Curve detail
Definition
| Name | B-233 (nist/B-233, secg/sect233r1, x962/ansit233r1) |
|---|---|
| Category | nist |
| Field | Binary |
| Field polynomial | $x^{233} + x^{74} + 1$ |
| Field bits | 233 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x000000000000000000000000000000000000000000000000000000000001 |
| Param $b$ | 0x0066647ede6c332c7f8c0923bb58213b333b20e9ce4281fe115f7d8f90ad |
| Generator $x$ | 0x00fac9dfcbac8313bb2139f1bb755fef65bc391f8b36f8f8eb7371fd558b |
| Generator $y$ | 0x01006a08a41903350678e58528bebf8a0beff867a7ca36716f7e01f81052 |
| Simulation seed | 0x74d59ff07f6b413d0ea14b344b20a2db049b50c3 |
Characteristics
| Order | 0x1000000000000000000000000000013e974e72f8a6922031d2603cfe0d7 |
| Cofactor | 0x2 |
| Trace $t$ | -0x27d2e9ce5f14d244063a4c079fc1ad |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x1000000000000000000000000000013e974e72f8a6922031d2603cfe0d7 |
| cofactor | 0x2 |
| $\text{discriminant}()$ | |
| cm_disc | -0x1ce0efeb2ea5a553eed56a0c8a4bca205c5305973c2190c7cfe6293b117 |
| factorization | ['0x7', '0x25f', '0x33ab00b639', '0xa488689bf7', '0xd69cf03f370ed4df158db869b94af7fca6d1'] |
| max_conductor | 0x1 |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x1ffffffffffffffffffffffffffffd82d1631a0eb2dbbf9c5b3f8603e54 |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x4000000000000000000000000000000000000000000000000000000000231f1014d15a5aac112a95f375b435dfa3acfa68c3de6f383019d6c4eea |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x29', '0x47', '0x2d06c584b475239ddb3d78bea8ac616cb0652a923b78ec3ba6a73bb1'] |
| (+)largest_factor_bitlen | 0xde |
| (-)factorization | ['0x3', '0x5', '0x5', '0x4f', '0x89', '0x15d', '0x841', '0x3ac7353f7f894a34875f936ab343ba06a56fb5b999ab54b95'] |
| (-)largest_factor_bitlen | 0xc2 |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0x3', '0x3', '0xb', '0x9ad', '0x128e01a4d', '0xebfc2a56a87fbafd3ec380f075d94c0fb56b001b0d1c07'] |
| (+)largest_factor_bitlen | 0xb8 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x7', '0x13', '0x17', '0x1d', '0x6d', '0x17c66c595a89a4a71', '0x70183c297838b71668240b5b573f4846b1941'] |
| (+)largest_factor_bitlen | 0x93 |
| (-)factorization | ['0x1faf', '0x8992188383f', '0x5a369e83852a1fff282c8032462010241feab4be03979'] |
| (-)largest_factor_bitlen | 0xb3 |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0x5', '0x2bcd8e198dc9', '0x141c9dc5409cc7', '0x7706f29e5cf3d5cd44a638bcec4116f99b'] |
| (+)largest_factor_bitlen | 0x87 |
| (-)factorization | ['0x3', '0x7', '0x7', '0x1b1', '0x268703', '0x36bb30baec1388c6bcffb2362d18741189a72e23af3fbd749f'] |
| (-)largest_factor_bitlen | 0xc6 |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x3', '0x43', '0x21ccf4218d2015155', '0x47413e4085f1f45565', '0x15a90b15fb846b44f3ce9777'] |
| (+)largest_factor_bitlen | 0x5d |
| (-)factorization | ['0x1f', '0x125ea3ae9', '0x1295212501b71', '0x3dee760aa8678137142c6f74217c1940afc0f3'] |
| (-)largest_factor_bitlen | 0x96 |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0xc0000000000000000000000000000eef17ad63a7ced98255dc82dbe8a15'] |
| (+)largest_factor_bitlen | 0xec |
| (-)factorization | ['0x5', '0xd', '0x1522061eff', '0x23c846c975b8444d1f425c3c9c628f51fc87aad1f9d94bdad'] |
| (-)largest_factor_bitlen | 0xc2 |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0x3', '0x3', '0x11', '0x295', '0xf596a7', '0x326fc4c1330e3a90c2c6b7ab223b260e6639013570596ee51'] |
| (-)largest_factor_bitlen | 0xc2 |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x1954681b7', '0x35e70639f41e4b9acb1048a1411b6bb4934f83f9e5795b6435d'] |
| (+)largest_factor_bitlen | 0xca |
| (-)factorization | ['0x23c1bf', '0x2d97d201', '0x7b68f1365b', '0x33eae5bdc3132f9637', '0x19b2f82e308a550e66d5'] |
| (-)largest_factor_bitlen | 0x4d |
| $\text{torsion_extension}(l=2)$ | |
| least | None |
| full | None |
| relative | None |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x2 |
| full | 0x4 |
| relative | 0x2 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x6 |
| full | 0x7 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x28 |
| full | 0x28 |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0xa8 |
| full | 0xa8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x27d2e9ce5f14d244063a4c079fc1ad |
