id	sid	tid	token	lemma	pos
cana-20	1	1	communications	communication	NOUN
cana-20	1	2	on	on	ADP
cana-20	1	3	applied	apply	VERB
cana-20	1	4	nonlinear	nonlinear	ADJ
cana-20	1	5	analysis	analysis	NOUN
cana-20	1	6	issn	issn	NOUN
cana-20	1	7	:	:	PUNCT
cana-20	1	8	1074	1074	NUM
cana-20	1	9	-	-	PUNCT
cana-20	1	10	133x	133x	NUM
cana-20	1	11	vol	vol	NOUN
cana-20	1	12	29	29	NUM
cana-20	1	13	no	no	NOUN
cana-20	1	14	.	.	NOUN
cana-20	1	15	2	2	NUM
cana-20	1	16	(	(	PUNCT
cana-20	1	17	2022	2022	NUM
cana-20	1	18	)	)	PUNCT
cana-20	1	19	10	10	NUM
cana-20	2	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-20	2	2	recent	recent	ADJ
cana-20	2	3	developments	development	NOUN
cana-20	2	4	in	in	ADP
cana-20	2	5	number	number	NOUN
cana-20	2	6	theory	theory	NOUN
cana-20	2	7	:	:	PUNCT
cana-20	2	8	from	from	ADP
cana-20	2	9	diophantine	diophantine	NOUN
cana-20	2	10	equations	equation	NOUN
cana-20	2	11	to	to	ADP
cana-20	2	12	cryptography	cryptography	PROPN
cana-20	2	13	douglas	douglas	PROPN
cana-20	2	14	r.	r.	PROPN
cana-20	2	15	anderson	anderson	PROPN
cana-20	2	16	division	division	PROPN
cana-20	2	17	of	of	ADP
cana-20	2	18	mathematics	mathematic	NOUN
cana-20	2	19	and	and	CCONJ
cana-20	2	20	computer	computer	NOUN
cana-20	2	21	science	science	NOUN
cana-20	2	22	,	,	PUNCT
cana-20	2	23	national	national	PROPN
cana-20	2	24	university	university	PROPN
cana-20	2	25	of	of	ADP
cana-20	2	26	singapore	singapore	PROPN
cana-20	2	27	,	,	PUNCT
cana-20	2	28	singapore	singapore	PROPN
cana-20	2	29	article	article	PROPN
cana-20	2	30	history	history	NOUN
cana-20	2	31	:	:	PUNCT
cana-20	2	32	received	receive	VERB
cana-20	2	33	:	:	PUNCT
cana-20	2	34	16	16	NUM
cana-20	2	35	-	-	SYM
cana-20	2	36	03	03	NUM
cana-20	2	37	-	-	PUNCT
cana-20	2	38	2022	2022	NUM
cana-20	2	39	revised	revise	VERB
cana-20	2	40	:	:	PUNCT
cana-20	2	41	30	30	NUM
cana-20	2	42	-	-	SYM
cana-20	2	43	05	05	NUM
cana-20	2	44	-	-	PUNCT
cana-20	2	45	2022	2022	NUM
cana-20	2	46	accepted	accept	VERB
cana-20	2	47	:	:	PUNCT
cana-20	2	48	28	28	NUM
cana-20	2	49	-	-	SYM
cana-20	2	50	06	06	NUM
cana-20	2	51	-	-	PUNCT
cana-20	2	52	2022	2022	NUM
cana-20	2	53	abstract	abstract	NOUN
cana-20	2	54	:	:	PUNCT
cana-20	2	55	number	number	NOUN
cana-20	2	56	theory	theory	NOUN
cana-20	2	57	,	,	PUNCT
cana-20	2	58	a	a	DET
cana-20	2	59	branch	branch	NOUN
cana-20	2	60	of	of	ADP
cana-20	2	61	mathematics	mathematic	NOUN
cana-20	2	62	with	with	ADP
cana-20	2	63	a	a	DET
cana-20	2	64	rich	rich	ADJ
cana-20	2	65	history	history	NOUN
cana-20	2	66	,	,	PUNCT
cana-20	2	67	continues	continue	VERB
cana-20	2	68	to	to	PART
cana-20	2	69	evolve	evolve	VERB
cana-20	2	70	with	with	ADP
cana-20	2	71	recent	recent	ADJ
cana-20	2	72	developments	development	NOUN
cana-20	2	73	.	.	PUNCT
cana-20	3	1	this	this	DET
cana-20	3	2	article	article	NOUN
cana-20	3	3	delves	delve	VERB
cana-20	3	4	into	into	ADP
cana-20	3	5	the	the	DET
cana-20	3	6	exciting	exciting	ADJ
cana-20	3	7	advancements	advancement	NOUN
cana-20	3	8	in	in	ADP
cana-20	3	9	number	number	NOUN
cana-20	3	10	theory	theory	NOUN
cana-20	3	11	,	,	PUNCT
cana-20	3	12	focusing	focus	VERB
cana-20	3	13	on	on	ADP
cana-20	3	14	topics	topic	NOUN
cana-20	3	15	such	such	ADJ
