Network Working Group S. Josefsson
Internet-Draft SJD AB
Intended status: Informational February 7, 2015
Expires: August 11, 2015
EdDSA and Ed25519
draft-josefsson-eddsa-ed25519-00
Abstract
The elliptic curve signature scheme EdDSA and one instance of it
called Ed25519 is described. An example implementation and test
vectors are provided.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. EdDSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.1. Encoding . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2. Keys . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.3. Sign . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.4. Verify . . . . . . . . . . . . . . . . . . . . . . . . . 4
4. Ed25519 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
5. Test Vectors for Ed25519 . . . . . . . . . . . . . . . . . . 9
6. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 10
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 10
8. Security Considerations . . . . . . . . . . . . . . . . . . . 10
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 10
9.1. Normative References . . . . . . . . . . . . . . . . . . 10
9.2. Informative References . . . . . . . . . . . . . . . . . 10
Appendix A. Ed25519 Python Library . . . . . . . . . . . . . . . 11
Appendix B. Library driver . . . . . . . . . . . . . . . . . . . 14
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 15
1. Introduction
The Edwards-curve Digital Signature Algorithm (EdDSA) is a variant of
Schnorr's signature system with Twisted Edwards curves. EdDSA needs
to be instantiated with certain parameters, and Ed25519 is described
in this document. To facilitate adoption in the Internet community
of Ed25519, this document describe the signature scheme in an
implementation-oriented way, and we provide sample code and test
vectors.
The advantages with EdDSA and Ed25519 include:
1. High-performance on a variety of platforms.
2. Does not require the use of a unique random number for each
signature.
3. Collision resilience, meaning that hash-function collisions do
not break this system.
4. More resilient to side-channel attacks.
5. Small public keys (32 bytes) and signatures (64 bytes).
For further background, see the original EdDSA paper [EDDSA].
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2. Notation
The following notation is used throughout the document:
GF(p) finite field with p elements
x^y x multiplied by itself y times
h_i the i'th byte of h
a || b (bit-)string a concatenated with (bit-)string b
3. EdDSA
EdDSA has seven parameters:
1. an integer b >= 10.
2. a cryptographic hash function H producing 2b-bit outputs.
3. a prime power q congruent to 1 modulo 4.
4. a (b-1)-bit encoding of elements of the finite field GF(q).
5. a non-square element d of GF(q)
6. a prime l between 2^(b-4) and 2^(b-3) satisfying lB=0 where nB
means the n'th multiple of B in the group E.
7. an element B != (0,1) of the set E = { (x,y) is a member of GF(q)
x GF(q) such that -x^2 + y^2 = 1 + dx^2y^2 }.
3.1. Encoding
An element (x,y) of E is encoded as a b-bit string called ENC(x,y)
which is the (b-1)-bit encoding of y concatenated with one bit that
is 1 if x is negative and 0 if x is not negative. Negative elements
of GF(q) are those x which the (b-1)-bit encoding of x is
lexicographically larger than the (b-1)-bit encoding of -x.
3.2. Keys
An EdDSA secret key is a b-bit string k. Let the hash H(k) = (h_0,
h_1, ..., h_(2b-1)) determine an integer a which is 2^(b-2) plus the
sum of m = 2^i * h_i for all i equal or larger than 3 and equal to or
less than b-3 such that m is a member of the set { 2^(b-2), 2^(b-2) +
8, ..., 2^(b-1) - 8 }. The EdDSA public key is ENC(A) = ENC(aB).
The bits h_b, ..., h_(2b-1) is used below during signing.
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3.3. Sign
The signature of a message M under a secret key k is the 2b-bit
string ENC(R) || ENC'(S), where ENC'(S) is defined as the b-bit
little-endian encoding of S. R and S are derived as follows. First
define r = H(h_b, ... h_(2b-1)), M) interpreting 2b-bit strings in
little-endian form as integers in {0, 1, ..., 2^(2b)-1}. Let R=rB
and S=(r+H(ENC(R) || ENC(A) || M)a) mod l.
3.4. Verify
To verify a signature ENC(R) || ENC'(S) on a message M under a public
key ENC(A), proceed as follows. Parse the inputs so that A and R is
an element of E, and S is a member of the set {0, 1, ..., l-1 }.
Compute H' = H(ENC(R) || ENC(A) || M) and check the group equation
8SB = 8R + 8H'A in E. Verification is rejected if parsing fails or
the group equation does not hold.
