Curve detail
Definition
| Name | P-384 (nist/P-384, secg/secp384r1, x962/ansip384r1) |
|---|---|
| Category | nist |
| Field | Prime (0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000ffffffff) |
| Field bits | 384 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000fffffffc |
| Param $b$ | 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef |
| Generator $x$ | 0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7 |
| Generator $y$ | 0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f |
| Simulation seed | 0xa335926aa319a27a1d00896a6773a4827acdac73 |
Characteristics
| Order | 0xffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf581a0db248b0a77aecec196accc52973 |
| Cofactor | 0x1 |
| $j$-invariant | 0x518a082af8607c03213839c2b6efae43d5cd8c65cd5865926ed9160babd0375237286aca1f65c23db03a303b04a2855b |
| Trace $t$ | 0x389cb27e0bc8d21fa7e5f24cb74f58851313e696333ad68d |
| Embedding degree $k$ | 0x1249249249249249249249249249249249249249249249248e3df34023bacc6b61b8b7d5e09ee764357e9419ea0e153f |
| CM discriminant | -0x3f37b11fed4c640df3189098260a4a0a693503fc3028613e2c0f691ba40196c5ef8997a25fabd8ebb6c58b7c5e64bf653 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0xffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf581a0db248b0a77aecec196accc52973 |
| cofactor | 0x1 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x1000000000000000000000000000000000000000000000000389cb27e0bc8d21ea7e5f24bb74f58851313e697333ad68d |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffdfffffffe0000000000000001fffffffc0c84ee012b39bf20ce76f67d9f5b5f5a6cafc03efd79ec1e3f096e43bfe6939f076685da0542714593a7483a19b409ad |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x5', '0xe17030e6c5', '0x8cdd59a4080a8f85', '0xd3531fe6f2b3c30f8f85f0f05c05b5c290e0d9513d7005824d9d762887ae8265ffe549'] |
| (-)largest_factor_bitlen | 0x118 |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | ['0x2', '0x5', '0x331a69', '0x1bb05e4ab7843eefbb', '0xde506aa4374d3d0940d8bed5c8cd9daddfb658b6c745474e79c9dd5a87928dca2a6943353'] |
| (+)largest_factor_bitlen | 0x124 |
| (-)factorization | ['0x2', '0x2', '0x2', '0x1d', '0x47ed', '0x3d274ed', '0x96854167', '0x53e2e6145682af1cd8045be4746ecc4e963b0ac537dca4672cbd287f915bba804f50dcd961179'] |
| (-)largest_factor_bitlen | 0x133 |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | ['0x13', '0x12c77', '0x25ca3', '0xea391', '0x44e5ef', '0x4ef48a6ce295ffba3e175e040293a5b12d7c19f94031040abb1dcf4e14e800e68f8ce2c4286cd'] |
| (+)largest_factor_bitlen | 0x133 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0x7', '0x2d7', '0x1aab', '0x19f709', '0x2e0c5d', '0x9ed03b8baf04175e5a72f2b04c55a3253b459917a5ccbc508688bce598519b6fae4690aa94b82065'] |
| (+)largest_factor_bitlen | 0x140 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x2', '0xb', '0x17', '0x6fd', '0xedb', '0x122b', '0x587b', '0x28e3e9849ee7', '0x8b508bd68f6f199855c925e88a7fff04f56447352acdf7f8d09e016598f43be1e75cc5f'] |
| (+)largest_factor_bitlen | 0x11c |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x3', '0x5', '0x9a3', '0x17468bc43b', '0x108da7d33099', '0x20050520fe8f7', '0x1916878704b1c1187cbfebac1ce6d4d3395d21e7caf3ea3b7d10e03ed0b93'] |
| (+)largest_factor_bitlen | 0xf1 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0xc |
| full | 0xc |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x28 |
| full | 0x28 |
| relative | 0x1 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x1c |
| full | 0x1c |
| relative | 0x1 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0xc |
| full | 0xc |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x389cb27e0bc8d21fa7e5f24cb74f58851313e696333ad68d |
| factorization | ['0x52f', '0x51fc8935', '0x111e8c17b', '0x1fdf3be8001af353285f891b7277d5'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0xf37b11fed4c640df3189098260a4a0a693503fc3028613e2c0f691ba40196c61f8997a28fabd8ebb6c58b7c2e64bf656 |
| factorization | ['0x2', '0x5', '0x11', '0x2dd16199787', '0x8009d9cb4667e508ef8ca08fa451f9ca5d83293c048a093427e116b0ce1b63a63b6415c84b8705dbe849'] |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x6e74a73035e92e43a8022e89005474f7af1a232d2c456185b5f4f81a6f641670273a87279eebcc0a7e6c11aeb20ce356ab3deb56fa621ed919b3951824db1d4846dffc801603bad1 |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | None |
