Curve detail
Definition
| Name | secp160r2 |
|---|---|
| Category | secg |
| Description | A randomly generated curve. [SEC2v1](https://www.secg.org/SEC2-Ver-1.0.pdf) states 'E was chosen verifiably at random as specified in ANSI X9.62 [1] from the seed'. |
| Field | Prime (0xfffffffffffffffffffffffffffffffeffffac73) |
| Field bits | 160 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0xfffffffffffffffffffffffffffffffeffffac70 |
| Param $b$ | 0xb4e134d3fb59eb8bab57274904664d5af50388ba |
| Generator $x$ | 0x52dcb034293a117e1f4ff11b30f7199d3144ce6d |
| Generator $y$ | 0xfeaffef2e331f296e071fa0df9982cfea7d43f2e |
| Simulation seed | 0xb99b99b099b323e02709a4d696e6768756151751 |
Characteristics
| Order | 0x100000000000000000000351ee786a818f3a1a16b |
| Cofactor | 0x1 |
| $j$-invariant | 0xb3e6e6ec627556928cb0ebd0f934efaddc70f163 |
| Trace $t$ | -0x351ee786a819f3a1f4f7 |
| Embedding degree $k$ | 0x800000000000000000001a8f73c3540c79d0d0b5 |
| CM discriminant | -0x3f4fa3067297190d69760e813bc691a3b0ce8eb7b |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x100000000000000000000351ee786a818f3a1a16b |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | -0x3f4fa3067297190d69760e813bc691a3b0ce8eb7b |
| factorization | ['0x6d3', '0x24a09d', '0x40d7b7898bc458ceae06869cd1862facd'] |
| max_conductor | 0x1 |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0xffffffffffffffffffffcae1187957e50c5db77d |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xfffffffffffffffffffffffffffffffdffff58e40b05cf98d68e6f29689f17ed43978cdd0e5b2915 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x2', '0x2', '0x3', '0x13', '0x35', '0x56c61f5a96543b516f6efec512bc83e382cc7'] |
| (+)largest_factor_bitlen | 0x93 |
| (-)factorization | ['0x2', '0x4d5', '0x511', '0x7b2b66d09aaae93f3', '0xaddff89dc11b9145ab'] |
| (-)largest_factor_bitlen | 0x48 |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0xcbb', '0x10dc5', '0x453b7f834733', '0x8d1fee403af6534900272b'] |
| (+)largest_factor_bitlen | 0x58 |
| (-)factorization | ['0x3', '0x3', '0x5', '0x2b', '0x9c2689', '0x6f0d40782fcb5ba03b947a09023f8bc3'] |
| (-)largest_factor_bitlen | 0x7f |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x2', '0x5', '0x7', '0x1d', '0x1910b', '0x46e75cd', '0x1ba2e357', '0xa3244639', '0xcac97dcc43f7'] |
| (+)largest_factor_bitlen | 0x30 |
| (-)factorization | ['0x2', '0x2', '0x2', '0x2', '0x2', '0x2', '0xb', '0x15d7', '0x2eb2c77a100c5026d', '0x461992bfe3a92a48259'] |
| (-)largest_factor_bitlen | 0x4b |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0x3', '0x11', '0x551', '0x1085', '0x1611d', '0x12f815476e33d', '0x23ca11c89f6fe208eb'] |
| (+)largest_factor_bitlen | 0x46 |
| (-)factorization | ['0x7', '0x264523', '0x4dc0159a7', '0x166d5234a9', '0x8faa086faa32a2421'] |
| (-)largest_factor_bitlen | 0x44 |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x2', '0x2', '0x2', '0xd', '0x49', '0x65', '0x7d05487', '0x189d87bb', '0x13ea0a0d2b', '0x74fced025d475'] |
| (+)largest_factor_bitlen | 0x33 |
| (-)factorization | ['0x2', '0x3', '0x29', '0x29', '0x13bdf', '0x65a8f', '0x424e46973ac62fe0f8cf26dc1ddb9'] |
| (-)largest_factor_bitlen | 0x73 |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0x1f', '0x17b', '0x1382e0b8f', '0x1b71f44d55e2e9b59b0509525b6dc9'] |
| (+)largest_factor_bitlen | 0x75 |
| (-)factorization | ['0x6000000000000000000013eb96d27f095b5c9c881'] |
| (-)largest_factor_bitlen | 0xa3 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x2', '0x3', '0x3', '0x3', '0xd4ccdd9d', '0x27ec06d387165d6808e7df6e6a30a019'] |
| (+)largest_factor_bitlen | 0x7e |
| (-)factorization | ['0x2', '0x2', '0x5', '0x5', '0x74420332661', '0x2775b9fcf567b2d9461ac3d682893'] |
| (-)largest_factor_bitlen | 0x72 |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x5', '0xb', '0x17f', '0x853', '0x3ce35', '0xc9234531582871db8fae0416f8d83f'] |
| (+)largest_factor_bitlen | 0x78 |
| (-)factorization | ['0x3', '0xd', '0x348348348348348348348e2db956b2e44c3b6951'] |
| (-)largest_factor_bitlen | 0x9e |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x14 |
| full | 0x14 |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x15 |
| full | 0x15 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x120 |
| full | 0x120 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x351ee786a819f3a1f4f7 |
