Curve detail
Definition
| Name | ed-512-mont |
|---|---|
| Category | nums |
| Description | Original nums curve from https://eprint.iacr.org/2014/130.pdf |
| Field | Prime (0xfe14ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff) |
| Field bits | 512 |
| Form | Twisted Edwards $ax^2 + y^2 = 1 + dx^2y^2$ |
| Param $a$ | 0xfe14fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffe |
| Param $d$ | 0x12a9c |
Characteristics
| Order | 0x3f853fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffcccfd59cdc33470d103060513f6def4d37d9af21b2b2701fa331487ecb8db605 |
| Cofactor | 0x4 |
| $j$-invariant | 0xcda9233a5135668daa95aecf615d3de755bef0f5e9be3c3c630af4a986c701d0f7e96570e595d6d5f14f57c807143c6f196221dc3c9b7ebdce5f445445871263 |
| Trace $t$ | 0xccc0a98c8f32e3cbbf3e7ebb024842cb2099437935363f81733ade04d1c927ec |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x3f853fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffcccfd59cdc33470d103060513f6def4d37d9af21b2b2701fa331487ecb8db605 |
| cofactor | 0x4 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0xfe15000000000000000000000000000000000000000000000000000000000000ccc0a98c8f32e3cbbf3e7ebb024842cb2099437935363f81733ade04d1c927ec |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0xfc2dadb8fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffcab6f9f36c75ac9a08af085ad39c599b5f232fa7bdee1c82596e59fcf57b03e0af5816923d94bc218107020b75a8068024a83219aad2688cc3a60a09f0ed1c194 |
| 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 | ['0x5', '0x35', '0x1eae7d95bc609a90e7d95bc609a90e7d95bc609a90e7d95bc609a90e7d95bc6081d758b03bfdb80e09c06c597a69444058cd9d0b725441c010ffa75d1fa9e21'] |
| (+)largest_factor_bitlen | 0x1f9 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=3)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=4)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=5)$ | |
| (+)factorization | ['0x683', '0xcebfe11e9d', '0xf192073a78e7cb16bea73a884df8e08aeeba2598539caed042e327ecb017344172753c61df6e0b2bb88502afcf1db0206969fd7a01f1ecfc2fa3'] |
| (+)largest_factor_bitlen | 0x1d0 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=6)$ | |
| (+)factorization | ['0x7', '0xd9c8db6db6db6db6db6db6db6db6db6db6db6db6db6db6db6db6db6db6db6db62bed25878542182cc9ca6ecd6bc20fe42d337d05d263c9910af21d2070c14b7f'] |
| (+)largest_factor_bitlen | 0x200 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | ['0x3', '0x5', '0x5', '0x5', '0xd0db4ae9022d', '0x5d03cf613c4ac819f44f76f19da965060596ba8b7b20388329e09ecdaf5f10c4660b48062b84b7ff5abf350e748ac43830f5a8835c963c4a367'] |
| (+)largest_factor_bitlen | 0x1cb |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{torsion_extension}(l=2)$ | |
| least | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0xc |
| full | 0xc |
| relative | 0x1 |
| $\text{torsion_extension}(l=7)$ | |
| least | 0x6 |
| full | 0x6 |
| relative | 0x1 |
| $\text{torsion_extension}(l=11)$ | |
| least | 0x5 |
| full | 0xa |
| relative | 0x2 |
| $\text{torsion_extension}(l=13)$ | |
| least | 0x4 |
| full | 0xc |
| relative | 0x3 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0xccc0a98c8f32e3cbbf3e7ebb024842cb2099437935363f81733ade04d1c927ec |
| factorization | ['0x2', '0x2', '0x3', '0xb', '0x126a7', '0x159014d509844af1bde66e11381bebca7a143d1dbddd73d43bb703ba2ed'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x5a5160c938a5365f750f7a52c63a664a0dcd0584211e37da691a6030a84fc1f50a7e96dc26b43de7ef8fdf48a57f97fdb57cde6552d97733c59f5f60f12e3e6f |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0x11374c6019e958e311459dd7ed700c023e790b8328d53b196ea35b03920935995e1e4b50afb121f6d3df75fea0ebeec6535c9e705caaeadcab009b4b62a0e4497e6d9472c1075c9d563ad9a4073f6bd15f68fd61aac008e72ec402e1b34824f68 |
