Curve detail
Definition
| Name | B-409 (nist/B-409, secg/sect409r1, x962/ansit409r1) |
|---|---|
| Category | nist |
| Field | Binary |
| Field polynomial | $x^{409} + x^{87} + 1$ |
| Field bits | 409 |
| Form | Weierstrass $y^2 = x^3 + ax + b$ |
| Param $a$ | 0x0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 |
| Param $b$ | 0x021a5c2c8ee9feb5c4b9a753b7b476b7fd6422ef1f3dd674761fa99d6ac27c8a9a197b272822f6cd57a55aa4f50ae317b13545f |
| Generator $x$ | 0x15d4860d088ddb3496b0c6064756260441cde4af1771d4db01ffe5b34e59703dc255a868a1180515603aeab60794e54bb7996a7 |
| Generator $y$ | 0x061b1cfab6be5f32bbfa78324ed106a7636b9c5a7bd198d0158aa4f5488d08f38514f1fdf4b4f40d2181b3681c364ba0273c706 |
| Simulation seed | 0x4099b5a457f9d69f79213d094c4bcd4d4262210b |
Characteristics
| Order | 0x10000000000000000000000000000000000000000000000000001e2aad6a612f33307be5fa47c3c9e052f838164cd37d9a21173 |
| Cofactor | 0x2 |
| Trace $t$ | -0x3c555ad4c25e6660f7cbf48f8793c0a5f0702c99a6fb34422e5 |
Traits
| $\text{cofactor}()$ | |
|---|---|
| order | 0x10000000000000000000000000000000000000000000000000001e2aad6a612f33307be5fa47c3c9e052f838164cd37d9a21173 |
| cofactor | 0x2 |
| $\text{discriminant}()$ | |
| cm_disc | None |
| factorization | None |
| max_conductor | None |
| $\text{twist_order}(deg=1)$ | |
| twist_cardinality | 0x1fffffffffffffffffffffffffffffffffffffffffffffffffffc3aaa52b3da1999f08340b70786c3f5a0f8fd3665904cbbdd1c |
| factorization | None |
| $\text{twist_order}(deg=2)$ | |
| twist_cardinality | 0x3fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffc0e381f092c9f9c2a50751ce3d057c61a99aa7fe1e0f1e63a674b156e8eb74093d67b900543bc7e10e3047b4c342184cc69a0da |
| factorization | None |
| $\text{kn_factorization}(k=1)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0xb', '0xb5', '0x425', '0x54b71f77696b4d1950d60621d19cc6f328a89ead3ef9818e09fa04752efe9fd020bf41f0d3a4fcf789268bac0e93b96bd'] |
| (-)largest_factor_bitlen | 0x183 |
| $\text{kn_factorization}(k=2)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=3)$ | |
| (+)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 | 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 | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{kn_factorization}(k=7)$ | |
| (+)factorization | NO DATA (timed out) |
| (+)largest_factor_bitlen | NO DATA (timed out) |
| (-)factorization | ['0x3', '0x11', '0x1d', '0x230b', '0x11b3e2856cb58c7f2b14d4b2e152da6d7c4dc7573c6c72cb17c2817ce15f46a1fcd0a955a36399970cfb96dcb3385735ad'] |
| (-)largest_factor_bitlen | 0x185 |
| $\text{kn_factorization}(k=8)$ | |
| (+)factorization | ['0x3', '0x12ac7', '0x545a31d', '0x6653c75', '0x22b23d9d1afde9841c8ce1d65efcfd063ec1fde340da71b0a2a11ff98782737a8322cdca8ab8f5a231242d'] |
| (+)largest_factor_bitlen | 0x156 |
| (-)factorization | NO DATA (timed out) |
| (-)largest_factor_bitlen | NO DATA (timed out) |
| $\text{torsion_extension}(l=2)$ | |
| least | None |
| full | None |
| relative | None |
| $\text{torsion_extension}(l=3)$ | |
| least | 0x8 |
| full | 0x8 |
| relative | 0x1 |
