Datasets:

itsalissonsilva
/

crypto-arithmetic-proofs-synthetic

id stringlengths 24 24	group_id stringlengths 24 24	split stringclasses 1 value	family stringclasses 5 values	modulus_prime int64 17 151	prompt stringlengths 22 81	answer_integer int64 1 21.3k	reference_certificate stringlengths 15 69	candidate_integer int64 1 21.3k	candidate_valid bool 2 classes	counterexample stringclasses 699 values	parameter_partition stringclasses 1 value
47ca7e71a20b8a360aa2827d	47ca7e71a20b8a360aa2827d	train	modular_inverse	53	Find the multiplicative inverse of 17 modulo prime 53.	25	17 * 25 = 425; remainder modulo 53 is 1.	25	true		train
a82c3e0c98ec80108e6802df	a82c3e0c98ec80108e6802df	train	modular_power	113	Compute 50^11 modulo 113.	25	Repeated squaring residues: [50, 14, 83, 109]	26	false	Expected 25, proposed 26.	train
510fe1d16ea02e58170ef8ee	510fe1d16ea02e58170ef8ee	train	modular_product	83	Compute (7 * 28) modulo 83.	30	196 = 2 * 83 + 30.	30	true		train
f9285d06193a0800062a4eff	f9285d06193a0800062a4eff	train	toy_rsa	43	Toy RSA only: p=43, q=107, e=5, ciphertext=3293. Recover the integer message.	1,300	n=4601; phi=4452; d=1781; e*d mod phi=1; ciphertext^d mod n=1300.	1,301	false	Expected 1300, proposed 1301.	train
d34e9bbccb835f725c6963c9	d34e9bbccb835f725c6963c9	train	bezout	29	Return u in a Bezout certificate 26u + 29v = 1 with 0 <= u < 29.	19	u=19; v=-17; 2619+29-17=1.	19	true		train
e75a8fac3b35c0854536f3c0	e75a8fac3b35c0854536f3c0	train	modular_inverse	139	Find the multiplicative inverse of 10 modulo prime 139.	14	10 * 14 = 140; remainder modulo 139 is 1.	15	false	Expected 14, proposed 15.	train
7e948ca03b4f0a783c286edb	7e948ca03b4f0a783c286edb	train	modular_power	43	Compute 7^16 modulo 43.	36	Repeated squaring residues: [7, 6, 36, 6, 36]	36	true		train
3cf5e782edd98de37737f829	3cf5e782edd98de37737f829	train	modular_product	97	Compute (62 * 25) modulo 97.	95	1550 = 15 * 97 + 95.	96	false	Expected 95, proposed 96.	train
c24e062f7feff94d52b5dabb	c24e062f7feff94d52b5dabb	train	toy_rsa	127	Toy RSA only: p=127, q=149, e=5, ciphertext=4086. Recover the integer message.	11,787	n=18923; phi=18648; d=11189; e*d mod phi=1; ciphertext^d mod n=11787.	11,787	true		train
6d12cd861e200d66363248d8	6d12cd861e200d66363248d8	train	bezout	131	Return u in a Bezout certificate 63u + 131v = 1 with 0 <= u < 131.	52	u=52; v=-25; 6352+131-25=1.	53	false	Expected 52, proposed 53.	train
334eed4db198acebd4654aa6	334eed4db198acebd4654aa6	train	modular_inverse	19	Find the multiplicative inverse of 15 modulo prime 19.	14	15 * 14 = 210; remainder modulo 19 is 1.	14	true		train
419504352ed64b3989137c4a	419504352ed64b3989137c4a	train	modular_power	103	Compute 55^64 modulo 103.	28	Repeated squaring residues: [55, 38, 2, 4, 16, 50, 28]	29	false	Expected 28, proposed 29.	train
599c64cda9da24c028868ad6	599c64cda9da24c028868ad6	train	modular_product	89	Compute (44 * 60) modulo 89.	59	2640 = 29 * 89 + 59.	59	true		train