| complement_bit_length | None |
| $\text{class_number}()$ | |
| upper | NO DATA (timed out) |
| lower | NO DATA (timed out) |
| $\text{small_prime_order}(l=2)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=3)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=5)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=7)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=11)$ | |
| order | None |
| complement_bit_length | None |
| $\text{small_prime_order}(l=13)$ | |
| order | None |
| complement_bit_length | None |
| $\text{division_polynomials}(l=2)$ | |
| factorization | [['0x3', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x4', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x6', '0x2']] |
| 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 | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x7 |
| full | 0x7 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x9 |
| relative | 0x9 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0x389cb27e0bc8d21fa7e5f24cb74f58851313e696333ad68d |
| trace_factorization | ['0x52f', '0x51fc8935', '0x111e8c17b', '0x1fdf3be8001af353285f891b7277d5'] |
| number_of_factors | 0x4 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0x389cb27e0bc8d21fa7e5f24cb74f58851313e696333ad68d |
| trace_factorization | NO DATA (timed out) |
| number_of_factors | NO DATA (timed out) |
| $\text{isogeny_neighbors}(l=2)$ | |
| 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 | 0xad |
| expected | 0xc0 |
| ratio | 1.10983 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x8f6a |
| expected | 0x8fa0 |
| ratio | 1.00147 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0x476ab0 |
| expected | 0x477040 |
| ratio | 1.0003 |
| $\text{square_4p1}()$ | |
| p | NO DATA (timed out) |
| order | NO DATA (timed out) |
| $\text{pow_distance}()$ | |
| distance | 0x389cb27e0bc8d220a7e5f24db74f58851313e695333ad68d |
| ratio | 2.8385064052386134e+58 |
| distance 32 | 0xd |
| distance 64 | 0xd |
| $\text{multiples_x}(k=1)$ | |
| Hx | 0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7 |
| bits | 0x180 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=2)$ | |
| Hx | 0xd36fed39ca71063a5163e8119a37aff10f6b86d50f02f1d324238d2b090d80670849550566396ff5778738c0b39b107a |
| bits | 0x180 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=3)$ | |
| Hx | 0x29f990b67e6b7cf3d5a9a0225fb89b7495ca54cc7308969d10ca98fab517c32823f21446dd2f436799f178999f82381 |
| bits | 0x17a |
| difference | 0x6 |
| ratio | 0.98438 |
| $\text{multiples_x}(k=4)$ | |
| Hx | 0x25019af71ca649bc202d447dcd173a9d0f1a3d59e0cd755442fda6de3aa542559389ab079669ab3223c79138fcea622b |
| bits | 0x17e |
| difference | 0x2 |
| ratio | 0.99479 |
| $\text{multiples_x}(k=5)$ | |
| Hx | 0xb0afd497cea5ae12e88e80b2c0088b37ea4f54dd789ca24f8de5c2d6edc29ffbb9565ead286d4bbbb3e8ac26b6a013fb |
| bits | 0x180 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=6)$ | |
| Hx | 0x2f0a44eac77252b165c1d3e289cb83c7ef5dbd67913f8c9a101e2650337b2fa57269d71ea816dddfdbf44c8c3307e618 |
| bits | 0x17e |
| difference | 0x2 |
| ratio | 0.99479 |
| $\text{multiples_x}(k=7)$ | |
| Hx | 0xaaacb769959743ed0600a862ea19492d31b27517e7434821056d619eda02c66d826ad1a6802f27ef262f8450c3225400 |
| bits | 0x180 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=8)$ | |
| Hx | 0xe87f2b075db1062dcac1a4b4f07785348d69d2ea7328b1050a6f38d605a8c17177921365efc13b72ed8354bd93d536af |
| bits | 0x180 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=9)$ | |
| Hx | 0x96b25b7b1f4c4d31c5eac8cf20eebc5dcafa6591c1361da8b15960d2d078d7cab094cfa013d28edfc10db0976799ba8b |
| bits | 0x180 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{multiples_x}(k=10)$ | |
| Hx | 0xb039c6db651f71f63a5d56cee745b083277616b368afea1b9ef62e49ed4f6e2f878fe27b6dfb84b57f0fa024fc2c707c |
| bits | 0x180 |
| difference | 0x0 |
| ratio | 1.0 |
| $\text{x962_invariant}()$ | |
| r | 0x79d1e655f868f02fff48dcdee14151ddb80643c1406d0ca10dfe6fc52009540a495e8042ea5f744f6e184667cc722483 |
| $\text{brainpool_overlap}()$ | |
| o | -0x33312fa7e23ee7e4988e056ce3f82d1a181d9c6efe81411103140893 |
| $\text{weierstrass}()$ | |
| a | 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000fffffffc |
| b | 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef |