Curve detail

Definition

Name secp192r1 (nist/P-192, secg/secp192r1, x962/ansix9p192r1)
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 (0xfffffffffffffffffffffffffffffffeffffffffffffffff)
Field bits 192
Form Weierstrass $y^2 = x^3 + ax + b$
Param $a$ 0xfffffffffffffffffffffffffffffffefffffffffffffffc
Param $b$ 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1
Generator $x$ 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012
Generator $y$ 0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811
Simulation seed 0x3045ae6fc8422f64ed579528d38120eae12196d5

Characteristics

Order 0xffffffffffffffffffffffff99def836146bc9b1b4d22831
Cofactor 0x1
$j$-invariant 0xfe40fc48ef4b5633d091b1a2707da063644b08bb896f560b
Trace $t$ 0x662107c8eb94364e4b2dd7cf
Embedding degree $k$ 0x1999999999999999999999998f6318d2353dfa91c5483738
CM discriminant -0x3d741a988e23c8df090918240db2143ab31994533214ca69b

Traits

$\text{cofactor}()$
order 0xffffffffffffffffffffffff99def836146bc9b1b4d22831
cofactor 0x1
$\text{discriminant}()$
cm_disc -0x3d741a988e23c8df090918240db2143ab31994533214ca69b
factorization ['0x5', '0xb', '0x1f', '0x93a1f51e9cc9e15c6cca9fcb46e383073c72e71b0bc1a3']
max_conductor 0x1
$\text{twist_order}(deg=1)$
twist_cardinality 0x1000000000000000000000000662107c7eb94364e4b2dd7cf
factorization None
$\text{twist_order}(deg=2)$
twist_cardinality 0xfffffffffffffffffffffffffffffffdfffffffffffffffc28be56771dc372106f6e7dbf24debc54ce66baccdeb35965
factorization None
$\text{kn_factorization}(k=1)$
(+)factorization ['0x2', '0x3', '0x29', '0x18b6203f8e753dc0066d', '0xac7dbf157f9a6497ad1aa8cb887']
(+)largest_factor_bitlen 0x6c
(-)factorization ['0x2', '0x2', '0x2', '0x2', '0x5', '0x955', '0x7e966779a05e328ce6915', '0xb186bf71c774922768ebedf']
(-)largest_factor_bitlen 0x5c
$\text{kn_factorization}(k=2)$
(+)factorization ['0x1abc9d3e40f', '0x702abd9380f5a97', '0x2bb49aeef2cab7f6c945179b']
(+)largest_factor_bitlen 0x5e
(-)factorization ['0x3', '0x3', '0x18fb205b', '0x246fc7574a80bfc68fa7a366f71e653536648fe1b']
(-)largest_factor_bitlen 0xa2
$\text{kn_factorization}(k=3)$
(+)factorization ['0x2', '0x2', '0x7', '0xd', '0xcdb4c9', '0x27f5aead', '0x10d25bc94502eb7ae65de5448fc94aae03']
(+)largest_factor_bitlen 0x85
(-)factorization ['0x2', '0xb', '0x107', '0x26a85', '0xe10633fe29115d2368e73b0481a3dc404ac9c3349']
(-)largest_factor_bitlen 0xa4
$\text{kn_factorization}(k=4)$
(+)factorization ['0x3', '0x5', '0x5', '0x95', '0xa3385f', '0x12ab9f3219c9b085d091', '0x1f87b24ff3792ed0faedd']
(+)largest_factor_bitlen 0x51
(-)factorization ['0x7', '0x17', '0x2b9b1', '0xb5e0563', '0x348eb688561844f87a3834543374b4900bd1']
(-)largest_factor_bitlen 0x8e
$\text{kn_factorization}(k=5)$
(+)factorization ['0x2', '0x27fffffffffffffffffffffff00ad6c87330d783c440d647b']
(+)largest_factor_bitlen 0xc2
(-)factorization ['0x2', '0x2', '0x3', '0x55103', '0x3b6a65857c58e9', '0x56723e36483839e1729771e43c3ccd']
(-)largest_factor_bitlen 0x77
$\text{kn_factorization}(k=6)$
(+)factorization ['0x1f', '0x2d14b', '0x5fc902d', '0x1dea28a3', '0x81bc537c9a8dd', '0x319a974787f8f5161']