9f78e213945de4fe0ac1ad79	9f78e213945de4fe0ac1ad79	train	toy_rsa	61	Toy RSA only: p=61, q=107, e=17, ciphertext=898. Recover the integer message.	3,235	n=6527; phi=6360; d=2993; e*d mod phi=1; ciphertext^d mod n=3235.	3,236	false	Expected 3235, proposed 3236.	train
39a381bad7532565d2b40eb8	39a381bad7532565d2b40eb8	train	bezout	61	Return u in a Bezout certificate 9u + 61v = 1 with 0 <= u < 61.	34	u=34; v=-5; 934+61-5=1.	34	true		train
16b34ae21b277c357dbf0144	16b34ae21b277c357dbf0144	train	modular_inverse	103	Find the multiplicative inverse of 34 modulo prime 103.	100	34 * 100 = 3400; remainder modulo 103 is 1.	101	false	Expected 100, proposed 101.	train
a9068a0544503d569a9c7314	a9068a0544503d569a9c7314	train	modular_power	19	Compute 15^4 modulo 19.	9	Repeated squaring residues: [15, 16, 9]	9	true		train
e1358f6ff6914dc94390d7f2	e1358f6ff6914dc94390d7f2	train	modular_product	53	Compute (26 * 52) modulo 53.	27	1352 = 25 * 53 + 27.	28	false	Expected 27, proposed 28.	train
8dbb6ae453db89921f59e293	8dbb6ae453db89921f59e293	train	toy_rsa	29	Toy RSA only: p=29, q=109, e=5, ciphertext=2741. Recover the integer message.	801	n=3161; phi=3024; d=605; e*d mod phi=1; ciphertext^d mod n=801.	801	true		train
c55327639cc596aabb3c1483	c55327639cc596aabb3c1483	train	bezout	31	Return u in a Bezout certificate 5u + 31v = 1 with 0 <= u < 31.	25	u=25; v=-4; 525+31-4=1.	26	false	Expected 25, proposed 26.	train
ecf43aa298ad5391b79d9b67	ecf43aa298ad5391b79d9b67	train	modular_inverse	149	Find the multiplicative inverse of 53 modulo prime 149.	45	53 * 45 = 2385; remainder modulo 149 is 1.	45	true		train
725dff7eb95b97b7d7789524	725dff7eb95b97b7d7789524	train	modular_power	37	Compute 34^17 modulo 37.	12	Repeated squaring residues: [34, 9, 7, 12, 33]	13	false	Expected 12, proposed 13.	train
069617c81fddde6267829e82	069617c81fddde6267829e82	train	modular_product	139	Compute (24 * 6) modulo 139.	5	144 = 1 * 139 + 5.	5	true		train
fb93391549c5127a7630dd90	fb93391549c5127a7630dd90	train	toy_rsa	43	Toy RSA only: p=43, q=71, e=17, ciphertext=2302. Recover the integer message.	2,167	n=3053; phi=2940; d=173; e*d mod phi=1; ciphertext^d mod n=2167.	2,168	false	Expected 2167, proposed 2168.	train
7446029c450f2472230480d3	7446029c450f2472230480d3	train	bezout	89	Return u in a Bezout certificate 30u + 89v = 1 with 0 <= u < 89.	3	u=3; v=-1; 303+89-1=1.	3	true		train
b96bf58f9f8b5c2d7155faa6	b96bf58f9f8b5c2d7155faa6	train	modular_inverse	59	Find the multiplicative inverse of 38 modulo prime 59.	14	38 * 14 = 532; remainder modulo 59 is 1.	15	false	Expected 14, proposed 15.	train
13e08d1728b6a47e758c172b	13e08d1728b6a47e758c172b	train	modular_power	41	Compute 15^31 modulo 41.	12	Repeated squaring residues: [15, 20, 31, 18, 37]	12	true		train
4ca506c37acd8fed36f7656d	4ca506c37acd8fed36f7656d	train	modular_product	73	Compute (18 * 66) modulo 73.	20	1188 = 16 * 73 + 20.	21	false	Expected 20, proposed 21.	train
e997d8652779eaf857bf9409	e997d8652779eaf857bf9409	train	toy_rsa	131	Toy RSA only: p=131, q=109, e=17, ciphertext=3994. Recover the integer message.	2,483	n=14279; phi=14040; d=7433; e*d mod phi=1; ciphertext^d mod n=2483.	2,483	true		train
