Curve detail
Definition
| Name | sect163r2 (nist/B-163, secg/sect163r2, x962/ansit163r2) |
|---|---|
| Category | secg |
| Description | A randomly generated curve. 'E was selected from S as specified in ANSI X9.62 [X9.62] in normal basis representation and converted into polynomial basis representation.' |
| Field | Binary |
| Field polynomial | $x^{163} + x^{7} + x^{6} + x^{3} + 1$ |
| Field bits | 163 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x000000000000000000000000000000000000000001 |
| Param $b$ | 0x020a601907b8c953ca1481eb10512f78744a3205fd |
| Generator $x$ | 0x03f0eba16286a2d57ea0991168d4994637e8343e36 |
| Generator $y$ | 0x00d51fbc6c71a0094fa2cdd545b11c5c0c797324f1 |
| Simulation seed | 0x85e25bfe5c86226cdb12016f7553f9d0e693a268 |
Characteristics
| Order | 0x40000000000000000000292fe77e70c12a4234c33 |
| Cofactor | 0x2 |
| $j$-invariant | 0xa1f9aabdf28d9e1ae61dfdf41a22011cfdd17e82 |
| Trace $t$ | -0x525fcefce182548469865 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x40000000000000000000292fe77e70c12a4234c33 |
| cofactor | 0x2 |
| $\text{discriminant}()$ | |
| cm_disc | -0x57e7b8ab87d2e65f2ddc9a202f211399d7e0be827 |
| factorization | ['0x599', '0x1d5447', '0xd581ad22b084fcf', '0xa45a43146d75947be7'] |
| max_conductor | 0x1 |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x7fffffffffffffffffffada031031e7dab7b9679c |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x40000000000000000000000000000000000000000a8184754782d19a0d22365dfd0deec66281f417da |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | ['0x5', '0x5', '0x53111d5f4d', '0x14005e422fc0f', '0xc9f53a968ea6840455'] |
| (+)largest_factor_bitlen | 0x48 |
| (-)factorization | ['0x3', '0x3', '0x7', '0x1eebd', '0x7a22a6381', '0x2341f07239a8313a92bf6425e77'] |
| (-)largest_factor_bitlen | 0x6a |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | ['0x3', '0x25', '0x49d', '0xd345e881', '0x8be8abf699', '0x11bc046f5b48629b7777'] |
| (+)largest_factor_bitlen | 0x4d |
| (-)factorization | ['0xd', '0x15d', '0xe71dd94ae03b95b2054e6418b4fd3eee2528a3'] |
| (-)largest_factor_bitlen | 0x98 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x18d2e69', '0x119c2ddd', '0xe0e00072c0de4319e05f5a8ecbf7'] |
| (+)largest_factor_bitlen | 0x70 |
| (-)factorization | ['0x13', '0x1435e50d79435e50d79442e6b4e48f656ba91e22b'] |
| (-)largest_factor_bitlen | 0xa1 |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0x4f', '0x1dcc1185', '0x7489862f5', '0x8a693955fe67', '0xe23b3c00e2c9'] |
| (+)largest_factor_bitlen | 0x30 |
| (-)factorization | ['0x3', '0x5', '0xb', '0xa88f', '0x4ef55171af5d51', '0xf47a05ae7e3e76264cacb5'] |
| (-)largest_factor_bitlen | 0x58 |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x3', '0x16db', '0x469d', '0x21d6e0640dff58a7e5acca1e6f8263d271b'] |
| (+)largest_factor_bitlen | 0x8a |
| (-)factorization | ['0x9c2b', '0x5c84bd', '0x4ea9dc15', '0x24e742f3ad246392306f7f1f7'] |
| (-)largest_factor_bitlen | 0x62 |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0x5', '0x7', '0x11', '0x17', '0x175', '0x5d916a569904ae0b', '0x1afa3685b5f6c8d68b707'] |
| (+)largest_factor_bitlen | 0x51 |
| (-)factorization | ['0x498aea9889527', '0x17ac0777146141', '0x70ef452f5749a825'] |
| (-)largest_factor_bitlen | 0x3f |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0xb', '0x97', '0x14a2b3', '0x2567660d7fa6fb775', '0x2dcd571e705d13d469'] |
| (+)largest_factor_bitlen | 0x46 |
| (-)factorization | ['0x3', '0x59f', '0x1c7f172e2c63', '0x47ed7c609b0f', '0x6a2dda8d8a53ea51'] |
| (-)largest_factor_bitlen | 0x3f |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x3', '0x71c71c71c71c71c71c72105529c456acbce77a3e9'] |
