Curve detail

Definition

Name sect163k1 (nist/K-163, secg/sect163k1, wtls/wap-wsg-idm-ecid-wtls3, x962/ansix9t163k1)
Category secg
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 $b$ 0x000000000000000000000000000000000000000001
Generator $x$ 0x02fe13c0537bbc11acaa07d793de4e6d5e5c94eee8
Generator $y$ 0x0289070fb05d38ff58321f2e800536d538ccdaa3d9

Characteristics

Order 0x4000000000000000000020108a2e0cc0d99f8a5ef
Cofactor 0x2
$j$-invariant 0x1
Trace $t$ -0x4021145c1981b33f14bdd

Traits

$\text{cofactor}()$
order 0x4000000000000000000020108a2e0cc0d99f8a5ef
cofactor 0x2
$\text{discriminant}()$
cm_disc -0x7
factorization ['0x7', '0xb249', '0xb249', '0x140e9', '0x140e9', '0x836207', '0x836207', '0x35e17a1', '0x35e17a1']
max_conductor 0x18240aafba82a33aca077
$\text{twist_order}(deg=1)$
twist_cardinality 0x7fffffffffffffffffffbfdeeba3e67e4cc0eb424
factorization None
$\text{twist_order}(deg=2)$
twist_cardinality 0x400000000000000000000000000000000000000000108e744e1df22e5c67883ca60ec07847fa953cca
factorization None
$\text{kn_factorization}(k=1)$
(+)factorization ['0x3', '0x3', '0xc1', '0x336b5bd94588ca1', '0x5debcf73233639a0bdc31107']
(+)largest_factor_bitlen 0x5f
(-)factorization ['0x5', '0x5', '0x5', '0x5', '0x11c3', '0xd05ac7', '0x3a074fa39ba1d7d90ccaf0e3e35d39']
(-)largest_factor_bitlen 0x76
$\text{kn_factorization}(k=2)$
(+)factorization ['0x355', '0x47fdb9', '0x711db4d8728ecb', '0x26a4feae182a2adc7593']
(+)largest_factor_bitlen 0x4e
(-)factorization ['0x3', '0x13', '0x17', '0x25', '0xb3', '0x1eea7aeea8b3c4f879ff03eb1aa21cf348bb']
(-)largest_factor_bitlen 0x8d
$\text{kn_factorization}(k=3)$
(+)factorization ['0x7', '0x7', '0x755', '0x3da30f31d', '0x1557014a9b', '0x3541346049fb5955b231']
(+)largest_factor_bitlen 0x4e
(-)factorization ['0x3dc79', '0xb5dd5', '0x351e03f', '0x12357d853', '0x3aeedec9f', '0xa0f3f3def']
(-)largest_factor_bitlen 0x24
$\text{kn_factorization}(k=4)$
(+)factorization ['0x3', '0x5', '0x25e11', '0x4303a3', '0x1aa3ceb0df', '0x21145c57f63061390e93d3']
(+)largest_factor_bitlen 0x56
(-)factorization ['0x7', '0xb', '0x4173a26cdc6960b5', '0x1a01ea454ae95a7d0dca4a867']
(-)largest_factor_bitlen 0x61
$\text{kn_factorization}(k=5)$
(+)factorization ['0x4f', '0x65', '0xf07', '0xb23cfc3', '0x30082077b', '0x7c5ac99d9', '0x15889e3f833b']
(+)largest_factor_bitlen 0x2d
(-)factorization ['0x3', '0xd', '0x11', '0x1cd', '0x2cf', '0x10dff', '0x159c1', '0x5e452ba6996d', '0x5d261a13248d00b']
(-)largest_factor_bitlen 0x3b
$\text{kn_factorization}(k=6)$
(+)factorization ['0xe3', '0x76f0f', '0x2029b40ecad', '0x39f5be16665cd0245c67356cd']
(+)largest_factor_bitlen 0x62
(-)factorization ['0x5', '0x1c45', '0xad1461f513b7cb65b', '0x80957b0df5d35cd914c31']
(-)largest_factor_bitlen 0x54
$\text{kn_factorization}(k=7)$
(+)factorization ['0x3', '0xb', '0xb', '0x89f', '0x118d477', '0x42d0b3a1bdb38e72b75b6a4496ba691']
(+)largest_factor_bitlen 0x7b
(-)factorization ['0x917ed', '0x259c1113', '0x29eafcab01b404155c61f259ebd397']
