Curve detail
Definition
| Name | secp160k1 |
|---|---|
| Category | secg |
| Description | A Koblitz curve. |
| Field | Prime (0xfffffffffffffffffffffffffffffffeffffac73) |
| Field bits | 160 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x0000000000000000000000000000000000000000 |
| Param $b$ | 0x0000000000000000000000000000000000000007 |
| Generator $x$ | 0x3b4c382ce37aa192a4019e763036f4f5dd4d7ebb |
| Generator $y$ | 0x938cf935318fdced6bc28286531733c3f03c4fee |
Characteristics
| Order | 0x100000000000000000001b8fa16dfab9aca16b6b3 |
| Cofactor | 0x1 |
| $j$-invariant | 0x0 |
| Trace $t$ | -0x1b8fa16dfab9bca170a3f |
| Embedding degree $k$ | 0x80000000000000000000dc7d0b6fd5cd650b5b59 |
| CM discriminant | -0x3 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x100000000000000000001b8fa16dfab9aca16b6b3 |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | -0x3 |
| factorization | ['0x3', '0x7', '0x7', '0x6661', '0x6661', '0x1dc4a03f', '0x1dc4a03f', '0x1cd6cc8cd', '0x1cd6cc8cd'] |
| max_conductor | 0x96341f1138933bc2f505 |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0xfffffffffffffffffffe4705e920546335e8a235 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xfffffffffffffffffffffffffffffffdffff58e6f79ca2f18ccde7e11b681bbccce197a45f005e45 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x2', '0x2', '0x139', '0x3cb', '0x8adf003', '0x1970bffb53c93339e2b56d277e935'] |
| (+)largest_factor_bitlen | 0x71 |
| (-)factorization | ['0x2', '0x3', '0x5', '0x2285', '0x2539c66d03622d505', '0x1b333c2460c275ae272f'] |
| (-)largest_factor_bitlen | 0x4d |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0x3', '0x3', '0x1a3', '0x9691f4f', '0x76c5a08dbcd', '0x7f5fb827418fdeec32d7'] |
| (+)largest_factor_bitlen | 0x4f |
| (-)factorization | ['0x7', '0x47', '0x97', '0x4fff', '0x2e1461', '0x15c91501', '0x1018e46d30b', '0x16aad6e70c7'] |
| (-)largest_factor_bitlen | 0x29 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x2', '0x13', '0x13', '0x17', '0x194b5a2bb', '0x4d890620f07', '0x18ba24514514f7792df'] |
| (+)largest_factor_bitlen | 0x49 |
| (-)factorization | ['0x2', '0x2', '0x2', '0x3b', '0x5b2cd', '0xddb45', '0x5467b7a1c9971501f9f0e95d6ba19'] |
| (-)largest_factor_bitlen | 0x73 |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0x5', '0xa585ccf', '0x13cbf393a8c6be402ab8efd05a53788f87'] |
| (+)largest_factor_bitlen | 0x85 |
| (-)factorization | ['0x3', '0x407e6f', '0x54ae0bd036ec1549f6cbbf29a36f3ebb277'] |
| (-)largest_factor_bitlen | 0x8b |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x2', '0x2', '0x2', '0x2', '0x2', '0x2', '0x2', '0x3', '0x7', '0x7', '0x11', '0xa99', '0x18bef583ca89d60503d4cfa65f761ae1c9'] |
| (+)largest_factor_bitlen | 0x85 |
| (-)factorization | ['0x2', '0xb', '0xd', '0x2bae771', '0x25020dd98dad', '0xb5705b4303d82cf52a305'] |
| (-)largest_factor_bitlen | 0x54 |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0xb', '0x1169', '0x2073d695132d', '0x3f44be8d8c150fa9726a884a75'] |
| (+)largest_factor_bitlen | 0x66 |
| (-)factorization | ['0x5', '0x5', '0x5bcd9c293', '0x26a4bccf6ec8db', '0x46f00bd2b1bee04619'] |
| (-)largest_factor_bitlen | 0x47 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x2', '0x2f', '0x53', '0xef', '0x6ccb', '0x47091d', '0x11d4d57', '0x8d3310c51', '0x364e0e093549'] |
| (+)largest_factor_bitlen | 0x2e |
| (-)factorization | ['0x2', '0x2', '0x3', '0x3', '0x862a2223b', '0x5efb2eefb73a27d9c7a63735c3abe83'] |
| (-)largest_factor_bitlen | 0x7b |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0xd', '0x5b56ce7f083', '0x9dd7fce8e3d', '0xeeb474a3091d684139'] |
| (+)largest_factor_bitlen | 0x48 |
| (-)factorization | ['0xed1', '0x6c07db951f', '0x1478d3cf5b0deea1ad5f6dba30c39'] |
| (-)largest_factor_bitlen | 0x71 |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x2 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0xf |
| full | 0xf |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x2 |
| full | 0x6 |
| relative | 0x3 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x60 |
| full | 0x60 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x1b8fa16dfab9bca170a3f |
| factorization | ['0x1349', '0x16ddbac4ec04232947'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x1f79ca2f18ccde7e11b681bbbcce0f08943bb4f0e |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x1aa871d2617546a4895161e0e1a9e466f0c51d91561fa39dcb9aee9741225 |