cana-20	3	16	as	as	ADP
cana-20	3	17	diophantine	diophantine	NOUN
cana-20	3	18	equations	equation	NOUN
cana-20	3	19	,	,	PUNCT
cana-20	3	20	prime	prime	ADJ
cana-20	3	21	numbers	number	NOUN
cana-20	3	22	,	,	PUNCT
cana-20	3	23	and	and	CCONJ
cana-20	3	24	their	their	PRON
cana-20	3	25	applications	application	NOUN
cana-20	3	26	in	in	ADP
cana-20	3	27	modern	modern	ADJ
cana-20	3	28	cryptography	cryptography	NOUN
cana-20	3	29	.	.	PUNCT
cana-20	4	1	keywords	keyword	NOUN
cana-20	4	2	:	:	PUNCT
cana-20	4	3	number	number	NOUN
cana-20	4	4	theory	theory	NOUN
cana-20	4	5	,	,	PUNCT
cana-20	4	6	cryptography	cryptography	NOUN
cana-20	4	7	.	.	PUNCT
cana-20	5	1	1	1	X
cana-20	5	2	.	.	X
cana-20	5	3	introduction	introduction	NOUN
cana-20	5	4	number	number	NOUN
cana-20	5	5	theory	theory	NOUN
cana-20	5	6	,	,	PUNCT
cana-20	5	7	one	one	NUM
cana-20	5	8	of	of	ADP
cana-20	5	9	the	the	DET
cana-20	5	10	oldest	old	ADJ
cana-20	5	11	branches	branch	NOUN
cana-20	5	12	of	of	ADP
cana-20	5	13	mathematics	mathematic	NOUN
cana-20	5	14	,	,	PUNCT
cana-20	5	15	has	have	AUX
cana-20	5	16	seen	see	VERB
cana-20	5	17	remarkable	remarkable	ADJ
cana-20	5	18	advancements	advancement	NOUN
cana-20	5	19	in	in	ADP
cana-20	5	20	recent	recent	ADJ
cana-20	5	21	years	year	NOUN
cana-20	5	22	.	.	PUNCT
cana-20	6	1	this	this	DET
cana-20	6	2	article	article	NOUN
cana-20	6	3	explores	explore	VERB
cana-20	6	4	some	some	PRON
cana-20	6	5	of	of	ADP
cana-20	6	6	the	the	DET
cana-20	6	7	most	most	ADV
cana-20	6	8	significant	significant	ADJ
cana-20	6	9	developments	development	NOUN
cana-20	6	10	in	in	ADP
cana-20	6	11	number	number	NOUN
cana-20	6	12	theory	theory	NOUN
cana-20	6	13	and	and	CCONJ
cana-20	6	14	their	their	PRON
cana-20	6	15	applications	application	NOUN
cana-20	6	16	,	,	PUNCT
cana-20	6	17	particularly	particularly	ADV
cana-20	6	18	in	in	ADP
cana-20	6	19	the	the	DET
cana-20	6	20	field	field	NOUN
cana-20	6	21	of	of	ADP
cana-20	6	22	cryptography	cryptography	NOUN
cana-20	6	23	.	.	PUNCT
cana-20	7	1	2	2	X
cana-20	7	2	.	.	X
cana-20	7	3	diophantine	diophantine	PROPN
cana-20	7	4	equations	equation	NOUN
cana-20	7	5	2.1	2.1	NUM
cana-20	7	6	fermat	fermat	PROPN
cana-20	7	7	's	's	PART
cana-20	7	8	last	last	ADJ
cana-20	7	9	theorem	theorem	ADJ
cana-20	7	10	fermat	fermat	PROPN
cana-20	7	11	's	's	PART
cana-20	7	12	last	last	ADJ
cana-20	7	13	theorem	theorem	NOUN
cana-20	7	14	,	,	PUNCT
cana-20	7	15	a	a	DET
cana-20	7	16	problem	problem	NOUN
cana-20	7	17	that	that	PRON
cana-20	7	18	baffled	baffle	VERB
cana-20	7	19	mathematicians	mathematician	NOUN
cana-20	7	20	for	for	ADP
cana-20	7	21	centuries	century	NOUN
cana-20	7	22	,	,	PUNCT
cana-20	7	23	was	be	AUX
cana-20	7	24	finally	finally	ADV
cana-20	7	25	proven	prove	VERB
cana-20	7	26	by	by	ADP
cana-20	7	27	andrew	andrew	PROPN
cana-20	7	28	wiles	wile	NOUN
cana-20	7	29	in	in	ADP
cana-20	7	30	1994	1994	NUM
cana-20	7	31	.	.	PUNCT
cana-20	8	1	this	this	DET
cana-20	8	2	groundbreaking	groundbreake	VERB
cana-20	8	3	achievement	achievement	NOUN
cana-20	8	4	demonstrated	demonstrate	VERB
cana-20	8	5	the	the	DET
cana-20	8	6	power	power	NOUN
cana-20	8	7	of	of	ADP
cana-20	8	8	modern	modern	ADJ
