"brainpoolP256r1"“ - For 256-Bit ECC Keys

This section describes 'brainpoolP256r1' elliptic curve domain parameters for generating 256-Bit ECC Keys as specified by RFC 5639.

What Is "brainpoolP256r1"? "brainpoolP256r1" is a specific elliptic curve and associated domain parameters selected and recommended in "RFC 5639 - Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation" at https://datatracker.ietf.org/doc/html/rfc5639.

The "P256r1" part of the "brainpoolP256r1" name indicates:

P     Field type = Prime field
256   Key size = 256
r     Curve type = Regular curve
1     Cofactor = 1

"brainpoolP256r1" domain parameters (p, a, b, G, n, h)

p: The modulo used to specify the reduced elliptic curve group:

p = 0xA9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377

p is congruent to 3 mod 4: p mod 4 = p%4 = 3
  This allows fast computation of y for a given x.

a: The first coefficient of the elliptic curve:

a = 0x7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9

b: The second coefficient of the elliptic curve:

b = 0x26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6

G: The generator (base point) of the subgroup:

G =(0x8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262,
    0x547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997)

n: The order of the subgroup:

n = 0xA9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7

h: The cofactor of the subgroup:

h = 1

Verify domain parameters with Python - G is on the curve.

herong> python

>>> p = 0xA9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377
>>> a = 0x7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9
>>> b = 0x26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6
>>> G =(0x8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262,
...     0x547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997)

>>> G[1]**2 % p == (G[0]**3 + a*G[0] + b) % p
True

Generate a "brainpoolP256r1" key pair with OpenSSL

herong> openssl ecparam -genkey -name brainpoolP256r1 \
  -out brainpoolP256r1.pem -param_enc explicit

herong> openssl ec -in brainpoolP256r1.pem -noout -text

Private-Key: (256 bit)
priv:
    30:cb:fd:6a:ce:88:9f:ee:b9:57:1c:c5:f0:64:7f:
    5c:b2:f7:48:83:07:20:e8:c4:0e:b3:77:08:d1:bb:
    22:1c
pub:
    04:62:48:3b:be:5d:77:64:59:d9:ff:28:69:5f:b2:
    e8:cb:62:14:4e:55:9b:67:78:f4:20:51:05:a1:8c:
    3d:bc:4d:30:06:4c:74:34:74:88:08:52:bb:46:21:
    7a:51:a4:55:66:8c:48:05:ec:5e:4a:ec:b4:39:63:
    1f:90:1f:bd:63
Field Type: prime-field
Prime:
    00:a9:fb:57:db:a1:ee:a9:bc:3e:66:0a:90:9d:83:
    8d:72:6e:3b:f6:23:d5:26:20:28:20:13:48:1d:1f:
    6e:53:77
A:
    7d:5a:09:75:fc:2c:30:57:ee:f6:75:30:41:7a:ff:
    e7:fb:80:55:c1:26:dc:5c:6c:e9:4a:4b:44:f3:30:
    b5:d9
B:
    26:dc:5c:6c:e9:4a:4b:44:f3:30:b5:d9:bb:d7:7c:
    bf:95:84:16:29:5c:f7:e1:ce:6b:cc:dc:18:ff:8c:
    07:b6
Generator (uncompressed):
    04:8b:d2:ae:b9:cb:7e:57:cb:2c:4b:48:2f:fc:81:
    b7:af:b9:de:27:e1:e3:bd:23:c2:3a:44:53:bd:9a:
    ce:32:62:54:7e:f8:35:c3:da:c4:fd:97:f8:46:1a:
    14:61:1d:c9:c2:77:45:13:2d:ed:8e:54:5c:1d:54:
    c7:2f:04:69:97
Order:
    00:a9:fb:57:db:a1:ee:a9:bc:3e:66:0a:90:9d:83:
    8d:71:8c:39:7a:a3:b5:61:a6:f7:90:1e:0e:82:97:
    48:56:a7
Cofactor:  1 (0x1)

The printed domain parameters (Prime, A, B, Generator, Order, Cofactor) match well with (p, a, b, G, n, h) specified by RFC 5639. Remember that OpenSSL prints "Generator" as 0x04<G.x><G.y>.

Exercise: Verify all "prime field" and "regular" curves: brainpoolP160r1, brainpoolP192r1, brainpoolP224r1, brainpoolP256r1, brainpoolP320r1, brainpoolP384r1, and brainpoolP512r1 specified by RFC 5639.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 EC Cryptography in Java

Standard Elliptic Curves

 What Are Standard Elliptic Curves

 "openssl ecparam -list_curves" - Curves Supported by OpenSSL

 "secp256r1" - For 256-Bit ECC Keys

 "secp256k1" - For 256-Bit ECC Keys

 "sect283r1" - For 256-Bit ECC Keys

"brainpoolP256r1"“ - For 256-Bit ECC Keys

 "brainpoolP256t1"“ - For 256-Bit ECC Keys

 Terminology

 References

 Full Version in PDF/EPUB