUNCT cana-1656 4 5 received receive VERB cana-1656 4 6 : : PUNCT cana-1656 4 7 16 16 NUM cana-1656 4 8 - - SYM cana-1656 4 9 07 07 NUM cana-1656 4 10 - - PUNCT cana-1656 4 11 2024 2024 NUM cana-1656 4 12 revised revise VERB cana-1656 4 13 : : PUNCT cana-1656 4 14 30 30 NUM cana-1656 4 15 - - SYM cana-1656 4 16 08 08 NUM cana-1656 4 17 - - PUNCT cana-1656 4 18 2024 2024 NUM cana-1656 4 19 accepted accept VERB cana-1656 4 20 : : PUNCT cana-1656 4 21 11 11 NUM cana-1656 4 22 - - SYM cana-1656 4 23 09 09 NUM cana-1656 4 24 - - PUNCT cana-1656 4 25 2024 2024 NUM cana-1656 4 26 abstract abstract NOUN cana-1656 4 27 : : PUNCT cana-1656 4 28 message message NOUN cana-1656 4 29 encryption encryption NOUN cana-1656 4 30 is be AUX cana-1656 4 31 a a DET cana-1656 4 32 process process NOUN cana-1656 4 33 of of ADP cana-1656 4 34 protection protection NOUN cana-1656 4 35 of of ADP cana-1656 4 36 the the DET cana-1656 4 37 original original ADJ cana-1656 4 38 data datum NOUN cana-1656 4 39 in in ADP cana-1656 4 40 transmission transmission NOUN cana-1656 4 41 . . PUNCT cana-1656 5 1 several several ADJ cana-1656 5 2 cryptosystems cryptosystem NOUN cana-1656 5 3 were be AUX cana-1656 5 4 designed design VERB cana-1656 5 5 in in ADP cana-1656 5 6 the the DET cana-1656 5 7 history history NOUN cana-1656 5 8 for for ADP cana-1656 5 9 different different ADJ cana-1656 5 10 environments environment NOUN cana-1656 5 11 . . PUNCT cana-1656 6 1 public public ADJ cana-1656 6 2 key key ADJ cana-1656 6 3 or or CCONJ cana-1656 6 4 asymmetric asymmetric ADJ cana-1656 6 5 cryptosystem cryptosystem NOUN cana-1656 6 6 has have AUX cana-1656 6 7 become become VERB cana-1656 6 8 more more ADV cana-1656 6 9 popular popular ADJ cana-1656 6 10 in in ADP cana-1656 6 11 recent recent ADJ cana-1656 6 12 era era NOUN cana-1656 6 13 where where SCONJ cana-1656 6 14 the the DET cana-1656 6 15 key key ADJ cana-1656 6 16 space space NOUN cana-1656 6 17 consists consist VERB cana-1656 6 18 both both CCONJ cana-1656 6 19 public public ADJ cana-1656 6 20 and and CCONJ cana-1656 6 21 private private ADJ cana-1656 6 22 keys key NOUN cana-1656 6 23 . . PUNCT cana-1656 7 1 elliptic elliptic ADJ cana-1656 7 2 curve curve NOUN cana-1656 7 3 cryptography cryptography NOUN cana-1656 7 4 dominates dominate VERB cana-1656 7 5 public public ADJ cana-1656 7 6 key key ADJ cana-1656 7 7 cryptosystems cryptosystem NOUN cana-1656 7 8 like like ADP cana-1656 7 9 diffie diffie PROPN cana-1656 7 10 hellman hellman PROPN cana-1656 7 11 key key PROPN cana-1656 7 12 exchange exchange PROPN cana-1656 7 13 and and CCONJ cana-1656 7 14 rsa rsa PROPN cana-1656 7 15 . . PUNCT cana-1656 8 1 ecc ecc PROPN cana-1656 8 2 is be AUX cana-1656 8 3 more more ADV cana-1656 8 4 powerful powerful ADJ cana-1656 8 5 and and CCONJ cana-1656 8 6 alternative alternative ADJ cana-1656 8 7 technique technique NOUN cana-1656 8 8 of of ADP cana-1656 8 9 rsa rsa NOUN cana-1656 8 10 , , PUNCT cana-1656 8 11 with with ADP cana-1656 8 12 a a DET cana-1656 8 13 simple simple ADJ cana-1656 8 14 replacement replacement NOUN cana-1656 8 15 of of ADP cana-1656 8 16 exponent exponent ADJ cana-1656 8 17 calculations calculation NOUN cana-1656 8 18 of of ADP cana-1656 8 19 large large ADJ cana-1656 8 20 prime prime ADJ cana-1656 8 21 numbers number NOUN cana-1656 8 22 to to ADP cana-1656 8 23 mathematical mathematical ADJ cana-1656 8 24 representation representation NOUN cana-1656 8 25 of of ADP cana-1656 8 26 elliptic elliptic ADJ cana-1656 8 27 curve curve NOUN cana-1656 8 28 points point NOUN cana-1656 8 29 . . PUNCT cana-1656 9 1 besides besides SCONJ cana-1656 9 2 the the DET cana-1656 9 3 computation computation NOUN cana-1656 9 4 ecc ecc PROPN cana-1656 9 5 works work VERB cana-1656 9 6 with with ADP cana-1656 9 7 shorter short ADJ cana-1656 9 8 key key ADJ cana-1656 9 9 length length NOUN cana-1656 9 10 than than ADP cana-1656 9 11 rsa rsa PROPN cana-1656 9 12 . . PUNCT cana-1656 10 1 the the DET cana-1656 10 2 present present ADJ cana-1656 10 3 paper paper NOUN cana-1656 10 4 describes describe VERB cana-1656 10 5 an an DET cana-1656 10 6 encryption encryption NOUN cana-1656 10 7 algorithm algorithm NOUN cana-1656 10 8 using use VERB cana-1656 10 9 elliptic elliptic ADJ cana-1656 10 10 curves curve NOUN cana-1656 10 11 over over ADP cana-1656 10 12 finite finite ADJ cana-1656 10 13 fields field NOUN cana-1656 10 14 . . PUNCT cana-1656 11 1 here here ADV cana-1656 11 2 a a DET cana-1656 11 3 group group NOUN cana-1656 11 4 of of ADP cana-1656 11 5 authentic authentic ADJ cana-1656 11 6 users user NOUN cana-1656 11 7 transmit transmit VERB cana-1656 11 8 . . PUNCT cana-1656 12 1 keywords keyword NOUN cana-1656 12 2 : : PUNCT cana-1656 12 3 elliptic elliptic ADJ cana-1656 12 4 curve curve NOUN cana-1656 12 5 cryptography cryptography NOUN cana-1656 12 6 , , PUNCT cana-1656 12 7 elliptic elliptic ADJ cana-1656 12 8 curve curve NOUN cana-1656 12 9 points point NOUN cana-1656 12 10 , , PUNCT cana-1656 12 11 encryption encryption NOUN cana-1656 12 12 decryption decryption NOUN cana-1656 12 13 , , PUNCT cana-1656 12 14 generator generator NOUN cana-1656 12 15 . . PUNCT cana-1656 13 1 1 1 X cana-1656 13 2 . . X cana-1656 13 3 introduction introduction NOUN cana-1656 13 4 : : PUNCT cana-1656 13 5 diffie diffie PROPN cana-1656 13 6 hellman hellman PROPN cana-1656 13 7 key key PROPN cana-1656 13 8 exchange exchange NOUN cana-1656 13 9 algorithm algorithm NOUN cana-1656 13 10 is be AUX cana-1656 13 11 a a DET cana-1656 13 12 compelling compelling ADJ cana-1656 13 13 protocol protocol NOUN cana-1656 13 14 which which PRON cana-1656 13 15 utilizes utilize VERB cana-1656 13 16 exponentiation exponentiation NOUN cana-1656 13 17 of of ADP cana-1656 13 18 huge huge ADJ cana-1656 13 19 groups group NOUN cana-1656 13 20 to to PART cana-1656 13 21 start start VERB cana-1656 13 22 an an DET cana-1656 13 23 encrypted encrypt VERB cana-1656 13 24 communication communication NOUN cana-1656 13 25 between between ADP cana-1656 13 26 two two NUM cana-1656 13 27 unknown unknown ADJ cana-1656 13 28 communicators communicator NOUN cana-1656 13 29 . . PUNCT cana-1656 14 1 it it PRON cana-1656 14 2 has have AUX cana-1656 14 3 been be AUX cana-1656 14 4 proposed propose VERB cana-1656 14 5 as as ADP cana-1656 14 6 an an DET cana-1656 14 7 internet internet NOUN cana-1656 14 8 standard standard NOUN cana-1656 14 9 and and CCONJ cana-1656 14 10 significantly significantly ADV cana-1656 14 11 enhances enhance VERB cana-1656 14 12 internet internet NOUN cana-1656 14 13 security security NOUN cana-1656 14 14 . . PUNCT cana-1656 15 1 however however ADV cana-1656 15 2 , , PUNCT cana-1656 15 3 the the DET cana-1656 15 4 algorithm algorithm NOUN cana-1656 15 5 affects affect VERB cana-1656 15 6 from from ADP cana-1656 15 7 a a DET cana-1656 15 8 security security NOUN cana-1656 15 9 flaw flaw NOUN cana-1656 15 10 known know VERB cana-1656 15 11 as as ADP cana-1656 15 12 a a DET cana-1656 15 13 man man NOUN cana-1656 15 14 - - PUNCT cana-1656 15 15 in in ADP cana-1656 15 16 - - PUNCT cana-1656 15 17 the the DET cana-1656 15 18 - - PUNCT cana-1656 15 19 middle middle ADJ cana-1656 15 20 attack attack NOUN cana-1656 15 21 . . PUNCT cana-1656 16 1 later later ADV cana-1656 16 2 , , PUNCT cana-1656 16 3 the the DET cana-1656 16 4 diffie diffie PROPN cana-1656 16 5 hellman hellman PROPN cana-1656 16 6 key key PROPN cana-1656 16 7 exchange exchange NOUN cana-1656 16 8 algorithm algorithm NOUN cana-1656 16 9 was be AUX cana-1656 16 10 modified modify VERB cana-1656 16 11 and and CCONJ cana-1656 16 12 published publish VERB cana-1656 16 13 by by ADP cana-1656 16 14 several several ADJ cana-1656 16 15 authors author NOUN cana-1656 16 16 , , PUNCT cana-1656 16 17 on on ADP cana-1656 16 18 the the DET cana-1656 16 19 other other ADJ cana-1656 16 20 hand hand NOUN cana-1656 16 21 , , PUNCT cana-1656 16 22 rsa rsa PROPN cana-1656 16 23 is be AUX cana-1656 16 24 a a DET cana-1656 16 25 public public ADJ cana-1656 16 26 key key ADJ cana-1656 16 27 cryptosystem cryptosystem NOUN cana-1656 16 28 comes come VERB cana-1656 16 29 from from ADP cana-1656 16 30 the the DET cana-1656 16 31 challenge challenge NOUN cana-1656 16 32 of of ADP cana-1656 16 33 factorizing factorize VERB cana-1656 16 34 big big ADJ cana-1656 16 35 integers integer NOUN cana-1656 16 36 into into ADP cana-1656 16 37 their their PRON cana-1656 16 38 prime prime NOUN cana-1656 16 39 . . PUNCT cana-1656 17 1 even even ADV cana-1656 17 2 if if SCONJ cana-1656 17 3 the the DET cana-1656 17 4 key key ADJ cana-1656 17 5 size size NOUN cana-1656 17 6 doubles double VERB cana-1656 17 7 , , PUNCT cana-1656 17 8 the the DET cana-1656 17 9 rsa rsa PROPN cana-1656 17 10 algorithm algorithm NOUN cana-1656 17 11 's 's PART cana-1656 17 12 strength strength NOUN cana-1656 17 13 will will AUX cana-1656 17 14 not not PART cana-1656 17 15 increase increase VERB cana-1656 17 16 although although SCONJ cana-1656 17 17 the the DET cana-1656 17 18 public public ADJ cana-1656 17 19 key key ADJ cana-1656 17 20 operations operation NOUN cana-1656 17 21 being be AUX cana-1656 17 22 faster fast ADJ cana-1656 17 23 . . PUNCT cana-1656 18 1 elliptic elliptic ADJ cana-1656 18 2 curve curve NOUN cana-1656 18 3 cryptography cryptography NOUN cana-1656 18 4 depends depend VERB cana-1656 18 5 of of ADP cana-1656 18 6 the the DET cana-1656 18 7 hardness hardness NOUN cana-1656 18 8 of of ADP cana-1656 18 9 elliptic elliptic ADJ cana-1656 18 10 curve curve NOUN cana-1656 18 11 discrete discrete ADJ cana-1656 18 12 logarithm logarithm NOUN cana-1656 18 13 problem problem NOUN cana-1656 18 14 ( ( PUNCT cana-1656 18 15 ecdlp ecdlp NOUN cana-1656 18 16 ) ) PUNCT cana-1656 18 17 . . PUNCT cana-1656 19 1 the the DET cana-1656 19 2 concept concept NOUN cana-1656 19 3 of of ADP cana-1656 19 4 elliptic elliptic ADJ cana-1656 19 5 curve curve NOUN cana-1656 19 6 utilization utilization NOUN cana-1656 19 7 in in ADP cana-1656 19 8 public public ADJ cana-1656 19 9 key key ADJ cana-1656 19 10 cryptography cryptography NOUN cana-1656 19 11 trailblazers trailblazer NOUN cana-1656 19 12 were be AUX cana-1656 19 13 koblitz koblitz ADJ cana-1656 19 14 and and CCONJ cana-1656 19 15 miller miller PROPN cana-1656 19 16 . . PUNCT cana-1656 20 1 the the DET cana-1656 20 2 main main ADJ cana-1656 20 3 motive motive NOUN cana-1656 20 4 for for ADP cana-1656 20 5 attentive attentive ADJ cana-1656 20 6 attraction attraction NOUN cana-1656 20 7 of of ADP cana-1656 20 8 the the DET cana-1656 20 9 researchers researcher NOUN cana-1656 20 10 towards towards ADP cana-1656 20 11 ecc ecc PROPN cana-1656 20 12 is be AUX cana-1656 20 13 yields yield NOUN cana-1656 20 14 the the DET cana-1656 20 15 level level NOUN cana-1656 20 16 of of ADP cana-1656 20 17 security security NOUN cana-1656 20 18 as as ADP cana-1656 20 19 rsa rsa PROPN cana-1656 20 20 . . PUNCT cana-1656 21 1 2 2 X cana-1656 21 2 . . X cana-1656 21 3 literature literature NOUN cana-1656 21 4 survey survey NOUN cana-1656 21 5 : : PUNCT cana-1656 21 6 numerous numerous ADJ cana-1656 21 7 researchers researcher NOUN cana-1656 21 8 employed employ VERB cana-1656 21 9 elliptic elliptic ADJ cana-1656 21 10 curves curve NOUN cana-1656 21 11 over over ADP cana-1656 21 12 finite finite ADJ cana-1656 21 13 fields field NOUN cana-1656 21 14 for for ADP cana-1656 21 15 security security NOUN cana-1656 21 16 applications application NOUN cana-1656 21 17 and and CCONJ cana-1656 21 18 researched research VERB cana-1656 21 19 the the DET cana-1656 21 20 benefits benefit NOUN cana-1656 21 21 of of ADP cana-1656 21 22 ecc ecc PROPN cana-1656 21 23 over over ADP cana-1656 21 24 traditional traditional ADJ cana-1656 21 25 cryptosystems cryptosystem NOUN cana-1656 21 26 . . PUNCT cana-1656 22 1 ecc ecc PROPN cana-1656 22 2 - - PUNCT cana-1656 22 3 based base VERB cana-1656 22 4 text text NOUN cana-1656 22 5 - - PUNCT cana-1656 22 6 based base VERB cana-1656 22 7 encryption encryption NOUN cana-1656 22 8 techniques technique NOUN cana-1656 22 9 were be AUX cana-1656 22 10 proposed propose VERB cana-1656 22 11 by by ADP cana-1656 22 12 several several ADJ cana-1656 22 13 authors author NOUN cana-1656 22 14 . . PUNCT cana-1656 23 1 when when SCONJ cana-1656 23 2 it it PRON cana-1656 23 3 comes come VERB cana-1656 23 4 to to ADP cana-1656 23 5 ( ( PUNCT cana-1656 23 6 secure secure VERB cana-1656 23 7 sockets socket NOUN cana-1656 23 8 layer layer NOUN cana-1656 23 9 ) ) PUNCT cana-1656 23 10 ssl ssl PROPN cana-1656 23 11 and and CCONJ cana-1656 23 12 tls tls PROPN cana-1656 23 13 certificates certificate NOUN cana-1656 23 14 , , PUNCT cana-1656 23 15 benjamin benjamin PROPN cana-1656 23 16 clement clement PROPN cana-1656 23 17 sebastian sebastian PROPN cana-1656 23 18 and and CCONJ cana-1656 23 19 ugar ugar ADJ cana-1656 23 20 alpay alpay ADJ cana-1656 23 21 cenar cenar PROPN cana-1656 23 22 [ [ X cana-1656 23 23 1 1 NUM cana-1656 23 24 ] ] PUNCT cana-1656 23 25 researched research VERB cana-1656 23 26 and and CCONJ cana-1656 23 27 compared compare VERB cana-1656 23 28 the the DET cana-1656 23 29 substantial substantial ADJ cana-1656 23 30 advantages advantage NOUN cana-1656 23 31 of of ADP cana-1656 23 32 ecc ecc PROPN cana-1656 23 33 over over ADP cana-1656 23 34 rsa rsa PROPN cana-1656 23 35 . . PUNCT cana-1656 24 1 elliptic elliptic ADJ cana-1656 24 2 curve curve NOUN cana-1656 24 3 cryptography cryptography NOUN cana-1656 24 4 was be AUX cana-1656 24 5 used use VERB cana-1656 24 6 to to PART cana-1656 24 7 implement implement VERB cana-1656 24 8 text text NOUN cana-1656 24 9 encryption encryption NOUN cana-1656 24 10 by by ADP cana-1656 24 11 l. l. PROPN cana-1656 24 12 dolendro dolendro PROPN cana-1656 24 13 and and CCONJ cana-1656 24 14 k. k. PROPN cana-1656 24 15 manglem manglem PROPN cana-1656 25 1 [ [ X cana-1656 25 2 2 2 NUM cana-1656 25 3 ] ] PUNCT cana-1656 25 4 . . PUNCT cana-1656 26 1 the the DET cana-1656 26 2 writers writer NOUN cana-1656 26 3 of of ADP cana-1656 26 4 this this DET cana-1656 26 5 chapter chapter NOUN cana-1656 26 6 provided provide VERB cana-1656 26 7 a a DET cana-1656 