Curve detail

Definition

Name secp224r1 (nist/P-224, secg/secp224r1, x962/ansip224r1)
Category secg
Description A randomly generated curve. [SEC2v1](https://www.secg.org/SEC2-Ver-1.0.pdf) states 'E was chosen verifiably at random as specified in ANSI X9.62 [1] from the seed'.
Field Prime (0xffffffffffffffffffffffffffffffff000000000000000000000001)
Field bits 224
Form Weierstrass $y^2 = x^3 + ax + b$
Param $a$ 0xfffffffffffffffffffffffffffffffefffffffffffffffffffffffe
Param $b$ 0xb4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4
Generator $x$ 0xb70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21
Generator $y$ 0xbd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34
Simulation seed 0xbd71344799d5c7fcdc45b59fa3b9ab8f6a948bc5

Characteristics

Order 0xffffffffffffffffffffffffffff16a2e0b8f03e13dd29455c5c2a3d
Cofactor 0x1
$j$-invariant 0xc55630164a29a17b465f4a532fa010aabe1ffc9fdea639fc47307d75
Trace $t$ 0xe95c1f470fc1ec22d6baa3a3d5c5
Embedding degree $k$ 0x5555555555555555555555555555078ba03da56a069f0dc1c9740e14
CM discriminant -0x5a245a6f599a2778760a3eacdf9b0b9c6f78b867c3f4a9a10611d7d3

