Curve detail
Definition
| Name | secp521r1 (nist/P-521, secg/secp521r1, x962/ansip521r1) |
|---|---|
| Category | secg |
| Field | Prime (0x01ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff) |
| Field bits | 521 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x01fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc |
| Param $b$ | 0x0051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00 |
| Generator $x$ | 0x00c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66 |
| Generator $y$ | 0x011839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650 |
| Simulation seed | 0xd09e8800291cb85396cc6717393284aaa0da64ba |
Characteristics
| Order | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386409 |
| Cofactor | 0x1 |
| $j$-invariant | 0x10dec9284c5dafd5ade693a4b878916bbf99b97b51a91684867740b8730466241c32a294645557b3915d9c1a5c1c9cf144f3b79a41c70f8bc76ed5928da5e3f69cd |
| Trace $t$ | 0x5ae79787c40d069948033feb708f65a2fc44a36477663b851449048e16ec79bf7 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386409 |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x2000000000000000000000000000000000000000000000000000000000000000005ae79787c40d069948033feb708f65a2fc44a36477663b851449048e16ec79bf7 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x3fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff82047a80e468e696d68ebfa3110e0f4b6380f4524b435156f31b586a3959ef33fcee957b70d566936e65b7af2f36f2b21d61a3b8f1d34c4028ce06b3dda1d070855 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0x2b', '0x66ba301', '0x13c81fb6736e3e75242c2de9acc352e3169c61c44e8c6c2cf7e12ae9c3b341c5efdcb3899ee4680618f0c83ce56163d9ac09b4eeb647684a0c0b675ac91'] |
| (-)largest_factor_bitlen | 0x1e9 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x2', '0x355', '0x2a70f', '0x9984391', '0xffac036294c5d', '0x91161e613b18e41dd8cf32a1d70d8095b7f56a9f1101fe28e00705aec070f6497dd068f7a6f89e4923e561cf497e38f4c1c18bd3'] |
| (-)largest_factor_bitlen | 0x1a0 |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x5', '0xd', '0x7f', '0x8cf3ed19d6d264f', '0x7359b1ba760e1e49086ab4e9a05fff83a6fd3e80c0148bbd06791aef9ab67f304b35432f0a4d588a1a571bf40aa008b13a0aefd9b8a9d7cd3'] |
| (-)largest_factor_bitlen | 0x1c3 |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x2', '0x6d', '0xcedfaf', '0xe8820fe50571039a27c079c258e045be22f7abcba1cf667b43e4b4515e0a8be57bcc8c8a577f755966a092e9ceadd5667551b8312be8850135e4697a0dd'] |
| (+)largest_factor_bitlen | 0x1ec |
| (-)factorization | ['0x2', '0x2', '0x3', '0xa3', '0x3c467f', '0x58f053808667a0c6ee11cf2b20f996ddfc32d77f753b61f25c66309bc8a5b9b70a47fe4cce740362d2a2dbe93cdf965799ccf5b6527f5ca2a0950a4c9ad'] |
| (-)largest_factor_bitlen | 0x1eb |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x25', '0x19747ba47b0f', '0x4daefdb99a5fdd', '0xe54e61bf7429c7330a407813b3489323154f5b5473f15a5c627d4f67adb7572d75ccc736a56f692ced3642c3983db2635bfe81b97'] |
| (+)largest_factor_bitlen | 0x1a4 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x2 |
| full | 0x5 |
| relative | 0x2 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x2 |
| full | 0x6 |
| relative | 0x3 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x3 |
| full | 0xc |
| relative | 0x4 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x5ae79787c40d069948033feb708f65a2fc44a36477663b851449048e16ec79bf7 |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x1dfb857f1b9719692971405ceef1f0b49c7f0badb4bcaea90ce4a795c6a610cc03116a848f2a996c919a4850d0c90d4de29e5c470e2cb3bfd731f94c225e2f8f7ae |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x16027f94603e1d28ad99046e74c34ba2833e41cd30163ae378b8cd8ab4b6c4416341885e21c4eb6e0df52f95f57acecaf3be5fa59f8de1a78c9a9babd5c457ba6434750187a39b6e7c768fecd62ba157113070d8c9450c0a84f684d903fc99a7b6eb |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x4 |
| complement_bit_length | 0x3 |
| $\text{class_number}()$ | |
| upper | NO DATA (timed out) |
| lower | NO DATA (timed out) |
| $\text{small_prime_order}(l=2)$ | |
| order | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd28c343c1df97cb35bfe600a47b84d2e81ddae4dc44ce23d75db7db8f489c3204 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=3)$ | |
| order | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386408 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe9461a1e0efcbe59adff300523dc269740eed726e226711ebaedbedc7a44e1902 |
| complement_bit_length | 0x3 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd28c343c1df97cb35bfe600a47b84d2e81ddae4dc44ce23d75db7db8f489c3204 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd28c343c1df97cb35bfe600a47b84d2e81ddae4dc44ce23d75db7db8f489c3204 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe9461a1e0efcbe59adff300523dc269740eed726e226711ebaedbedc7a44e1902 |