(-)largest_factor_bitlen 0x76
$\text{kn_factorization}(k=8)$
(+)factorization ['0xd', '0x16b72ea8ed', '0xb86f9f5b26e17e9', '0x4d0292f1bd95f9fc1']
(+)largest_factor_bitlen 0x43
(-)factorization ['0x3', '0x3', '0x3', '0x3', '0x53', '0x13d', '0xf267b', '0x11b5b64b6b7', '0x1e0b5638846b0f4ef9c7cd']
(-)largest_factor_bitlen 0x55
$\text{torsion_extension}(l=2)$
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 0x3
full 0x7
relative 0x2
$\text{torsion_extension}(l=11)$
least 0x5
full 0xa
relative 0x2
$\text{torsion_extension}(l=13)$
least 0xa8
full 0xa8
relative 0x1
$\text{torsion_extension}(l=17)$
least 0x90
full 0x90
relative 0x1
$\text{conductor}(deg=2)$
ratio_sqrt 0x4021145c1981b33f14bdd
factorization ['0x24b44688a7', '0x1bf47d9a13db']
$\text{conductor}(deg=3)$
ratio_sqrt 0x8108e744e1df22e5c67883ca60ec07847fa953cc9
factorization ['0xf47', '0x46b724db9', '0x80c412a1dba8a1', '0x3cc9f207cab08a07']
$\text{conductor}(deg=4)$
ratio_sqrt 0x425c0c19b7c9af4ae75ac74fcfdc82a9284e23b16872bc714f5a4fdd5c85
factorization ['0x3', '0x28d', '0x98cf', '0x14957b', '0x10e3402d', '0x24b44688a7', '0x1bf47d9a13db', '0x2ab5b1740880911506e9f']
$\text{embedding}()$
embedding_degree_complement 0x146
complement_bit_length 0x9
$\text{class_number}()$
upper 0x2
lower 0x0
$\text{small_prime_order}(l=2)$
order None
complement_bit_length None
$\text{small_prime_order}(l=3)$
order 0x4000000000000000000020108a2e0cc0d99f8a5ee
complement_bit_length 0x1
$\text{small_prime_order}(l=5)$
order 0x4000000000000000000020108a2e0cc0d99f8a5ee
complement_bit_length 0x1
$\text{small_prime_order}(l=7)$
order 0x4000000000000000000020108a2e0cc0d99f8a5ee
complement_bit_length 0x1
$\text{small_prime_order}(l=11)$
order 0xaaaaaaaaaaaaaaaaaaab002c1b25775799a970fd
complement_bit_length 0x3
$\text{small_prime_order}(l=13)$
order 0x200000000000000000001008451706606ccfc52f7
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 [['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 0x1
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 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 0x1
full 0x7
relative 0x7
$\text{isogeny_extension}(l=11)$
least 0x1
full 0xa
relative 0xa
$\text{isogeny_extension}(l=13)$
least 0xe
full 0xe
relative 0x1
$\text{isogeny_extension}(l=17)$
least 0x9
full 0x9
relative 0x1
$\text{isogeny_extension}(l=19)$
least 0x14
full 0x14
relative 0x1
$\text{trace_factorization}(deg=1)$
trace -0x4021145c1981b33f14bdd
trace_factorization ['0x24b44688a7', '0x1bf47d9a13db']
number_of_factors 0x2
$\text{trace_factorization}(deg=2)$
trace -0x4021145c1981b33f14bdd
trace_factorization ['0x3', '0x28d', '0x98cf', '0x14957b', '0x10e3402d', '0x2ab5b1740880911506e9f']
number_of_factors 0x6
$\text{isogeny_neighbors}(l=2)$
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 0x4021145c1981b33f14bde
ratio 2.4129797985049123e+24
distance 32 0x2
distance 64 0x1e
$\text{multiples_x}(k=1)$
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 0x1
$\text{weierstrass}()$
a 0x1
b 0x1