d6792f43e58bfff64ee9248c	d6792f43e58bfff64ee9248c	train	bezout	19	Return u in a Bezout certificate 10u + 19v = 1 with 0 <= u < 19.	2	u=2; v=-1; 102+19-1=1.	3	false	Expected 2, proposed 3.	train
4ffd8c28789a088a168e160e	4ffd8c28789a088a168e160e	train	modular_inverse	151	Find the multiplicative inverse of 16 modulo prime 151.	85	16 * 85 = 1360; remainder modulo 151 is 1.	85	true		train
14063f7d1c9b5d68b4c0cc8f	14063f7d1c9b5d68b4c0cc8f	train	modular_power	17	Compute 13^12 modulo 17.	1	Repeated squaring residues: [13, 16, 1, 1]	2	false	Expected 1, proposed 2.	train
634a7d72b33b2e7c44cec8ab	634a7d72b33b2e7c44cec8ab	train	modular_product	23	Compute (11 * 8) modulo 23.	19	88 = 3 * 23 + 19.	19	true		train
0158034e7074c1b9f9a8f56a	0158034e7074c1b9f9a8f56a	train	toy_rsa	61	Toy RSA only: p=61, q=127, e=17, ciphertext=5806. Recover the integer message.	6,660	n=7747; phi=7560; d=3113; e*d mod phi=1; ciphertext^d mod n=6660.	6,661	false	Expected 6660, proposed 6661.	train
928012861a2c56f57fb35475	928012861a2c56f57fb35475	train	bezout	83	Return u in a Bezout certificate 47u + 83v = 1 with 0 <= u < 83.	53	u=53; v=-30; 4753+83-30=1.	53	true		train
f3924052da15742e39034a0d	f3924052da15742e39034a0d	train	modular_inverse	41	Find the multiplicative inverse of 21 modulo prime 41.	2	21 * 2 = 42; remainder modulo 41 is 1.	3	false	Expected 2, proposed 3.	train
f7479de323288b9e43086726	f7479de323288b9e43086726	train	modular_power	103	Compute 73^45 modulo 103.	31	Repeated squaring residues: [73, 76, 8, 64, 79, 61]	31	true		train
d9192cf47965fae30f649f44	d9192cf47965fae30f649f44	train	modular_product	101	Compute (43 * 24) modulo 101.	22	1032 = 10 * 101 + 22.	23	false	Expected 22, proposed 23.	train
f4405e61b5a3eeef9b9314da	f4405e61b5a3eeef9b9314da	train	toy_rsa	79	Toy RSA only: p=79, q=19, e=5, ciphertext=216. Recover the integer message.	106	n=1501; phi=1404; d=281; e*d mod phi=1; ciphertext^d mod n=106.	106	true		train
c5d9c28b8b4418f46e68d31f	c5d9c28b8b4418f46e68d31f	train	bezout	17	Return u in a Bezout certificate 6u + 17v = 1 with 0 <= u < 17.	3	u=3; v=-1; 63+17-1=1.	4	false	Expected 3, proposed 4.	train
98cf01275edd72a9f30ff989	98cf01275edd72a9f30ff989	train	modular_inverse	89	Find the multiplicative inverse of 65 modulo prime 89.	63	65 * 63 = 4095; remainder modulo 89 is 1.	63	true		train
a145ca6e6532fbf0491d4154	a145ca6e6532fbf0491d4154	train	modular_power	97	Compute 37^10 modulo 97.	31	Repeated squaring residues: [37, 11, 24, 91]	32	false	Expected 31, proposed 32.	train
b2de63b7d98b65b7a242e08e	b2de63b7d98b65b7a242e08e	train	modular_product	31	Compute (29 * 20) modulo 31.	22	580 = 18 * 31 + 22.	22	true		train
ad7081385c954aa28ffe2451	ad7081385c954aa28ffe2451	train	toy_rsa	43	Toy RSA only: p=43, q=137, e=5, ciphertext=5636. Recover the integer message.	2,434	n=5891; phi=5712; d=2285; e*d mod phi=1; ciphertext^d mod n=2434.	2,435	false	Expected 2434, proposed 2435.	train