| complement_bit_length | 0x3 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x2', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x1', '0x2'], ['0x5', '0x2']] |
| len | 0x2 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=3)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=5)$ | |
| crater_degree | 0x1 |
| depth | 0x0 |
| $\text{volcano}(l=7)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=17)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=19)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x1 |
| full | 0x5 |
| relative | 0x5 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x1 |
| full | 0x6 |
| relative | 0x6 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0xc |
| relative | 0xc |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x8 |
| relative | 0x8 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x9 |
| relative | 0x9 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x5ae79787c40d069948033feb708f65a2fc44a36477663b851449048e16ec79bf7 |
| trace_factorization | NO DATA (timed out) |
| number_of_factors | NO DATA (timed out) |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x5ae79787c40d069948033feb708f65a2fc44a36477663b851449048e16ec79bf7 |
| trace_factorization | ['0x49', '0x1a69f', '0x83aa9a8f04031434daba5f461074b62a1792839cf06d7974aebcfaa9241376907b64a251b39db287b87d565bcba7c4c8e644ce888049f3db90fe35344de9b'] |
| number_of_factors | 0x3 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x1 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x110 |
| expected | 0x104 |
| ratio | 0.95588 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x108a7 |
| expected | 0x10892 |
| ratio | 0.99969 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xb2cb77 |
| expected | 0xb2caaa |
| ratio | 0.99998 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x5ae79787c40d069948033feb708f65a2fc44a36477663b851449048e16ec79bf7 |
| ratio | 1.0434765804216952e+79 |
| distance 32 | 0x9 |
| distance 64 | 0x9 |
| $\text{multiples_x}(k=1)$ | |
| Hx | 0xc6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66 |
| bits | 0x208 |
| difference | 0x1 |
| ratio | 0.99808 |
| $\text{multiples_x}(k=2)$ | |
| Hx | 0x7c1bb67bc4f1a47a2cab98f6832fd9681fd803a639451943b35eeb82b705fd41327338840f7b531313f188de7e42bb46b68e0fa5cb05b53558c1ca8e31d783223f |
| bits | 0x207 |
| difference | 0x2 |
| ratio | 0.99616 |
| $\text{multiples_x}(k=3)$ | |
| Hx | 0xc48bdc22bbb449823c9c435543a4ebfa60c00b449009e3eae035d927a4754bc4a75abb90a46cfab5325f627632082100a69e590ab282c12a3745115366a2742967 |
| bits | 0x208 |
| difference | 0x1 |
| ratio | 0.99808 |
| $\text{multiples_x}(k=4)$ | |
| Hx | 0x677bb651a20522b7468af5c46cb7bfe6a882893db47d48f962821419e425220da12876a3846797b153e27f64d6b9587d823924f352e64cd42da76e69be121ba7fe |
| bits | 0x207 |
| difference | 0x2 |
| ratio | 0.99616 |
| $\text{multiples_x}(k=5)$ | |
| Hx | 0x1ffd220fefb9111893fc2a145221350770c13cac1787d8c7228aa5a4173ccc6f1ff6ba264c4c679c239b7ec8af18e1981d3ad5a9d29505e99e7c0cdf1042d64c9b5 |
| bits | 0x209 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=6)$ | |
| Hx | 0x8fa01492c9547609d2e96f68ebce291877f916ca637b5ae571faf89ecb2985be7da168883a7926288f4f6b505499f6e00a101b1e2125f360f6a597f50ba7c5ed4b |
| bits | 0x208 |
| difference | 0x1 |
| ratio | 0.99808 |
| $\text{multiples_x}(k=7)$ | |
| Hx | 0x2f7142d4fe819ac33f272dd18a774418e282f20ab2f191194d37974c5230691750deee735d3ece84d8fcf72a8e6d2477a7380008b9908909cbf1abda59c54b0255 |
| bits | 0x206 |
| difference | 0x3 |
| ratio | 0.99424 |
| $\text{multiples_x}(k=8)$ | |
| Hx | 0xa8156c2288d85c653adbc20de39200bd2879cec09a360862a744ab4b71cf31361c206a1789868e93a32025d53507b9f0964ffcaa291dffa480c92dc3c26c1e6e5e |
| bits | 0x208 |
| difference | 0x1 |
| ratio | 0.99808 |
| $\text{multiples_x}(k=9)$ | |
| Hx | 0x2c4e8b477643b55e7ab3eeb26c3cf54ad591a632b733f8745d8118aaa4f45c579142af8bc5318ff40351e2d6251a563d068b5391e8933b758f82f4266c02da6f61 |
| bits | 0x206 |
| difference | 0x3 |
| ratio | 0.99424 |
| $\text{multiples_x}(k=10)$ | |
| Hx | 0x806446e89eab673252bb8ac2d30e5afcd10f0369550c547a09cb76ebbc0eae0cd531b0c796f87068004af00d91e15041b4e06223b77d27baa1d24e2231084428a5 |
| bits | 0x208 |
| difference | 0x1 |
| ratio | 0.99808 |
| $\text{x962_invariant}()$ | |
| r | 0xb48bfa5f420a34949539d2bdfc264eeeeb077688e44fbf0ad8f6d0edb37bd6b533281000518e19f1b9ffbe0fe9ed8a3c2200b8f875e523868c70c1e5bf55bad637 |
| $\text{brainpool_overlap}()$ | |
| o | 0xae6ac1469e71e365e06d65de5f497abf115d258da4664cea0c474b766e710ef61ea9e6c6ae13816c84e9ad3f3f |
| $\text{weierstrass}()$ | |
| a | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc |
| b | 0x51953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00 |