26 8 conventional conventional ADJ cana-1656 26 9 communications communication NOUN cana-1656 26 10 on on ADP cana-1656 26 11 applied apply VERB cana-1656 26 12 nonlinear nonlinear ADJ cana-1656 26 13 analysis analysis NOUN cana-1656 26 14 issn issn NOUN cana-1656 26 15 : : PUNCT cana-1656 26 16 1074 1074 NUM cana-1656 26 17 - - PUNCT cana-1656 26 18 133x 133x NUM cana-1656 26 19 vol vol NOUN cana-1656 26 20 32 32 NUM cana-1656 26 21 no no NOUN cana-1656 26 22 . . NOUN cana-1656 26 23 1 1 NUM cana-1656 26 24 ( ( PUNCT cana-1656 26 25 2025 2025 NUM cana-1656 26 26 ) ) PUNCT cana-1656 27 1 353 353 NUM cana-1656 27 2 https://internationalpubls.com https://internationalpubls.com X cana-1656 27 3 way way NOUN cana-1656 27 4 of of ADP cana-1656 27 5 mapping map VERB cana-1656 27 6 the the DET cana-1656 27 7 characters character NOUN cana-1656 27 8 to to PART cana-1656 27 9 affine affine VERB cana-1656 27 10 points point NOUN cana-1656 27 11 in in ADP cana-1656 27 12 the the DET cana-1656 27 13 curve curve NOUN cana-1656 27 14 by by ADP cana-1656 27 15 matching match VERB cana-1656 27 16 the the DET cana-1656 27 17 ascii ascii ADJ cana-1656 27 18 values value NOUN cana-1656 27 19 of of ADP cana-1656 27 20 the the DET cana-1656 27 21 plain plain ADJ cana-1656 27 22 text text NOUN cana-1656 27 23 characters character NOUN cana-1656 27 24 . . PUNCT cana-1656 28 1 a a DET cana-1656 28 2 proposed propose VERB cana-1656 28 3 encryption encryption NOUN cana-1656 28 4 method method NOUN cana-1656 28 5 using use VERB cana-1656 28 6 elliptic elliptic ADJ cana-1656 28 7 curve curve NOUN cana-1656 28 8 cryptography cryptography NOUN cana-1656 28 9 was be AUX cana-1656 28 10 made make VERB cana-1656 28 11 by by ADP cana-1656 28 12 renee renee PROPN cana-1656 28 13 brady brady PROPN cana-1656 28 14 et et PROPN cana-1656 28 15 al al PROPN cana-1656 28 16 . . PUNCT cana-1656 29 1 [ [ X cana-1656 29 2 3 3 NUM cana-1656 29 3 ] ] PUNCT cana-1656 29 4 . . PUNCT cana-1656 30 1 they they PRON cana-1656 30 2 used use VERB cana-1656 30 3 unicode unicode NOUN cana-1656 30 4 to to PART cana-1656 30 5 encrypt encrypt VERB cana-1656 30 6 the the DET cana-1656 30 7 text text NOUN cana-1656 30 8 in in ADP cana-1656 30 9 that that DET cana-1656 30 10 study study NOUN cana-1656 30 11 and and CCONJ cana-1656 30 12 explored explore VERB cana-1656 30 13 the the DET cana-1656 30 14 drawbacks drawback NOUN cana-1656 30 15 of of ADP cana-1656 30 16 sending send VERB cana-1656 30 17 text text NOUN cana-1656 30 18 communications communication NOUN cana-1656 30 19 through through ADP cana-1656 30 20 platforms platform NOUN cana-1656 30 21 like like ADP cana-1656 30 22 twitter twitter NOUN cana-1656 30 23 . . PUNCT cana-1656 31 1 the the DET cana-1656 31 2 text text NOUN cana-1656 31 3 is be AUX cana-1656 31 4 encoded encode VERB cana-1656 31 5 using use VERB cana-1656 31 6 the the DET cana-1656 31 7 koblitz koblitz ADJ cana-1656 31 8 method method NOUN cana-1656 31 9 as as ADP cana-1656 31 10 a a DET cana-1656 31 11 point point NOUN cana-1656 31 12 on on ADP cana-1656 31 13 an an DET cana-1656 31 14 elliptic elliptic ADJ cana-1656 31 15 curve curve NOUN cana-1656 31 16 . . PUNCT cana-1656 32 1 for for ADP cana-1656 32 2 users user NOUN cana-1656 32 3 of of ADP cana-1656 32 4 online online ADJ cana-1656 32 5 social social ADJ cana-1656 32 6 networks network NOUN cana-1656 32 7 , , PUNCT cana-1656 32 8 s. s. PROPN cana-1656 32 9 thiraviya thiraviya PROPN cana-1656 32 10 and and CCONJ cana-1656 32 11 s. s. PROPN cana-1656 32 12 britto britto PROPN cana-1656 33 1 [ [ X cana-1656 33 2 4 4 X cana-1656 33 3 ] ] PUNCT cana-1656 33 4 suggested suggest VERB cana-1656 33 5 better well ADJ cana-1656 33 6 elliptic elliptic ADJ cana-1656 33 7 curve curve NOUN cana-1656 33 8 cryptography cryptography NOUN cana-1656 33 9 . . PUNCT cana-1656 34 1 they they PRON cana-1656 34 2 used use VERB cana-1656 34 3 encryption encryption NOUN cana-1656 34 4 , , PUNCT cana-1656 34 5 decryption decryption NOUN cana-1656 34 6 time time NOUN cana-1656 34 7 analysis analysis NOUN cana-1656 34 8 , , PUNCT cana-1656 34 9 plain plain ADJ cana-1656 34 10 text text NOUN cana-1656 34 11 , , PUNCT cana-1656 34 12 and and CCONJ cana-1656 34 13 cipher cipher ADJ cana-1656 34 14 text text NOUN cana-1656 34 15 size size NOUN cana-1656 34 16 analysis analysis NOUN cana-1656 34 17 in in ADP cana-1656 34 18 that that DET cana-1656 34 19 paper paper NOUN cana-1656 34 20 . . PUNCT cana-1656 35 1 a. a. NOUN cana-1656 35 2 shamir shamir PROPN cana-1656 36 1 [ [ X cana-1656 36 2 5 5 NUM cana-1656 36 3 ] ] PUNCT cana-1656 36 4 was be AUX cana-1656 36 5 the the DET cana-1656 36 6 first first ADJ cana-1656 36 7 to to PART cana-1656 36 8 suggest suggest VERB cana-1656 36 9 an an DET cana-1656 36 10 identity identity NOUN cana-1656 36 11 - - PUNCT cana-1656 36 12 based base VERB cana-1656 36 13 cryptosystem cryptosystem NOUN cana-1656 36 14 as as ADP cana-1656 36 15 an an DET cana-1656 36 16 alternative alternative NOUN cana-1656 36 17 to to ADP cana-1656 36 18 conventional conventional ADJ cana-1656 36 19 public public ADJ cana-1656 36 20 key key ADJ cana-1656 36 21 infrastructure infrastructure NOUN cana-1656 36 22 for for ADP cana-1656 36 23 resolving resolve VERB cana-1656 36 24 authenticity authenticity NOUN cana-1656 36 25 problems problem NOUN cana-1656 36 26 . . PUNCT cana-1656 37 1 a a DET cana-1656 37 2 pre pre ADJ cana-1656 37 3 - - ADJ cana-1656 37 4 communication communication ADJ cana-1656 37 5 requirement requirement NOUN cana-1656 37 6 is be AUX cana-1656 37 7 part part NOUN cana-1656 37 8 of of ADP cana-1656 37 9 the the DET cana-1656 37 10 authentication authentication NOUN cana-1656 37 11 system system NOUN cana-1656 37 12 described describe VERB cana-1656 37 13 by by ADP cana-1656 37 14 aziz aziz PROPN cana-1656 37 15 and and CCONJ cana-1656 37 16 diffie diffie PROPN cana-1656 38 1 [ [ X cana-1656 38 2 6 6 NUM cana-1656 38 3 ] ] PUNCT cana-1656 38 4 . . PUNCT cana-1656 39 1 the the DET cana-1656 39 2 use use NOUN cana-1656 39 3 of of ADP cana-1656 39 4 a a DET cana-1656 39 5 nonsingular nonsingular ADJ cana-1656 39 6 matrix matrix NOUN cana-1656 39 7 to to PART cana-1656 39 8 map map VERB cana-1656 39 9 the the DET cana-1656 39 10 identical identical ADJ cana-1656 39 11 characters character NOUN cana-1656 39 12 in in ADP cana-1656 39 13 the the DET cana-1656 39 14 message message NOUN cana-1656 39 15 to to ADP cana-1656 39 16 different different ADJ cana-1656 39 17 positions position NOUN cana-1656 39 18 on on ADP cana-1656 39 19 the the DET cana-1656 39 20 curve curve NOUN cana-1656 39 21 was be AUX cana-1656 39 22 recommended recommend VERB cana-1656 39 23 by by ADP cana-1656 39 24 f. f. PROPN cana-1656 39 25 amounas amounas PROPN cana-1656 39 26 and and CCONJ cana-1656 39 27 e.h e.h PROPN cana-1656 39 28 . . PROPN cana-1656 39 29 ei ei PROPN cana-1656 39 30 kinani kinani PROPN cana-1656 40 1 [ [ X cana-1656 40 2 7 7 NUM cana-1656 40 3 ] ] PUNCT cana-1656 40 4 . . PUNCT cana-1656 41 1 elliptic elliptic ADJ cana-1656 41 2 curves curve NOUN cana-1656 41 3 were be AUX cana-1656 41 4 suggested suggest VERB cana-1656 41 5 for for ADP cana-1656 41 6 encryption encryption NOUN cana-1656 41 7 and and CCONJ cana-1656 41 8 decryption decryption NOUN cana-1656 41 9 by by ADP cana-1656 41 10 kolhekar kolhekar NOUN cana-1656 41 11 m. m. NOUN cana-1656 41 12 and and CCONJ cana-1656 41 13 jadhav jadhav PROPN cana-1656 41 14 a. a. PROPN cana-1656 42 1 [ [ X cana-1656 42 2 8 8 NUM cana-1656 42 3 ] ] PUNCT cana-1656 42 4 , , PUNCT cana-1656 42 5 a. a. NOUN cana-1656 42 6 gera gera PROPN cana-1656 42 7 and and CCONJ cana-1656 42 8 k. k. PROPN cana-1656 42 9 agrawal agrawal PROPN cana-1656 43 1 [ [ X cana-1656 43 2 9 9 NUM cana-1656 43 3 ] ] PUNCT cana-1656 43 4 . . PUNCT cana-1656 44 1 elliptic elliptic ADJ cana-1656 44 2 curve curve PROPN cana-1656 44 3 arithmetic arithmetic VERB cana-1656 44 4 an an DET cana-1656 44 5 irreducible irreducible ADJ cana-1656 44 6 cubic cubic NOUN cana-1656 44 7 with with ADP cana-1656 44 8 a a DET cana-1656 44 9 flex flex ADJ cana-1656 44 10 can can AUX cana-1656 44 11 be be AUX cana-1656 44 12 affinely affinely ADV cana-1656 44 13 transformed transform VERB cana-1656 44 14 into into ADP cana-1656 44 15 a a DET cana-1656 44 16 weierstrass weierstrass NOUN cana-1656 44 17 equation equation NOUN cana-1656 44 18 : : PUNCT cana-1656 44 19 𝑄2 𝑄2 PROPN cana-1656 44 20 + + CCONJ cana-1656 44 21 π‘Ž1𝑃𝑄 π‘Ž1𝑃𝑄 NOUN cana-1656 44 22 + + CCONJ cana-1656 44 23 π‘Ž3𝑄 π‘Ž3𝑄 NOUN cana-1656 44 24 = = SYM cana-1656 44 25 𝑃3 𝑃3 NOUN cana-1656 45 1 + + CCONJ cana-1656 45 2 π‘Ž2𝑃2 π‘Ž2𝑃2 PRON cana-1656 45 3 + + NUM cana-1656 45 4 π‘Ž4𝑃 π‘Ž4𝑃 NOUN cana-1656 45 5 + + CCONJ cana-1656 45 6 π‘Ž6 π‘Ž6 NOUN cana-1656 45 7 , , PUNCT cana-1656 45 8 where where SCONJ cana-1656 45 9 π‘Ž1π‘Ž2 π‘Ž1π‘Ž2 X cana-1656 45 10 , , PUNCT cana-1656 45 11 π‘Ž3,π‘Ž4 π‘Ž3,π‘Ž4 PROPN cana-1656 45 12 , , PUNCT cana-1656 45 13 π‘Ž6 π‘Ž6 NOUN cana-1656 45 14 are be AUX cana-1656 45 15 real real ADJ cana-1656 45 16 and and CCONJ cana-1656 45 17 p∈ p∈ PROPN cana-1656 45 18 𝑅 𝑅 PROPN cana-1656 45 19 , , PUNCT cana-1656 45 20 q∈ q∈ VERB cana-1656 45 21 𝑅.the 𝑅.the DET cana-1656 45 22 equationy2 equationy2 NOUN cana-1656 46 1 = = SYM cana-1656 46 2 x3 x3 VERB cana-1656 46 3 + + PUNCT cana-1656 46 4 ax ax ADJ cana-1656 46 5 + + NOUN cana-1656 46 6 b b NOUN cana-1656 46 7 , , PUNCT cana-1656 46 8 4a3 4a3 NUM cana-1656 46 9 + + PROPN cana-1656 46 10 27b2β‰  27b2β‰  NUM cana-1656 46 11 0 0 NUM cana-1656 46 12 over over ADP cana-1656 46 13 the the DET cana-1656 46 14 finite finite ADJ cana-1656 46 15 field field NOUN cana-1656 46 16 fq fq PROPN cana-1656 46 17 defines define VERB cana-1656 46 18 an an DET cana-1656 46 19 elliptic elliptic ADJ cana-1656 46 20 curve curve NOUN cana-1656 46 21 for for ADP cana-1656 46 22 cryptography cryptography NOUN cana-1656 46 23 purposes purpose NOUN cana-1656 46 24 . . PUNCT cana-1656 47 1 the the DET cana-1656 47 2 points point NOUN cana-1656 47 3 on on ADP cana-1656 47 4 the the DET cana-1656 47 5 curve curve NOUN cana-1656 47 6 taking take VERB cana-1656 47 7 ∞ ∞ PROPN cana-1656 47 8 , , PUNCT cana-1656 47 9 infinity infinity NOUN cana-1656 47 10 point point NOUN cana-1656 47 11 as as SCONJ cana-1656 47 12 identity identity NOUN cana-1656 47 13 element element NOUN cana-1656 47 14 an an DET cana-1656 47 15 abelian abelian ADJ cana-1656 47 16 group group NOUN cana-1656 47 17 with with ADP cana-1656 47 18 respect respect NOUN cana-1656 47 19 to to ADP cana-1656 47 20 point point NOUN cana-1656 47 21 addition addition NOUN cana-1656 47 22 is be AUX cana-1656 47 23 formed form VERB cana-1656 47 24 point point NOUN cana-1656 47 25 addition addition NOUN cana-1656 47 26 on on ADP cana-1656 47 27 the the DET cana-1656 47 28 curve curve NOUN cana-1656 47 29 : : PUNCT cana-1656 47 30 if if SCONJ cana-1656 47 31 ΖΏ(a1,b1 ΖΏ(a1,b1 NUM cana-1656 47 32 ) ) PUNCT cana-1656 47 33 and and CCONJ cana-1656 47 34 Ι‹(a2,b2 Ι‹(a2,b2 VERB cana-1656 47 35 ) ) PUNCT cana-1656 47 36 are be AUX cana-1656 47 37 two two NUM cana-1656 47 38 points point NOUN cana-1656 47 39 on on ADP cana-1656 47 40 the the DET cana-1656 47 41 curve curve NOUN cana-1656 47 42 , , PUNCT cana-1656 47 43 ΖΏ ΖΏ PUNCT cana-1656 48 1 + + PUNCT cana-1656 48 2 Ι‹ Ι‹ PRON cana-1656 48 3 is be AUX cana-1656 48 4 obtained obtain VERB cana-1656 48 5 by by ADP cana-1656 48 6 taking take VERB cana-1656 48 7 the the DET cana-1656 48 8 image image NOUN cana-1656 48 9 or or CCONJ cana-1656 48 10 reflection reflection NOUN cana-1656 48 11 of of ADP cana-1656 48 12 the the DET cana-1656 48 13 point point NOUN cana-1656 48 14 obtained obtain VERB cana-1656 48 15 by by ADP cana-1656 48 16 intersecting intersect VERB cana-1656 48 17 the the DET cana-1656 48 18 line line NOUN cana-1656 48 19 through through ADP cana-1656 48 20 ΖΏ ΖΏ PROPN cana-1656 48 21 , , PUNCT cana-1656 48 22 Ι‹ Ι‹ X cana-1656 48 23 with with ADP cana-1656 48 24 the the DET cana-1656 48 25 curve curve NOUN cana-1656 48 26 . . PUNCT cana-1656 49 1 the the DET cana-1656 49 2 reflection reflection NOUN cana-1656 49 3 is be AUX cana-1656 49 4 considered consider VERB cana-1656 49 5 about about ADP cana-1656 49 6 x x NOUN cana-1656 49 7 - - NOUN cana-1656 49 8 axis axis ADJ cana-1656 49 9 . . PUNCT cana-1656 50 1 for for ADP cana-1656 50 2 opposite opposite ADJ cana-1656 50 3 points point NOUN cana-1656 50 4 ΖΏ ΖΏ PROPN cana-1656 50 5 and and CCONJ cana-1656 50 6 – – PUNCT cana-1656 50 7 ΖΏ ΖΏ PROPN cana-1656 50 8 the the DET cana-1656 50 9 vertical vertical ADJ cana-1656 50 10 line line NOUN cana-1656 50 11 is be AUX cana-1656 50 12 intersected intersect VERB cana-1656 50 13 through through ADP cana-1656 50 14 the the DET cana-1656 50 15 curve curve NOUN cana-1656 50 16 . . PUNCT cana-1656 51 1 the the DET cana-1656 51 2 identity identity NOUN cana-1656 51 3 is be AUX cana-1656 51 4 ΖΏ ΖΏ PROPN cana-1656 51 5 + + CCONJ cana-1656 51 6 ( ( PUNCT cana-1656 51 7 -ΖΏ -ΖΏ PUNCT cana-1656 51 8 ) ) PUNCT cana-1656 51 9 = = SYM cana-1656 51 10 ∞. ∞. PROPN cana-1656 51 11 for for ADP cana-1656 51 12 ΖΏ(a1,b1),Ι‹(a2,b2 ΖΏ(a1,b1),Ι‹(a2,b2 PRON cana-1656 51 13 ) ) PUNCT cana-1656 51 14 ΖΏ ΖΏ PUNCT cana-1656 52 1 + + PUNCT cana-1656 52 2 Ι‹ Ι‹ PROPN cana-1656 52 3 is be AUX cana-1656 52 4 ɍ(a3,b3 ɍ(a3,b3 NOUN cana-1656 52 5 ) ) PUNCT cana-1656 52 6 π‘Ž3 π‘Ž3 NOUN cana-1656 52 7 = = SYM cana-1656 52 8 ( ( PUNCT cana-1656 52 9 𝑏2βˆ’π‘1 𝑏2βˆ’π‘1 PROPN cana-1656 52 10 π‘Ž2βˆ’π‘Ž1 π‘Ž2βˆ’π‘Ž1 CCONJ cana-1656 52 11 ) ) PUNCT cana-1656 52 12 2 2 NUM cana-1656 52 13 βˆ’ βˆ’ NOUN cana-1656 53 1 π‘Ž1 π‘Ž1 INTJ cana-1656 53 2 βˆ’ βˆ’ NOUN cana-1656 54 1 π‘Ž1 π‘Ž1 INTJ cana-1656 54 2 βˆ’ βˆ’ PROPN cana-1656 54 3 π‘Ž2 π‘Ž2 NOUN cana-1656 54 4 𝑏3 𝑏3 NOUN cana-1656 54 5 = = SYM cana-1656 54 6 ( ( PUNCT cana-1656 54 7 𝑏2βˆ’π‘1 𝑏2βˆ’π‘1 PROPN cana-1656 54 8 π‘Ž2βˆ’π‘Ž1 π‘Ž2βˆ’π‘Ž1 CCONJ cana-1656 54 9 ) ) PUNCT cana-1656 54 10 ( ( PUNCT cana-1656 54 11 π‘Ž2 π‘Ž2 NOUN cana-1656 54 12 βˆ’ βˆ’ PROPN cana-1656 54 13 π‘Ž3 π‘Ž3 PROPN cana-1656 54 14 ) ) PUNCT cana-1656 54 15 βˆ’ βˆ’ ADP cana-1656 54 16 𝑏1 𝑏1 