4. Ed25519
Ed25519 is EdDSA instantiated with b=256, H being SHA-512 [RFC4634],
q is the prime 2^255-19, the 255-bit encoding of GF(2^255-19) being
the little-endian encoding of {0, 1, ..., 2^255-20}, l is the prime
2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed, d = -121665/121666 which
is a member of GF(q), and B is the unique point (x, 4/5) in E for
which x is positive. The curve q, prime l, d and B follows from
[I-D.irtf-cfrg-curves].
The rest of this section describes how Ed25519 can be implemented in
Python (version 3.2 or later) for illustration. See appendix A for
the complete implementation and appendix B for a test-driver to run
it through some test vectors.
First some preliminaries that will be needed.
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import hashlib
def sha512(s):
return hashlib.sha512(s).digest()
# Base field Z_p
p = 2**255 - 19
def modp_inv(x):
return pow(x, p-2, p)
# Curve constant
d = -121665 * modp_inv(121666) % p
# Group order
q = 2**252 + 27742317777372353535851937790883648493
def sha512_modq(s):
return int.from_bytes(sha512(s), "little") % q
Then follows functions to perform point operations.
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# Points are represented as tuples (X, Y, Z, T) of extended coordinates,
# with x = X/Z, y = Y/Z, x*y = T/Z
def point_add(P, Q):
A = (P[1]-P[0])*(Q[1]-Q[0]) % p
B = (P[1]+P[0])*(Q[1]+Q[0]) % p
C = 2 * P[3] * Q[3] * d % p
D = 2 * P[2] * Q[2] % p
E = B-A
F = D-C
G = D+C
H = B+A
return (E*F, G*H, F*G, E*H)
# Computes Q = s * Q
def point_mul(s, P):
Q = (0, 1, 1, 0) # Neutral element
while s > 0:
# Is there any bit-set predicate?
if s & 1:
Q = point_add(Q, P)
P = point_add(P, P)
s >>= 1
return Q
def point_equal(P, Q):
# x1 / z1 == x2 / z2 <==> x1 * z2 == x2 * z1
if (P[0] * Q[2] - Q[0] * P[2]) % p != 0:
return False
if (P[1] * Q[2] - Q[1] * P[2]) % p != 0:
return False
return True
Now follows functions for point compression.
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# Square root of -1
modp_sqrt_m1 = pow(2, (p-1) // 4, p)
# Compute corresponding x coordinate, with low bit corresponding to sign,
# or return None on failure
def recover_x(y, sign):
x2 = (y*y-1) * modp_inv(d*y*y+1)
if x2 == 0:
if sign:
return None
else:
return 0
# Compute square root of x2
x = pow(x2, (p+3) // 8, p)
if (x*x - x2) % p != 0:
x = x * modp_sqrt_m1 % p
if (x*x - x2) % p != 0:
return None
if (x & 1) != sign:
x = p - x
return x
# Base point
g_y = 4 * modp_inv(5) % p
g_x = recover_x(g_y, 0)
G = (g_x, g_y, 1, g_x * g_y % p)
def point_compress(P):
zinv = modp_inv(P[2])
x = P[0] * zinv % p
y = P[1] * zinv % p
return int.to_bytes(y | ((x & 1) << 255), 32, "little")
def point_decompress(s):
if len(s) != 32:
raise Exception("Invalid input length for decompression")
y = int.from_bytes(s, "little")
sign = y >> 255
y &= (1 << 255) - 1
x = recover_x(y, sign)
if x is None:
return None
else:
return (x, y, 1, x*y % p)
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These are functions for manipulating the secret.
def secret_expand(secret):
if len(secret) != 32:
raise Exception("Bad size of private key")
h = sha512(secret)
a = int.from_bytes(h[:32], "little")
a &= (1 << 254) - 8
a |= (1 << 254)
return (a, h[32:])
def secret_to_public(secret):
(a, dummy) = secret_expand(secret)
return point_compress(point_mul(a, G))
The signature function works as below.
def sign(secret, msg):
a, prefix = secret_expand(secret)
A = point_compress(point_mul(a, G))
r = sha512_modq(prefix + msg)
R = point_mul(r, G)
Rs = point_compress(R)
h = sha512_modq(Rs + A + msg)
s = (r + h * a) % q
return Rs + int.to_bytes(s, 32, "little")
And finally the verification function.
def verify(public, msg, signature):
if len(public) != 32:
raise Exception("Bad public-key length")
if len(signature) != 64:
Exception("Bad signature length")
A = point_decompress(public)
if not A:
return False
Rs = signature[:32]
R = point_decompress(Rs)
if not R:
return False
s = int.from_bytes(signature[32:], "little")
h = sha512_modq(Rs + public + msg)
sB = point_mul(s, G)
hA = point_mul(h, A)
return point_equal(sB, point_add(R, hA))
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5. Test Vectors for Ed25519
Below is a sequence of octets with test vectors for the the Ed25519
signature algorithm. The octets are hex encoded and whitespace is
inserted for readability. Private keys are 64 bytes, public keys 32
bytes, message of arbitrary length, and signatures are 64 bytes.
