Curve detail
Definition
| Name | P-192 (nist/P-192, secg/secp192r1, x962/ansix9p192r1) |
|---|---|
| Category | nist |
| Field | Prime (0xfffffffffffffffffffffffffffffffeffffffffffffffff) |
| Field bits | 192 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0xfffffffffffffffffffffffffffffffefffffffffffffffc |
| Param $b$ | 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1 |
| Generator $x$ | 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012 |
| Generator $y$ | 0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811 |
| Simulation seed | 0x3045ae6fc8422f64ed579528d38120eae12196d5 |
Characteristics
| Order | 0xffffffffffffffffffffffff99def836146bc9b1b4d22831 |
| Cofactor | 0x1 |
| $j$-invariant | 0xfe40fc48ef4b5633d091b1a2707da063644b08bb896f560b |
| Trace $t$ | 0x662107c8eb94364e4b2dd7cf |
| Embedding degree $k$ | 0x1999999999999999999999998f6318d2353dfa91c5483738 |
| CM discriminant | -0x3d741a988e23c8df090918240db2143ab31994533214ca69b |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0xffffffffffffffffffffffff99def836146bc9b1b4d22831 |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | -0x3d741a988e23c8df090918240db2143ab31994533214ca69b |
| factorization | ['0x5', '0xb', '0x1f', '0x93a1f51e9cc9e15c6cca9fcb46e383073c72e71b0bc1a3'] |
| max_conductor | 0x1 |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x1000000000000000000000000662107c7eb94364e4b2dd7cf |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xfffffffffffffffffffffffffffffffdfffffffffffffffc28be56771dc372106f6e7dbf24debc54ce66baccdeb35965 |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | - |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | - |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0x1abc9d3e40f', '0x702abd9380f5a97', '0x2bb49aeef2cab7f6c945179b'] |
| (+)largest_factor_bitlen | 0x5e |
| (-)factorization | ['0x3', '0x3', '0x18fb205b', '0x246fc7574a80bfc68fa7a366f71e653536648fe1b'] |
| (-)largest_factor_bitlen | 0xa2 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x2', '0x2', '0x7', '0xd', '0xcdb4c9', '0x27f5aead', '0x10d25bc94502eb7ae65de5448fc94aae03'] |
| (+)largest_factor_bitlen | 0x85 |
| (-)factorization | ['0x2', '0xb', '0x107', '0x26a85', '0xe10633fe29115d2368e73b0481a3dc404ac9c3349'] |
| (-)largest_factor_bitlen | 0xa4 |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0x3', '0x5', '0x5', '0x95', '0xa3385f', '0x12ab9f3219c9b085d091', '0x1f87b24ff3792ed0faedd'] |
| (+)largest_factor_bitlen | 0x51 |
| (-)factorization | ['0x7', '0x17', '0x2b9b1', '0xb5e0563', '0x348eb688561844f87a3834543374b4900bd1'] |
| (-)largest_factor_bitlen | 0x8e |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x2', '0x27fffffffffffffffffffffff00ad6c87330d783c440d647b'] |
| (+)largest_factor_bitlen | 0xc2 |
| (-)factorization | ['0x2', '0x2', '0x3', '0x55103', '0x3b6a65857c58e9', '0x56723e36483839e1729771e43c3ccd'] |
| (-)largest_factor_bitlen | 0x77 |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0x1f', '0x2d14b', '0x5fc902d', '0x1dea28a3', '0x81bc537c9a8dd', '0x319a974787f8f5161'] |
| (+)largest_factor_bitlen | 0x42 |
| (-)factorization | ['0x5', '0xe8bd57', '0x151e6ef4e64694c82213e456fc6194bfa44c33ad8c7'] |
| (-)largest_factor_bitlen | 0xa9 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x2', '0x2', '0x2', '0x3', '0x3', '0x3', '0x11', '0x11', '0x13', '0x1d', '0xa7', '0xa4fccfa5', '0x12c3c6ddf', '0xbdf81e85d5b5', '0x954ea7ca9f8e7'] |
| (+)largest_factor_bitlen | 0x34 |
| (-)factorization | ['0x2', '0x35', '0x15bb', '0xd70a96a017b', '0xed181f258eedb6dffa32f1023007c3ff7'] |
| (-)largest_factor_bitlen | 0x84 |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0xb', '0x11f27ba464ab', '0xa5fb59b51d42c08af32c4ce5873744f7589b1'] |
| (+)largest_factor_bitlen | 0x94 |
| (-)factorization | ['0x3', '0x53', '0xc1', '0x60f7d', '0x69abdbdaf', '0x18b140b6cb507', '0x2d368df2675e8d182d93'] |
| (-)largest_factor_bitlen | 0x4e |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x4 |
| full | 0x5 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x30 |
| full | 0x30 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x5 |
| full | 0xb |
| relative | 0x2 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x4 |
| full | 0xc |
| relative | 0x3 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x24 |
| full | 0x24 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x662107c8eb94364e4b2dd7cf |
| factorization | ['0x59', '0xb3', '0x17f', '0x17156f011', '0xc2a4e727d3'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0xd741a988e23c8df090918240db2143ae31994533214ca69e |