ADJ cana-1656 54 17 point point NOUN cana-1656 54 18 multiplication multiplication NOUN cana-1656 54 19 or or CCONJ cana-1656 54 20 point point NOUN cana-1656 54 21 doubling double VERB cana-1656 54 22 on on ADP cana-1656 54 23 the the DET cana-1656 54 24 curve curve NOUN cana-1656 54 25 : : PUNCT cana-1656 54 26 for for ADP cana-1656 54 27 the the DET cana-1656 54 28 point point NOUN cana-1656 54 29 ΖΏ ΖΏ X cana-1656 54 30 ( ( PUNCT cana-1656 54 31 a1 a1 PROPN cana-1656 54 32 , , PUNCT cana-1656 54 33 b1 b1 NOUN cana-1656 54 34 ) ) PUNCT cana-1656 54 35 , , PUNCT cana-1656 54 36 2 2 NUM cana-1656 54 37 ΖΏ ΖΏ X cana-1656 54 38 is be AUX cana-1656 54 39 the the DET cana-1656 54 40 image image NOUN cana-1656 54 41 of of ADP cana-1656 54 42 the the DET cana-1656 54 43 point point NOUN cana-1656 54 44 obtained obtain VERB cana-1656 54 45 by by ADP cana-1656 54 46 intersecting intersect VERB cana-1656 54 47 the the DET cana-1656 54 48 tangent tangent NOUN cana-1656 54 49 through through ADP cana-1656 54 50 ΖΏ ΖΏ PROPN cana-1656 54 51 with with ADP cana-1656 54 52 the the DET cana-1656 54 53 curve curve NOUN cana-1656 54 54 . . PUNCT cana-1656 55 1 the the DET cana-1656 55 2 image image NOUN cana-1656 55 3 is be AUX cana-1656 55 4 about about ADP cana-1656 55 5 xaxis xaxis ADJ cana-1656 56 1 [ [ X cana-1656 56 2 1,2 1,2 NUM cana-1656 56 3 ] ] PUNCT cana-1656 56 4 . . PUNCT cana-1656 57 1 for for ADP cana-1656 57 2 ΖΏ ΖΏ PROPN cana-1656 57 3 β‰  β‰  PROPN cana-1656 57 4 ΖΏ ΖΏ PROPN cana-1656 57 5 , , PUNCT cana-1656 57 6 2 2 NUM cana-1656 57 7 ΖΏ ΖΏ NOUN cana-1656 57 8 is be AUX cana-1656 57 9 given give VERB cana-1656 57 10 by by ADP cana-1656 57 11 ( ( PUNCT cana-1656 57 12 a3 a3 NOUN cana-1656 57 13 , , PUNCT cana-1656 57 14 b3 b3 PROPN cana-1656 57 15 ) ) PUNCT cana-1656 57 16 π‘Ž3 π‘Ž3 NOUN cana-1656 58 1 = = PUNCT cana-1656 58 2 ( ( PUNCT cana-1656 58 3 3π‘Ž1 3π‘Ž1 NUM cana-1656 58 4 2+π‘Ž 2+π‘Ž NUM cana-1656 58 5 2𝑏1 2𝑏1 NUM cana-1656 58 6 ) ) PUNCT cana-1656 58 7 2 2 NUM cana-1656 58 8 βˆ’ βˆ’ NUM cana-1656 58 9 2π‘Ž1 2π‘Ž1 NUM cana-1656 58 10 𝑏3 𝑏3 NOUN cana-1656 58 11 = = SYM cana-1656 58 12 ( ( PUNCT cana-1656 58 13 3π‘Ž1 3π‘Ž1 NUM cana-1656 58 14 2 2 NUM cana-1656 58 15 + + CCONJ cana-1656 58 16 π‘Ž π‘Ž PROPN cana-1656 58 17 2𝑏1 2𝑏1 NUM cana-1656 58 18 ) ) PUNCT cana-1656 58 19 ( ( PUNCT cana-1656 58 20 π‘Ž1 π‘Ž1 ADP cana-1656 58 21 βˆ’ βˆ’ PROPN cana-1656 58 22 π‘Ž3 π‘Ž3 PROPN cana-1656 58 23 ) ) PUNCT cana-1656 58 24 βˆ’ βˆ’ ADP cana-1656 58 25 𝑏1 𝑏1 ADJ cana-1656 58 26 communications communication NOUN cana-1656 58 27 on on ADP cana-1656 58 28 applied apply VERB cana-1656 58 29 nonlinear nonlinear ADJ cana-1656 58 30 analysis analysis NOUN cana-1656 58 31 issn issn NOUN cana-1656 58 32 : : PUNCT cana-1656 58 33 1074 1074 NUM cana-1656 58 34 - - PUNCT cana-1656 58 35 133x 133x NUM cana-1656 58 36 vol vol NOUN cana-1656 58 37 32 32 NUM cana-1656 58 38 no no NOUN cana-1656 58 39 . . NOUN cana-1656 58 40 1 1 NUM cana-1656 58 41 ( ( PUNCT cana-1656 58 42 2025 2025 NUM cana-1656 58 43 ) ) PUNCT cana-1656 58 44 354 354 NUM cana-1656 58 45 https://internationalpubls.com https://internationalpubls.com X cana-1656 58 46 figure figure NOUN cana-1656 58 47 1 1 NUM cana-1656 58 48 : : PUNCT cana-1656 58 49 elliptic elliptic ADJ cana-1656 58 50 curve curve NOUN cana-1656 58 51 arithmetic arithmetic ADJ cana-1656 58 52 discrete discrete ADJ cana-1656 58 53 logarithmic logarithmic ADJ cana-1656 58 54 problem problem NOUN cana-1656 58 55 on on ADP cana-1656 58 56 elliptic elliptic ADJ cana-1656 58 57 curve curve NOUN cana-1656 58 58 ( ( PUNCT cana-1656 58 59 ecdlp ecdlp NOUN cana-1656 58 60 ): ): PUNCT cana-1656 58 61 above above ADP cana-1656 58 62 figure figure NOUN cana-1656 58 63 1 1 NUM cana-1656 58 64 , , PUNCT cana-1656 58 65 elliptic elliptic ADJ cana-1656 58 66 curve curve NOUN cana-1656 58 67 cryptography cryptography NOUN cana-1656 58 68 is be AUX cana-1656 58 69 safeguarded safeguard VERB cana-1656 58 70 by by ADP cana-1656 58 71 a a DET cana-1656 58 72 hard hard ADJ cana-1656 58 73 problem problem NOUN cana-1656 58 74 called call VERB cana-1656 58 75 elliptic elliptic ADJ cana-1656 58 76 curve curve NOUN cana-1656 58 77 discrete discrete ADJ cana-1656 58 78 logarithmic logarithmic ADJ cana-1656 58 79 problem problem NOUN cana-1656 58 80 ( ( PUNCT cana-1656 58 81 ecdlp ecdlp NOUN cana-1656 58 82 ) ) PUNCT cana-1656 59 1 [ [ X cana-1656 59 2 4,5,6 4,5,6 NUM cana-1656 59 3 ] ] X cana-1656 59 4 . . PUNCT cana-1656 60 1 for for ADP cana-1656 60 2 two two NUM cana-1656 60 3 given give VERB cana-1656 60 4 points point NOUN cana-1656 60 5 p p X cana-1656 60 6 , , PUNCT cana-1656 60 7 q q NOUN cana-1656 60 8 on on ADP cana-1656 60 9 the the DET cana-1656 60 10 curve curve NOUN cana-1656 60 11 such such DET cana-1656 60 12 that that DET cana-1656 60 13 q q NOUN cana-1656 61 1 = = X cana-1656 61 2 xp xp INTJ cana-1656 61 3 for for ADP cana-1656 61 4 β€˜ ' PUNCT cana-1656 61 5 x x NOUN cana-1656 61 6 ’ ' PUNCT cana-1656 61 7 a a DET cana-1656 61 8 big big ADJ cana-1656 61 9 random random ADJ cana-1656 61 10 number number NOUN cana-1656 61 11 it it PRON cana-1656 61 12 is be AUX cana-1656 61 13 difficult difficult ADJ cana-1656 61 14 to to PART cana-1656 61 15 find find VERB cana-1656 61 16 x x SYM cana-1656 61 17 = = PRON cana-1656 61 18 π‘™π‘œπ‘”π‘q π‘™π‘œπ‘”π‘q NOUN cana-1656 61 19 for for ADP cana-1656 61 20 given give VERB cana-1656 61 21 p p PROPN cana-1656 61 22 , , PUNCT cana-1656 61 23 q. q. PROPN cana-1656 61 24 methodology methodology PROPN cana-1656 61 25 : : PUNCT cana-1656 61 26 an an DET cana-1656 61 27 elliptic elliptic ADJ cana-1656 61 28 curve curve NOUN cana-1656 61 29 𝐸𝑝(π‘Ž 𝐸𝑝(π‘Ž NOUN cana-1656 61 30 , , PUNCT cana-1656 61 31 𝑏 𝑏 NOUN cana-1656 61 32 ) ) PUNCT cana-1656 61 33 , , PUNCT cana-1656 61 34 with with ADP cana-1656 61 35 prime prime NOUN cana-1656 61 36 p p NOUN cana-1656 61 37 having have VERB cana-1656 61 38 as as ADV cana-1656 61 39 many many ADJ cana-1656 61 40 points point NOUN cana-1656 61 41 as as ADP cana-1656 61 42 possible possible ADJ cana-1656 61 43 and and CCONJ cana-1656 61 44 generators generator NOUN cana-1656 61 45 are be AUX cana-1656 61 46 selected select VERB cana-1656 61 47 by by ADP cana-1656 61 48 one one NUM cana-1656 61 49 of of ADP cana-1656 61 50 the the DET cana-1656 61 51 members member NOUN cana-1656 61 52 of of ADP cana-1656 61 53 asset asset NOUN cana-1656 61 54 of of ADP cana-1656 61 55 communicating communicate VERB cana-1656 61 56 team team NOUN cana-1656 61 57 . . PUNCT cana-1656 62 1 in in ADP cana-1656 62 2 curve curve NOUN cana-1656 62 3 selection selection NOUN cana-1656 62 4 the the DET cana-1656 62 5 following follow VERB cana-1656 62 6 main main ADJ cana-1656 62 7 features feature NOUN cana-1656 62 8 should should AUX cana-1656 62 9 be be AUX cana-1656 62 10 carefully carefully ADV cana-1656 62 11 observed observe VERB cana-1656 62 12 . . PUNCT cana-1656 63 1 i.the i.the DET cana-1656 63 2 order order NOUN cana-1656 63 3 of of ADP cana-1656 63 4 the the DET cana-1656 63 5 curve curve NOUN cana-1656 63 6 𝐸𝑝(π‘Ž 𝐸𝑝(π‘Ž PROPN cana-1656 63 7 , , PUNCT cana-1656 63 8 𝑏 𝑏 NOUN cana-1656 63 9 ) ) PUNCT cana-1656 63 10 must must AUX cana-1656 63 11 not not PART cana-1656 63 12 contain contain VERB cana-1656 63 13 small small ADJ cana-1656 63 14 prime prime ADJ cana-1656 63 15 factors factor NOUN cana-1656 63 16 . . PUNCT cana-1656 64 1 ii.the ii.the DET cana-1656 64 2 curve curve NOUN cana-1656 64 3 is be AUX cana-1656 64 4 supposed suppose VERB cana-1656 64 5 to to PART cana-1656 64 6 be be AUX cana-1656 64 7 not not PART cana-1656 64 8 super super ADV cana-1656 64 9 singular singular NOUN cana-1656 64 10 . . PUNCT cana-1656 65 1 iii iii X cana-1656 65 2 . . PUNCT cana-1656 66 1 it it PRON cana-1656 66 2 is be AUX cana-1656 66 3 non non NOUN cana-1656 66 4 anomalous anomalous ADJ cana-1656 66 5 ; ; PUNCT cana-1656 66 6 order order NOUN cana-1656 66 7 is be AUX cana-1656 66 8 not not PART cana-1656 66 9 equal equal ADJ cana-1656 66 10 to to ADP cana-1656 66 11 p p PROPN cana-1656 66 12 # # NOUN cana-1656 66 13 𝐸𝑝(π‘Ž 𝐸𝑝(π‘Ž NOUN cana-1656 66 14 , , PUNCT cana-1656 66 15 𝑏 𝑏 NOUN cana-1656 66 16 ) ) PUNCT cana-1656 66 17 β‰ p β‰ p NOUN cana-1656 66 18 in in ADP cana-1656 66 19 𝐸𝑝(π‘Ž 𝐸𝑝(π‘Ž NOUN cana-1656 66 20 , , PUNCT cana-1656 66 21 𝑏 𝑏 NOUN cana-1656 66 22 ) ) PUNCT cana-1656 66 23 with with ADP cana-1656 66 24 p p PRON cana-1656 66 25 as as ADP cana-1656 66 26 prime prime ADJ cana-1656 66 27 , , PUNCT cana-1656 66 28 to to PART cana-1656 66 29 have have VERB cana-1656 66 30 the the DET cana-1656 66 31 cyclic cyclic ADJ cana-1656 66 32 nature nature NOUN cana-1656 66 33 of of ADP cana-1656 66 34 the the DET cana-1656 66 35 group group NOUN cana-1656 66 36 and and CCONJ cana-1656 66 37 to to PART cana-1656 66 38 treat treat VERB cana-1656 66 39 each each DET cana-1656 66 40 point point NOUN cana-1656 66 41 as as ADP cana-1656 66 42 a a DET cana-1656 66 43 generator generator NOUN cana-1656 66 44 . . PUNCT cana-1656 67 1 all all DET cana-1656 67 2 the the DET cana-1656 67 3 points point NOUN cana-1656 67 4 on on ADP cana-1656 67 5 the the DET cana-1656 67 6 curve curve NOUN cana-1656 67 7 are be AUX cana-1656 67 8 mapped map VERB cana-1656 67 9 to to ADP cana-1656 67 10 alphabet alphabet PROPN cana-1656 67 11 , , PUNCT cana-1656 67 12 numeral numeral ADJ cana-1656 67 13 and and CCONJ cana-1656 67 14 alpha alpha ADJ cana-1656 67 15 numeral numeral ADJ cana-1656 67 16 characters character NOUN cana-1656 67 17 and and CCONJ cana-1656 67 18 a a DET cana-1656 67 19 common common ADJ cana-1656 67 20 look look NOUN cana-1656 67 21 up up ADP cana-1656 67 22 table table NOUN cana-1656 67 23 available available ADJ cana-1656 67 24 for for ADP cana-1656 67 25 all all DET cana-1656 67 26 the the DET cana-1656 67 27 members member NOUN cana-1656 67 28 . . PUNCT cana-1656 68 1 this this DET cana-1656 68 2 procedure procedure NOUN cana-1656 68 3 is be AUX cana-1656 68 4 set set VERB cana-1656 68 5 up up ADP cana-1656 68 6 phase phase NOUN cana-1656 68 7 for for SCONJ cana-1656 68 8 the the DET cana-1656 68 9 group group NOUN cana-1656 68 10 set set VERB cana-1656 68 11 up up ADP cana-1656 68 12 phase phase NOUN cana-1656 68 13 trusted trust VERB cana-1656 68 14 external external ADJ cana-1656 68 15 services service NOUN cana-1656 68 16 can can AUX cana-1656 68 17 be be AUX cana-1656 68 18 availed avail VERB cana-1656 68 19 . . PUNCT cana-1656 69 1 individual individual PROPN cana-1656 69 2 set set VERB cana-1656 69 3 up up ADP cana-1656 69 4 phase phase NOUN cana-1656 69 5 : : PUNCT cana-1656 69 6 the the DET cana-1656 69 7 curve curve NOUN cana-1656 69 8 generator generator NOUN cana-1656 69 9 and and CCONJ cana-1656 69 10 common common ADJ cana-1656 69 11 look look VERB cana-1656 69 12 up up ADP cana-1656 69 13 table table NOUN cana-1656 69 14 are be AUX cana-1656 69 15 available available ADJ cana-1656 69 16 for for ADP cana-1656 69 17 all all DET cana-1656 69 18 the the DET cana-1656 69 19 members member NOUN cana-1656 69 20 of of ADP cana-1656 69 21 the the DET cana-1656 69 22 community community NOUN cana-1656 69 23 . . PUNCT cana-1656 70 1 every every DET cana-1656 70 2 authentic authentic ADJ cana-1656 70 3 member member NOUN cana-1656 70 4 of of ADP cana-1656 70 5 the the DET cana-1656 70 6 group group NOUN cana-1656 70 7 can can AUX cana-1656 70 8 share share VERB cana-1656 70 9 messages message NOUN cana-1656 70 10 with with ADP cana-1656 70 11 every every DET cana-1656 70 12 other other ADJ cana-1656 70 13 member member NOUN cana-1656 70 14 . . PUNCT cana-1656 71 1 for for ADP cana-1656 71 2 that that PRON cana-1656 71 3 a a DET cana-1656 71 4 pair pair NOUN cana-1656 71 5 of of ADP cana-1656 71 6 authentic authentic ADJ cana-1656 71 7 users user NOUN cana-1656 71 8 of of ADP cana-1656 71 9 the the DET cana-1656 71 10 group group NOUN cana-1656 71 11 has have VERB cana-1656 71 12 to to PART cana-1656 71 13 arrange arrange VERB cana-1656 71 14 a a DET cana-1656 71 15 pre pre NOUN cana-1656 71 16 setup setup NOUN cana-1656 71 17 . . PUNCT cana-1656 72 1 if if SCONJ cana-1656 72 2 two two NUM cana-1656 72 3 users user NOUN cana-1656 72 4 robert robert PROPN cana-1656 72 5 and and CCONJ cana-1656 72 6 james james PROPN cana-1656 72 7 of of ADP cana-1656 72 8 the the DET cana-1656 72 9 group group NOUN cana-1656 72 10 want want VERB cana-1656 72 11 to to PART cana-1656 72 12 transmit transmit VERB cana-1656 72 13 messages message NOUN cana-1656 72 14 before before ADP cana-1656 72 15 transmission transmission NOUN cana-1656 72 16 , , PUNCT cana-1656 72 17 they they PRON cana-1656 72 18 set set VERB cana-1656 72 19 up up ADP cana-1656 72 20 private private ADJ cana-1656 72 21 and and CCONJ cana-1656 72 22 public public ADJ cana-1656 72 23 key key ADJ cana-1656 72 24 system system NOUN cana-1656 72 25 . . PUNCT cana-1656 73 1 robert robert PROPN cana-1656 73 2 choose choose VERB cana-1656 73 3 two two NUM cana-1656 73 4 random random ADJ cana-1656 73 5 points point NOUN cana-1656 73 6 r1 r1 NOUN cana-1656 73 7 , , PUNCT cana-1656 73 8 r2 r2 PROPN cana-1656 73 9 on on ADP cana-1656 73 10 the the DET cana-1656 73 11 public public ADJ cana-1656 73 12 curve curve NOUN cana-1656 73 13 and and CCONJ cana-1656 73 14 two two NUM cana-1656 73 15 large large ADJ cana-1656 73 16 random random ADJ cana-1656 73 17 prime prime ADJ cana-1656 73 18 numbers number NOUN cana-1656 73 19 ∝1 ∝1 NOUN cana-1656 73 20 , , PUNCT cana-1656 73 21 ∝2 ∝2 NOUN cana-1656 73 22 ( ( PUNCT cana-1656 73 23 less less ADJ cana-1656 73 24 than than ADP cana-1656 73 25 the the DET cana-1656 73 26 order order NOUN cana-1656 73 27 ) ) PUNCT cana-1656 73 28 . . PUNCT cana-1656 74 1 he he PRON cana-1656 74 2 computes compute VERB cana-1656 74 3 r3 r3 NOUN cana-1656 74 4 = = SYM cana-1656 74 5 ∝1𝑅2+∝2r1 ∝1𝑅2+∝2r1 NOUN cana-1656 74 6 r4 r4 NOUN cana-1656 74 7 = = SYM cana-1656 74 8 ( ( PUNCT cana-1656 74 9 ∝1∝2 ∝1∝2 PROPN cana-1656 74 10 ) ) PUNCT cana-1656 74 11 r3 r3 NOUN cana-1656 74 12 he he PRON cana-1656 74 13 communicates communicate VERB cana-1656 74 14 r3 r3 PROPN cana-1656 