| factorization | ['0x13a41', '0x21523', '0x2fd9f9f1', '0x5356d07c67dd1f'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x431f1014d15a5aac112a95f375b435dfa3acfa68c3de6f383019d6c4ee9 |
| factorization | ['0x1f', '0x67', '0x6493f', '0x1a2f969', '0x4e4770526bb9ef', '0x1b5ee72faa6035be44929fcfca4510f9'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x576ab01f2a5c578169c19adbbbfab4ab8d19924e46a189fe04786b0c0c72d6284a58318d1520b35f6b65fc75 |
| factorization | ['0x3', '0x5', '0xb', '0x11', '0x17', '0x13a41', '0x21523', '0x2fd9f9f1', '0x113a47f519', '0x5356d07c67dd1f', '0xa91b8c273890688473f', '0x32290e7ca88a80f30da5815a225'] |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x2 |
| complement_bit_length | 0x2 |
| $\text{class_number}()$ | |
| upper | 0x45057c97fa2b6c87d7f2484225adab9 |
| lower | 0x40e919 |
| $\text{small_prime_order}(l=2)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=3)$ | |
| order | 0x800000000000000000000000000009f4ba7397c53491018e9301e7f06b |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x1000000000000000000000000000013e974e72f8a6922031d2603cfe0d6 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0x1000000000000000000000000000013e974e72f8a6922031d2603cfe0d6 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x800000000000000000000000000009f4ba7397c53491018e9301e7f06b |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x800000000000000000000000000009f4ba7397c53491018e9301e7f06b |
| complement_bit_length | 0x2 |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x1', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x4', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x1', '0x2'], ['0x2', '0x1'], ['0x4', '0x2']] |
| len | 0x3 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=3)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=5)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=7)$ | |
| crater_degree | 0x1 |
| 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 | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=19)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{isogeny_extension}(l=2)$ | |
| least | None |
| full | None |
| relative | None |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x1 |
| full | 0x4 |
| relative | 0x4 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x1 |
| full | 0x7 |
| relative | 0x7 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0xe |
| full | 0xe |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x4 |
| relative | 0x4 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x12 |
| relative | 0x12 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | -0x27d2e9ce5f14d244063a4c079fc1ad |
| trace_factorization | ['0x13a41', '0x21523', '0x2fd9f9f1', '0x5356d07c67dd1f'] |
| number_of_factors | 0x4 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | -0x27d2e9ce5f14d244063a4c079fc1ad |
| trace_factorization | ['0x3', '0x5', '0xb', '0x11', '0x17', '0x113a47f519', '0xa91b8c273890688473f', '0x32290e7ca88a80f30da5815a225'] |
| number_of_factors | 0x8 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x3 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x2 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0xe9 |
| expected | 0x75 |
| ratio | 0.50215 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x6994 |
| expected | 0x353e |
| ratio | 0.50429 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x1fc184 |
| expected | 0x10158c |
| ratio | 0.50649 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | NO DATA (timed out) |
| $\text{pow_distance}()$ | |
| distance | 0x27d2e9ce5f14d244063a4c079fc1ae |
| ratio | 6.675532331333225e+34 |
| distance 32 | 0xe |
| distance 64 | 0x12 |
| $\text{multiples_x}(k=1)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=2)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=3)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=4)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=5)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=6)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=7)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=8)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=9)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{multiples_x}(k=10)$ | |
| Hx | None |
| bits | None |
| difference | None |
| ratio | None |
| $\text{x962_invariant}()$ | |
| r | 0x0 |
| $\text{brainpool_overlap}()$ | |
| o | None |
| $\text{weierstrass}()$ | |
| a | None |
| b | None |