| (+)largest_factor_bitlen | 0xa3 |
| (-)factorization | ['0x7', '0x1d', '0xb3cfcd713', '0x72e826220654ed2b92c56b6e6229b3f'] |
| (-)largest_factor_bitlen | 0x7b |
| $\text{torsion_extension}(l=2)$ | |
| least | None |
| full | None |
| relative | None |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x18 |
| full | 0x18 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x5 |
| full | 0xa |
| relative | 0x2 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0xa8 |
| full | 0xa8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x30 |
| full | 0x30 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x525fcefce182548469865 |
| factorization | ['0x5', '0x1d', '0x17141', '0x181a9', '0x78f49', '0x8da765'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x128184754782d19a0d22365dfd0deec66281f417d9 |
| factorization | ['0x11', '0x17', '0x32989', '0x3d4e2aad0fbb92c8ca6421316dff18971e7'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x3616afc51c362550fcbd45148799e7e752b88b52087fe8aa9cbbd6c0cc409d |
| factorization | ['0x3', '0x3', '0x5', '0x7', '0x1d', '0x2b', '0x3a9', '0x17141', '0x181a9', '0x78f49', '0x8da765', '0x44061fffb', '0x1580162e81', '0xc279032db4af7be1f7'] |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x1 |
| complement_bit_length | 0x1 |
| $\text{class_number}()$ | |
| upper | 0x54042c4b1aae11f5f95a40 |
| lower | 0x5ff8 |
| $\text{small_prime_order}(l=2)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=3)$ | |
| order | 0x200000000000000000001497f3bf386095211a619 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=5)$ | |
| order | 0x40000000000000000000292fe77e70c12a4234c32 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=7)$ | |
| order | 0x40000000000000000000292fe77e70c12a4234c32 |
| complement_bit_length | 0x1 |
| $\text{small_prime_order}(l=11)$ | |
| order | 0x200000000000000000001497f3bf386095211a619 |
| complement_bit_length | 0x2 |
| $\text{small_prime_order}(l=13)$ | |
| order | 0x200000000000000000001497f3bf386095211a619 |
| 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 | [['0x4', '0x3']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x2 |
| 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 | 0x2 |
| 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 | None |
| full | None |
| relative | None |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0xa |
| relative | 0xa |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0xe |
| full | 0xe |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x12 |
| relative | 0x12 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | -0x525fcefce182548469865 |
| trace_factorization | ['0x5', '0x1d', '0x17141', '0x181a9', '0x78f49', '0x8da765'] |
| number_of_factors | 0x6 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | -0x525fcefce182548469865 |
| trace_factorization | ['0x3', '0x3', '0x7', '0x2b', '0x3a9', '0x44061fffb', '0x1580162e81', '0xc279032db4af7be1f7'] |
| number_of_factors | 0x7 |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x3 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x0 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0xa3 |
| expected | 0x52 |
| ratio | 0.50307 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x3393 |
| expected | 0x1a1b |
| ratio | 0.50617 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xacfd1 |
| expected | 0x581b2 |
| ratio | 0.50932 |
| $\text{square_4p1}()$ | |
| p | 0x1 |
| order | 0x1 |
| $\text{pow_distance}()$ | |
| distance | 0x525fcefce182548469866 |
| ratio | 1.8785300127524536e+24 |
| distance 32 | 0x6 |
| distance 64 | 0x1a |
| $\text{multiples_x}(k=1)$ | |
| 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 |