74 15 , , PUNCT cana-1656 74 16 r4 r4 VERB cana-1656 74 17 to to ADP cana-1656 74 18 james james PROPN cana-1656 74 19 . . PUNCT cana-1656 75 1 communications communication NOUN cana-1656 75 2 on on ADP cana-1656 75 3 applied apply VERB cana-1656 75 4 nonlinear nonlinear ADJ cana-1656 75 5 analysis analysis NOUN cana-1656 75 6 issn issn NOUN cana-1656 75 7 : : PUNCT cana-1656 75 8 1074 1074 NUM cana-1656 75 9 - - PUNCT cana-1656 75 10 133x 133x NUM cana-1656 75 11 vol vol NOUN cana-1656 75 12 32 32 NUM cana-1656 75 13 no no NOUN cana-1656 75 14 . . NOUN cana-1656 75 15 1 1 NUM cana-1656 75 16 ( ( PUNCT cana-1656 75 17 2025 2025 NUM cana-1656 75 18 ) ) PUNCT cana-1656 75 19 355 355 NUM cana-1656 75 20 https://internationalpubls.com https://internationalpubls.com X cana-1656 75 21 similarly similarly ADV cana-1656 75 22 , , PUNCT cana-1656 75 23 james james PROPN cana-1656 75 24 selects select VERB cana-1656 75 25 two two NUM cana-1656 75 26 random random ADJ cana-1656 75 27 points point NOUN cana-1656 75 28 j1 j1 PROPN cana-1656 75 29 , , PUNCT cana-1656 75 30 j2 j2 NOUN cana-1656 75 31 and and CCONJ cana-1656 75 32 large large ADJ cana-1656 75 33 random random ADJ cana-1656 75 34 numbers number NOUN cana-1656 75 35 𝛽1 𝛽1 NOUN cana-1656 75 36 , , PUNCT cana-1656 75 37 𝛽2(less 𝛽2(less ADJ cana-1656 75 38 than than ADP cana-1656 75 39 the the DET cana-1656 75 40 order order NOUN cana-1656 75 41 ) ) PUNCT cana-1656 75 42 , , PUNCT cana-1656 75 43 computes compute VERB cana-1656 75 44 j3 j3 PROPN cana-1656 75 45 = = SYM cana-1656 75 46 𝛽1𝐽2 𝛽1𝐽2 PROPN cana-1656 75 47 + + NUM cana-1656 75 48 𝛽2𝐽1 𝛽2𝐽1 PROPN cana-1656 75 49 j4 j4 NOUN cana-1656 75 50 = = SYM cana-1656 75 51 ( ( PUNCT cana-1656 75 52 𝛽1𝛽2)j3 𝛽1𝛽2)j3 X cana-1656 75 53 and and CCONJ cana-1656 75 54 communicates communicate NOUN cana-1656 75 55 to to PART cana-1656 75 56 robert robert PROPN cana-1656 75 57 . . PUNCT cana-1656 76 1 the the DET cana-1656 76 2 secret secret ADJ cana-1656 76 3 key key NOUN cana-1656 76 4 set set NOUN cana-1656 76 5 for for ADP cana-1656 76 6 robert robert PROPN cana-1656 76 7 is be AUX cana-1656 76 8 ( ( PUNCT cana-1656 76 9 r1 r1 PROPN cana-1656 76 10 , , PUNCT cana-1656 76 11 r2 r2 PROPN cana-1656 76 12 , , PUNCT cana-1656 76 13 ∝1 ∝1 NOUN cana-1656 76 14 , , PUNCT cana-1656 76 15 ∝2 ∝2 PROPN cana-1656 76 16 ) ) PUNCT cana-1656 76 17 the the DET cana-1656 76 18 public public ADJ cana-1656 76 19 key key NOUN cana-1656 76 20 set set NOUN cana-1656 76 21 for for SCONJ cana-1656 76 22 robert robert PROPN cana-1656 76 23 to to PART cana-1656 76 24 communicate communicate VERB cana-1656 76 25 is be AUX cana-1656 76 26 ( ( PUNCT cana-1656 76 27 r3 r3 PROPN cana-1656 76 28 , , PUNCT cana-1656 76 29 r4 r4 NOUN cana-1656 76 30 ) ) PUNCT cana-1656 76 31 in in ADP cana-1656 76 32 the the DET cana-1656 76 33 same same ADJ cana-1656 76 34 way way NOUN cana-1656 76 35 key key ADJ cana-1656 76 36 set set NOUN cana-1656 76 37 for for ADP cana-1656 76 38 james james PROPN cana-1656 76 39 is be AUX cana-1656 76 40 ( ( PUNCT cana-1656 76 41 j1 j1 PROPN cana-1656 76 42 , , PUNCT cana-1656 76 43 j2 j2 PROPN cana-1656 76 44 , , PUNCT cana-1656 76 45 𝛽1 𝛽1 NOUN cana-1656 76 46 , , PUNCT cana-1656 76 47 𝛽2 𝛽2 PROPN cana-1656 76 48 ) ) PUNCT cana-1656 76 49 . . PUNCT cana-1656 77 1 public public ADJ cana-1656 77 2 key key NOUN cana-1656 77 3 set set NOUN cana-1656 77 4 to to PART cana-1656 77 5 communicate communicate VERB cana-1656 77 6 with with ADP cana-1656 77 7 robert robert PROPN cana-1656 77 8 is be AUX cana-1656 77 9 ( ( PUNCT cana-1656 77 10 j3 j3 PROPN cana-1656 77 11 , , PUNCT cana-1656 77 12 j4 j4 PROPN cana-1656 77 13 ) ) PUNCT cana-1656 77 14 . . PUNCT cana-1656 78 1 encryption encryption NOUN cana-1656 78 2 : : PUNCT cana-1656 78 3 if if SCONJ cana-1656 78 4 james james PROPN cana-1656 78 5 wants want VERB cana-1656 78 6 send send VERB cana-1656 78 7 message message NOUN cana-1656 78 8 to to ADP cana-1656 78 9 robert robert PROPN cana-1656 78 10 , , PUNCT cana-1656 78 11 he he PRON cana-1656 78 12 uses use VERB cana-1656 78 13 his his PRON cana-1656 78 14 secret secret ADJ cana-1656 78 15 keys key NOUN cana-1656 78 16 and and CCONJ cana-1656 78 17 robert robert PROPN cana-1656 78 18 ’s ’s PART cana-1656 78 19 public public ADJ cana-1656 78 20 keys key NOUN cana-1656 78 21 for for ADP cana-1656 78 22 encrypting encrypt VERB cana-1656 78 23 the the DET cana-1656 78 24 messages message NOUN cana-1656 78 25 . . PUNCT cana-1656 79 1 encryption encryption NOUN cana-1656 79 2 stages stage NOUN cana-1656 79 3 are be AUX cana-1656 79 4 1 1 NUM cana-1656 79 5 . . PUNCT cana-1656 80 1 consider consider VERB cana-1656 80 2 message message NOUN cana-1656 80 3 m m AUX cana-1656 80 4 consisting consist VERB cana-1656 80 5 of of ADP cana-1656 80 6 characters character NOUN cana-1656 80 7 π‘š1 π‘š1 VERB cana-1656 80 8 , , PUNCT cana-1656 80 9 π‘š2 π‘š2 NOUN cana-1656 80 10 , , PUNCT cana-1656 80 11 π‘š3 π‘š3 NOUN cana-1656 80 12 - - PUNCT cana-1656 80 13 -π‘šπ‘›. -π‘šπ‘›. NOUN cana-1656 80 14 all all DET cana-1656 80 15 the the DET cana-1656 80 16 characters character NOUN cana-1656 80 17 are be AUX cana-1656 80 18 mapped map VERB cana-1656 80 19 to to ADP cana-1656 80 20 elliptic elliptic ADJ cana-1656 80 21 curve curve NOUN cana-1656 80 22 points point NOUN cana-1656 80 23 with with ADP cana-1656 80 24 the the DET cana-1656 80 25 help help NOUN cana-1656 80 26 of of ADP cana-1656 80 27 common common ADJ cana-1656 80 28 look look VERB cana-1656 80 29 up up ADP cana-1656 80 30 table table NOUN cana-1656 80 31 p1 p1 NOUN cana-1656 80 32 , , PUNCT cana-1656 80 33 p2 p2 NOUN cana-1656 80 34 , , PUNCT cana-1656 80 35 p3 p3 PROPN cana-1656 80 36 - - PROPN cana-1656 80 37 --𝑃𝑛. --𝑃𝑛. PROPN cana-1656 80 38 2 2 NUM cana-1656 80 39 . . PUNCT cana-1656 80 40 an an DET cana-1656 80 41 additional additional ADJ cana-1656 80 42 parameter parameter NOUN cana-1656 80 43 𝛾 𝛾 ADP cana-1656 80 44 other other ADJ cana-1656 80 45 than than ADP cana-1656 80 46 secret secret ADJ cana-1656 80 47 key key ADJ cana-1656 80 48 set set NOUN cana-1656 80 49 is be AUX cana-1656 80 50 chosen choose VERB cana-1656 80 51 by by ADP cana-1656 80 52 james james PROPN cana-1656 80 53 to to PART cana-1656 80 54 encrypt encrypt VERB cana-1656 80 55 the the DET cana-1656 80 56 plaintext plaintext NOUN cana-1656 80 57 . . PUNCT cana-1656 81 1 here here ADV cana-1656 81 2 𝛾 𝛾 PROPN cana-1656 81 3 is be AUX cana-1656 81 4 a a DET cana-1656 81 5 big big ADJ cana-1656 81 6 number number NOUN cana-1656 81 7 smaller small ADJ cana-1656 81 8 than than ADP cana-1656 81 9 the the DET cana-1656 81 10 order order NOUN cana-1656 81 11 of of ADP cana-1656 81 12 the the DET cana-1656 81 13 curve curve NOUN cana-1656 81 14 different different ADJ cana-1656 81 15 for for ADP cana-1656 81 16 different different ADJ cana-1656 81 17 plaintext plaintext NOUN cana-1656 81 18 characters character NOUN cana-1656 81 19 . . PUNCT cana-1656 82 1 3 3 X cana-1656 82 2 . . X cana-1656 82 3 he he PRON cana-1656 82 4 encrypts encrypt VERB cana-1656 82 5 the the DET cana-1656 82 6 first first ADJ cana-1656 82 7 point point NOUN cana-1656 82 8 p1 p1 NOUN cana-1656 82 9 using use VERB cana-1656 82 10 𝛾1as 𝛾1as NOUN cana-1656 82 11 e11 e11 X cana-1656 82 12 = = PUNCT cana-1656 82 13 𝛾1r3 𝛾1r3 NOUN cana-1656 82 14 e12 e12 NOUN cana-1656 82 15 = = SYM cana-1656 82 16 p1 p1 PROPN cana-1656 82 17 + + PRON cana-1656 82 18 𝛾1r4 𝛾1r4 NOUN cana-1656 82 19 so so ADV cana-1656 82 20 , , PUNCT cana-1656 82 21 the the DET cana-1656 82 22 point point NOUN cana-1656 82 23 p1 p1 NOUN cana-1656 82 24 of of ADP cana-1656 82 25 the the DET cana-1656 82 26 message message NOUN cana-1656 82 27 is be AUX cana-1656 82 28 converted convert VERB cana-1656 82 29 to to ADP cana-1656 82 30 pair pair NOUN cana-1656 82 31 of of ADP cana-1656 82 32 points point NOUN cana-1656 82 33 ( ( PUNCT cana-1656 82 34 e11 e11 X cana-1656 82 35 , , PUNCT cana-1656 82 36 e12 e12 NOUN cana-1656 82 37 ) ) PUNCT cana-1656 82 38 . . PUNCT cana-1656 83 1 4 4 X cana-1656 83 2 . . X cana-1656 83 3 similar similar ADJ cana-1656 83 4 series series NOUN cana-1656 83 5 of of ADP cana-1656 83 6 stages stage NOUN cana-1656 83 7 2 2 NUM cana-1656 83 8 , , PUNCT cana-1656 83 9 3 3 NUM cana-1656 83 10 is be AUX cana-1656 83 11 applied apply VERB cana-1656 83 12 for for ADP cana-1656 83 13 all all DET cana-1656 83 14 the the DET cana-1656 83 15 plaintext plaintext NOUN cana-1656 83 16 elliptic elliptic ADJ cana-1656 83 17 curve curve NOUN cana-1656 83 18 points point NOUN cana-1656 83 19 p1 p1 NOUN cana-1656 83 20 , , PUNCT cana-1656 83 21 p2 p2 NOUN cana-1656 83 22 , , PUNCT cana-1656 83 23 p3 p3 PROPN cana-1656 83 24 - - PUNCT cana-1656 83 25 --𝑃𝑛. --𝑃𝑛. PROPN cana-1656 83 26 ( ( PUNCT cana-1656 83 27 en1 en1 PROPN cana-1656 83 28 , , PUNCT cana-1656 83 29 en2 en2 PROPN cana-1656 83 30 ) ) PUNCT cana-1656 83 31 is be AUX cana-1656 83 32 cipher cipher ADJ cana-1656 83 33 pair pair NOUN cana-1656 83 34 and and CCONJ cana-1656 83 35 𝛾nis 𝛾nis VERB cana-1656 83 36 large large ADJ cana-1656 83 37 random random ADJ cana-1656 83 38 number number NOUN cana-1656 83 39 for for ADP cana-1656 83 40 n= n= ADJ cana-1656 83 41 1,2,3 1,2,3 NUM cana-1656 83 42 - - NUM cana-1656 83 43 - - PUNCT cana-1656 83 44 , , PUNCT cana-1656 83 45 where where SCONJ cana-1656 83 46 en1= en1= ADJ cana-1656 83 47 𝛾nr3 𝛾nr3 NOUN cana-1656 83 48 en2= en2= PROPN cana-1656 83 49 𝑃𝑛 𝑃𝑛 PROPN cana-1656 83 50 + + CCONJ cana-1656 83 51 𝛾nr4 𝛾nr4 PROPN cana-1656 83 52 the the DET cana-1656 83 53 cipher cipher ADJ cana-1656 83 54 pair pair NOUN cana-1656 83 55 for for ADP cana-1656 83 56 p1 p1 PROPN cana-1656 83 57 is be AUX cana-1656 83 58 ( ( PUNCT cana-1656 83 59 e11 e11 X cana-1656 83 60 , , PUNCT cana-1656 83 61 e12 e12 NOUN cana-1656 83 62 ) ) PUNCT cana-1656 83 63 , , PUNCT cana-1656 83 64 p2 p2 PROPN cana-1656 83 65 is be AUX cana-1656 83 66 ( ( PUNCT cana-1656 83 67 e21 e21 PROPN cana-1656 83 68 , , PUNCT cana-1656 83 69 e22 e22 NOUN cana-1656 83 70 ) ) PUNCT cana-1656 83 71 , , PUNCT cana-1656 83 72 p3is p3is X cana-1656 83 73 ( ( PUNCT cana-1656 83 74 e31 e31 PROPN cana-1656 83 75 , , PUNCT cana-1656 83 76 e32 e32 PROPN cana-1656 83 77 ) ) PUNCT cana-1656 83 78 , , PUNCT cana-1656 83 79 ----- ----- PROPN cana-1656 83 80 , , PUNCT cana-1656 83 81 𝑃𝑛 𝑃𝑛 PROPN cana-1656 83 82 is be AUX cana-1656 83 83 ( ( PUNCT cana-1656 83 84 en1 en1 PROPN cana-1656 83 85 , , PUNCT cana-1656 83 86 en2 en2 PROPN cana-1656 83 87 ) ) PUNCT cana-1656 83 88 . . PUNCT cana-1656 84 1 5 5 X cana-1656 84 2 . . X cana-1656 84 3 all all DET cana-1656 84 4 the the DET cana-1656 84 5 points point NOUN cana-1656 84 6 e11 e11 ADJ cana-1656 84 7 , , PUNCT cana-1656 84 8 e12 e12 NOUN cana-1656 84 9 , , PUNCT cana-1656 84 10 e21 e21 PROPN cana-1656 84 11 , , PUNCT cana-1656 84 12 e22 e22 PROPN cana-1656 84 13 , , PUNCT cana-1656 84 14 e31 e31 PROPN cana-1656 84 15 , , PUNCT cana-1656 84 16 e32 e32 PROPN cana-1656 84 17 , , PUNCT cana-1656 84 18 ---- ---- PROPN cana-1656 84 19 , , PUNCT cana-1656 84 20 en1 en1 VERB cana-1656 84 21 , , PUNCT cana-1656 84 22 en2 en2 PROPN cana-1656 84 23 are be AUX cana-1656 84 24 reverse reverse ADJ cana-1656 84 25 mapped map VERB cana-1656 84 26 to to ADP cana-1656 84 27 text text NOUN cana-1656 84 28 characters character NOUN cana-1656 84 29 , , PUNCT cana-1656 84 30 which which PRON cana-1656 84 31 is be AUX cana-1656 84 32 the the DET cana-1656 84 33 cipher cipher ADJ cana-1656 84 34 text text NOUN cana-1656 84 35 c c NOUN cana-1656 84 36 communicated communicate VERB cana-1656 84 37 to to PART cana-1656 84 38 robert robert VERB cana-1656 84 39 in in ADP cana-1656 84 40 a a DET cana-1656 84 41 public public ADJ cana-1656 84 42 channel channel NOUN cana-1656 84 43 . . PUNCT cana-1656 85 1 decryption decryption NOUN cana-1656 85 2 : : PUNCT cana-1656 85 3 robert robert PROPN cana-1656 85 4 converts convert VERB cana-1656 85 5 the the DET cana-1656 85 6 cipher cipher ADJ cana-1656 85 7 text text NOUN cana-1656 85 8 characters character NOUN cana-1656 85 9 into into ADP cana-1656 85 10 points point NOUN cana-1656 85 11 on on ADP cana-1656 85 12 the the DET cana-1656 85 13 curve curve NOUN cana-1656 85 14 and and CCONJ cana-1656 85 15 1 1 NUM cana-1656 85 16 . . PUNCT cana-1656 85 17 considers consider VERB cana-1656 85 18 the the DET cana-1656 85 19 points point NOUN cana-1656 85 20 pair pair VERB cana-1656 85 21 wise wise ADJ cana-1656 85 22 ( ( PUNCT cana-1656 85 23 e11 e11 X cana-1656 85 24 , , PUNCT cana-1656 85 25 e12 e12 NOUN cana-1656 85 26 ) ) PUNCT cana-1656 85 27 , , PUNCT cana-1656 85 28 ( ( PUNCT cana-1656 85 29 e21 e21 PROPN cana-1656 85 30 , , PUNCT cana-1656 85 31 e22 e22 NOUN cana-1656 85 32 ) ) PUNCT cana-1656 85 33 , , PUNCT cana-1656 85 34 ( ( PUNCT cana-1656 85 35 e31 e31 X cana-1656 85 36 , , PUNCT cana-1656 85 37 e32 e32 NOUN cana-1656 85 38 ) ) PUNCT cana-1656 85 39 --(en1 --(en1 NOUN cana-1656 85 40 , , PUNCT cana-1656 85 41 en2 en2 PROPN cana-1656 85 42 ) ) PUNCT cana-1656 85 43 . . PUNCT cana-1656 86 1 2 2 X cana-1656 86 2 . . X cana-1656 86 3 decrypts decrypt NOUN cana-1656 86 4 each each DET cana-1656 86 5 pair pair NOUN cana-1656 86 6 using use VERB cana-1656 86 7 his his PRON cana-1656 86 8 own own ADJ cana-1656 86 9 secret secret ADJ cana-1656 86 10 keys key NOUN cana-1656 86 11 and and CCONJ cana-1656 86 12 james james PROPN cana-1656 86 13 ’s ’s PART cana-1656 86 14 public public ADJ cana-1656 86 15 keys key NOUN cana-1656 86 16 to to PART cana-1656 86 17 get get VERB cana-1656 86 18 original original ADJ cana-1656 86 19 message message NOUN cana-1656 86 20 . . PUNCT cana-1656 87 1 p1= p1= NOUN cana-1656 87 2 e12-∝1∝2e11 e12-∝1∝2e11 PROPN cana-1656 87 3 p2= p2= PROPN cana-1656 87 4 e22-∝1∝2e21 e22-∝1∝2e21 PROPN cana-1656 87 5 ------------------- ------------------- PUNCT cana-1656 87 6 𝑃𝑛= 𝑃𝑛= ADJ cana-1656 87 7 en2-∝1∝2en1 en2-∝1∝2en1 ADJ cana-1656 87 8 proof proof NOUN cana-1656 87 9 of of ADP cana-1656 87 10 the the DET cana-1656 87 11 decryption decryption NOUN cana-1656 87 12 𝑃𝑛= 𝑃𝑛= PROPN cana-1656 87 13 en2-∝1∝2en1 en2-∝1∝2en1 ADV cana-1656 88 1 = = PUNCT cana-1656 88 2 𝑃𝑛 𝑃𝑛 NOUN cana-1656 88 3 + + CCONJ cana-1656 88 4 𝛾nr4∝1∝2𝛾nr3 𝛾nr4∝1∝2𝛾nr3 NOUN cana-1656 88 5 communications communication NOUN cana-1656 88 6 on on ADP cana-1656 88 7 applied apply VERB cana-1656 88 8 nonlinear nonlinear ADJ cana-1656 88 9 analysis analysis NOUN cana-1656 88 10 issn issn NOUN cana-1656 88 11 : : PUNCT cana-1656 88 12 1074 1074 NUM cana-1656 88 13 - - PUNCT cana-1656 88 14 133x 133x NUM cana-1656 88 15 vol vol NOUN cana-1656 88 16 32 32 NUM cana-1656 88 17 no no NOUN cana-1656 88 18 . . NOUN cana-1656 88 19 1 1 NUM cana-1656 88 20 ( ( PUNCT cana-1656 88 21 2025 2025 NUM cana-1656 88 22 ) ) PUNCT cana-1656 88 23 356 356 NUM cana-1656 88 24 https://internationalpubls.com https://internationalpubls.com X cana-1656 88 25 = = PUNCT cana-1656 88 26 𝑃𝑛 𝑃𝑛 PROPN cana-1656 88 27 + + CCONJ cana-1656 88 28 𝛾n(∝1∝2 𝛾n(∝1∝2 ADJ cana-1656 88 29 ) ) PUNCT cana-1656 88 30 r3∝1∝2𝛾nr3 r3∝1∝2𝛾nr3 NOUN cana-1656 89 1 = = SYM cana-1656 89 2 𝑃𝑛 𝑃𝑛 PROPN cana-1656 89 3 + + CCONJ cana-1656 89 4 𝛾n(∝1∝2 𝛾n(∝1∝2 ADJ cana-1656 89 5 ) ) PUNCT cana-1656 89 6 r3∝1∝2𝛾nr3 r3∝1∝2𝛾nr3 NOUN cana-1656 90 1 = = SYM cana-1656 90 2 𝑃𝑛 𝑃𝑛 PROPN cana-1656 90 3 if if SCONJ cana-1656 90 4 robert robert PROPN cana-1656 90 5 wants want VERB cana-1656 90 6 to to PART cana-1656 90 7 send send VERB cana-1656 90 8 massage massage NOUN cana-1656 90 9 to to ADP cana-1656 90 10 james james PROPN cana-1656 90 11 , , PUNCT cana-1656 90 12 he he PRON cana-1656 90 13 uses use VERB cana-1656 90 14 james james PROPN cana-1656 90 15 ’s ’s PART cana-1656 90 16 public public ADJ cana-1656 90 17 keys key NOUN cana-1656 90 18 for for ADP cana-1656 90 19 encryption encryption NOUN cana-1656 90 20 en1= en1= NOUN cana-1656 90 21 𝛾nj3 𝛾nj3 PROPN cana-1656 90 22 en2= en2= PROPN cana-1656 90 23 𝑃𝑛+ 𝑃𝑛+ PROPN cana-1656 90 24 𝛾nj4 𝛾nj4 NOUN cana-1656 90 25 james james PROPN cana-1656 90 26 decrypts decrypt VERB cana-1656 90 27 the the DET cana-1656 90 28 pair pair NOUN cana-1656 90 29 to to ADP cana-1656 90 30 𝑃𝑛as 𝑃𝑛as PROPN cana-1656 90 31 𝑃𝑛= 𝑃𝑛= PROPN cana-1656 90 32 en2𝛽1𝛽2 en2𝛽1𝛽2 PROPN cana-1656 90 33 en1 en1 PROPN cana-1656 90 34 example example NOUN cana-1656 90 35 : : PUNCT cana-1656 90 36 consider consider VERB cana-1656 90 37 the the DET cana-1656 90 38 elliptic elliptic ADJ cana-1656 90 39 curve curve NOUN cana-1656 90 40 𝐸31(5,3 𝐸31(5,3 PROPN cana-1656 90 41 ) ) PUNCT cana-1656 90 42 and and CCONJ cana-1656 90 43 there there PRON cana-1656 90 44 are be VERB cana-1656 90 45 40 40 NUM cana-1656 90 46 number number NOUN cana-1656 90 47 of of ADP cana-1656 90 48 points point NOUN cana-1656 90 49 on on ADP cana-1656 90 50 this this DET cana-1656 90 51 curve curve NOUN cana-1656 90 52 . . PUNCT cana-1656 91 1 take take VERB cana-1656 91 2 any any DET cana-1656 91 3 point point NOUN cana-1656 91 4 on on ADP cana-1656 91 5 this this DET cana-1656 91 6 curve curve NOUN cana-1656 91 7 let let VERB cana-1656 91 8 it it PRON cana-1656 91 9 be be AUX cana-1656 91 10 g= g= NOUN cana-1656 91 11 ( ( PUNCT cana-1656 91 12 1,3 1,3 NUM cana-1656 91 13 ) ) PUNCT cana-1656 91 14 . . PUNCT cana-1656 92 1 now now ADV cana-1656 92 2 we we PRON cana-1656 92 3 calculate calculate VERB cana-1656 92 4 2g,3 2g,3 PROPN cana-1656 92 5 g g NOUN cana-1656 92 6 , , PUNCT cana-1656 92 7 ---40 ---40 PROPN cana-1656 92 8 g. g. NOUN cana-1656 92 9 create create VERB cana-1656 92 10 a a DET cana-1656 92 11 table table NOUN cana-1656 92 12 by by ADP cana-1656 92 13 using use VERB cana-1656 92 14 points point NOUN cana-1656 92 15 on on ADP cana-1656 92 16 the the DET cana-1656 92 17 curve curve NOUN cana-1656 92 18 and and CCONJ cana-1656 92 19 assign assign VERB cana-1656 92 20 character character NOUN cana-1656 92 21 values value NOUN cana-1656 92 22 , , PUNCT cana-1656 92 23 special special ADJ cana-1656 92 24 symbols symbol NOUN cana-1656 92 25 and and CCONJ cana-1656 92 26 numerals numeral NOUN cana-1656 92 27 to to ADP cana-1656 92 28 those those DET cana-1656 92 29 points point NOUN cana-1656 92 30 . . PUNCT cana-1656 93 1 for for ADP cana-1656 93 2 given give VERB cana-1656 93 3 message message NOUN cana-1656 93 4 compare compare VERB cana-1656 93 5 each each DET cana-1656 93 6 letter letter NOUN cana-1656 93 7 with with ADP cana-1656 93 8 that that DET cana-1656 93 9 table table NOUN cana-1656 93 10 . . PUNCT cana-1656 94 1 take take VERB cana-1656 94 2 the the DET cana-1656 94 3 message message NOUN cana-1656 94 4 mrpg mrpg PROPN cana-1656 94 5 college college NOUN cana-1656 94 6 vizianagaram vizianagaram NOUN cana-1656 94 7 by by ADP cana-1656 94 8 comparing compare VERB cana-1656 94 9 each each DET cana-1656 94 10 letter letter NOUN cana-1656 94 11 with with ADP cana-1656 94 12 the the DET cana-1656 94 13 table table NOUN cana-1656 94 14 , , PUNCT cana-1656 94 15 we we PRON cana-1656 94 16 get get VERB cana-1656 94 17 13 13 NUM cana-1656 94 18 g g NOUN cana-1656 94 19 18 18 NUM cana-1656 94 20 g g NOUN cana-1656 94 21 16 16 NUM cana-1656 94 22 g g NOUN cana-1656 94 23 7 7 NUM cana-1656 94 24 g g NOUN cana-1656 94 25 28 28 NUM cana-1656 94 26 g g NOUN cana-1656 94 27 3 3 NUM cana-1656 94 28 g g NOUN cana-1656 94 29 15 15 NUM cana-1656 94 30 g g NOUN cana-1656 94 31 12 12 NUM cana-1656 94 32 g g NOUN cana-1656 94 33 12 12 NUM cana-1656 94 34 g g NOUN cana-1656 94 35 5 5 NUM cana-1656 94 36 g g NOUN cana-1656 94 37 7 7 NUM cana-1656 94 38 g g NOUN cana-1656 94 39 5 5 NUM cana-1656 94 40 g g NOUN cana-1656 94 41 28 28 NUM cana-1656 94 42 g g NOUN cana-1656 94 43 22 22 NUM cana-1656 94 44 g g NOUN cana-1656 94 45 9 9 NUM cana-1656 94 46 g g NOUN cana-1656 94 47 26 26 NUM cana-1656 94 48 g g NOUN cana-1656 94 49 9 9 NUM cana-1656 94 50 g g NOUN cana-1656 94 51 1 1 NUM cana-1656 94 52 g g NOUN cana-1656 94 53 7 7 NUM cana-1656 94 54 g g NOUN cana-1656 94 55 1 1 NUM cana-1656 94 56 g g NOUN cana-1656 94 57 18 18 NUM cana-1656 94 58 g g NOUN cana-1656 94 59 1 1 NUM cana-1656 94 60 g g NOUN cana-1656 94 61 13 13 NUM cana-1656 94 62 g g NOUN cana-1656 94 63 here here ADV cana-1656 94 64 r1= r1= NOUN cana-1656 94 65 ( ( PUNCT cana-1656 94 66 3,13 3,13 NUM cana-1656 94 67 ) ) PUNCT cana-1656 94 68 r2= r2= PUNCT cana-1656 94 69 ( ( PUNCT cana-1656 94 70 4,5 4,5 NUM cana-1656 94 71 ) ) PUNCT cana-1656 94 72 and and CCONJ cana-1656 94 73 ∝1= ∝1= VERB cana-1656 94 74 13 13 NUM cana-1656 94 75 ∝2= ∝2= NOUN cana-1656 94 76 17 17 NUM cana-1656 94 77 r3 r3 NOUN cana-1656 94 78 = = SYM cana-1656 94 79 13(4,5 13(4,5 NUM cana-1656 94 80 ) ) PUNCT cana-1656 94 81 + + CCONJ cana-1656 94 82 17(3,13 17(3,13 NUM cana-1656 94 83 ) ) PUNCT cana-1656 94 84 r3 r3 PROPN cana-1656 94 85 = = SYM cana-1656 94 86 104g+ 104g+ NUM cana-1656 94 87 306 306 NUM cana-1656 94 88 g g NOUN cana-1656 94 89 r3= r3= NOUN cana-1656 94 90 410 410 NUM cana-1656 94 91 g g NOUN cana-1656 94 92 ( ( PUNCT cana-1656 94 93 mod mod PROPN cana-1656 94 94 31 31 NUM cana-1656 94 95 ) ) PUNCT cana-1656 94 96 = = PUNCT cana-1656 94 97 7 7 NUM cana-1656 94 98 g g NOUN cana-1656 94 99 r4= r4= PROPN cana-1656 94 100 ( ( PUNCT cana-1656 94 101 13 13 NUM cana-1656 94 102 * * NUM cana-1656 94 103 17 17 NUM cana-1656 94 104 ) ) PUNCT cana-1656 94 105 7g= 7g= NUM cana-1656 94 106 28 28 NUM cana-1656 94 107 g g PROPN cana-1656 94 108 j1= j1= NOUN cana-1656 94 109 ( ( PUNCT cana-1656 94 110 6,1 6,1 NUM cana-1656 94 111 ) ) PUNCT cana-1656 94 112 j2= j2= NOUN cana-1656 94 113 ( ( PUNCT cana-1656 94 114 7,3 7,3 NUM cana-1656 94 115 ) ) PUNCT cana-1656 94 116 𝛽1=19 𝛽1=19 PROPN cana-1656 94 117 𝛽1=23 𝛽1=23 PROPN cana-1656 94 118 j3=19(7,3 j3=19(7,3 PROPN cana-1656 94 119 ) ) PUNCT cana-1656 95 1 + + CCONJ cana-1656 96 1 23(6,1 23(6,1 NUM cana-1656 96 2 ) ) PUNCT cana-1656 96 3 j3= j3= PROPN cana-1656 96 4 589g+299 589g+299 NUM cana-1656 96 5 g g PROPN cana-1656 96 6 j3=888 j3=888 PROPN cana-1656 96 7 g g PROPN cana-1656 96 8 ( ( PUNCT cana-1656 96 9 mod mod PROPN cana-1656 96 10 31 31 NUM cana-1656 96 11 ) ) PUNCT cana-1656 96 12 = = SYM cana-1656 96 13 20 20 NUM cana-1656 96 14 g g NOUN cana-1656 96 15 j4= j4= NOUN cana-1656 96 16 ( ( PUNCT cana-1656 96 17 19 19 NUM cana-1656 96 18 * * NOUN cana-1656 96 19 23)20g=29 23)20g=29 NUM cana-1656 96 20 g g NOUN cana-1656 96 21 we we PRON cana-1656 96 22 have have VERB cana-1656 96 23 en1= en1= ADJ cana-1656 96 24 𝛾nj3 𝛾nj3 NOUN cana-1656 96 25 and and CCONJ cana-1656 96 26 en2= en2= PROPN cana-1656 96 27 𝑃𝑛+ 𝑃𝑛+ PROPN cana-1656 96 28 𝛾nj4 𝛾nj4 NOUN cana-1656 96 29 also also ADV cana-1656 96 30 , , PUNCT cana-1656 96 31 we we PRON cana-1656 96 32 consider consider VERB cana-1656 96 33 different different ADJ cana-1656 96 34 𝛾nfor 𝛾nfor ADP cana-1656 96 35 different different ADJ cana-1656 96 36 characters character NOUN cana-1656 96 37 in in ADP cana-1656 96 38 our our PRON cana-1656 96 39 message message NOUN cana-1656 96 40 here here ADV cana-1656 96 41 we we PRON cana-1656 96 42 take take VERB cana-1656 96 43 𝛾n=6 𝛾n=6 ADV cana-1656 96 44 , , PUNCT cana-1656 96 45 2 2 NUM cana-1656 96 46 , , PUNCT cana-1656 96 47 4 4 NUM cana-1656 96 48 , , PUNCT cana-1656 96 49 9 9 NUM cana-1656 96 50 , , PUNCT cana-1656 96 51 11 11 NUM cana-1656 96 52 , , PUNCT cana-1656 96 53 8 8 NUM cana-1656 96 54 , , PUNCT cana-1656 96 55 5 5 NUM cana-1656 96 56 , , PUNCT cana-1656 96 57 3 3 NUM cana-1656 96 58 , , PUNCT cana-1656 96 59 7 7 NUM cana-1656 96 60 , , PUNCT cana-1656 96 61 10 10 NUM cana-1656 96 62 , , PUNCT cana-1656 96 63 12 12 NUM cana-1656 96 64 , , PUNCT cana-1656 96 65 14 14 NUM cana-1656 96 66 , , PUNCT cana-1656 96 67 15 15 NUM cana-1656 96 68 , , PUNCT cana-1656 96 69 21 21 NUM cana-1656 96 70 , , PUNCT cana-1656 96 71 16 16 NUM cana-1656 96 72 , , PUNCT cana-1656 96 73 17 17 NUM cana-1656 96 74 , , PUNCT cana-1656 96 75 22 22 NUM cana-1656 96 76 , , PUNCT cana-1656 96 77 24 24 NUM cana-1656 96 78 , , PUNCT cana-1656 96 79 19 19 NUM cana-1656 96 80 , , PUNCT cana-1656 96 81 18 18 NUM cana-1656 96 82 , , PUNCT cana-1656 96 83 25 25 NUM cana-1656 96 84 , , PUNCT cana-1656 96 85 27 27 NUM cana-1656 96 86 , , PUNCT cana-1656 96 87 29 29 NUM cana-1656 96 88 , , PUNCT cana-1656 96 89 20 20 NUM cana-1656 96 90 , , PUNCT cana-1656 96 91 23 23 NUM cana-1656 96 92 for for ADP cana-1656 96 93 n= n= ADJ cana-1656 96 94 1 1 NUM cana-1656 96 95 , , PUNCT cana-1656 96 96 2 2 NUM cana-1656 96 97 , , PUNCT cana-1656 96 98 3 3 NUM cana-1656 96 99 , , PUNCT cana-1656 96 100 ---25 ---25 PROPN cana-1656 96 101 . . PUNCT cana-1656 97 1 then then ADV cana-1656 97 2 by by ADP cana-1656 97 3 using use VERB cana-1656 97 4 above above ADJ cana-1656 97 5 formula formula NOUN cana-1656 97 6 we we PRON cana-1656 97 7 calculate calculate VERB cana-1656 97 8 ( ( PUNCT cana-1656 97 9 e11 e11 X cana-1656 97 10 , , PUNCT cana-1656 97 11 e12 e12 NOUN cana-1656 97 12 ) ) PUNCT cana-1656 97 13 = = PUNCT cana-1656 98 1 ( ( PUNCT cana-1656 98 2 27,1 27,1 NUM cana-1656 98 3 ) ) PUNCT cana-1656 98 4 ( ( PUNCT cana-1656 98 5 e21 e21 PROPN cana-1656 98 6 , , PUNCT cana-1656 98 7 e22 e22 NOUN cana-1656 98 8 ) ) PUNCT cana-1656 98 9 = = SYM cana-1656 98 10 ( ( PUNCT cana-1656 98 11 9,14 9,14 NUM cana-1656 98 12 ) ) PUNCT cana-1656 98 13 ( ( PUNCT cana-1656 98 14 e31 e31 X cana-1656 98 15 , , PUNCT cana-1656 98 16 e32 e32 PROPN cana-1656 98 17 ) ) PUNCT cana-1656 98 18 = = SYM cana-1656 98 19 ( ( PUNCT cana-1656 98 20 18,8 18,8 NUM cana-1656 98 21 ) ) PUNCT cana-1656 98 22 ( ( PUNCT cana-1656 98 23 e41 e41 NOUN cana-1656 98 24 , , PUNCT cana-1656 98 25 e42 e42 NOUN cana-1656 98 26 ) ) PUNCT cana-1656 98 27 = = SYM cana-1656 98 28 ( ( PUNCT cana-1656 98 29 25,20 25,20 NUM cana-1656 98 30 ) ) PUNCT cana-1656 98 31 ( ( PUNCT cana-1656 98 32 e51 e51 PROPN cana-1656 98 33 , , PUNCT cana-1656 98 34 e52 e52 NOUN cana-1656 98 35 ) ) PUNCT cana-1656 99 1 = = PUNCT cana-1656 99 2 ( ( PUNCT cana-1656 99 3 5,12 5,12 NUM cana-1656 99 4 ) ) PUNCT cana-1656 99 5 ( ( PUNCT cana-1656 99 6 e61 e61 PROPN cana-1656 99 7 , , PUNCT cana-1656 99 8 e62 e62 NOUN cana-1656 99 9 ) ) PUNCT cana-1656 99 10 = = SYM cana-1656 99 11 ( ( PUNCT cana-1656 99 12 3,12 3,12 NUM cana-1656 99 13 ) ) PUNCT cana-1656 99 14 ---(en1 ---(en1 PROPN cana-1656 99 15 , , PUNCT cana-1656 99 16 en2 en2 PROPN cana-1656 99 17 ) ) PUNCT cana-1656 99 18 = = PUNCT cana-1656 99 19 ( ( PUNCT cana-1656 99 20 26,29 26,29 NUM cana-1656 99 21 ) ) PUNCT cana-1656 99 22 , , PUNCT cana-1656 99 23 for for ADP cana-1656 99 24 n= n= ADJ cana-1656 99 25 1 1 NUM cana-1656 99 26 , , PUNCT cana-1656 99 27 2 2 NUM cana-1656 99 28 , , PUNCT cana-1656 99 29 3 3 NUM cana-1656 99 30 , , PUNCT cana-1656 99 31 ---25 ---25 PROPN cana-1656 99 32 . . PUNCT cana-1656 100 1 now now ADV cana-1656 100 2 compare compare VERB cana-1656 100 3 each each DET cana-1656 100 4 point point NOUN cana-1656 100 5 with with ADP cana-1656 100 6 the the DET cana-1656 100 7 table table NOUN cana-1656 100 8 to to PART cana-1656 100 9 get get VERB cana-1656 100 10 the the DET cana-1656 100 11 cipher cipher ADJ cana-1656 100 12 text text NOUN cana-1656 100 13 communications communication NOUN cana-1656 100 14 on on ADP cana-1656 100 15 applied apply VERB cana-1656 100 16 nonlinear nonlinear ADJ cana-1656 100 17 analysis analysis NOUN cana-1656 100 18 issn issn NOUN cana-1656 100 19 : : PUNCT cana-1656 100 20 1074 1074 NUM cana-1656 100 21 - - PUNCT cana-1656 100 22 133x 133x NUM cana-1656 100 23 vol vol NOUN cana-1656 100 24 32 32 NUM cana-1656 100 25 no no NOUN cana-1656 100 26 . . NOUN cana-1656 100 27 1 1 NUM cana-1656 100 28 ( ( PUNCT cana-1656 100 29 2025 2025 NUM cana-1656 100 30 ) ) PUNCT cana-1656 100 31 357 357 NUM cana-1656 100 32 https://internationalpubls.com https://internationalpubls.com X cana-1656 100 33 ainrhytelclge#fpwnahu#qkjhjwf.oohgs.dsmivv ainrhytelclge#fpwnahu#qkjhjwf.oohgs.dsmivv PROPN cana-1656 100 34 wz wz ADP cana-1656 100 35 # # NOUN cana-1656 100 36 example example NOUN cana-1656 100 37 for for ADP cana-1656 100 38 repeated repeat VERB cana-1656 100 39 words word NOUN cana-1656 100 40 take take VERB cana-1656 100 41 the the DET cana-1656 100 42 message message NOUN cana-1656 100 43 gitamgitamgitam gitamgitamgitam NOUN cana-1656 100 44 here here ADV cana-1656 100 45 we we PRON cana-1656 