| factorization | NO DATA (timed out) |
| $\text{embedding}()$ | |
| embedding_degree_complement | 0x4 |
| complement_bit_length | 0x3 |
| $\text{class_number}()$ | |
| 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 | [['0x1', '0x1'], ['0x2', '0x1']] |
| len | 0x2 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x2', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0x6', '0x2']] |
| len | 0x1 |
| $\text{volcano}(l=2)$ | |
| crater_degree | 0x0 |
| depth | 0x1 |
| $\text{volcano}(l=3)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=5)$ | |
| crater_degree | 0x0 |
| depth | 0x0 |
| $\text{volcano}(l=7)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=11)$ | |
| crater_degree | 0x2 |
| depth | 0x0 |
| $\text{volcano}(l=13)$ | |
| crater_degree | 0x2 |
| 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 | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x2 |
| full | 0x2 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x3 |
| full | 0x3 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=7)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=11)$ | |
| least | 0x1 |
| full | 0x2 |
| relative | 0x2 |
| $\text{isogeny_extension}(l=13)$ | |
| least | 0x1 |
| full | 0x3 |
| relative | 0x3 |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x4 |
| relative | 0x4 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x6 |
| relative | 0x6 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | 0xccc0a98c8f32e3cbbf3e7ebb024842cb2099437935363f81733ade04d1c927ec |
| trace_factorization | ['0x2', '0x2', '0x3', '0xb', '0x126a7', '0x159014d509844af1bde66e11381bebca7a143d1dbddd73d43bb703ba2ed'] |
| number_of_factors | 0x5 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | 0xccc0a98c8f32e3cbbf3e7ebb024842cb2099437935363f81733ade04d1c927ec |
| trace_factorization | NO DATA (timed out) |
| number_of_factors | NO DATA (timed out) |
| $\text{isogeny_neighbors}(l=2)$ | |
| len | 0x1 |
| $\text{isogeny_neighbors}(l=3)$ | |
| len | 0x0 |
| $\text{isogeny_neighbors}(l=5)$ | |
| len | 0x0 |
| $\text{q_torsion}()$ | |
| Q_torsion | 0x1 |
| $\text{hamming_x}(weight=1)$ | |
| x_coord_count | 0x115 |
| expected | 0x100 |
| ratio | 0.92419 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x1009a |
| expected | 0xff80 |
| ratio | 0.99571 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xa9b5d8 |
| expected | 0xa9ab00 |
| ratio | 0.99975 |
| $\text{square_4p1}()$ | |
| p | NO DATA (timed out) |
| order | NO DATA (timed out) |
| $\text{pow_distance}()$ | |
| distance | 0x1eb000000000000000000000000000000000000000000000000000000000000ccc0a98c8f32e3cbbf3e7ebb024842cb2099437935363f81733ade04d1c927ec |
| ratio | 132.47454 |
| distance 32 | 0xc |
| distance 64 | 0x14 |
| $\text{multiples_x}(k=1)$ | |
| 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 | 0xdae4a1b74d6d4a0d2a1eced9efd278598ea89efec8ec27a4a282badcaad87cd86bd2a8e80e1fe97acc34c8de463d17be35c60dbbbe2aa2255f94763d74eee28e |
| $\text{brainpool_overlap}()$ | |
| o | -0x10c5f9556cfa235d979056eec06293a5f484c46db6d3a0d4566aab69d87bd1c20609044e8b38d8befa9c3408 |
| $\text{weierstrass}()$ | |
| a | 0x633346b019e85111b1fcea3b7f96b114e027b70e8ebcd0e7bfd5937be4724240191c0292730a9e7858d4fafb45522d522c497949f919898ad1234a49e0a925c1 |
| b | 0x1f82ca3d2ccfb6d97c02992ed97e9638678f62e60fa89bcf9bbcd8bc04c54b0bff228dd95c5c2308db4559c98540873e39890c872985408148643f9348129a22 |