69adb1b010d04d22263a795c	69adb1b010d04d22263a795c	train	bezout	23	Return u in a Bezout certificate 19u + 23v = 1 with 0 <= u < 23.	17	u=17; v=-14; 1917+23-14=1.	17	true		train
e8df6e8b11f96e217795bc0a	e8df6e8b11f96e217795bc0a	train	modular_inverse	71	Find the multiplicative inverse of 37 modulo prime 71.	48	37 * 48 = 1776; remainder modulo 71 is 1.	49	false	Expected 48, proposed 49.	train
58d0fcdac56e5dec71f10c46	58d0fcdac56e5dec71f10c46	train	modular_power	41	Compute 6^38 modulo 41.	8	Repeated squaring residues: [6, 36, 25, 10, 18, 37]	8	true		train
4e37a138063ddc822df268d6	4e37a138063ddc822df268d6	train	modular_product	101	Compute (34 * 92) modulo 101.	98	3128 = 30 * 101 + 98.	99	false	Expected 98, proposed 99.	train
75995d9065ca8f2d2015ad31	75995d9065ca8f2d2015ad31	train	toy_rsa	151	Toy RSA only: p=151, q=131, e=17, ciphertext=1561. Recover the integer message.	5,356	n=19781; phi=19500; d=18353; e*d mod phi=1; ciphertext^d mod n=5356.	5,356	true		train
8b6699b953c808afffac8f7b	8b6699b953c808afffac8f7b	train	bezout	127	Return u in a Bezout certificate 11u + 127v = 1 with 0 <= u < 127.	104	u=104; v=-9; 11104+127-9=1.	105	false	Expected 104, proposed 105.	train
e49bfa04c50f8ec4d3685d9e	e49bfa04c50f8ec4d3685d9e	train	modular_inverse	109	Find the multiplicative inverse of 81 modulo prime 109.	35	81 * 35 = 2835; remainder modulo 109 is 1.	35	true		train
4effe66ecd5ad3288fdcf133	4effe66ecd5ad3288fdcf133	train	modular_power	137	Compute 84^129 modulo 137.	111	Repeated squaring residues: [84, 69, 103, 60, 38, 74, 133, 16]	112	false	Expected 111, proposed 112.	train
bb1aaee0e7fae241e9dcd283	bb1aaee0e7fae241e9dcd283	train	modular_product	83	Compute (36 * 12) modulo 83.	17	432 = 5 * 83 + 17.	17	true		train
95d4e0455351c879a972bcde	95d4e0455351c879a972bcde	train	toy_rsa	137	Toy RSA only: p=137, q=127, e=5, ciphertext=935. Recover the integer message.	10,332	n=17399; phi=17136; d=13709; e*d mod phi=1; ciphertext^d mod n=10332.	10,333	false	Expected 10332, proposed 10333.	train
41cacc8504c39b7db564e998	41cacc8504c39b7db564e998	train	bezout	137	Return u in a Bezout certificate 61u + 137v = 1 with 0 <= u < 137.	9	u=9; v=-4; 619+137-4=1.	9	true		train
29ebf419b077bdf87440858a	29ebf419b077bdf87440858a	train	modular_inverse	113	Find the multiplicative inverse of 53 modulo prime 113.	32	53 * 32 = 1696; remainder modulo 113 is 1.	33	false	Expected 32, proposed 33.	train
4e7a4a338e5d699101bd2a7a	4e7a4a338e5d699101bd2a7a	train	modular_power	19	Compute 9^5 modulo 19.	16	Repeated squaring residues: [9, 5, 6]	16	true		train
454c76a8929eb0084a3ff299	454c76a8929eb0084a3ff299	train	modular_product	139	Compute (64 * 86) modulo 139.	83	5504 = 39 * 139 + 83.	84	false	Expected 83, proposed 84.	train
f74a845938f92d26f12b1506	f74a845938f92d26f12b1506	train	toy_rsa	43	Toy RSA only: p=43, q=103, e=5, ciphertext=2845. Recover the integer message.	2,918	n=4429; phi=4284; d=857; e*d mod phi=1; ciphertext^d mod n=2918.	2,918	true		train