cana-20	8	9	mathematical	mathematical	ADJ
cana-20	8	10	techniques	technique	NOUN
cana-20	8	11	,	,	PUNCT
cana-20	8	12	including	include	VERB
cana-20	8	13	elliptic	elliptic	ADJ
cana-20	8	14	curves	curve	NOUN
cana-20	8	15	and	and	CCONJ
cana-20	8	16	modular	modular	ADJ
cana-20	8	17	forms	form	NOUN
cana-20	8	18	.	.	PUNCT
cana-20	9	1	2.2	2.2	NUM
cana-20	9	2	elliptic	elliptic	ADJ
cana-20	9	3	curve	curve	NOUN
cana-20	9	4	cryptography	cryptography	NOUN
cana-20	9	5	(	(	PUNCT
cana-20	9	6	ecc	ecc	PROPN
cana-20	9	7	)	)	PUNCT
cana-20	9	8	advancements	advancement	NOUN
cana-20	9	9	in	in	ADP
cana-20	9	10	elliptic	elliptic	ADJ
cana-20	9	11	curve	curve	NOUN
cana-20	9	12	theory	theory	NOUN
cana-20	9	13	have	have	AUX
cana-20	9	14	not	not	PART
cana-20	9	15	only	only	ADV
cana-20	9	16	contributed	contribute	VERB
cana-20	9	17	to	to	ADP
cana-20	9	18	solving	solve	VERB
cana-20	9	19	diophantine	diophantine	NOUN
cana-20	9	20	equations	equation	NOUN
cana-20	9	21	but	but	CCONJ
cana-20	9	22	also	also	ADV
cana-20	9	23	led	lead	VERB
cana-20	9	24	to	to	ADP
cana-20	9	25	the	the	DET
cana-20	9	26	development	development	NOUN
cana-20	9	27	of	of	ADP
cana-20	9	28	elliptic	elliptic	ADJ
cana-20	9	29	curve	curve	NOUN
cana-20	9	30	cryptography	cryptography	NOUN
cana-20	9	31	(	(	PUNCT
cana-20	9	32	ecc	ecc	PROPN
cana-20	9	33	)	)	PUNCT
cana-20	9	34	.	.	PUNCT
cana-20	10	1	ecc	ecc	PROPN
cana-20	10	2	is	be	AUX
cana-20	10	3	now	now	ADV
cana-20	10	4	a	a	DET
cana-20	10	5	cornerstone	cornerstone	NOUN
cana-20	10	6	of	of	ADP
cana-20	10	7	modern	modern	ADJ
cana-20	10	8	cryptography	cryptography	NOUN
cana-20	10	9	due	due	ADP
cana-20	10	10	to	to	ADP
cana-20	10	11	its	its	PRON
cana-20	10	12	efficiency	efficiency	NOUN
cana-20	10	13	and	and	CCONJ
cana-20	10	14	security	security	NOUN
cana-20	10	15	.	.	PUNCT
cana-20	11	1	3	3	X
cana-20	11	2	.	.	X
cana-20	11	3	prime	prime	ADJ
cana-20	11	4	numbers	number	NOUN
cana-20	11	5	3.1	3.1	NUM
cana-20	11	6	twin	twin	ADJ
cana-20	11	7	prime	prime	ADJ
cana-20	11	8	conjecture	conjecture	NOUN
cana-20	11	9	recent	recent	ADJ
cana-20	11	10	progress	progress	NOUN
cana-20	11	11	has	have	AUX
cana-20	11	12	been	be	AUX
cana-20	11	13	made	make	VERB
cana-20	11	14	in	in	ADP
cana-20	11	15	understanding	understand	VERB
cana-20	11	16	twin	twin	ADJ
cana-20	11	17	primes	prime	NOUN
cana-20	11	18	,	,	PUNCT
cana-20	11	19	which	which	PRON
cana-20	11	20	are	be	AUX
cana-20	11	21	pairs	pair	NOUN
cana-20	11	22	of	of	ADP
cana-20	11	23	primes	prime	NOUN
cana-20	11	24	with	with	ADP
cana-20	11	25	a	a	DET
cana-20	11	26	difference	difference	NOUN
cana-20	11	27	of	of	ADP
cana-20	11	28	two	two	NUM
cana-20	11	29	.	.	PUNCT
cana-20	12	1	yitang	yitang	PROPN
cana-20	12	2	zhang	zhang	PROPN
cana-20	12	3	's	's	PART
cana-20	12	4	work	work	NOUN
cana-20	12	5	in	in	ADP
cana-20	12	6	2013	2013	NUM
cana-20	12	7	brought	bring	VERB
cana-20	12	8	us	we	PRON
cana-20	12	9	closer	close	ADV
cana-20	12	10	to	to	ADP
cana-20	12	11	proving	prove	VERB
cana-20	12	12	the	the	DET
cana-20	12	13	twin	twin	ADJ
cana-20	12	14	prime	prime	ADJ
cana-20	12	15	conjecture	conjecture	NOUN
cana-20	12	16	,	,	PUNCT
cana-20	12	17	highlighting	highlight	VERB
cana-20	12	18	the	the	DET
cana-20	12	19	ongoing	ongoing	ADJ
cana-20	12	20	importance	importance	NOUN
cana-20	12	21	of	of	ADP
cana-20	12	22	prime	prime	ADJ
cana-20	12	23	number	number	NOUN
cana-20	12	24	research	research	NOUN
cana-20	12	25	.	.	PUNCT
cana-20	13	1	communications	communication	NOUN
cana-20	13	2	on	on	ADP
cana-20	13	3	applied	apply	VERB
cana-20	13	4	nonlinear	nonlinear	ADJ
cana-20	13	5	analysis	analysis	NOUN
cana-20	13	6	issn	issn	NOUN
cana-20	13	7	:	:	PUNCT
cana-20	13	8	1074	1074	NUM
cana-20	13	9	-	-	PUNCT
cana-20	13	10	133x	133x	NUM
cana-20	13	11	vol	vol	NOUN
cana-20	13	12	29	29	NUM
cana-20	13	13	no	no	NOUN
cana-20	13	14	.	.	NOUN
cana-20	13	15	2	2	NUM
cana-20	13	16	(	(	PUNCT
cana-20	13	17	2022	2022	NUM
cana-20	13	18	)	)	PUNCT
cana-20	13	19	11	11	NUM
cana-20	13	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-20	13	21	3.2	3.2	NUM
cana-20	13	22	prime	prime	NOUN
cana-20	13	23	gap	gap	NOUN
cana-20	13	24	conjecture	conjecture	VERB
cana-20	13	25	the	the	DET
cana-20	13	26	prime	prime	ADJ
cana-20	13	27	gap	gap	NOUN
cana-20	13	28	conjecture	conjecture	NOUN
cana-20	13	29	,	,	PUNCT
cana-20	13	30	which	which	PRON
cana-20	13	31	deals	deal	VERB
cana-20	13	32	with	with	ADP
cana-20	13	33	the	the	DET
cana-20	13	34	distribution	distribution	NOUN
cana-20	13	35	of	of	ADP
cana-20	13	36	prime	prime	ADJ
cana-20	13	37	numbers	number	NOUN
cana-20	13	38	,	,	PUNCT
cana-20	13	39	remains	remain	VERB
cana-20	13	40	an	an	DET
cana-20	13	41	active	active	ADJ
cana-20	13	42	area	area	NOUN
cana-20	13	43	of	of	ADP
cana-20	13	44	research	research	NOUN
cana-20	13	45	,	,	PUNCT
cana-20	13	46	with	with	ADP
cana-20	13	47	mathematicians	mathematician	NOUN
cana-20	13	48	exploring	explore	VERB
cana-20	13	49	the	the	DET
cana-20	13	50	existence	existence	NOUN
cana-20	13	51	of	of	ADP
cana-20	13	52	large	large	ADJ
cana-20	13	53	prime	prime	ADJ
cana-20	13	54	gaps	gap	NOUN
cana-20	13	55	.	.	PUNCT
cana-20	14	1	4	4	X
cana-20	14	2	.	.	X
cana-20	14	3	cryptography	cryptography	NOUN
cana-20	14	4	4.1	4.1	NUM
cana-20	14	5	rsa	rsa	PROPN
cana-20	14	6	encryption	encryption	NOUN
cana-20	14	7	number	number	NOUN
cana-20	14	8	theory	theory	NOUN
cana-20	14	9	plays	play	VERB
cana-20	14	10	a	a	DET
cana-20	14	11	fundamental	fundamental	ADJ
cana-20	14	12	role	role	NOUN
cana-20	14	13	in	in	ADP
cana-20	14	14	rsa	rsa	PROPN
cana-20	14	15	encryption	encryption	NOUN
cana-20	14	16	,	,	PUNCT
cana-20	14	17	a	a	DET
cana-20	14	18	widely	widely	ADV
cana-20	14	19	used	use	VERB
cana-20	14	20	cryptographic	cryptographic	ADJ
cana-20	14	21	method	method	NOUN
cana-20	14	22	that	that	PRON
cana-20	14	23	relies	rely	VERB
cana-20	14	24	on	on	ADP
cana-20	14	25	the	the	DET
cana-20	14	26	difficulty	difficulty	NOUN
cana-20	14	27	of	of	ADP
cana-20	14	28	factoring	factor	VERB
cana-20	14	29	large	large	ADJ
cana-20	14	30	composite	composite	ADJ
cana-20	14	31	numbers	number	NOUN
cana-20	14	32	.	.	PUNCT
cana-20	15	1	recent	recent	ADJ
cana-20	15	2	developments	development	NOUN
cana-20	15	3	in	in	ADP
cana-20	15	4	factoring	factoring	NOUN
cana-20	15	5	algorithms	algorithm	NOUN
cana-20	15	6	have	have	AUX
cana-20	15	7	raised	raise	VERB
cana-20	15	8	concerns	concern	NOUN
cana-20	15	9	about	about	ADP
cana-20	15	10	the	the	DET
cana-20	15	11	security	security	NOUN
cana-20	15	12	of	of	ADP
cana-20	15	13	rsa	rsa	NOUN
cana-20	15	14	,	,	PUNCT
cana-20	15	15	leading	lead	VERB
cana-20	15	16	to	to	ADP
cana-20	15	17	the	the	DET
cana-20	15	18	exploration	exploration	NOUN
cana-20	15	19	of	of	ADP
cana-20	15	20	alternative	alternative	ADJ
cana-20	15	21	cryptographic	cryptographic	ADJ
cana-20	15	22	schemes	scheme	NOUN
cana-20	15	23	.	.	PUNCT
cana-20	16	1	4.2	4.2	NUM
cana-20	16	2	post	post	ADJ
cana-20	16	3	-	-	ADJ
cana-20	16	4	quantum	quantum	ADJ
cana-20	16	5	cryptography	cryptography	NOUN
cana-20	16	6	the	the	DET
cana-20	16	7	advent	advent	NOUN
cana-20	16	8	of	of	ADP
cana-20	16	9	quantum	quantum	ADJ
cana-20	16	10	computers	computer	NOUN
cana-20	16	11	poses	pose	VERB
cana-20	16	12	a	a	DET
cana-20	16	13	threat	threat	NOUN
cana-20	16	14	to	to	ADP
cana-20	16	15	traditional	traditional	ADJ
cana-20	16	16	cryptographic	cryptographic	ADJ
cana-20	16	17	systems	system	NOUN
cana-20	16	18	like	like	ADP
cana-20	16	19	rsa	rsa	PROPN
cana-20	16	20	.	.	PUNCT
cana-20	17	1	researchers	researcher	NOUN
cana-20	17	2	are	be	AUX
cana-20	17	3	actively	actively	ADV
cana-20	17	4	developing	develop	VERB
cana-20	17	5	post	post	ADJ
cana-20	17	6	-	-	ADJ
cana-20	17	7	quantum	quantum	ADJ
cana-20	17	8	cryptography	cryptography	NOUN
cana-20	17	9	methods	method	NOUN
cana-20	17	10	based	base	VERB
cana-20	17	11	on	on	ADP
cana-20	17	12	number	number	NOUN
cana-20	17	13	theory	theory	NOUN
cana-20	17	14	,	,	PUNCT
cana-20	17	15	such	such	ADJ
cana-20	17	16	as	as	ADP
cana-20	17	17	lattice	lattice	NOUN
cana-20	17	18	-	-	PUNCT
cana-20	17	19	based	base	VERB
cana-20	17	20	cryptography	cryptography	NOUN
cana-20	17	21	and	and	CCONJ
cana-20	17	22	code	code	NOUN
cana-20	17	23	-	-	PUNCT
cana-20	17	24	based	base	VERB
cana-20	17	25	cryptography	cryptography	NOUN
cana-20	17	26	.	.	PUNCT
cana-20	18	1	5	5	X
cana-20	18	2	.	.	X
cana-20	18	3	applications	application	NOUN
cana-20	18	4	beyond	beyond	ADP
cana-20	18	5	cryptography	cryptography	NOUN
cana-20	18	6	number	number	NOUN
cana-20	18	7	theory	theory	NOUN
cana-20	18	8	finds	find	VERB
cana-20	18	9	applications	application	NOUN
cana-20	18	10	beyond	beyond	ADP
cana-20	18	11	cryptography	cryptography	NOUN
cana-20	18	12	.	.	PUNCT
cana-20	19	1	it	it	PRON
cana-20	19	2	is	be	AUX
cana-20	19	3	used	use	VERB
cana-20	19	4	in	in	ADP
cana-20	19	5	coding	code	VERB
cana-20	19	6	theory	theory	NOUN
cana-20	19	7	,	,	PUNCT
cana-20	19	8	error	error	NOUN
cana-20	19	9	detection	detection	NOUN
cana-20	19	10	and	and	CCONJ
cana-20	19	11	correction	correction	NOUN
cana-20	19	12	,	,	PUNCT
cana-20	19	13	and	and	CCONJ
cana-20	19	14	even	even	ADV
cana-20	19	15	quantum	quantum	ADJ
cana-20	19	16	computing	computing	NOUN
cana-20	19	17	,	,	PUNCT
cana-20	19	18	where	where	SCONJ
cana-20	19	19	integer	integer	NOUN
cana-20	19	20	factorization	factorization	NOUN
cana-20	19	21	problems	problem	NOUN
cana-20	19	22	are	be	AUX
cana-20	19	23	pivotal	pivotal	ADJ
cana-20	19	24	.	.	PUNCT
cana-20	20	1	6	6	NUM
cana-20	20	2	.	.	NOUN
cana-20	20	3	significance	significance	NOUN
cana-20	20	4	and	and	CCONJ
cana-20	20	5	future	future	ADJ
cana-20	20	6	directions	direction	NOUN
cana-20	20	7	recent	recent	ADJ
cana-20	20	8	developments	development	NOUN
cana-20	20	9	in	in	ADP
cana-20	20	10	number	number	NOUN
cana-20	20	11	theory	theory	NOUN
cana-20	20	12	underline	underline	VERB
cana-20	20	13	its	its	PRON
cana-20	20	14	enduring	endure	VERB
cana-20	20	15	importance	importance	NOUN
cana-20	20	16	in	in	ADP
cana-20	20	17	mathematics	mathematic	NOUN
cana-20	20	18	and	and	CCONJ
cana-20	20	19	its	its	PRON
cana-20	20	20	practical	practical	ADJ
cana-20	20	21	applications	application	NOUN
cana-20	20	22	in	in	ADP
cana-20	20	23	cryptography	cryptography	NOUN
cana-20	20	24	and	and	CCONJ
cana-20	20	25	beyond	beyond	ADP
cana-20	20	26	.	.	PUNCT
cana-20	21	1	future	future	ADJ
cana-20	21	2	directions	direction	NOUN
cana-20	21	3	include	include	VERB
cana-20	21	4	ongoing	ongoing	ADJ
cana-20	21	5	research	research	NOUN
cana-20	21	6	in	in	ADP
cana-20	21	7	prime	prime	ADJ
cana-20	21	8	numbers	number	NOUN
cana-20	21	9	,	,	PUNCT
cana-20	21	10	exploring	explore	VERB
cana-20	21	11	new	new	ADJ
cana-20	21	12	cryptographic	cryptographic	ADJ
cana-20	21	13	methods	method	NOUN
cana-20	21	14	,	,	PUNCT
cana-20	21	15	and	and	CCONJ
cana-20	21	16	preparing	prepare	VERB
cana-20	21	17	for	for	ADP
cana-20	21	18	the	the	DET
cana-20	21	19	challenges	challenge	NOUN
cana-20	21	20	posed	pose	VERB
cana-20	21	21	by	by	ADP
cana-20	21	22	quantum	quantum	NOUN
cana-20	21	23	computing	computing	NOUN
cana-20	21	24	.	.	PUNCT
cana-20	22	1	7	7	X
cana-20	22	2	.	.	X
cana-20	22	3	conclusion	conclusion	NOUN
cana-20	22	4	number	number	NOUN
cana-20	22	5	theory	theory	NOUN
cana-20	22	6	,	,	PUNCT
cana-20	22	7	a	a	DET
cana-20	22	8	field	field	NOUN
cana-20	22	9	with	with	ADP
cana-20	22	10	a	a	DET
cana-20	22	11	rich	rich	ADJ
cana-20	22	12	history	history	NOUN
cana-20	22	13	,	,	PUNCT
cana-20	22	14	continues	continue	VERB
cana-20	22	15	to	to	PART
cana-20	22	16	evolve	evolve	VERB
cana-20	22	17	and	and	CCONJ
cana-20	22	18	make	make	VERB
cana-20	22	19	significant	significant	ADJ
cana-20	22	20	contributions	contribution	NOUN
cana-20	22	21	to	to	ADP
cana-20	22	22	mathematics	mathematic	NOUN
cana-20	22	23	and	and	CCONJ
cana-20	22	24	cryptography	cryptography	NOUN
cana-20	22	25	.	.	PUNCT
cana-20	23	1	recent	recent	ADJ
cana-20	23	2	developments	development	NOUN
cana-20	23	3	in	in	ADP
cana-20	23	4	solving	solve	VERB
cana-20	23	5	diophantine	diophantine	NOUN
cana-20	23	6	equations	equation	NOUN
cana-20	23	7	,	,	PUNCT
cana-20	23	8	understanding	understand	VERB
cana-20	23	9	prime	prime	ADJ
cana-20	23	10	numbers	number	NOUN
cana-20	23	11	,	,	PUNCT
cana-20	23	12	and	and	CCONJ
cana-20	23	13	advancing	advance	VERB
cana-20	23	14	cryptography	cryptography	NOUN
cana-20	23	15	underscore	underscore	NOUN
cana-20	23	16	the	the	DET
cana-20	23	17	vitality	vitality	NOUN
cana-20	23	18	of	of	ADP
cana-20	23	19	number	number	NOUN
cana-20	23	20	theory	theory	NOUN
cana-20	23	21	in	in	ADP
cana-20	23	22	the	the	DET
cana-20	23	23	modern	modern	ADJ
cana-20	23	24	age	age	NOUN
cana-20	23	25	.	.	PUNCT
cana-20	24	1	references	reference	NOUN
cana-20	24	2	:	:	PUNCT
cana-20	25	1	[	[	X
cana-20	25	2	1	1	NUM