100 46 take take AUX cana-1656 100 47 𝛾n= 𝛾n= ADJ cana-1656 100 48 7 7 NUM cana-1656 100 49 , , PUNCT cana-1656 100 50 3 3 NUM cana-1656 100 51 , , PUNCT cana-1656 100 52 5 5 NUM cana-1656 100 53 , , PUNCT cana-1656 100 54 10 10 NUM cana-1656 100 55 , , PUNCT cana-1656 100 56 12 12 NUM cana-1656 100 57 , , PUNCT cana-1656 100 58 9 9 NUM cana-1656 100 59 , , PUNCT cana-1656 100 60 6 6 NUM cana-1656 100 61 , , PUNCT cana-1656 100 62 4 4 NUM cana-1656 100 63 , , PUNCT cana-1656 100 64 8 8 NUM cana-1656 100 65 , , PUNCT cana-1656 100 66 11 11 NUM cana-1656 100 67 , , PUNCT cana-1656 100 68 13 13 NUM cana-1656 100 69 , , PUNCT cana-1656 100 70 15 15 NUM cana-1656 100 71 , , PUNCT cana-1656 100 72 16 16 NUM cana-1656 100 73 , , PUNCT cana-1656 100 74 22 22 NUM cana-1656 100 75 , , PUNCT cana-1656 100 76 17 17 NUM cana-1656 100 77 , , PUNCT cana-1656 100 78 for for ADP cana-1656 100 79 n= n= ADJ cana-1656 100 80 1 1 NUM cana-1656 100 81 , , PUNCT cana-1656 100 82 2 2 NUM cana-1656 100 83 , , PUNCT cana-1656 100 84 3 3 NUM cana-1656 100 85 , , PUNCT cana-1656 100 86 ---15 ---15 PUNCT cana-1656 100 87 . . PUNCT cana-1656 101 1 then then ADV cana-1656 101 2 by by ADP cana-1656 101 3 using use VERB cana-1656 101 4 above above ADJ cana-1656 101 5 formula formula NOUN cana-1656 101 6 we we PRON cana-1656 101 7 calculate calculate VERB cana-1656 101 8 ( ( PUNCT cana-1656 101 9 e11 e11 X cana-1656 101 10 , , PUNCT cana-1656 101 11 e12 e12 NOUN cana-1656 101 12 ) ) PUNCT cana-1656 101 13 = = PUNCT cana-1656 102 1 ( ( PUNCT cana-1656 102 2 17,5 17,5 NUM cana-1656 102 3 ) ) PUNCT cana-1656 102 4 ( ( PUNCT cana-1656 102 5 e21 e21 PROPN cana-1656 102 6 , , PUNCT cana-1656 102 7 e22 e22 NOUN cana-1656 102 8 ) ) PUNCT cana-1656 102 9 = = SYM cana-1656 102 10 ( ( PUNCT cana-1656 102 11 19,14 19,14 ADV cana-1656 102 12 ) ) PUNCT cana-1656 102 13 ( ( PUNCT cana-1656 102 14 e31 e31 X cana-1656 102 15 , , PUNCT cana-1656 102 16 e32 e32 PROPN cana-1656 102 17 ) ) PUNCT cana-1656 102 18 = = PUNCT cana-1656 102 19 ( ( PUNCT cana-1656 102 20 18,1 18,1 NUM cana-1656 102 21 ) ) PUNCT cana-1656 102 22 ( ( PUNCT cana-1656 102 23 e41 e41 NOUN cana-1656 102 24 , , PUNCT cana-1656 102 25 e42 e42 NOUN cana-1656 102 26 ) ) PUNCT cana-1656 102 27 = = SYM cana-1656 102 28 ( ( PUNCT cana-1656 102 29 36,4 36,4 NUM cana-1656 102 30 ) ) PUNCT cana-1656 102 31 ( ( PUNCT cana-1656 102 32 e51 e51 PROPN cana-1656 102 33 , , PUNCT cana-1656 102 34 e52 e52 NOUN cana-1656 102 35 ) ) PUNCT cana-1656 102 36 = = PUNCT cana-1656 102 37 ( ( PUNCT cana-1656 102 38 35,33 35,33 NUM cana-1656 102 39 ) ) PUNCT cana-1656 102 40 ( ( PUNCT cana-1656 102 41 e61 e61 PROPN cana-1656 102 42 , , PUNCT cana-1656 102 43 e62 e62 NOUN cana-1656 102 44 ) ) PUNCT cana-1656 102 45 = = SYM cana-1656 102 46 ( ( PUNCT cana-1656 102 47 16,22 16,22 NOUN cana-1656 102 48 ) ) PUNCT cana-1656 102 49 ---(en1 ---(en1 PROPN cana-1656 102 50 , , PUNCT cana-1656 102 51 en2 en2 PROPN cana-1656 102 52 ) ) PUNCT cana-1656 102 53 = = PUNCT cana-1656 102 54 ( ( PUNCT cana-1656 102 55 12,14 12,14 ADV cana-1656 102 56 ) ) PUNCT cana-1656 102 57 , , PUNCT cana-1656 102 58 n= n= ADJ cana-1656 102 59 1 1 NUM cana-1656 102 60 , , PUNCT cana-1656 102 61 2 2 NUM cana-1656 102 62 , , PUNCT cana-1656 102 63 3 3 NUM cana-1656 102 64 , , PUNCT cana-1656 102 65 ---15 ---15 PUNCT cana-1656 102 66 . . PUNCT cana-1656 103 1 now now ADV cana-1656 103 2 compare compare VERB cana-1656 103 3 each each DET cana-1656 103 4 point point NOUN cana-1656 103 5 with with ADP cana-1656 103 6 the the DET cana-1656 103 7 table table NOUN cana-1656 103 8 to to PART cana-1656 103 9 get get VERB cana-1656 103 10 the the DET cana-1656 103 11 cipher cipher ADJ cana-1656 103 12 text text NOUN cana-1656 103 13 qesnra5d42pv7s8m6 qesnra5d42pv7s8m6 PROPN cana-1656 103 14 oenom322$xln oenom322$xln ADJ cana-1656 103 15 table table NOUN cana-1656 103 16 1 1 NUM cana-1656 103 17 : : PUNCT cana-1656 103 18 table table NOUN cana-1656 103 19 for for ADP cana-1656 103 20 time time NOUN cana-1656 103 21 calculation calculation NOUN cana-1656 103 22 message message NOUN cana-1656 103 23 size size NOUN cana-1656 103 24 in in ADP cana-1656 103 25 kb kb PROPN cana-1656 103 26 execution execution NOUN cana-1656 103 27 time time NOUN cana-1656 103 28 in in ADP cana-1656 103 29 sec sec PROPN cana-1656 103 30 . . PROPN cana-1656 104 1 0.1 0.1 NUM cana-1656 104 2 0.03816 0.03816 NUM cana-1656 104 3 0.2 0.2 NUM cana-1656 104 4 0.04918 0.04918 NUM cana-1656 104 5 0.3 0.3 NUM cana-1656 104 6 0.06110 0.06110 NUM cana-1656 104 7 0.4 0.4 NUM cana-1656 104 8 0.07234 0.07234 NUM cana-1656 104 9 0.5 0.5 NUM cana-1656 104 10 0.08318 0.08318 NUM cana-1656 104 11 0.6 0.6 NUM cana-1656 104 12 0.09812 0.09812 NUM cana-1656 104 13 0.7 0.7 NUM cana-1656 104 14 0.1121 0.1121 NUM cana-1656 104 15 0.8 0.8 NUM cana-1656 104 16 0.2314 0.2314 NUM cana-1656 104 17 0.9 0.9 NUM cana-1656 104 18 0.3145 0.3145 NUM cana-1656 104 19 1 1 NUM cana-1656 104 20 0.4285 0.4285 NUM cana-1656 104 21 execution execution NOUN cana-1656 104 22 time time NOUN cana-1656 104 23 graph graph NOUN cana-1656 104 24 : : PUNCT cana-1656 104 25 figure figure NOUN cana-1656 104 26 2 2 NUM cana-1656 104 27 : : PUNCT cana-1656 104 28 execution execution NOUN cana-1656 104 29 time time NOUN cana-1656 104 30 graph graph NOUN cana-1656 104 31 communications communication NOUN cana-1656 104 32 on on ADP cana-1656 104 33 applied apply VERB cana-1656 104 34 nonlinear nonlinear ADJ cana-1656 104 35 analysis analysis NOUN cana-1656 104 36 issn issn NOUN cana-1656 104 37 : : PUNCT cana-1656 104 38 1074 1074 NUM cana-1656 104 39 - - PUNCT cana-1656 104 40 133x 133x NUM cana-1656 104 41 vol vol NOUN cana-1656 104 42 32 32 NUM cana-1656 104 43 no no NOUN cana-1656 104 44 . . NOUN cana-1656 104 45 1 1 NUM cana-1656 104 46 ( ( PUNCT cana-1656 104 47 2025 2025 NUM cana-1656 104 48 ) ) PUNCT cana-1656 104 49 358 358 NUM cana-1656 104 50 https://internationalpubls.com https://internationalpubls.com X cana-1656 104 51 figure figure NOUN cana-1656 104 52 3 3 NUM cana-1656 104 53 : : PUNCT cana-1656 104 54 conventional conventional ADJ cana-1656 104 55 ecc ecc PROPN cana-1656 104 56 for for ADP cana-1656 104 57 different different ADJ cana-1656 104 58 sizes size NOUN cana-1656 104 59 of of ADP cana-1656 104 60 data datum NOUN cana-1656 104 61 ( ( PUNCT cana-1656 104 62 number number NOUN cana-1656 104 63 of of ADP cana-1656 104 64 characters character NOUN cana-1656 104 65 ) ) PUNCT cana-1656 104 66 figure figure VERB cana-1656 104 67 4 4 NUM cana-1656 104 68 : : PUNCT cana-1656 104 69 total total ADJ cana-1656 104 70 time time NOUN cana-1656 104 71 required require VERB cana-1656 104 72 for for ADP cana-1656 104 73 conventional conventional ADJ cana-1656 104 74 ecc ecc PROPN cana-1656 104 75 & & CCONJ cana-1656 104 76 proposed propose VERB cana-1656 104 77 method method NOUN cana-1656 104 78 the the DET cana-1656 104 79 above above ADJ cana-1656 104 80 table table NOUN cana-1656 104 81 1 1 NUM cana-1656 104 82 and and CCONJ cana-1656 104 83 figures figure NOUN cana-1656 104 84 2,3,4 2,3,4 NUM cana-1656 104 85 show show NOUN cana-1656 104 86 encryption encryption NOUN cana-1656 104 87 and and CCONJ cana-1656 104 88 decryption decryption NOUN cana-1656 104 89 time time NOUN cana-1656 104 90 of of ADP cana-1656 104 91 the the DET cana-1656 104 92 present present ADJ cana-1656 104 93 technique technique NOUN cana-1656 104 94 and and CCONJ cana-1656 104 95 conventional conventional ADJ cana-1656 104 96 ecc ecc PROPN cana-1656 104 97 for for ADP cana-1656 104 98 different different ADJ cana-1656 104 99 sizes size NOUN cana-1656 104 100 of of ADP cana-1656 104 101 data datum NOUN cana-1656 104 102 ( ( PUNCT cana-1656 104 103 number number NOUN cana-1656 104 104 of of ADP cana-1656 104 105 characters character NOUN cana-1656 104 106 ) ) PUNCT cana-1656 104 107 . . PUNCT cana-1656 105 1 present present ADJ cana-1656 105 2 technique technique NOUN cana-1656 105 3 shows show VERB cana-1656 105 4 approximately approximately ADV cana-1656 105 5 same same ADJ cana-1656 105 6 execution execution NOUN cana-1656 105 7 time time NOUN cana-1656 105 8 when when SCONJ cana-1656 105 9 compared compare VERB cana-1656 105 10 with with ADP cana-1656 105 11 conventional conventional ADJ cana-1656 105 12 elliptic elliptic ADJ cana-1656 105 13 curve curve NOUN cana-1656 105 14 cryptography cryptography NOUN cana-1656 105 15 . . PUNCT cana-1656 106 1 so so ADV cana-1656 106 2 , , PUNCT cana-1656 106 3 the the DET cana-1656 106 4 computational computational ADJ cana-1656 106 5 cost cost NOUN cana-1656 106 6 is be AUX cana-1656 106 7 not not PART cana-1656 106 8 more more ADJ cana-1656 106 9 than than ADP cana-1656 106 10 conventional conventional ADJ cana-1656 106 11 ecc ecc PROPN cana-1656 106 12 and and CCONJ cana-1656 106 13 the the DET cana-1656 106 14 present present ADJ cana-1656 106 15 algorithm algorithm NOUN cana-1656 106 16 is be AUX cana-1656 106 17 as as ADV cana-1656 106 18 fast fast ADJ cana-1656 106 19 as as ADP cana-1656 106 20 conventional conventional ADJ cana-1656 106 21 ecc ecc PROPN cana-1656 106 22 algorithm algorithm NOUN cana-1656 106 23 . . PUNCT cana-1656 107 1 security security NOUN cana-1656 107 2 analysis analysis NOUN cana-1656 107 3 : : PUNCT cana-1656 107 4 here here ADV cana-1656 107 5 𝛾 𝛾 ADP cana-1656 107 6 value value NOUN cana-1656 107 7 is be AUX cana-1656 107 8 different different ADJ cana-1656 107 9 for for ADP cana-1656 107 10 encrypting encrypt VERB cana-1656 107 11 different different ADJ cana-1656 107 12 message message NOUN cana-1656 107 13 characters character NOUN cana-1656 107 14 . . PUNCT cana-1656 108 1 due due ADP cana-1656 108 2 to to ADP cana-1656 108 3 these these DET cana-1656 108 4 same same ADJ cana-1656 108 5 characters character NOUN cana-1656 108 6 in in ADP cana-1656 108 7 the the DET cana-1656 108 8 original original ADJ cana-1656 108 9 message message NOUN cana-1656 108 10 will will AUX cana-1656 108 11 be be AUX cana-1656 108 12 mapped map VERB cana-1656 108 13 to to ADP cana-1656 108 14 different different ADJ cana-1656 108 15 cipher cipher ADJ cana-1656 108 16 characters character NOUN cana-1656 108 17 . . PUNCT cana-1656 109 1 in in ADP cana-1656 109 2 the the DET cana-1656 109 3 first first ADJ cana-1656 109 4 example example NOUN cana-1656 109 5 given give VERB cana-1656 109 6 above above ADP cana-1656 109 7 there there PRON cana-1656 109 8 are be VERB cana-1656 109 9 repeated repeat VERB cana-1656 109 10 characters character NOUN cana-1656 109 11 ( ( PUNCT cana-1656 109 12 m m PROPN cana-1656 109 13 , , PUNCT cana-1656 109 14 r r NOUN cana-1656 109 15 , , PUNCT cana-1656 109 16 l l NOUN cana-1656 109 17 , , PUNCT cana-1656 109 18 a a DET cana-1656 109 19 , , PUNCT cana-1656 109 20 g g NOUN cana-1656 109 21 ) ) PUNCT cana-1656 109 22 in in ADP cana-1656 109 23 the the DET cana-1656 109 24 original original ADJ cana-1656 109 25 message message NOUN cana-1656 109 26 . . PUNCT cana-1656 110 1 first first ADJ cana-1656 110 2 β€˜ ' PUNCT cana-1656 110 3 r r NOUN cana-1656 110 4 ’ ' PUNCT cana-1656 110 5 letter letter NOUN cana-1656 110 6 is be AUX cana-1656 110 7 mapped map VERB cana-1656 110 8 to to PART cana-1656 110 9 β€˜ ' PUNCT cana-1656 110 10 ai ai VERB cana-1656 110 11 ’ ' PUNCT cana-1656 110 12 whereas whereas SCONJ cana-1656 110 13 second second ADJ cana-1656 110 14 β€˜ ' PUNCT cana-1656 110 15 r r NOUN cana-1656 110 16 ’ ' PUNCT cana-1656 110 17 letter letter NOUN cana-1656 110 18 is be AUX cana-1656 110 19 mapped map VERB cana-1656 110 20 to to ADP cana-1656 110 21 β€˜ ' PUNCT cana-1656 110 22 iv iv NUM cana-1656 110 23 ’ ' PUNCT cana-1656 110 24 . . PUNCT cana-1656 111 1 similarly similarly ADV cana-1656 111 2 , , PUNCT cana-1656 111 3 the the DET cana-1656 111 4 same same ADJ cana-1656 111 5 character character NOUN cana-1656 111 6 β€˜ ' PUNCT cana-1656 111 7 l l NOUN cana-1656 111 8 ’ ' PUNCT cana-1656 111 9 repeated repeat VERB cana-1656 111 10 side side NOUN cana-1656 111 11 by by ADP cana-1656 111 12 side side NOUN cana-1656 111 13 is be AUX cana-1656 111 14 mapped map VERB cana-1656 111 15 to to ADP cana-1656 111 16 β€˜ ' PUNCT cana-1656 111 17 fp fp NOUN cana-1656 111 18 ’ ' PUNCT cana-1656 111 19 and and CCONJ cana-1656 111 20 β€˜ ' PUNCT cana-1656 111 21 wn wn NOUN cana-1656 111 22 ’ ' PUNCT cana-1656 111 23 in in ADP cana-1656 111 24 the the DET cana-1656 111 25 cipher cipher ADJ cana-1656 111 26 text text NOUN cana-1656 111 27 . . PUNCT cana-1656 112 1 since since SCONJ cana-1656 112 2 different different ADJ cana-1656 112 3 random random ADJ cana-1656 112 4 numbers number NOUN cana-1656 112 5 are be AUX cana-1656 112 6 used use VERB cana-1656 112 7 for for ADP cana-1656 112 8 𝛾(less 𝛾(less NOUN cana-1656 112 9 than than ADP cana-1656 112 10 generator generator NOUN cana-1656 112 11 ) ) PUNCT cana-1656 112 12 the the DET cana-1656 112 13 present present ADJ cana-1656 112 14 encryption encryption NOUN cana-1656 112 15 has have AUX cana-1656 112 16 achieved achieve VERB cana-1656 112 17 a a DET cana-1656 112 18 good good ADJ cana-1656 112 19 required require VERB cana-1656 112 20 feature feature NOUN cana-1656 112 21 to to PART cana-1656 112 22 face face VERB cana-1656 112 23 several several ADJ cana-1656 112 24 active active ADJ cana-1656 112 25 and and CCONJ cana-1656 112 26 passive passive ADJ cana-1656 112 27 attacks attack NOUN cana-1656 112 28 . . PUNCT cana-1656 113 1 in in ADP cana-1656 113 2 the the DET cana-1656 113 3 history history NOUN cana-1656 113 4 of of ADP cana-1656 113 5 cryptography cryptography NOUN cana-1656 113 6 several several ADJ cana-1656 113 7 authors author NOUN cana-1656 113 8 suggested suggest VERB cana-1656 113 9 multiple multiple ADJ cana-1656 113 10 encryptions encryption NOUN cana-1656 113 11 using use VERB cana-1656 113 12 the the DET cana-1656 113 13 same same ADJ cana-1656 113 14 key key NOUN cana-1656 113 15 . . PUNCT cana-1656 114 1 the the DET cana-1656 114 2 proposed propose VERB cana-1656 114 3 method method NOUN cana-1656 114 4 is be AUX cana-1656 114 5 as as ADP cana-1656 114 6 starts start NOUN cana-1656 