| factorization | ['0x3', '0x7', '0xb', '0x1382d19', '0x3046ae6a8219'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0xf4fa3067297190d69760e813bc691a3e0ce9e622 |
| factorization | ['0x2', '0x5', '0xb5', '0x19b5', '0xb8c1', '0x798bb', '0x19932512001d', '0x275fc22aa0abf0b'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x67f4466c4d44c3a96b920435d2b2842041a82cc3290d22106ecbb3c771c3 |
| factorization | ['0x3', '0x7', '0xb', '0x239', '0x50b', '0xb47', '0x1382d19', '0x97781d11', '0x3046ae6a8219', '0x6b2c789e2c923c0c03239ade1'] |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x2 |
| complement_bit_length | 0x2 |
| $\text{class_number}()$ | |
| upper | 0x4717e1e65bd3bdf7318d7b |
| lower | 0xda |
| $\text{small_prime_order}(l=2)$ | |
| order | 0x100000000000000000000351ee786a818f3a1a16a |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=3)$ | |
| order | 0x800000000000000000001a8f73c3540c79d0d0b5 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x100000000000000000000351ee786a818f3a1a16a |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0x100000000000000000000351ee786a818f3a1a16a |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x100000000000000000000351ee786a818f3a1a16a |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x100000000000000000000351ee786a818f3a1a16a |
| complement_bit_length | 0x1 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x2', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x3', '0x4']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=3)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=5)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=7)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x0 |
| 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 | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x7 |
| full | 0x7 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x12 |
| full | 0x12 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x9 |
| relative | 0x9 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | -0x351ee786a819f3a1f4f7 |
| trace_factorization | ['0x3', '0x7', '0xb', '0x1382d19', '0x3046ae6a8219'] |
| number_of_factors | 0x5 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | -0x351ee786a819f3a1f4f7 |
| trace_factorization | ['0x239', '0x50b', '0xb47', '0x97781d11', '0x6b2c789e2c923c0c03239ade1'] |
| number_of_factors | 0x5 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x0 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x58 |
| expected | 0x50 |
| ratio | 0.90909 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x1956 |
| expected | 0x1928 |
| ratio | 0.99291 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x5329c |
| expected | 0x53548 |
| ratio | 1.00201 |
| $\text{square_4p1}()$ | |
| p | 0x3 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x351ee786a818f3a1a16b |
| ratio | 5.826069526844493e+24 |
| distance 32 | 0xb |
| distance 64 | 0x15 |
| $\text{multiples_x}(k=1)$ | |
| Hx | 0x52dcb034293a117e1f4ff11b30f7199d3144ce6d |
| bits | 0x9f |
| difference | 0x2 |
| ratio | 0.98758 |
| $\text{multiples_x}(k=2)$ | |
| Hx | 0x4a96b5688ef573284664698e81e1510fae4e8422 |
| bits | 0x9f |
| difference | 0x2 |
| ratio | 0.98758 |
| $\text{multiples_x}(k=3)$ | |
| Hx | 0x1421c8c619daec4731e83038bbed610761c0d911 |
| bits | 0x9d |
| difference | 0x4 |
| ratio | 0.97516 |
| $\text{multiples_x}(k=4)$ | |
| Hx | 0xff7d1cec7bee4058971a9823505705b146764206 |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=5)$ | |
| Hx | 0xf6e377f2a6d723d301261d5c05d0d2c731a25a1d |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=6)$ | |
| Hx | 0xf4dfb91e3ae6d32bbeb4efe9e50e05bc6021ca2a |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=7)$ | |
| Hx | 0xb5ac65e674bf133a87d43b49ff5cf8095aad7ee2 |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=8)$ | |
| Hx | 0x95c8e3e6d91806a62575820c2896779a27432bd4 |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=9)$ | |
| Hx | 0xaa25e2b10286f97d27a8ae11229d6aaa5eba1153 |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{multiples_x}(k=10)$ | |
| Hx | 0xb31ff6619745e512942048b1fbd1d89f5b21046b |
| bits | 0xa0 |
| difference | 0x1 |
| ratio | 0.99379 |
| $\text{x962_invariant}()$ | |
| r | 0x5f1a5a70d2a4b9f1f0c9293f148d4b79b99060d2 |
| $\text{brainpool_overlap}()$ | |
| o | None |
| $\text{weierstrass}()$ | |
| a | 0xfffffffffffffffffffffffffffffffeffffac70 |
| b | 0xb4e134d3fb59eb8bab57274904664d5af50388ba |