cana-20	25	3	]	]	PUNCT
cana-20	25	4	ribenboim	ribenboim	NOUN
cana-20	25	5	,	,	PUNCT
cana-20	25	6	p.	p.	NOUN
cana-20	25	7	(	(	PUNCT
cana-20	25	8	2012	2012	NUM
cana-20	25	9	)	)	PUNCT
cana-20	25	10	.	.	PUNCT
cana-20	26	1	fermat	fermat	PROPN
cana-20	26	2	's	's	PART
cana-20	26	3	last	last	ADJ
cana-20	26	4	theorem	theorem	NOUN
cana-20	26	5	for	for	ADP
cana-20	26	6	amateurs	amateur	NOUN
cana-20	26	7	.	.	PUNCT
cana-20	27	1	springer	springer	NOUN
cana-20	27	2	.	.	PUNCT
cana-20	28	1	[	[	X
cana-20	28	2	2	2	NUM
cana-20	28	3	]	]	PUNCT
cana-20	28	4	silverman	silverman	NOUN
cana-20	28	5	,	,	PUNCT
cana-20	28	6	j.	j.	PROPN
cana-20	28	7	h.	h.	PROPN
cana-20	28	8	(	(	PUNCT
cana-20	28	9	2009	2009	NUM
cana-20	28	10	)	)	PUNCT
cana-20	28	11	.	.	PUNCT
cana-20	29	1	the	the	DET
cana-20	29	2	arithmetic	arithmetic	NOUN
cana-20	29	3	of	of	ADP
cana-20	29	4	elliptic	elliptic	ADJ
cana-20	29	5	curves	curve	NOUN
cana-20	29	6	.	.	PUNCT
cana-20	30	1	springer	springer	NOUN
cana-20	30	2	.	.	PUNCT
cana-20	31	1	[	[	X
cana-20	31	2	3	3	NUM
cana-20	31	3	]	]	X
cana-20	31	4	zhang	zhang	PROPN
cana-20	31	5	,	,	PUNCT
cana-20	31	6	y.	y.	PROPN
cana-20	31	7	(	(	PUNCT
cana-20	31	8	2013	2013	NUM
cana-20	31	9	)	)	PUNCT
cana-20	31	10	.	.	PUNCT
cana-20	32	1	bounded	bound	VERB
cana-20	32	2	gaps	gap	NOUN
cana-20	32	3	between	between	ADP
cana-20	32	4	primes	prime	NOUN
cana-20	32	5	.	.	PUNCT
cana-20	33	1	annals	annal	NOUN
cana-20	33	2	of	of	ADP
cana-20	33	3	mathematics	mathematic	NOUN
cana-20	33	4	,	,	PUNCT
cana-20	33	5	179(3	179(3	NUM
cana-20	33	6	)	)	PUNCT
cana-20	33	7	,	,	PUNCT
cana-20	33	8	1121	1121	NUM
cana-20	33	9	-	-	SYM
cana-20	33	10	1174	1174	NUM
cana-20	33	11	.	.	PUNCT
cana-20	34	1	[	[	X
cana-20	34	2	4	4	NUM
cana-20	34	3	]	]	X
cana-20	34	4	hardy	hardy	ADJ
cana-20	34	5	,	,	PUNCT
cana-20	34	6	g.	g.	PROPN
cana-20	34	7	h.	h.	PROPN
cana-20	34	8	,	,	PUNCT
cana-20	34	9	&	&	CCONJ
cana-20	34	10	wright	wright	PROPN
cana-20	34	11	,	,	PUNCT
cana-20	34	12	e.	e.	PROPN
cana-20	34	13	m.	m.	PROPN
cana-20	34	14	(	(	PUNCT
cana-20	34	15	2008	2008	NUM
cana-20	34	16	)	)	PUNCT
cana-20	34	17	.	.	PUNCT
cana-20	35	1	an	an	DET
cana-20	35	2	introduction	introduction	NOUN
cana-20	35	3	to	to	ADP
cana-20	35	4	the	the	DET
cana-20	35	5	theory	theory	NOUN
cana-20	35	6	of	of	ADP
cana-20	35	7	numbers	number	NOUN
cana-20	35	8	(	(	PUNCT
cana-20	35	9	6th	6th	ADJ
cana-20	35	10	ed	ed	NOUN
cana-20	35	11	.	.	PUNCT
cana-20	35	12	)	)	PUNCT
cana-20	35	13	.	.	PUNCT
cana-20	36	1	oxford	oxford	PROPN
cana-20	36	2	university	university	PROPN
cana-20	36	3	press	press	NOUN
cana-20	36	4	.	.	PUNCT
cana-20	37	1	[	[	X
cana-20	37	2	5	5	NUM
cana-20	37	3	]	]	PUNCT
cana-20	37	4	mollin	mollin	NOUN
cana-20	37	5	,	,	PUNCT
cana-20	37	6	r.	r.	PROPN
cana-20	37	7	a.	a.	PROPN
cana-20	37	8	(	(	PUNCT
cana-20	37	9	2006	2006	NUM
cana-20	37	10	)	)	PUNCT
cana-20	37	11	.	.	PUNCT
cana-20	38	1	an	an	DET
cana-20	38	2	introduction	introduction	NOUN
cana-20	38	3	to	to	ADP
cana-20	38	4	cryptography	cryptography	NOUN
cana-20	38	5	(	(	PUNCT
cana-20	38	6	2nd	2nd	ADJ
cana-20	38	7	ed	ed	NOUN
cana-20	38	8	.	.	PUNCT
cana-20	38	9	)	)	PUNCT
cana-20	38	10	.	.	PUNCT
cana-20	39	1	chapman	chapman	NOUN