114 7 as as ADP cana-1656 114 8 available available ADJ cana-1656 114 9 multiple multiple ADJ cana-1656 114 10 encryption encryption NOUN cana-1656 114 11 algorithms algorithm NOUN cana-1656 114 12 with with ADP cana-1656 114 13 comparatively comparatively ADV cana-1656 114 14 less less ADV cana-1656 114 15 computational computational ADJ cana-1656 114 16 overhead overhead NOUN cana-1656 114 17 . . PUNCT cana-1656 115 1 fig fig NOUN cana-1656 115 2 6 6 NUM cana-1656 115 3 shows show VERB cana-1656 115 4 the the DET cana-1656 115 5 repetition repetition NOUN cana-1656 115 6 of of ADP cana-1656 115 7 the the DET cana-1656 115 8 english english ADJ cana-1656 115 9 alphabets alphabet NOUN cana-1656 115 10 in in ADP cana-1656 115 11 the the DET cana-1656 115 12 cipher cipher ADJ cana-1656 115 13 text text NOUN cana-1656 115 14 for for ADP cana-1656 115 15 1 1 NUM cana-1656 115 16 kb kb PROPN cana-1656 115 17 of of ADP cana-1656 115 18 data data PROPN cana-1656 115 19 . . PUNCT cana-1656 116 1 in in ADP cana-1656 116 2 ecc ecc PROPN cana-1656 116 3 the the DET cana-1656 116 4 cipher cipher ADJ cana-1656 116 5 text text NOUN cana-1656 116 6 space space NOUN cana-1656 116 7 twice twice ADV cana-1656 116 8 than than ADP cana-1656 116 9 the the DET cana-1656 116 10 message message NOUN cana-1656 116 11 space space NOUN cana-1656 116 12 . . PUNCT cana-1656 117 1 so so ADV cana-1656 117 2 , , PUNCT cana-1656 117 3 english english ADJ cana-1656 117 4 alphabets alphabet NOUN cana-1656 117 5 repetition repetition NOUN cana-1656 117 6 will will AUX cana-1656 117 7 not not PART cana-1656 117 8 give give VERB cana-1656 117 9 any any DET cana-1656 117 10 information information NOUN cana-1656 117 11 about about ADP cana-1656 117 12 the the DET cana-1656 117 13 cipher cipher ADJ cana-1656 117 14 text text NOUN cana-1656 117 15 , , PUNCT cana-1656 117 16 because because SCONJ cana-1656 117 17 each each DET cana-1656 117 18 character character NOUN cana-1656 117 19 maps map VERB cana-1656 117 20 to to ADP cana-1656 117 21 two two NUM cana-1656 117 22 cipher cipher ADJ cana-1656 117 23 characters character NOUN cana-1656 117 24 . . PUNCT cana-1656 118 1 so so ADV cana-1656 118 2 , , PUNCT cana-1656 118 3 linear linear ADJ cana-1656 118 4 cryptanalysis cryptanalysis NOUN cana-1656 118 5 and and CCONJ cana-1656 118 6 differential differential NOUN cana-1656 118 7 cryptanalysis cryptanalysis NOUN cana-1656 118 8 are be AUX cana-1656 118 9 tough tough ADJ cana-1656 118 10 to to PART cana-1656 118 11 execute execute VERB cana-1656 118 12 for for ADP cana-1656 118 13 the the DET cana-1656 118 14 present present ADJ cana-1656 118 15 cipher cipher ADJ cana-1656 118 16 . . PUNCT cana-1656 119 1 this this DET cana-1656 119 2 hard hard ADJ cana-1656 119 3 problem problem NOUN cana-1656 119 4 protects protect VERB cana-1656 119 5 ecc ecc PROPN cana-1656 119 6 against against ADP cana-1656 119 7 several several ADJ cana-1656 119 8 active active ADJ cana-1656 119 9 and and CCONJ cana-1656 119 10 passive passive ADJ cana-1656 119 11 attacks attack NOUN cana-1656 119 12 . . PUNCT cana-1656 120 1 in in ADP cana-1656 120 2 pollard pollard PROPN cana-1656 120 3 – – PUNCT cana-1656 120 4 rho rho ADJ cana-1656 120 5 attack attack NOUN cana-1656 120 6 ecdlp ecdlp NOUN cana-1656 120 7 is be AUX cana-1656 120 8 cracked crack VERB cana-1656 120 9 by by ADP cana-1656 120 10 exhaustive exhaustive ADJ cana-1656 120 11 search search NOUN cana-1656 120 12 of of ADP cana-1656 120 13 cipher cipher ADJ cana-1656 120 14 text text NOUN cana-1656 120 15 points point NOUN cana-1656 120 16 ( ( PUNCT cana-1656 120 17 points point NOUN cana-1656 120 18 on on ADP cana-1656 120 19 elliptic elliptic ADJ cana-1656 120 20 curve curve NOUN cana-1656 120 21 ) ) PUNCT cana-1656 120 22 . . PUNCT cana-1656 121 1 but but CCONJ cana-1656 121 2 in in ADP cana-1656 121 3 the the DET cana-1656 121 4 present present ADJ cana-1656 121 5 technique technique NOUN cana-1656 121 6 conventional conventional ADJ cana-1656 121 7 elliptic elliptic ADJ cana-1656 121 8 curve curve NOUN cana-1656 121 9 cryptographic cryptographic ADJ cana-1656 121 10 algorithm algorithm NOUN cana-1656 121 11 ( ( PUNCT cana-1656 121 12 similar similar ADJ cana-1656 121 13 to to ADP cana-1656 121 14 elgamal elgamal ADJ cana-1656 121 15 encryption encryption NOUN cana-1656 121 16 ) ) PUNCT cana-1656 121 17 is be AUX cana-1656 121 18 modified modify VERB cana-1656 121 19 by by ADP cana-1656 121 20 considering consider VERB cana-1656 121 21 more more ADJ cana-1656 121 22 number number NOUN cana-1656 121 23 of of ADP cana-1656 121 24 points point NOUN cana-1656 121 25 and and CCONJ cana-1656 121 26 big big ADJ cana-1656 121 27 random random ADJ cana-1656 121 28 numbers number NOUN cana-1656 121 29 to to PART cana-1656 121 30 construct construct VERB cana-1656 121 31 public public ADJ cana-1656 121 32 and and CCONJ cana-1656 121 33 secret secret ADJ cana-1656 121 34 keys key NOUN cana-1656 121 35 . . PUNCT cana-1656 122 1 in in ADP cana-1656 122 2 addition addition NOUN cana-1656 122 3 , , PUNCT cana-1656 122 4 each each DET cana-1656 122 5 message message NOUN cana-1656 122 6 character character NOUN cana-1656 122 7 is be AUX cana-1656 122 8 encrypted encrypt VERB cana-1656 122 9 using use VERB cana-1656 122 10 a a DET cana-1656 122 11 parameter parameter NOUN cana-1656 122 12 𝛾 𝛾 NOUN cana-1656 122 13 ( ( PUNCT cana-1656 122 14 random random ADJ cana-1656 122 15 number number NOUN cana-1656 122 16 less less ADJ cana-1656 122 17 than than ADP cana-1656 122 18 generator generator NOUN cana-1656 122 19 ) ) PUNCT cana-1656 122 20 . . PUNCT cana-1656 123 1 this this DET cana-1656 123 2 𝛾acts 𝛾act NOUN cana-1656 123 3 as as ADP cana-1656 123 4 a a DET cana-1656 123 5 safe safe ADJ cana-1656 123 6 guard guard NOUN cana-1656 123 7 for for ADP cana-1656 123 8 the the DET cana-1656 123 9 cipher cipher ADJ cana-1656 123 10 text text NOUN cana-1656 123 11 . . PUNCT cana-1656 124 1 pollard pollard PROPN cana-1656 124 2 – – PUNCT cana-1656 124 3 rho rho ADJ cana-1656 124 4 and and CCONJ cana-1656 124 5 pohlighellman pohlighellman ADJ cana-1656 124 6 attacks attack NOUN cana-1656 124 7 are be AUX cana-1656 124 8 quite quite ADV cana-1656 124 9 impossible impossible ADJ cana-1656 124 10 . . PUNCT cana-1656 125 1 𝛾values 𝛾value NOUN cana-1656 125 2 are be AUX cana-1656 125 3 chosen choose VERB cana-1656 125 4 by by ADP cana-1656 125 5 encryptor encryptor NOUN cana-1656 125 6 for for ADP cana-1656 125 7 different different ADJ cana-1656 125 8 message message NOUN cana-1656 125 9 characters character NOUN cana-1656 125 10 . . PUNCT cana-1656 126 1 these these DET cana-1656 126 2 𝛾values 𝛾value NOUN cana-1656 126 3 are be AUX cana-1656 126 4 inherent inherent ADJ cana-1656 126 5 in in ADP cana-1656 126 6 the the DET cana-1656 126 7 cipher cipher ADJ cana-1656 126 8 text text NOUN cana-1656 126 9 . . PUNCT cana-1656 127 1 communications communication NOUN cana-1656 127 2 on on ADP cana-1656 127 3 applied apply VERB cana-1656 127 4 nonlinear nonlinear ADJ cana-1656 127 5 analysis analysis NOUN cana-1656 127 6 issn issn NOUN cana-1656 127 7 : : PUNCT cana-1656 127 8 1074 1074 NUM cana-1656 127 9 - - PUNCT cana-1656 127 10 133x 133x NUM cana-1656 127 11 vol vol NOUN cana-1656 127 12 32 32 NUM cana-1656 127 13 no no NOUN cana-1656 127 14 . . NOUN cana-1656 127 15 1 1 NUM cana-1656 127 16 ( ( PUNCT cana-1656 127 17 2025 2025 NUM cana-1656 127 18 ) ) PUNCT cana-1656 128 1 359 359 NUM cana-1656 128 2 https://internationalpubls.com https://internationalpubls.com X cana-1656 128 3 even even ADV cana-1656 128 4 for for ADP cana-1656 128 5 the the DET cana-1656 128 6 repeated repeat VERB cana-1656 128 7 words word NOUN cana-1656 128 8 message message NOUN cana-1656 128 9 ( ( PUNCT cana-1656 128 10 ssss-----sususu------ ssss-----sususu------ PROPN cana-1656 128 11 ) ) PUNCT cana-1656 128 12 different different ADJ cana-1656 128 13 𝛾 𝛾 ADP cana-1656 128 14 values value NOUN cana-1656 128 15 are be AUX cana-1656 128 16 used use VERB cana-1656 128 17 to to ADP cana-1656 128 18 encrypting encrypt VERB cana-1656 128 19 the the DET cana-1656 128 20 characters character NOUN cana-1656 128 21 . . PUNCT cana-1656 129 1 this this PRON cana-1656 129 2 is be AUX cana-1656 129 3 the the DET cana-1656 129 4 basic basic ADJ cana-1656 129 5 strength strength NOUN cana-1656 129 6 of of ADP cana-1656 129 7 the the DET cana-1656 129 8 present present ADJ cana-1656 129 9 cipher cipher ADJ cana-1656 129 10 to to PART cana-1656 129 11 endure endure VERB cana-1656 129 12 and and CCONJ cana-1656 129 13 avoid avoid VERB cana-1656 129 14 pollard pollard NOUN cana-1656 129 15 – – PUNCT cana-1656 129 16 rho rho ADJ cana-1656 129 17 attack attack NOUN cana-1656 129 18 . . PUNCT cana-1656 130 1 underlying underlie VERB cana-1656 130 2 𝛾 𝛾 ADP cana-1656 130 3 values value NOUN cana-1656 130 4 is be AUX cana-1656 130 5 helping help VERB cana-1656 130 6 the the DET cana-1656 130 7 present present ADJ cana-1656 130 8 cipher cipher ADJ cana-1656 130 9 to to PART cana-1656 130 10 overcome overcome VERB cana-1656 130 11 the the DET cana-1656 130 12 weakness weakness NOUN cana-1656 130 13 of of ADP cana-1656 130 14 ecc ecc PROPN cana-1656 130 15 . . PUNCT cana-1656 131 1 so so ADV cana-1656 131 2 , , PUNCT cana-1656 131 3 the the DET cana-1656 131 4 ecc ecc PROPN cana-1656 131 5 algorithm algorithm NOUN cana-1656 131 6 presented present VERB cana-1656 131 7 here here ADV cana-1656 131 8 is be AUX cana-1656 131 9 as as ADV cana-1656 131 10 safe safe ADJ cana-1656 131 11 as as ADP cana-1656 131 12 conventional conventional ADJ cana-1656 131 13 public public ADJ cana-1656 131 14 key key ADJ cana-1656 131 15 cryptographic cryptographic ADJ cana-1656 131 16 techniques technique NOUN cana-1656 131 17 with with ADP cana-1656 131 18 small small ADJ cana-1656 131 19 key key ADJ cana-1656 131 20 size size NOUN cana-1656 131 21 . . PUNCT cana-1656 132 1 conclusion conclusion NOUN cana-1656 132 2 : : PUNCT cana-1656 132 3 the the DET cana-1656 132 4 process process NOUN cana-1656 132 5 of of ADP cana-1656 132 6 utilizing utilize VERB cana-1656 132 7 the the DET cana-1656 132 8 concept concept NOUN cana-1656 132 9 of of ADP cana-1656 132 10 elliptical elliptical ADJ cana-1656 132 11 curve curve NOUN cana-1656 132 12 cryptography cryptography NOUN cana-1656 132 13 ( ( PUNCT cana-1656 132 14 ecc ecc PROPN cana-1656 132 15 ) ) PUNCT cana-1656 132 16 delineated delineate VERB cana-1656 132 17 to to PART cana-1656 132 18 restraint restraint VERB cana-1656 132 19 intrusions intrusion NOUN cana-1656 132 20 in in ADP cana-1656 132 21 the the DET cana-1656 132 22 communication communication NOUN cana-1656 132 23 of of ADP cana-1656 132 24 sensitive sensitive ADJ cana-1656 132 25 information information NOUN cana-1656 132 26 securely securely ADV cana-1656 132 27 . . PUNCT cana-1656 133 1 the the DET cana-1656 133 2 encryption encryption NOUN cana-1656 133 3 technique technique NOUN cana-1656 133 4 proposed propose VERB cana-1656 133 5 here here ADV cana-1656 133 6 offers offer VERB cana-1656 133 7 same same ADJ cana-1656 133 8 level level NOUN cana-1656 133 9 of of ADP cana-1656 133 10 security security NOUN cana-1656 133 11 as as ADP cana-1656 133 12 the the DET cana-1656 133 13 conventional conventional ADJ cana-1656 133 14 public public ADJ cana-1656 133 15 key key ADJ cana-1656 133 16 cryptographic cryptographic ADJ cana-1656 133 17 techniques technique NOUN cana-1656 133 18 as as SCONJ cana-1656 133 19 it it PRON cana-1656 133 20 is be AUX cana-1656 133 21 highly highly ADV cana-1656 133 22 difficult difficult ADJ cana-1656 133 23 to to PART cana-1656 133 24 calculate calculate VERB cana-1656 133 25 the the DET cana-1656 133 26 secret secret ADJ cana-1656 133 27 keys key NOUN cana-1656 133 28 of of ADP cana-1656 133 29 both both CCONJ cana-1656 133 30 the the DET cana-1656 133 31 legitimate legitimate ADJ cana-1656 133 32 entities entity NOUN cana-1656 133 33 . . PUNCT cana-1656 134 1 references reference NOUN cana-1656 134 2 : : PUNCT cana-1656 135 1 [ [ X cana-1656 135 2 1 1 X cana-1656 135 3 ] ] PUNCT cana-1656 135 4 benjamin benjamin PROPN cana-1656 135 5 clement clement PROPN cana-1656 135 6 sebastin sebastin PROPN cana-1656 135 7 , , PUNCT cana-1656 135 8 ugaralpaycenar ugaralpaycenar ADJ cana-1656 135 9 β€œ " PUNCT cana-1656 135 10 advantage advantage NOUN cana-1656 135 11 of of ADP cana-1656 135 12 using use VERB cana-1656 135 13 elliptic elliptic ADJ cana-1656 135 14 curve curve NOUN cana-1656 135 15 cryptography cryptography NOUN cana-1656 135 16 in in ADP cana-1656 135 17 ssl ssl PROPN cana-1656 135 18 / / SYM cana-1656 135 19 tla tla NOUN cana-1656 135 20 , , PUNCT cana-1656 135 21 https://koclab.cs.ucsb.edu/teaching/ecc/project/2015projects/cenar+sebastian https://koclab.cs.ucsb.edu/teaching/ecc/project/2015projects/cenar+sebastian ADJ cana-1656 135 22 [ [ X cana-1656 135 23 2 2 NUM cana-1656 135 24 ] ] PUNCT cana-1656 135 25 laiphrakpam laiphrakpam NOUN cana-1656 135 26 dolendro dolendro PROPN cana-1656 135 27 singh singh PROPN cana-1656 135 28 and and CCONJ cana-1656 135 29 khumanthem khumanthem PROPN cana-1656 135 30 manglem manglem PROPN cana-1656 135 31 singh singh PROPN cana-1656 135 32 β€œ " PUNCT cana-1656 135 33 implementation implementation NOUN cana-1656 135 34 of of ADP cana-1656 135 35 text text NOUN cana-1656 135 36 encryption encryption NOUN cana-1656 135 37 using use VERB cana-1656 135 38 elliptic elliptic ADJ cana-1656 135 39 curve curve NOUN cana-1656 135 40 cryptography cryptography NOUN cana-1656 135 41 ” " PUNCT cana-1656 135 42 , , PUNCT cana-1656 135 43 procedia procedia NOUN cana-1656 135 44 computer computer NOUN cana-1656 135 45 science science NOUN cana-1656 135 46 54 54 NUM cana-1656 135 47 ( ( PUNCT cana-1656 135 48 2015 2015 NUM cana-1656 135 49 ) ) PUNCT cana-1656 135 50 73 73 NUM cana-1656 135 51 - - SYM cana-1656 135 52 82 82 NUM cana-1656 135 53 , , PUNCT cana-1656 135 54 www www NOUN cana-1656 135 55 . . PUNCT cana-1656 136 1 sciencedirect.com sciencedirect.com X cana-1656 137 1 [ [ X cana-1656 137 2 3 3 NUM cana-1656 137 3 ] ] PUNCT cana-1656 137 4 reneβ€Ÿe reneβ€Ÿe ADJ cana-1656 137 5 brady brady PROPN cana-1656 137 6 , , PUNCT cana-1656 137 7 naleceia naleceia PROPN cana-1656 137 8 david david PROPN cana-1656 137 9 and and CCONJ cana-1656 137 10 anna anna PROPN cana-1656 137 11 tracy tracy PROPN cana-1656 137 12 , , PUNCT cana-1656 137 13 β€œ " PUNCT cana-1656 137 14 encrypting encrypt VERB cana-1656 137 15 with with ADP cana-1656 137 16 elliptic elliptic ADJ cana-1656 137 17 curve curve NOUN cana-1656 137 18 cryptography cryptography NOUN cana-1656 137 19 , , PUNCT cana-1656 137 20 www.math.purdue.edu//encrypting www.math.purdue.edu//encrypte VERB cana-1656 137 21 [ [ X cana-1656 137 22 4 4 X cana-1656 137 23 ] ] PUNCT cana-1656 137 24 s. s. PROPN cana-1656 137 25 thiraviya thiraviya PROPN cana-1656 137 26 regina regina PROPN cana-1656 137 27 rajam rajam PROPN cana-1656 137 28 and and CCONJ cana-1656 137 29 s. s. PROPN cana-1656 137 30 britto britto PROPN cana-1656 137 31 ramesh ramesh PROPN cana-1656 137 32 kumar kumar PROPN cana-1656 137 33 , , PUNCT cana-1656 137 34 β€œ " PUNCT cana-1656 137 35 enhanced enhance VERB cana-1656 137 36 elliptic elliptic ADJ cana-1656 137 37 curve curve NOUN cana-1656 137 38 cryptography cryptography NOUN cana-1656 137 39 ” " PUNCT cana-1656 137 40 , , PUNCT cana-1656 137 41 indian indian ADJ cana-1656 137 42 journal journal NOUN cana-1656 137 43 of of ADP cana-1656 137 44 science science NOUN cana-1656 137 45 and and CCONJ cana-1656 137 46 technology technology NOUN cana-1656 137 47 , , PUNCT cana-1656 137 48 vol vol NOUN cana-1656 137 49 . . NOUN cana-1656 137 50 8 8 NUM cana-1656 137 51 ( ( PUNCT cana-1656 137 52 26 26 NUM cana-1656 137 53 ) ) PUNCT cana-1656 137 54 , , PUNCT cana-1656 137 55 oct oct PROPN cana-1656 137 56 2015 2015 NUM cana-1656 137 57 . . PUNCT cana-1656 138 1 a. a. PROPN cana-1656 138 2 shamir shamir PROPN cana-1656 138 3 , , PUNCT cana-1656 138 4 β€œ " PUNCT cana-1656 138 5 identitybased identitybase VERB cana-1656 138 6 cryptosystems cryptosystem NOUN cana-1656 138 7 and and CCONJ cana-1656 138 8 signature signature NOUN cana-1656 138 9 schemes scheme NOUN cana-1656 138 10 ” " PUNCT cana-1656 138 11 , , PUNCT cana-1656 138 12 in in ADP cana-1656 138 13 proc proc NOUN cana-1656 138 14 . . PUNCT cana-1656 139 1 crypto crypto PROPN cana-1656 139 2 1984 1984 NUM cana-1656 139 3 , , PUNCT cana-1656 139 4 lncs lncs NOUN cana-1656 139 5 vol vol NOUN cana-1656 139 6 . . PROPN cana-1656 139 7 196 196 NUM cana-1656 139 8 , , PUNCT cana-1656 139 9 pp pp ADJ cana-1656 139 10 . . PUNCT cana-1656 140 1 47 47 NUM cana-1656 140 2 - - SYM cana-1656 140 3 53 53 NUM cana-1656 140 4 , , PUNCT cana-1656 140 5 springer springer NOUN cana-1656 140 6 1984 1984 NUM cana-1656 140 7 . . PUNCT cana-1656 141 1 [ [ X cana-1656 141 2 5 5 NUM cana-1656 141 3 ] ] X cana-1656 141 4 aziz aziz PROPN cana-1656 141 5 & & CCONJ cana-1656 141 6 w. w. PROPN cana-1656 141 7 diffie diffie PROPN cana-1656 141 8 , , PUNCT cana-1656 141 9 β€œ " PUNCT cana-1656 141 10 a a DET cana-1656 141 11 secure secure ADJ cana-1656 141 12 communications communication NOUN cana-1656 141 13 protocol protocol NOUN cana-1656 141 14 to to PART cana-1656 141 15 prevent prevent VERB cana-1656 141 16 unauthorized unauthorized ADJ cana-1656 141 17 access access NOUN cana-1656 141 18 : : PUNCT cana-1656 141 19 privacy privacy NOUN cana-1656 141 20 and and CCONJ cana-1656 141 21 authentication authentication NOUN cana-1656 141 22 for for ADP cana-1656 141 23 wireless wireless ADJ cana-1656 141 24 local local ADJ cana-1656 141 25 area area NOUN cana-1656 141 26 networks network NOUN cana-1656 141 27 , , PUNCT cana-1656 141 28 ieee ieee NOUN cana-1656 141 29 personal personal ADJ cana-1656 141 30 communications communication NOUN cana-1656 141 31 , , PUNCT cana-1656 141 32 pages page NOUN cana-1656 141 33 25 25 NUM cana-1656 141 34 - - SYM cana-1656 141 35 31 31 NUM cana-1656 141 36 , , PUNCT cana-1656 141 37 first first ADJ cana-1656 141 38 quarter quarter NOUN cana-1656 141 39 1994 1994 NUM cana-1656 141 40 . . PUNCT cana-1656 142 1 [ [ X cana-1656 142 2 6 6 NUM cana-1656 142 3 ] ] X cana-1656 142 4 f. f. PROPN cana-1656 142 5 amounas amounas PROPN cana-1656 142 6 and and CCONJ cana-1656 142 7 e.h e.h PROPN cana-1656 142 8 . . PROPN cana-1656 142 9 ei ei PROPN cana-1656 142 10 kinani kinani PROPN cana-1656 142 11 , , PUNCT cana-1656 142 12 fast fast ADJ cana-1656 142 13 mapping mapping NOUN cana-1656 142 14 method method NOUN cana-1656 142 15 based base VERB cana-1656 142 16 on on ADP cana-1656 142 17 matrix matrix NOUN cana-1656 142 18 approach approach NOUN cana-1656 142 19 for for ADP cana-1656 142 20 elliptic elliptic ADJ cana-1656 142 21 curve curve NOUN cana-1656 142 22 cryptography cryptography NOUN cana-1656 142 23 , , PUNCT cana-1656 142 24 int int PROPN cana-1656 142 25 j j PROPN cana-1656 142 26 of of ADP cana-1656 142 27 information information NOUN cana-1656 142 28 and and CCONJ cana-1656 142 29 network network NOUN cana-1656 142 30 security security NOUN cana-1656 142 31 1 1 NUM cana-1656 142 32 , , PUNCT cana-1656 142 33 54–59 54–59 NUM cana-1656 142 34 ( ( PUNCT cana-1656 142 35 2012 2012 NUM cana-1656 142 36 ) ) PUNCT cana-1656 142 37 . . PUNCT cana-1656 143 1 [ [ X cana-1656 143 2 7 7 NUM cana-1656 143 3 ] ] X cana-1656 143 4 meghakohelekar meghakohelekar NOUN cana-1656 143 5 , , PUNCT cana-1656 143 6 anita anita PROPN cana-1656 143 7 jadhav jadhav PROPN cana-1656 143 8 , , PUNCT cana-1656 143 9 β€œ " PUNCT cana-1656 143 10 implementation implementation NOUN cana-1656 143 11 of of ADP cana-1656 143 12 elliptic elliptic ADJ cana-1656 143 13 curve curve NOUN cana-1656 143 14 cryptography cryptography NOUN cana-1656 143 15 on on ADP cana-1656 143 16 text text NOUN cana-1656 143 17 and and CCONJ cana-1656 143 18 image image NOUN cana-1656 143 19 ” " PUNCT cana-1656 143 20 , , PUNCT cana-1656 143 21 international international ADJ cana-1656 143 22 journal journal NOUN cana-1656 143 23 of of ADP cana-1656 143 24 enterprise enterprise NOUN cana-1656 143 25 computing computing NOUN cana-1656 143 26 and and CCONJ cana-1656 143 27 business business NOUN cana-1656 143 28 systems system NOUN cana-1656 143 29 , , PUNCT cana-1656 143 30 vol vol NOUN cana-1656 143 31 . . NOUN cana-1656 143 32 1 1 NUM cana-1656 143 33 issue issue NOUN cana-1656 143 34 2 2 NUM cana-1656 143 35 july july NOUN cana-1656 143 36 2011 2011 NUM cana-1656 144 1 [ [ X cana-1656 144 2 8 8 NUM cana-1656 144 3 ] ] PUNCT cana-1656 144 4 k. k. PROPN cana-1656 144 5 agrawal agrawal PROPN cana-1656 144 6 and and CCONJ cana-1656 144 7 a. a. NOUN cana-1656 144 8 gera gera PROPN cana-1656 144 9 , , PUNCT cana-1656 144 10 elliptic elliptic ADJ cana-1656 144 11 curve curve NOUN cana-1656 144 12 cryptography cryptography NOUN cana-1656 144 13 with with ADP cana-1656 144 14 hill hill NOUN cana-1656 144 15 cipher cipher ADJ cana-1656 144 16 generation generation NOUN cana-1656 144 17 for for ADP cana-1656 144 18 secure secure ADJ cana-1656 144 19 text text NOUN cana-1656 144 20 cryptosystem cryptosystem NOUN cana-1656 144 21 . . PUNCT cana-1656 145 1 int int PROPN cana-1656 145 2 j j PROPN cana-1656 145 3 of of ADP cana-1656 145 4 computer computer NOUN cana-1656 145 5 applications application NOUN cana-1656 145 6 106 106 NUM cana-1656 145 7 , , PUNCT cana-1656 145 8 18–24 18–24 NUM cana-1656 145 9 ( ( PUNCT cana-1656 145 10 2014 2014 NUM cana-1656 145 11 . . PUNCT cana-1656 146 1 [ [ X cana-1656 146 2 9 9 NUM cana-1656 146 3 ] ] X cana-1656 146 4 koblitz koblitz ADJ cana-1656 146 5 n. n. PROPN cana-1656 146 6 , , PUNCT cana-1656 146 7 β€œ " PUNCT cana-1656 146 8 elliptic elliptic ADJ cana-1656 146 9 curve curve NOUN cana-1656 146 10 cryptosystems cryptosystem NOUN cana-1656 146 11 , , PUNCT cana-1656 146 12 mathematics mathematic NOUN cana-1656 146 13 of of ADP cana-1656 146 14 computation computation NOUN cana-1656 146 15 ” " PUNCT cana-1656 146 16 , , PUNCT cana-1656 146 17 vol vol NOUN cana-1656 146 18 . . PROPN cana-1656 146 19 48 48 NUM cana-1656 146 20 , , PUNCT cana-1656 146 21 no.177 no.177 PROPN cana-1656 146 22 , , PUNCT cana-1656 146 23 pp pp X cana-1656 146 24 . . PUNCT cana-1656 147 1 203 203 NUM cana-1656 147 2 - - SYM cana-1656 147 3 209 209 NUM cana-1656 147 4 , , PUNCT cana-1656 147 5 january january PROPN cana-1656 147 6 1987.v 1987.v NUM cana-1656 148 1 [ [ X cana-1656 148 2 10 10 NUM cana-1656 148 3 ] ] X cana-1656 148 4 miller miller PROPN cana-1656 148 5 v. v. PROPN cana-1656 148 6 , , PUNCT cana-1656 148 7 β€œ " PUNCT cana-1656 148 8 uses use NOUN cana-1656 148 9 of of ADP cana-1656 148 10 elliptic elliptic ADJ cana-1656 148 11 curves curve NOUN cana-1656 148 12 in in ADP cana-1656 148 13 cryptography cryptography NOUN cana-1656 148 14 ” " PUNCT cana-1656 148 15 . . PUNCT cana-1656 149 1 in in ADP cana-1656 149 2 advances advance NOUN cana-1656 149 3 in in ADP cana-1656 149 4 cryptography(crypto cryptography(crypto PROPN cana-1656 149 5 1985 1985 NUM cana-1656 149 6 ) ) PUNCT cana-1656 149 7 , , PUNCT cana-1656 149 8 springer springer NOUN cana-1656 149 9 lncs lncs PROPN cana-1656 149 10 , , PUNCT cana-1656 149 11 1985 1985 NUM cana-1656 149 12 , , PUNCT cana-1656 149 13 vol vol NOUN cana-1656 149 14 . . PROPN cana-1656 149 15 218 218 NUM cana-1656 149 16 , , PUNCT cana-1656 149 17 pp pp ADV cana-1656 149 18 417 417 NUM cana-1656 149 19 - - SYM cana-1656 149 20 4 4 NUM cana-1656 149 21 26 26 NUM cana-1656 149 22 . . PUNCT cana-1656 150 1 [ [ X cana-1656 150 2 11 11 NUM cana-1656 150 3 ] ] X cana-1656 150 4 maurer maurer PROPN cana-1656 150 5 u. u. PROPN cana-1656 150 6 , , PUNCT cana-1656 150 7 a. a. NOUN cana-1656 150 8 menzes menzes PROPN cana-1656 150 9 and and CCONJ cana-1656 150 10 e. e. PROPN cana-1656 150 11 teske teske PROPN cana-1656 150 12 , , PUNCT cana-1656 150 13 β€œ " PUNCT cana-1656 150 14 analysis analysis NOUN cana-1656 150 15 of of ADP cana-1656 150 16 ghs ghs PROPN cana-1656 150 17 weil weil PROPN cana-1656 150 18 decent decent ADJ cana-1656 150 19 attack attack NOUN cana-1656 150 20 on on ADP cana-1656 150 21 theecdlp theecdlp NOUN cana-1656 150 22 over over ADP cana-1656 150 23 characteristic characteristic ADJ cana-1656 150 24 two two NUM cana-1656 150 25 fields field NOUN cana-1656 150 26 of of ADP cana-1656 150 27 composite composite ADJ cana-1656 150 28 degree degree NOUN cana-1656 150 29 ” " PUNCT cana-1656 150 30 . . PUNCT cana-1656 151 1 lms lm NOUN cana-1656 151 2 journal journal PROPN cana-1656 151 3 of of ADP cana-1656 151 4 computation computation NOUN cana-1656 151 5 and and CCONJ cana-1656 151 6 mathematics mathematic NOUN cana-1656 151 7 , , PUNCT cana-1656 151 8 5:127 5:127 NUM cana-1656 151 9 - - SYM cana-1656 151 10 174 174 NUM cana-1656 151 11 , , PUNCT cana-1656 151 12 2002 2002 NUM cana-1656 151 13 . . PUNCT cana-1656 152 1 [ [ X cana-1656 152 2 12 12 NUM cana-1656 152 3 ] ] X cana-1656 152 4 t.n t.n PROPN cana-1656 152 5 . . PROPN cana-1656 152 6 shankar shankar PROPN cana-1656 152 7 and and CCONJ cana-1656 152 8 g. g. PROPN cana-1656 152 9 sahoo sahoo PROPN cana-1656 152 10 , , PUNCT cana-1656 152 11 cryptography cryptography NOUN cana-1656 152 12 with with ADP cana-1656 152 13 elliptic elliptic ADJ cana-1656 152 14 curves curve NOUN cana-1656 152 15 , , PUNCT cana-1656 152 16 int int PROPN cana-1656 152 17 j j PROPN cana-1656 152 18 of of ADP cana-1656 152 19 computer computer NOUN cana-1656 152 20 science science NOUN cana-1656 152 21 and and CCONJ cana-1656 152 22 applications application NOUN cana-1656 152 23 2 2 NUM cana-1656 152 24 , , PUNCT cana-1656 152 25 38 38 NUM cana-1656 152 26 – – SYM cana-1656 152 27 42 42 NUM cana-1656 152 28 ( ( PUNCT cana-1656 152 29 2009 2009 NUM cana-1656 152 30 ) ) PUNCT cana-1656 152 31 . . PUNCT cana-1656 153 1 [ [ X cana-1656 153 2 13 13 NUM cana-1656 153 3 ] ] X cana-1656 153 4 arron arron PROPN cana-1656 153 5 blumenfeld blumenfeld PROPN cana-1656 153 6 , , PUNCT cana-1656 153 7 β€œ " PUNCT cana-1656 153 8 discrete discrete ADJ cana-1656 153 9 logarithms logarithm NOUN cana-1656 153 10 on on ADP cana-1656 153 11 elliptic elliptic ADJ cana-1656 153 12 curves curve NOUN cana-1656 153 13 ” " PUNCT cana-1656 153 14 , , PUNCT cana-1656 153 15 201115.menzes 201115.menzes NUM cana-1656 153 16 a. a. NOUN cana-1656 153 17 , , PUNCT cana-1656 153 18 and and CCONJ cana-1656 153 19 vanstone vanstone PROPN cana-1656 153 20 s. s. PROPN cana-1656 153 21 β€œ " PUNCT cana-1656 153 22 handbook handbook NOUN cana-1656 153 23 of of ADP cana-1656 153 24 applied applied ADJ cana-1656 153 25 cryptography cryptography NOUN cana-1656 153 26 ” " PUNCT cana-1656 153 27 , , PUNCT cana-1656 153 28 the the DET cana-1656 153 29 crc crc NOUN cana-1656 153 30 - - PUNCT cana-1656 153 31 pressseries pressserie NOUN cana-1656 153 32 of of ADP cana-1656 153 33 discrete discrete ADJ cana-1656 153 34 mathematics mathematic NOUN cana-1656 153 35 and and CCONJ cana-1656 153 36 its its PRON cana-1656 153 37 applications application NOUN cana-1656 153 38 crc crc NOUN cana-1656 153 39 - - PUNCT cana-1656 153 40 press,1997 press,1997 NOUN cana-1656 153 41 . . PUNCT cana-1656 154 1 [ [ X cana-1656 154 2 14 14 NUM cana-1656 154 3 ] ] X cana-1656 154 4 miyaji miyaji NOUN cana-1656 154 5 , , PUNCT cana-1656 154 6 nakabayashi nakabayashi PROPN cana-1656 154 7 and and CCONJ cana-1656 154 8 takano takano PROPN cana-1656 154 9 β€œ " PUNCT cana-1656 154 10 elliptic elliptic ADJ cana-1656 154 11 curves curve NOUN cana-1656 154 12 with with ADP cana-1656 154 13 low low ADJ cana-1656 154 14 embedding embedding NOUN cana-1656 154 15 degree degree NOUN cana-1656 154 16 ” " PUNCT cana-1656 154 17 , , PUNCT cana-1656 154 18 journal journal NOUN cana-1656 154 19 of of ADP cana-1656 154 20 cryptology cryptology NOUN cana-1656 154 21 , , PUNCT cana-1656 154 22 2006 2006 NUM cana-1656 154 23 , , PUNCT cana-1656 154 24 volume volume NOUN cana-1656 154 25 19 19 NUM cana-1656 154 26 , , PUNCT cana-1656 154 27 number number NOUN cana-1656 154 28 4 4 NUM cana-1656 154 29 , , PUNCT cana-1656 154 30 pages page NOUN cana-1656 154 31 553 553 NUM cana-1656 154 32 - - SYM cana-1656 154 33 562 562 NUM cana-1656 154 34 . . PUNCT