(+)largest_factor_bitlen 0x42
(-)factorization ['0x5', '0xe8bd57', '0x151e6ef4e64694c82213e456fc6194bfa44c33ad8c7']
(-)largest_factor_bitlen 0xa9
$\text{kn_factorization}(k=7)$
(+)factorization ['0x2', '0x2', '0x2', '0x3', '0x3', '0x3', '0x11', '0x11', '0x13', '0x1d', '0xa7', '0xa4fccfa5', '0x12c3c6ddf', '0xbdf81e85d5b5', '0x954ea7ca9f8e7']
(+)largest_factor_bitlen 0x34
(-)factorization ['0x2', '0x35', '0x15bb', '0xd70a96a017b', '0xed181f258eedb6dffa32f1023007c3ff7']
(-)largest_factor_bitlen 0x84
$\text{kn_factorization}(k=8)$
(+)factorization ['0xb', '0x11f27ba464ab', '0xa5fb59b51d42c08af32c4ce5873744f7589b1']
(+)largest_factor_bitlen 0x94
(-)factorization ['0x3', '0x53', '0xc1', '0x60f7d', '0x69abdbdaf', '0x18b140b6cb507', '0x2d368df2675e8d182d93']
(-)largest_factor_bitlen 0x4e
$\text{torsion_extension}(l=2)$
least 0x3
full 0x3
relative 0x1
$\text{torsion_extension}(l=3)$
least 0x8
full 0x8
relative 0x1
$\text{torsion_extension}(l=5)$
least 0x4
full 0x5
relative 0x1
$\text{torsion_extension}(l=7)$
least 0x30
full 0x30
relative 0x1
$\text{torsion_extension}(l=11)$
least 0x5
full 0xb
relative 0x2
$\text{torsion_extension}(l=13)$
least 0x4
full 0xc
relative 0x3
$\text{torsion_extension}(l=17)$
least 0x24
full 0x24
relative 0x1
$\text{conductor}(deg=2)$
ratio_sqrt 0x662107c8eb94364e4b2dd7cf
factorization ['0x59', '0xb3', '0x17f', '0x17156f011', '0xc2a4e727d3']
$\text{conductor}(deg=3)$
ratio_sqrt 0xd741a988e23c8df090918240db2143ae31994533214ca69e
factorization ['0x2', '0xd', '0x6443', '0x50a217d', '0x7112677918206154a5', '0x97f311a604bf144d9c9']
$\text{conductor}(deg=4)$
ratio_sqrt 0xbc00f7580d62980b31c69dc340e7bb4fa23282012035dfa349cfbe6ebb0f166e958193f3
factorization ['0x3', '0x3', '0x25', '0x59', '0xb3', '0x17f', '0x841', '0xede15f', '0x17156f011', '0xc2a4e727d3', '0x3eee68c777f', '0xc0274b7e2959f32be737a6e0931']
$\text{embedding}()$
embedding_degree_complement 0xa
complement_bit_length 0x4
$\text{class_number}()$
upper 0x53dc64eef1d84973106b4b1a2f
lower 0x12df
$\text{small_prime_order}(l=2)$
order 0x7fffffffffffffffffffffffccef7c1b0a35e4d8da691418
complement_bit_length 0x2
$\text{small_prime_order}(l=3)$
order 0xffffffffffffffffffffffff99def836146bc9b1b4d22830
complement_bit_length 0x1
$\text{small_prime_order}(l=5)$
order 0x3fffffffffffffffffffffffe677be0d851af26c6d348a0c
complement_bit_length 0x3
$\text{small_prime_order}(l=7)$
order 0x7fffffffffffffffffffffffccef7c1b0a35e4d8da691418
complement_bit_length 0x2
$\text{small_prime_order}(l=11)$
order 0x7fffffffffffffffffffffffccef7c1b0a35e4d8da691418
complement_bit_length 0x2
$\text{small_prime_order}(l=13)$
order 0x3fffffffffffffffffffffffe677be0d851af26c6d348a0c
complement_bit_length 0x3
$\text{division_polynomials}(l=2)$
factorization [['0x3', '0x1']]
len 0x1
$\text{division_polynomials}(l=3)$
factorization [['0x4', '0x1']]
len 0x1
$\text{division_polynomials}(l=5)$
factorization [['0x2', '0x1'], ['0xa', '0x1']]
len 0x2
$\text{volcano}(l=2)$
crater_degree 0x0
depth 0x0
$\text{volcano}(l=3)$
crater_degree 0x0