Traits

$\text{cofactor}()$
order 0xffffffffffffffffffffffffffff16a2e0b8f03e13dd29455c5c2a3d
cofactor 0x1
$\text{discriminant}()$
cm_disc -0x5a245a6f599a2778760a3eacdf9b0b9c6f78b867c3f4a9a10611d7d3
factorization ['0x3', '0x3', '0x3', '0x1d', '0x4f', '0x1d63', '0x9fdf', '0x2b062b9a5', '0x116ae6b402d187ad6d60180e0c71d806f74e1b']
max_conductor 0x3
$\text{twist_order}(deg=1)$
twist_cardinality 0x10000000000000000000000000000e95b1f470fc1ec22d6baa3a3d5c7
factorization None
$\text{twist_order}(deg=2)$
twist_cardinality 0xfffffffffffffffffffffffffffffffe000000000000000000000000d4b8d216d9949cc3d9a3cbec238c977c14c1845a1c660956c95f6999
factorization None
$\text{kn_factorization}(k=1)$
(+)factorization ['0x2', '0x7', '0x13', '0x9568bc0929', '0x1a624ff20e2655d749c94cdb716d800e1f8930cc5731b']
(+)largest_factor_bitlen 0xb1
(-)factorization ['0x2', '0x2', '0x3', '0x3', '0x3', '0x3', '0x3', '0x3', '0x5', '0x11', '0x869', '0x80c63bd8a8b2da59fd7b637139afe83defb38e5bc128c6a53']
(-)largest_factor_bitlen 0xc4
$\text{kn_factorization}(k=2)$
(+)factorization ['0x3', '0xd', '0x1336e4d', '0xaee94a305e40da568f6391466ba5e50cd27cae95b7d4fc841']
(+)largest_factor_bitlen 0xc4
(-)factorization ['0x9d3d', '0x626b29d', '0x13b348105a13a389e9', '0x6e101d732abdfbf7fe492856b2169']
(-)largest_factor_bitlen 0x73
$\text{kn_factorization}(k=3)$
(+)factorization ['0x2', '0x2', '0x2', '0xb', '0xd3', '0x101', '0x133', '0x2626d', '0x14d96427df', '0x43ff087a953d', '0xaa8358128e5b75a7be49fb7b']
(+)largest_factor_bitlen 0x60
(-)factorization ['0x2', '0xc4d201', '0x25aceb4d', '0xd41c576e0aa81ab471be9ad9ac8270c5b40422c3347']
(-)largest_factor_bitlen 0xac
$\text{kn_factorization}(k=4)$
(+)factorization ['0x5', '0x517', '0x943', '0xa69a98d30f41', '0x2c39d0d4d30ce8f', '0x26a4312f3f5aeb86564d0f9b3']
(+)largest_factor_bitlen 0x62
(-)factorization ['0x3', '0x1d', '0x3b', '0x4a9', '0xbe65d', '0xebc20b8d14b41f190fb1f0803526920bcea3f01464ca93']
(-)largest_factor_bitlen 0xb8
$\text{kn_factorization}(k=5)$
(+)factorization ['0x2', '0x3', '0x53', '0x29a1', '0xfce5ea2e5c2e9a911e7c0e1fc05959494c8af92ca27d0e8cc01']
(+)largest_factor_bitlen 0xcc
(-)factorization ['0x2', '0x2', '0x2', '0x2', '0x8b', '0x2f9', '0x12b8a9fb', '0xa506ea179f', '0x41b606eee9fded96546f810bc7614f0d5ed']
(-)largest_factor_bitlen 0x8b
$\text{kn_factorization}(k=6)$
(+)factorization NO DATA (timed out)
(+)largest_factor_bitlen -
(-)factorization ['0x5', '0x7', '0x146c5', '0x2526e3', '0xece88e757a6120ff2ba5b6724d29d064eaf1ccb25a6981']
(-)largest_factor_bitlen 0xb8
$\text{kn_factorization}(k=7)$
(+)factorization ['0x2', '0x2', '0x83f', '0x1c82920f2e540477', '0x1e7d5726088d13be4d88759ff3d55206e3d3593']
(+)largest_factor_bitlen 0x99
(-)factorization ['0x2', '0x3', '0x58ab', '0x35e4d99b177a23265d358b58fada4b29b4ed78e714a2ef0db41d5']
(-)largest_factor_bitlen 0xd2
$\text{kn_factorization}(k=8)$
(+)factorization ['0x3', '0x3', '0x7', '0x18d', '0x14f65911bcf51ea284eb0c45b9652f642a303ebe33cd0a598c97f3']
(+)largest_factor_bitlen 0xd5
(-)factorization ['0xb', '0xc5', '0x2ac9941', '0x5a78df241344d30f6cfbbb30f357b6a2b94c31c432f2b0d1']
(-)largest_factor_bitlen 0xbf
$\text{torsion_extension}(l=2)$
least 0x3
full 0x3
relative 0x1
$\text{torsion_extension}(l=3)$
least 0x2
full 0x2
relative 0x1
$\text{torsion_extension}(l=5)$
least 0x6
full 0x6
relative 0x1
$\text{torsion_extension}(l=7)$
least 0x18
full 0x18
relative 0x1
$\text{torsion_extension}(l=11)$
least 0x2
full 0xa
relative 0x5
$\text{torsion_extension}(l=13)$
least 0x15
full 0x15
relative 0x1
$\text{torsion_extension}(l=17)$
least 0x6
full 0x6
relative 0x1
$\text{conductor}(deg=2)$
ratio_sqrt 0xe95c1f470fc1ec22d6baa3a3d5c5
factorization ['0x10cf47d', '0xd02408239', '0x111315cc1bc431']
$\text{conductor}(deg=3)$
ratio_sqrt 0x2b472dea266b633c265c3413dc736882eb3e7ba5e399f6a936a09668
factorization ['0x2', '0x2', '0x2', '0x3', '0x5', '0x11', '0x6cd', '0xcc6fc306ee700a82e8977fe595a80bcb4d68169694b496c2ff']
$\text{conductor}(deg=4)$
ratio_sqrt 0x110cf7aef4ed682e9bfeb6a8dfbb49937d016a8f64941a761308fbc8fda1b3765a2debf96b0f946941bcd
factorization ['0x7', '0x13', '0xd3', '0x1171d', '0x10cf47d', '0xd02408239', '0x111315cc1bc431', '0x281085c2dac033a46989f97fa9bf763b423306d56284bfe23']