cana-20	39	2	and	and	CCONJ
cana-20	39	3	hall	hall	PROPN
cana-20	39	4	/	/	SYM
cana-20	39	5	crc	crc	NOUN
cana-20	39	6	.	.	PUNCT
cana-20	40	1	[	[	X
cana-20	40	2	6	6	NUM
cana-20	40	3	]	]	X
cana-20	40	4	washington	washington	PROPN
cana-20	40	5	,	,	PUNCT
cana-20	40	6	l.	l.	PROPN
cana-20	40	7	c.	c.	PROPN
cana-20	40	8	(	(	PUNCT
cana-20	40	9	2008	2008	NUM
cana-20	40	10	)	)	PUNCT
cana-20	40	11	.	.	PUNCT
cana-20	41	1	elliptic	elliptic	ADJ
cana-20	41	2	curves	curve	NOUN
cana-20	41	3	:	:	PUNCT
cana-20	41	4	number	number	NOUN
cana-20	41	5	theory	theory	NOUN
cana-20	41	6	and	and	CCONJ
cana-20	41	7	cryptography	cryptography	NOUN
cana-20	41	8	(	(	PUNCT
cana-20	41	9	2nd	2nd	ADJ
cana-20	41	10	ed	ed	NOUN
cana-20	41	11	.	.	PUNCT
cana-20	41	12	)	)	PUNCT
cana-20	41	13	.	.	PUNCT
cana-20	42	1	crc	crc	PROPN
cana-20	42	2	press	press	PROPN
cana-20	42	3	.	.	PUNCT
cana-20	43	1	[	[	X
cana-20	43	2	7	7	NUM
cana-20	43	3	]	]	X
cana-20	43	4	shor	shor	NOUN
cana-20	43	5	,	,	PUNCT
cana-20	43	6	p.	p.	PROPN
cana-20	43	7	w.	w.	PROPN
cana-20	43	8	(	(	PUNCT
cana-20	43	9	1994	1994	NUM
cana-20	43	10	)	)	PUNCT
cana-20	43	11	.	.	PUNCT
cana-20	44	1	algorithms	algorithm	NOUN
cana-20	44	2	for	for	ADP
cana-20	44	3	quantum	quantum	ADJ
cana-20	44	4	computation	computation	NOUN
cana-20	44	5	:	:	PUNCT
cana-20	44	6	discrete	discrete	ADJ
cana-20	44	7	logarithms	logarithm	NOUN
cana-20	44	8	and	and	CCONJ
cana-20	44	9	factoring	factoring	NOUN
cana-20	44	10	.	.	PUNCT
cana-20	45	1	proceedings	proceeding	NOUN
cana-20	45	2	of	of	ADP
cana-20	45	3	the	the	DET
cana-20	45	4	35th	35th	ADJ
cana-20	45	5	annual	annual	ADJ
cana-20	45	6	symposium	symposium	NOUN
cana-20	45	7	on	on	ADP
cana-20	45	8	foundations	foundation	NOUN
cana-20	45	9	of	of	ADP
cana-20	45	10	computer	computer	NOUN
cana-20	45	11	science	science	NOUN
cana-20	45	12	,	,	PUNCT
cana-20	45	13	124	124	NUM
cana-20	45	14	-	-	SYM
cana-20	45	15	134	134	NUM
cana-20	45	16	.	.	PUNCT
cana-20	46	1	communications	communication	NOUN
cana-20	46	2	on	on	ADP
cana-20	46	3	applied	apply	VERB
cana-20	46	4	nonlinear	nonlinear	ADJ
cana-20	46	5	analysis	analysis	NOUN
cana-20	46	6	issn	issn	NOUN
cana-20	46	7	:	:	PUNCT
cana-20	46	8	1074	1074	NUM
cana-20	46	9	-	-	PUNCT
cana-20	46	10	133x	133x	NUM
cana-20	46	11	vol	vol	NOUN
cana-20	46	12	29	29	NUM
cana-20	46	13	no	no	NOUN
cana-20	46	14	.	.	NOUN
cana-20	46	15	2	2	NUM
cana-20	46	16	(	(	PUNCT
cana-20	46	17	2022	2022	NUM
cana-20	46	18	)	)	PUNCT
cana-20	46	19	12	12	NUM
cana-20	46	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-20	47	1	[	[	X
cana-20	47	2	8	8	NUM
cana-20	47	3	]	]	PUNCT
cana-20	47	4	silverman	silverman	NOUN
cana-20	47	5	,	,	PUNCT
cana-20	47	6	j.	j.	PROPN
cana-20	47	7	h.	h.	PROPN
cana-20	47	8	,	,	PUNCT
cana-20	47	9	&	&	CCONJ
cana-20	47	10	tate	tate	PROPN
cana-20	47	11	,	,	PUNCT
cana-20	47	12	j.	j.	PROPN
cana-20	47	13	(	(	PUNCT
cana-20	47	14	1994	1994	NUM
cana-20	47	15	)	)	PUNCT
cana-20	47	16	.	.	PUNCT
cana-20	48	1	rational	rational	ADJ
cana-20	48	2	points	point	NOUN
cana-20	48	3	on	on	ADP
cana-20	48	4	elliptic	elliptic	ADJ
cana-20	48	5	curves	curve	NOUN
cana-20	48	6	.	.	PUNCT
cana-20	49	1	springer	springer	NOUN
cana-20	49	2	.	.	PUNCT