c9948b0d57ce06a36050cf7f	c9948b0d57ce06a36050cf7f	train	bezout	127	Return u in a Bezout certificate 3u + 127v = 1 with 0 <= u < 127.	85	u=85; v=-2; 385+127-2=1.	86	false	Expected 85, proposed 86.	train
cb309736e1fa079f31e28a87	cb309736e1fa079f31e28a87	train	modular_inverse	83	Find the multiplicative inverse of 82 modulo prime 83.	82	82 * 82 = 6724; remainder modulo 83 is 1.	82	true		train
3bd99e9696ac5f1b4fac190e	3bd99e9696ac5f1b4fac190e	train	modular_power	101	Compute 32^40 modulo 101.	1	Repeated squaring residues: [32, 14, 95, 36, 84, 87]	2	false	Expected 1, proposed 2.	train
89ad54a7a0e808553a6f4efa	89ad54a7a0e808553a6f4efa	train	modular_product	113	Compute (12 * 21) modulo 113.	26	252 = 2 * 113 + 26.	26	true		train
96642105c1b15568850db3a1	96642105c1b15568850db3a1	train	toy_rsa	137	Toy RSA only: p=137, q=89, e=3, ciphertext=5858. Recover the integer message.	4,062	n=12193; phi=11968; d=7979; e*d mod phi=1; ciphertext^d mod n=4062.	4,063	false	Expected 4062, proposed 4063.	train
9071d0c2b06cbb59d0df2774	9071d0c2b06cbb59d0df2774	train	bezout	107	Return u in a Bezout certificate 104u + 107v = 1 with 0 <= u < 107.	71	u=71; v=-69; 10471+107-69=1.	71	true		train
1a1d250d4837f6ca349788af	1a1d250d4837f6ca349788af	train	modular_inverse	113	Find the multiplicative inverse of 74 modulo prime 113.	84	74 * 84 = 6216; remainder modulo 113 is 1.	85	false	Expected 84, proposed 85.	train
98a768936f1659c958ee2708	98a768936f1659c958ee2708	train	modular_power	103	Compute 55^73 modulo 103.	83	Repeated squaring residues: [55, 38, 2, 4, 16, 50, 28]	83	true		train
a733bae460a5a237e4bccde4	a733bae460a5a237e4bccde4	train	modular_product	17	Compute (13 * 13) modulo 17.	16	169 = 9 * 17 + 16.	17	false	Expected 16, proposed 17.	train
f405d1f1b8b5234c3f32121d	f405d1f1b8b5234c3f32121d	train	toy_rsa	47	Toy RSA only: p=47, q=43, e=5, ciphertext=578. Recover the integer message.	1,093	n=2021; phi=1932; d=773; e*d mod phi=1; ciphertext^d mod n=1093.	1,093	true		train
ebc3c67ca913d2f79e127b41	ebc3c67ca913d2f79e127b41	train	bezout	43	Return u in a Bezout certificate 15u + 43v = 1 with 0 <= u < 43.	23	u=23; v=-8; 1523+43-8=1.	24	false	Expected 23, proposed 24.	train
ac81f1c30b0e3b89088291ca	ac81f1c30b0e3b89088291ca	train	modular_inverse	97	Find the multiplicative inverse of 34 modulo prime 97.	20	34 * 20 = 680; remainder modulo 97 is 1.	20	true		train
5b48d0113edaec538fa6453f	5b48d0113edaec538fa6453f	train	modular_power	137	Compute 76^126 modulo 137.	4	Repeated squaring residues: [76, 22, 73, 123, 59, 56, 122]	5	false	Expected 4, proposed 5.	train
85f3ef512e34ce5eee1ab061	85f3ef512e34ce5eee1ab061	train	modular_product	71	Compute (31 * 37) modulo 71.	11	1147 = 16 * 71 + 11.	11	true		train
6c44150f562bfac0cb89fe39	6c44150f562bfac0cb89fe39	train	toy_rsa	73	Toy RSA only: p=73, q=67, e=5, ciphertext=961. Recover the integer message.	3,450	n=4891; phi=4752; d=1901; e*d mod phi=1; ciphertext^d mod n=3450.	3,451	false	Expected 3450, proposed 3451.	train
ff9a91344a16e0d90d09da9a	ff9a91344a16e0d90d09da9a	train	bezout	151	Return u in a Bezout certificate 131u + 151v = 1 with 0 <= u < 151.	83	u=83; v=-72; 13183+151-72=1.	83	true		train
a251cd0e0bcb7f72b8037fa7	a251cd0e0bcb7f72b8037fa7	train	modular_inverse	89	Find the multiplicative inverse of 19 modulo prime 89.	75	19 * 75 = 1425; remainder modulo 89 is 1.	76	false	Expected 75, proposed 76.	train
77c274a884a99132ec4dfde3	77c274a884a99132ec4dfde3	train	modular_power	79	Compute 26^32 modulo 79.	25	Repeated squaring residues: [26, 44, 40, 20, 5, 25]	25	true		train
6c3569cb60d88ad48b652c71	6c3569cb60d88ad48b652c71	train	modular_product	59	Compute (47 * 46) modulo 59.	38	2162 = 36 * 59 + 38.	39	false	Expected 38, proposed 39.	train
523c139e2ce8609b347a20a7	523c139e2ce8609b347a20a7	train	toy_rsa	53	Toy RSA only: p=53, q=47, e=3, ciphertext=1547. Recover the integer message.	452	n=2491; phi=2392; d=1595; e*d mod phi=1; ciphertext^d mod n=452.	452	true		train
3df98e3b94020d39e455d1ee	3df98e3b94020d39e455d1ee	train	bezout	139	Return u in a Bezout certificate 114u + 139v = 1 with 0 <= u < 139.	50	u=50; v=-41; 11450+139-41=1.	51	false	Expected 50, proposed 51.	train
1d681037a86eec91aabf41c9	1d681037a86eec91aabf41c9	train	modular_inverse	43	Find the multiplicative inverse of 29 modulo prime 43.	3	29 * 3 = 87; remainder modulo 43 is 1.	3	true		train
7815eb9b3ae404b960b84ee5	7815eb9b3ae404b960b84ee5	train	modular_power	17	Compute 13^4 modulo 17.	1	Repeated squaring residues: [13, 16, 1]	2	false	Expected 1, proposed 2.	train
b4ff17373fa1436c6a5f9050	b4ff17373fa1436c6a5f9050	train	modular_product	101	Compute (21 * 46) modulo 101.	57	966 = 9 * 101 + 57.	57	true		train
eff9246dffc79126c5733eba	eff9246dffc79126c5733eba	train	toy_rsa	43	Toy RSA only: p=43, q=31, e=17, ciphertext=291. Recover the integer message.	921	n=1333; phi=1260; d=593; e*d mod phi=1; ciphertext^d mod n=921.	922	false	Expected 921, proposed 922.	train
3b1892adac887215fe732c68	3b1892adac887215fe732c68	train	bezout	67	Return u in a Bezout certificate 39u + 67v = 1 with 0 <= u < 67.	55	u=55; v=-32; 3955+67-32=1.	55	true		train
c9b94ec168f8d6bcc78d7feb	c9b94ec168f8d6bcc78d7feb	train	modular_inverse	137	Find the multiplicative inverse of 83 modulo prime 137.	104	83 * 104 = 8632; remainder modulo 137 is 1.	105	false	Expected 104, proposed 105.	train
17c6acf40f8b2aea004e8b34	17c6acf40f8b2aea004e8b34	train	modular_power	59	Compute 42^46 modulo 59.	9	Repeated squaring residues: [42, 53, 36, 57, 4, 16]	9	true		train
4b9605a2424f0796e6c0a3a5	4b9605a2424f0796e6c0a3a5	train	modular_product	137	Compute (96 * 96) modulo 137.	37	9216 = 67 * 137 + 37.	38	false	Expected 37, proposed 38.	train
1e5dbba24fd966e6c76591d4	1e5dbba24fd966e6c76591d4	train	toy_rsa	29	Toy RSA only: p=29, q=73, e=5, ciphertext=1848. Recover the integer message.	48	n=2117; phi=2016; d=1613; e*d mod phi=1; ciphertext^d mod n=48.	48	true		train
4ea08d6e74bdc1ee094b5f26	4ea08d6e74bdc1ee094b5f26	train	bezout	103	Return u in a Bezout certificate 79u + 103v = 1 with 0 <= u < 103.	30	u=30; v=-23; 7930+103-23=1.	31	false	Expected 30, proposed 31.	train