| factorization | ['0x49', '0x12c1', '0x1349', '0x3871', '0x3bf5', '0xbe4bd8ace351', '0x4b647ecc2fe6f6357', '0x16ddbac4ec04232947'] |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x2 |
| complement_bit_length | 0x2 |
| $\text{class_number}()$ | |
| upper | 0x2 |
| lower | 0x0 |
| $\text{small_prime_order}(l=2)$ | |
| order | 0x333333333333333333338b6537c655855b9e248a |
| complement_bit_length | 0x3 |
| $\text{small_prime_order}(l=3)$ | |
| order | 0x100000000000000000001b8fa16dfab9aca16b6b2 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x80000000000000000000dc7d0b6fd5cd650b5b59 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0x100000000000000000001b8fa16dfab9aca16b6b2 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x111111111111111111112e77129771d71e8a0c2e |
| complement_bit_length | 0x4 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x100000000000000000001b8fa16dfab9aca16b6b2 |
| complement_bit_length | 0x1 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x1', '0x1'], ['0x3', '0x1']] |
| len | 0x2 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x3', '0x4']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=3)$ | |
| crater_degree | 0x1 |
| depth | 0x0 |
| $\text{volcano}(l=5)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=7)$ | |
| crater_degree | 0x1 |
| depth | 0x1 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=17)$ | |
| crater_degree | 0x0 |
| 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 | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x1 |
| full | 0x1 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x9 |
| relative | 0x9 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | -0x1b8fa16dfab9bca170a3f |
| trace_factorization | ['0x1349', '0x16ddbac4ec04232947'] |
| number_of_factors | 0x2 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | -0x1b8fa16dfab9bca170a3f |
| trace_factorization | ['0x49', '0x12c1', '0x3871', '0x3bf5', '0xbe4bd8ace351', '0x4b647ecc2fe6f6357'] |
| number_of_factors | 0x6 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x3 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x4 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x6 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x56 |
| expected | 0x50 |
| ratio | 0.93023 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x18cd |
| expected | 0x1928 |
| ratio | 1.01433 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x5348e |
| expected | 0x53548 |
| ratio | 1.00055 |
| $\text{square_4p1}()$ | |
| p | 0x3 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x1b8fa16dfab9aca16b6b3 |
| ratio | 7.018168111136911e+23 |
| distance 32 | 0xd |
| distance 64 | 0xd |
| $\text{multiples_x}(k=1)$ | |
| Hx | 0x3b4c382ce37aa192a4019e763036f4f5dd4d7ebb |
| bits | 0x9e |
| difference | 0x3 |
| ratio | 0.98137 |
| $\text{multiples_x}(k=2)$ | |
| Hx | 0x48ce563f89a0ed9414f5aa28ad0d96d6795f9c62 |
| bits | 0x9f |
| difference | 0x2 |
| ratio | 0.98758 |
| $\text{multiples_x}(k=3)$ | |
| Hx | 0x22f058ad4d86af34604e1de3b760d76706b8e67 |
| bits | 0x9a |
| difference | 0x7 |
| ratio | 0.95652 |
| $\text{multiples_x}(k=4)$ | |
| Hx | 0x78c0859f91740ae3cb01b5a1770f1432e2afd93d |
| bits | 0x9f |
| difference | 0x2 |
| ratio | 0.98758 |
| $\text{multiples_x}(k=5)$ | |
| Hx | 0x7b9173b2e08e135046078e1ffe624139c83f2e6b |
| bits | 0x9f |
| difference | 0x2 |
| ratio | 0.98758 |
| $\text{multiples_x}(k=6)$ | |
| Hx | 0x9f1bf45413940d5c6c998f23efe68287ff89f949 |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=7)$ | |
| Hx | 0xd29ec7d5e7369a09ef5e5f80d3a2059e81b39dc1 |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=8)$ | |
| Hx | 0xcf0ae83d47d45582b0612d3a8b4a4d2e2d176b60 |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=9)$ | |
| Hx | 0xa8a65a2d54a3bddf7121a90c22a8ddb816eaeee |
| bits | 0x9c |
| difference | 0x5 |
| ratio | 0.96894 |
| $\text{multiples_x}(k=10)$ | |
| Hx | 0x6c383e0d76cdceeaa4fe4d7f9cd2989cfb7befc0 |
| bits | 0x9f |
| difference | 0x2 |
| ratio | 0.98758 |
| $\text{x962_invariant}()$ | |
| r | 0x0 |
| $\text{brainpool_overlap}()$ | |
| o | None |
| $\text{weierstrass}()$ | |
| a | 0x0 |
| b | 0x7 |