$\text{embedding}()$
embedding_degree_complement 0x3
complement_bit_length 0x2
$\text{class_number}()$
upper 0x1d213bba0af9fafc740843cd5a4b66
lower 0x73b3ccf53
$\text{small_prime_order}(l=2)$
order 0xffffffffffffffffffffffffffff16a2e0b8f03e13dd29455c5c2a3c
complement_bit_length 0x1
$\text{small_prime_order}(l=3)$
order 0x444444444444444444444444444406094cfe1dee6bb27167d45cd81
complement_bit_length 0x6
$\text{small_prime_order}(l=5)$
order 0x3fffffffffffffffffffffffffffc5a8b82e3c0f84f74a5157170a8f
complement_bit_length 0x3
$\text{small_prime_order}(l=7)$
order 0x5555555555555555555555555555078ba03da56a069f0dc1c9740e14
complement_bit_length 0x2
$\text{small_prime_order}(l=11)$
order 0xffffffffffffffffffffffffffff16a2e0b8f03e13dd29455c5c2a3c
complement_bit_length 0x1
$\text{small_prime_order}(l=13)$
order 0xffffffffffffffffffffffffffff16a2e0b8f03e13dd29455c5c2a3c
complement_bit_length 0x1
$\text{division_polynomials}(l=2)$
factorization [['0x3', '0x1']]
len 0x1
$\text{division_polynomials}(l=3)$
factorization [['0x1', '0x4']]
len 0x1
$\text{division_polynomials}(l=5)$
factorization [['0x3', '0x4']]
len 0x1
$\text{volcano}(l=2)$
crater_degree 0x0
depth 0x0
$\text{volcano}(l=3)$
crater_degree 0x1
depth 0x1
$\text{volcano}(l=5)$
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 0x0
depth 0x0
$\text{isogeny_extension}(l=2)$
least 0x3
full 0x3
relative 0x1
$\text{isogeny_extension}(l=3)$
least 0x1
full 0x1
relative 0x1
$\text{isogeny_extension}(l=5)$
least 0x3
full 0x3
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 0x7
full 0x7
relative 0x1
$\text{isogeny_extension}(l=17)$
least 0x3
full 0x3
relative 0x1
$\text{isogeny_extension}(l=19)$
least 0x4
full 0x4
relative 0x1
$\text{trace_factorization}(deg=1)$
trace 0xe95c1f470fc1ec22d6baa3a3d5c5
trace_factorization ['0x10cf47d', '0xd02408239', '0x111315cc1bc431']
number_of_factors 0x3
$\text{trace_factorization}(deg=2)$
trace 0xe95c1f470fc1ec22d6baa3a3d5c5
trace_factorization ['0x7', '0x13', '0xd3', '0x1171d', '0x281085c2dac033a46989f97fa9bf763b423306d56284bfe23']
number_of_factors 0x5
$\text{isogeny_neighbors}(l=2)$
len 0x0
$\text{isogeny_neighbors}(l=3)$
len 0x4
$\text{isogeny_neighbors}(l=5)$
len 0x0
$\text{q_torsion}()$
Q_torsion 0x1
$\text{hamming_x}(weight=1)$
x_coord_count 0x74
expected 0x70
ratio 0.96552
$\text{hamming_x}(weight=2)$
x_coord_count 0x3051
expected 0x30c8
ratio 1.00962
$\text{hamming_x}(weight=3)$
x_coord_count 0xe1979
expected 0xe19d0
ratio 1.00009
$\text{square_4p1}()$
p 0x1
order 0x1
$\text{pow_distance}()$
distance 0xe95d1f470fc1ec22d6baa3a3d5c3
ratio 5.695948695216996e+33
distance 32 0x3
distance 64 0x3
$\text{multiples_x}(k=1)$
Hx 0xb70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21
bits 0xe0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=2)$
Hx 0xe7f240285c2d03a7ee519efb8da70f8ff7292c0df5e20b89668cddda
bits 0xe0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=3)$
Hx 0x425f807b9381adc919e50b3851f4276553bb9597d489bbeba89de9d7
bits 0xdf
difference 0x1
ratio 0.99554
$\text{multiples_x}(k=4)$
Hx 0x86fc8e8a9c3d122a84329d2629916ef4db9d8c29d26d172899914428
bits 0xe0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=5)$
Hx 0x9c499c0e91159fb87dba6cd8f675c825b637887c2c2357a08127fa88
bits 0xe0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=6)$
Hx 0x5964e26c650fcf40b73feb64928347a5ef12259499960efe1dfbb565
bits 0xdf
difference 0x1
ratio 0.99554
$\text{multiples_x}(k=7)$
Hx 0x4d98db3d7df2fa1171d958689a1770015a574488ae13a68c51608d0
bits 0xdb
difference 0x5
ratio 0.97768
$\text{multiples_x}(k=8)$
Hx 0x381661205d9eb3dbab18a5c394c554bd7f288c597e85bfae45784183
bits 0xde
difference 0x2
ratio 0.99107
$\text{multiples_x}(k=9)$
Hx 0xdceb004162ce3af1eaedfe2c4136803910f8a86290c47ca8b0c560bf
bits 0xe0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=10)$
Hx 0x25a1f92088a170fb7bf2a515eee8cb8dd77849c34773c33d32232a1a
bits 0xde
difference 0x2
ratio 0.99107
$\text{x962_invariant}()$
r 0x5b056c7e11dd68f40469ee7f3c7a7d74f7d121116506d031218291fb
$\text{brainpool_overlap}()$
o -0x34050a850c04b3ad
$\text{weierstrass}()$
a 0xfffffffffffffffffffffffffffffffefffffffffffffffffffffffe
b 0xb4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4