cf1d9b61a26bfd45735da33e	cf1d9b61a26bfd45735da33e	train	modular_inverse	79	Find the multiplicative inverse of 3 modulo prime 79.	53	3 * 53 = 159; remainder modulo 79 is 1.	53	true		train
2b1393672eb3f53a89990b3c	2b1393672eb3f53a89990b3c	train	modular_power	83	Compute 6^81 modulo 83.	14	Repeated squaring residues: [6, 36, 51, 28, 37, 41, 21]	15	false	Expected 14, proposed 15.	train
bd4b2b275f213c7b23e80658	bd4b2b275f213c7b23e80658	train	modular_product	17	Compute (5 * 16) modulo 17.	12	80 = 4 * 17 + 12.	12	true		train
0ec0fbdc2d134e76a7e40e55	0ec0fbdc2d134e76a7e40e55	train	toy_rsa	31	Toy RSA only: p=31, q=71, e=17, ciphertext=941. Recover the integer message.	229	n=2201; phi=2100; d=1853; e*d mod phi=1; ciphertext^d mod n=229.	230	false	Expected 229, proposed 230.	train
4a9b0c4e6ce89a663e5786c4	4a9b0c4e6ce89a663e5786c4	train	bezout	101	Return u in a Bezout certificate 70u + 101v = 1 with 0 <= u < 101.	13	u=13; v=-9; 7013+101-9=1.	13	true		train
1f15c695d4fc9dd54c5bb814	1f15c695d4fc9dd54c5bb814	train	modular_inverse	101	Find the multiplicative inverse of 29 modulo prime 101.	7	29 * 7 = 203; remainder modulo 101 is 1.	8	false	Expected 7, proposed 8.	train
c2d6867334121dfdbe24cc60	c2d6867334121dfdbe24cc60	train	modular_power	107	Compute 81^46 modulo 107.	105	Repeated squaring residues: [81, 34, 86, 13, 62, 99]	105	true		train
1af6b0983588974c932704c5	1af6b0983588974c932704c5	train	modular_product	17	Compute (2 * 6) modulo 17.	12	12 = 0 * 17 + 12.	13	false	Expected 12, proposed 13.	train
4e2097eb0e1570d789f9fec2	4e2097eb0e1570d789f9fec2	train	toy_rsa	107	Toy RSA only: p=107, q=19, e=5, ciphertext=1880. Recover the integer message.	1,823	n=2033; phi=1908; d=1145; e*d mod phi=1; ciphertext^d mod n=1823.	1,823	true		train
69670ace28bfb3f84fa2d941	69670ace28bfb3f84fa2d941	train	bezout	79	Return u in a Bezout certificate 45u + 79v = 1 with 0 <= u < 79.	72	u=72; v=-41; 4572+79-41=1.	73	false	Expected 72, proposed 73.	train

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Verified toy cryptographic arithmetic

8,000 exact integer tasks: inverses, repeated squaring, modular products, toy RSA and Bezout certificates. Each task has a correct integer answer and executable arithmetic certificate, plus a balanced correct/incorrect candidate for verifier training. Incorrect candidates have a concrete counterexample. The proof is a computation certificate, not an unrestricted natural-language proof.

Prime parameter ranges are disjoint across splits; query hashes are unique. All toy RSA primes are deliberately tiny and provide no security. No real keys, credentials or operational targets. This dataset evaluates arithmetic and certificate checking, not cryptographic security expertise.

Rebuild with python build_exact_science.py --kind crypto. Source code is Apache-2.0; original generated records are CC BY 4.0. Seed 20261031. No teacher-model labels.

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