| $\text{torsion_extension}(l=5)$ | |
| least | 0x18 |
| full | 0x18 |
| 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 | 0x2 |
| full | 0xc |
| relative | 0x6 |
| $\text{torsion_extension}(l=17)$ | |
| least | 0x10 |
| full | 0x10 |
| relative | 0x1 |
| $\text{conductor}(deg=2)$ | |
| ratio_sqrt | 0x3c555ad4c25e6660f7cbf48f8793c0a5f0702c99a6fb34422e5 |
| factorization | ['0x95', '0x137', '0x2d182b', '0x1e467a4d360e1bee93741257567e3ec1527cd06565'] |
| $\text{conductor}(deg=3)$ | |
| ratio_sqrt | 0x1f1c7e0f6d36063d5af8ae31c2fa839e56655801e1f0e19c598b4ea917148bf6c29846ffabc4381ef1cfb84b3cbde7b33965f27 |
| factorization | NO DATA (timed out) |
| $\text{conductor}(deg=4)$ | |
| ratio_sqrt | 0xedfb865f08c9acd1f0d7590eab52441a96d1ce64eaa474c9d19a0922a9dc2b5234aaaec96e033fd546c4b055c2ce52f54b85222386bb9a84bad597e6bc2cee32da79c41a0fdc5f171a9824be3 |
| 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 | [['0x1', '0x2']] |
| len | 0x1 |
| $\text{division_polynomials}(l=3)$ | |
| factorization | [['0x4', '0x1']] |
| len | 0x1 |
| $\text{division_polynomials}(l=5)$ | |
| factorization | [['0xc', '0x1']] |
| 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 | 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 | None |
| full | None |
| relative | None |
| $\text{isogeny_extension}(l=3)$ | |
| least | 0x4 |
| full | 0x4 |
| relative | 0x1 |
| $\text{isogeny_extension}(l=5)$ | |
| least | 0x6 |
| full | 0x6 |
| 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 | 0x1 |
| full | 0xc |
| relative | 0xc |
| $\text{isogeny_extension}(l=17)$ | |
| least | 0x1 |
| full | 0x4 |
| relative | 0x4 |
| $\text{isogeny_extension}(l=19)$ | |
| least | 0x1 |
| full | 0x12 |
| relative | 0x12 |
| $\text{trace_factorization}(deg=1)$ | |
| trace | -0x3c555ad4c25e6660f7cbf48f8793c0a5f0702c99a6fb34422e5 |
| trace_factorization | ['0x95', '0x137', '0x2d182b', '0x1e467a4d360e1bee93741257567e3ec1527cd06565'] |
| number_of_factors | 0x4 |
| $\text{trace_factorization}(deg=2)$ | |
| trace | -0x3c555ad4c25e6660f7cbf48f8793c0a5f0702c99a6fb34422e5 |
| trace_factorization | ['0x3', '0x7', '0x11', '0x9eb', '0x1709', '0x62db10d33f', '0xaa59df6443', '0x1a59a3e8155', '0x1e7121f8d4d7869a9', '0x18a426ee3ea94f59464c73', '0x28e55ef7fabcbdba8bbaf36c549b'] |
| number_of_factors | 0xb |
| $\text{isogeny_neighbors}(l=2)$ | |
| 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 | 0x199 |
| expected | 0xcd |
| ratio | 0.50122 |
| $\text{hamming_x}(weight=2)$ | |
| x_coord_count | 0x145ec |
| expected | 0xa3c2 |
| ratio | 0.50244 |
| $\text{hamming_x}(weight=3)$ | |
| x_coord_count | 0xacb8bc |
| expected | 0x56ff54 |
| ratio | 0.50369 |
| $\text{square_4p1}()$ | |
| p | NO DATA (timed out) |
| order | NO DATA (timed out) |
| $\text{pow_distance}()$ | |
| distance | 0x3c555ad4c25e6660f7cbf48f8793c0a5f0702c99a6fb34422e6 |
| ratio | 2.1818814621476744e+62 |
| 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 |
| $\text{brainpool_overlap}()$ | |
| o | None |
| $\text{weierstrass}()$ | |
| a | None |
| b | None |