| factorization | ['0x2', '0xd', '0x6443', '0x50a217d', '0x7112677918206154a5', '0x97f311a604bf144d9c9'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0xbc00f7580d62980b31c69dc340e7bb4fa23282012035dfa349cfbe6ebb0f166e958193f3 |
| factorization | ['0x3', '0x3', '0x25', '0x59', '0xb3', '0x17f', '0x841', '0xede15f', '0x17156f011', '0xc2a4e727d3', '0x3eee68c777f', '0xc0274b7e2959f32be737a6e0931'] |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0xa |
| complement_bit_length | 0x4 |
| $\text{class_number}()$ | |
| upper | 0x53dc64eef1d84973106b4b1a2f |
| lower | 0x12df |
| $\text{small_prime_order}(l=2)$ | |
| order | 0x7fffffffffffffffffffffffccef7c1b0a35e4d8da691418 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=3)$ | |
| order | 0xffffffffffffffffffffffff99def836146bc9b1b4d22830 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x3fffffffffffffffffffffffe677be0d851af26c6d348a0c |
| complement_bit_length | 0x3 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0x7fffffffffffffffffffffffccef7c1b0a35e4d8da691418 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x7fffffffffffffffffffffffccef7c1b0a35e4d8da691418 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x3fffffffffffffffffffffffe677be0d851af26c6d348a0c |
| complement_bit_length | 0x3 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x4', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x2', '0x1'], ['0xa', '0x1']] |
| 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 | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x1 |
| 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 | 0x0 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x1 |
| full | 0x5 |
| relative | 0x5 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0xb |
| relative | 0xb |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x9 |
| full | 0x9 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x14 |
| full | 0x14 |
| relative | 0x1 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x662107c8eb94364e4b2dd7cf |
| trace_factorization | ['0x59', '0xb3', '0x17f', '0x17156f011', '0xc2a4e727d3'] |
| number_of_factors | 0x5 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x662107c8eb94364e4b2dd7cf |
| trace_factorization | ['0x3', '0x3', '0x25', '0x841', '0xede15f', '0x3eee68c777f', '0xc0274b7e2959f32be737a6e0931'] |
| number_of_factors | 0x6 |
| $\text{isogeny_neighbors}(l=2)$ | |
| 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 | 0x65 |
| expected | 0x60 |
| ratio | 0.9505 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x23f6 |
| expected | 0x23d0 |
| ratio | 0.99587 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x8dad8 |
| expected | 0x8dc20 |
| ratio | 1.00057 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x662107c9eb94364e4b2dd7cf |
| ratio | 1.985959386609862e+29 |
| distance 32 | 0xf |
| distance 64 | 0xf |
| $\text{multiples_x}(k=1)$ | |
| Hx | 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012 |
| bits | 0xbd |
| difference | 0x3 |
| ratio | 0.98438 |
| $\text{multiples_x}(k=2)$ | |
| Hx | 0x7b4603cc4ac847264022b07144c25277f2ad8fbe9224728f |
| bits | 0xbf |
| difference | 0x1 |
| ratio | 0.99479 |
| $\text{multiples_x}(k=3)$ | |
| Hx | 0x9697d1870247164b841077ca1a546084dfba2653ae29cb51 |
| bits | 0xc0 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=4)$ | |
| Hx | 0xf8330bf5b681db4682a149bf076bf8d2c64d67ee2e7d25e6 |
| bits | 0xc0 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=5)$ | |
| Hx | 0x11fecb910f2730d6aa3c242d2800ee1a1dbd449f487ee5ff |
| bits | 0xbd |
| difference | 0x3 |
| ratio | 0.98438 |
| $\text{multiples_x}(k=6)$ | |
| Hx | 0xa8d9ab9cdce3820d9ccc046ee9f1224b40c2d3ef3068a365 |
| bits | 0xc0 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=7)$ | |
| Hx | 0xaa3c3adb52e99783db0fe26f55cf53d442c0da5443b0dfc6 |
| bits | 0xc0 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=8)$ | |
| Hx | 0xbd2115367b7fabc16c47bbc5df5df0a565b9ad36fde9eba5 |
| bits | 0xc0 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=9)$ | |
| Hx | 0xbaa8b91724a3adc05cee5a6f20697b7a92b2a61d6243c81f |
| bits | 0xc0 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=10)$ | |
| Hx | 0x5c2cdc18eb53508e269d3dd00456200bf955863f635fa7a4 |
| bits | 0xbf |
| difference | 0x1 |
| ratio | 0.99479 |
| $\text{x962_invariant}()$ | |
| r | 0x3099d2bbbfcb2538542dcd5fb078b6ef5f3d6fe2c745de65 |
| $\text{brainpool_overlap}()$ | |
| o | 0x1bdefae3 |
| $\text{weierstrass}()$ | |
| a | 0xfffffffffffffffffffffffffffffffefffffffffffffffc |
| b | 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1 |