depth 0x0
$\text{volcano}(l=5)$
crater_degree 0x1
depth 0x0
$\text{volcano}(l=7)$
crater_degree 0x0
depth 0x0
$\text{volcano}(l=11)$
crater_degree 0x1
depth 0x0
$\text{volcano}(l=13)$
crater_degree 0x2
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 0x4
full 0x4
relative 0x1
$\text{isogeny_extension}(l=5)$
least 0x1
full 0x5
relative 0x5
$\text{isogeny_extension}(l=7)$
least 0x8
full 0x8
relative 0x1
$\text{isogeny_extension}(l=11)$
least 0x1
full 0xb
relative 0xb
$\text{isogeny_extension}(l=13)$
least 0x1
full 0x3
relative 0x3
$\text{isogeny_extension}(l=17)$
least 0x9
full 0x9
relative 0x1
$\text{isogeny_extension}(l=19)$
least 0x14
full 0x14
relative 0x1
$\text{trace_factorization}(deg=1)$
trace 0x662107c8eb94364e4b2dd7cf
trace_factorization ['0x59', '0xb3', '0x17f', '0x17156f011', '0xc2a4e727d3']
number_of_factors 0x5
$\text{trace_factorization}(deg=2)$
trace 0x662107c8eb94364e4b2dd7cf
trace_factorization ['0x3', '0x3', '0x25', '0x841', '0xede15f', '0x3eee68c777f', '0xc0274b7e2959f32be737a6e0931']
number_of_factors 0x6
$\text{isogeny_neighbors}(l=2)$
len 0x0
$\text{isogeny_neighbors}(l=3)$
len 0x0
$\text{isogeny_neighbors}(l=5)$
len 0x1
$\text{q_torsion}()$
Q_torsion 0x1
$\text{hamming_x}(weight=1)$
x_coord_count 0x65
expected 0x60
ratio 0.9505
$\text{hamming_x}(weight=2)$
x_coord_count 0x23f6
expected 0x23d0
ratio 0.99587
$\text{hamming_x}(weight=3)$
x_coord_count 0x8dad8
expected 0x8dc20
ratio 1.00057
$\text{square_4p1}()$
p 0x1
order 0x1
$\text{pow_distance}()$
distance 0x662107c9eb94364e4b2dd7cf
ratio 1.985959386609862e+29
distance 32 0xf
distance 64 0xf
$\text{multiples_x}(k=1)$
Hx 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012
bits 0xbd
difference 0x3
ratio 0.98438
$\text{multiples_x}(k=2)$
Hx 0x7b4603cc4ac847264022b07144c25277f2ad8fbe9224728f
bits 0xbf
difference 0x1
ratio 0.99479
$\text{multiples_x}(k=3)$
Hx 0x9697d1870247164b841077ca1a546084dfba2653ae29cb51
bits 0xc0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=4)$
Hx 0xf8330bf5b681db4682a149bf076bf8d2c64d67ee2e7d25e6
bits 0xc0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=5)$
Hx 0x11fecb910f2730d6aa3c242d2800ee1a1dbd449f487ee5ff
bits 0xbd
difference 0x3
ratio 0.98438
$\text{multiples_x}(k=6)$
Hx 0xa8d9ab9cdce3820d9ccc046ee9f1224b40c2d3ef3068a365
bits 0xc0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=7)$
Hx 0xaa3c3adb52e99783db0fe26f55cf53d442c0da5443b0dfc6
bits 0xc0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=8)$
Hx 0xbd2115367b7fabc16c47bbc5df5df0a565b9ad36fde9eba5
bits 0xc0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=9)$
Hx 0xbaa8b91724a3adc05cee5a6f20697b7a92b2a61d6243c81f
bits 0xc0
difference 0x0
ratio 1.0
$\text{multiples_x}(k=10)$
Hx 0x5c2cdc18eb53508e269d3dd00456200bf955863f635fa7a4
bits 0xbf
difference 0x1
ratio 0.99479
$\text{x962_invariant}()$
r 0x3099d2bbbfcb2538542dcd5fb078b6ef5f3d6fe2c745de65
$\text{brainpool_overlap}()$
o 0x1bdefae3
$\text{weierstrass}()$
a 0xfffffffffffffffffffffffffffffffefffffffffffffffc
b 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1