-----
PRIVATE KEY:
9d61b19deffd5a60ba844af492ec2cc4
4449c5697b326919703bac031cae7f60
d75a980182b10ab7d54bfed3c964073a
0ee172f3daa62325af021a68f707511a
PUBLIC KEY:
d75a980182b10ab7d54bfed3c964073a
0ee172f3daa62325af021a68f707511a
MESSAGE (length 0 bytes):
SIGNATURE:
e5564300c360ac729086e2cc806e828a
84877f1eb8e5d974d873e06522490155
5fb8821590a33bacc61e39701cf9b46b
d25bf5f0595bbe24655141438e7a100b
-----
PRIVATE KEY:
4ccd089b28ff96da9db6c346ec114e0f
5b8a319f35aba624da8cf6ed4fb8a6fb
3d4017c3e843895a92b70aa74d1b7ebc
9c982ccf2ec4968cc0cd55f12af4660c
PUBLIC KEY:
3d4017c3e843895a92b70aa74d1b7ebc
9c982ccf2ec4968cc0cd55f12af4660c
MESSAGE (length 1 byte):
72
SIGNATURE:
92a009a9f0d4cab8720e820b5f642540
a2b27b5416503f8fb3762223ebdb69da
085ac1e43e15996e458f3613d0f11d8c
387b2eaeb4302aeeb00d291612bb0c00
-----
PRIVATE KEY:
c5aa8df43f9f837bedb7442f31dcb7b1
66d38535076f094b85ce3a2e0b4458f7
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fc51cd8e6218a1a38da47ed00230f058
0816ed13ba3303ac5deb911548908025
PUBLIC KEY:
fc51cd8e6218a1a38da47ed00230f058
0816ed13ba3303ac5deb911548908025
MESSAGE (length 2 bytes):
af82
SIGNATURE:
6291d657deec24024827e69c3abe01a3
0ce548a284743a445e3680d7db5ac3ac
18ff9b538d16f290ae67f760984dc659
4a7c15e9716ed28dc027beceea1ec40a
-----
6. Acknowledgements
The Python code was written by Niels Moeller.
7. IANA Considerations
None.
8. Security Considerations
TBA.
9. References
9.1. Normative References
[RFC4634] Eastlake, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and HMAC-SHA)", RFC 4634, July 2006.
[I-D.irtf-cfrg-curves]
Langley, A., Salz, R., and S. Turner, "Elliptic Curves for
Security", draft-irtf-cfrg-curves-01 (work in progress),
January 2015.
9.2. Informative References
[EDDSA] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B.
Yang, "High-speed high-security signatures", WWW
http://ed25519.cr.yp.to/ed25519-20110926.pdf, September
2011.
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Appendix A. Ed25519 Python Library
Below is an example implementation of Ed25519 written in Python,
version 3.2 or higher is required.
# Loosely based on the public domain code at
# http://ed25519.cr.yp.to/software.html
#
# Needs python-3.2
import hashlib
def sha512(s):
return hashlib.sha512(s).digest()
# Base field Z_p
p = 2**255 - 19
def modp_inv(x):
return pow(x, p-2, p)
# Curve constant
d = -121665 * modp_inv(121666) % p
# Group order
q = 2**252 + 27742317777372353535851937790883648493
def sha512_modq(s):
return int.from_bytes(sha512(s), "little") % q
# Points are represented as tuples (X, Y, Z, T) of extended coordinates,
# with x = X/Z, y = Y/Z, x*y = T/Z
def point_add(P, Q):
A = (P[1]-P[0])*(Q[1]-Q[0]) % p
B = (P[1]+P[0])*(Q[1]+Q[0]) % p
C = 2 * P[3] * Q[3] * d % p
D = 2 * P[2] * Q[2] % p
E = B-A
F = D-C
G = D+C
H = B+A
return (E*F, G*H, F*G, E*H)
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# Computes Q = s * Q
def point_mul(s, P):
Q = (0, 1, 1, 0) # Neutral element
while s > 0:
# Is there any bit-set predicate?
if s & 1:
Q = point_add(Q, P)
P = point_add(P, P)
s >>= 1
return Q
def point_equal(P, Q):
# x1 / z1 == x2 / z2 <==> x1 * z2 == x2 * z1
if (P[0] * Q[2] - Q[0] * P[2]) % p != 0:
return False
if (P[1] * Q[2] - Q[1] * P[2]) % p != 0:
return False
return True
# Square root of -1
modp_sqrt_m1 = pow(2, (p-1) // 4, p)
# Compute corresponding x coordinate, with low bit corresponding to sign,
# or return None on failure
def recover_x(y, sign):
x2 = (y*y-1) * modp_inv(d*y*y+1)
if x2 == 0:
if sign:
return None
else:
return 0
# Compute square root of x2
x = pow(x2, (p+3) // 8, p)
if (x*x - x2) % p != 0:
x = x * modp_sqrt_m1 % p
if (x*x - x2) % p != 0:
return None
if (x & 1) != sign:
x = p - x
return x
# Base point
g_y = 4 * modp_inv(5) % p
g_x = recover_x(g_y, 0)
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G = (g_x, g_y, 1, g_x * g_y % p)
def point_compress(P):
zinv = modp_inv(P[2])
x = P[0] * zinv % p
y = P[1] * zinv % p
return int.to_bytes(y | ((x & 1) << 255), 32, "little")
def point_decompress(s):
if len(s) != 32:
raise Exception("Invalid input length for decompression")
y = int.from_bytes(s, "little")
sign = y >> 255
y &= (1 << 255) - 1
x = recover_x(y, sign)
if x is None:
return None
else:
return (x, y, 1, x*y % p)
def secret_expand(secret):
if len(secret) != 32:
raise Exception("Bad size of private key")
h = sha512(secret)
a = int.from_bytes(h[:32], "little")
a &= (1 << 254) - 8
a |= (1 << 254)
return (a, h[32:])
def secret_to_public(secret):
(a, dummy) = secret_expand(secret)
return point_compress(point_mul(a, G))
def sign(secret, msg):
a, prefix = secret_expand(secret)
A = point_compress(point_mul(a, G))
r = sha512_modq(prefix + msg)
R = point_mul(r, G)
Rs = point_compress(R)
h = sha512_modq(Rs + A + msg)
s = (r + h * a) % q
return Rs + int.to_bytes(s, 32, "little")
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def verify(public, msg, signature):
if len(public) != 32:
raise Exception("Bad public-key length")
if len(signature) != 64:
Exception("Bad signature length")
A = point_decompress(public)
if not A:
return False
Rs = signature[:32]
R = point_decompress(Rs)
if not R:
return False
s = int.from_bytes(signature[32:], "little")
h = sha512_modq(Rs + public + msg)
sB = point_mul(s, G)
hA = point_mul(h, A)
return point_equal(sB, point_add(R, hA))
Appendix B. Library driver
Below is a command-line tool that uses the library above to perform
computations, for interactive use or for self-checking.
import sys
import binascii
from ed25519 import *
def point_valid(P):
zinv = modp_inv(P[2])
x = P[0] * zinv % p
y = P[1] * zinv % p
assert (x*y - P[3]*zinv) % p == 0
return (-x*x + y*y - 1 - d*x*x*y*y) % p == 0
assert point_valid(G)
Z = (0, 1, 1, 0)
assert point_valid(Z)
assert point_equal(Z, point_add(Z, Z))
assert point_equal(G, point_add(Z, G))
assert point_equal(Z, point_mul(0, G))
assert point_equal(G, point_mul(1, G))
assert point_equal(point_add(G, G), point_mul(2, G))
for i in range(0, 100):
assert point_valid(point_mul(i, G))
assert point_equal(Z, point_mul(q, G))
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def munge_string(s, pos, change):
return (s[:pos] +
int.to_bytes(s[pos] ^ change, 1, "little") +
s[pos+1:])
# Read a file in the format of
# http://ed25519.cr.yp.to/python/sign.input
lineno = 0
while True:
line = sys.stdin.readline()
if not line:
break
lineno = lineno + 1
print(lineno)
fields = line.split(":")
secret = (binascii.unhexlify(fields[0]))[:32]
public = binascii.unhexlify(fields[1])
msg = binascii.unhexlify(fields[2])
signature = binascii.unhexlify(fields[3])[:64]
assert public == secret_to_public(secret)
assert signature == sign(secret, msg)
assert verify(public, msg, signature)
if len(msg) == 0:
bad_msg = b"x"
else:
bad_msg = munge_string(msg, len(msg) // 3, 4)
assert not verify(public, bad_msg, signature)
bad_signature = munge_string(signature, 20, 8)
assert not verify(public, msg, bad_signature)
bad_signature = munge_string(signature, 40, 16)
assert not verify(public, msg, bad_signature)
Author's Address
Simon Josefsson
SJD AB
Email: simon@josefsson.org
URI: http://josefsson.org/
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