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// Copyright 2011 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// Package ecdsa implements the Elliptic Curve Digital Signature Algorithm, as
// defined in FIPS 186-3.
//
// This implementation derives the nonce from an AES-CTR CSPRNG keyed by
// ChopMD(256, SHA2-512(priv.D || entropy || hash)). The CSPRNG key is IRO by
// a result of Coron; the AES-CTR stream is IRO under standard assumptions.
package ecdsa
// References:
// [NSA]: Suite B implementer's guide to FIPS 186-3,
// http://www.nsa.gov/ia/_files/ecdsa.pdf
// [SECG]: SECG, SEC1
// http://www.secg.org/sec1-v2.pdf
import (
"crypto"
"crypto/aes"
"crypto/cipher"
"crypto/elliptic"
"crypto/sha512"
"encoding/asn1"
"errors"
"io"
"math/big"
"crypto/internal/randutil"
)
// A invertible implements fast inverse mod Curve.Params().N
type invertible interface {
// Inverse returns the inverse of k in GF(P)
Inverse(k *big.Int) *big.Int
}
// combinedMult implements fast multiplication S1*g + S2*p (g - generator, p - arbitrary point)
type combinedMult interface {
CombinedMult(bigX, bigY *big.Int, baseScalar, scalar []byte) (x, y *big.Int)
}
const (
aesIV = "IV for ECDSA CTR"
)
// PublicKey represents an ECDSA public key.
type PublicKey struct {
elliptic.Curve
X, Y *big.Int
}
// PrivateKey represents an ECDSA private key.
type PrivateKey struct {
PublicKey
D *big.Int
}
type ecdsaSignature struct {
R, S *big.Int
}
// Public returns the public key corresponding to priv.
func (priv *PrivateKey) Public() crypto.PublicKey {
return &priv.PublicKey
}
// Sign signs digest with priv, reading randomness from rand. The opts argument
// is not currently used but, in keeping with the crypto.Signer interface,
// should be the hash function used to digest the message.
//
// This method implements crypto.Signer, which is an interface to support keys
// where the private part is kept in, for example, a hardware module. Common
// uses should use the Sign function in this package directly.
func (priv *PrivateKey) Sign(rand io.Reader, digest []byte, opts crypto.SignerOpts) ([]byte, error) {
r, s, err := Sign(rand, priv, digest)
if err != nil {
return nil, err
}
return asn1.Marshal(ecdsaSignature{r, s})
}
var one = new(big.Int).SetInt64(1)
// randFieldElement returns a random element of the field underlying the given
// curve using the procedure given in [NSA] A.2.1.
func randFieldElement(c elliptic.Curve, rand io.Reader) (k *big.Int, err error) {
params := c.Params()
b := make([]byte, params.BitSize/8+8)
_, err = io.ReadFull(rand, b)
if err != nil {
return
}
k = new(big.Int).SetBytes(b)
n := new(big.Int).Sub(params.N, one)
k.Mod(k, n)
k.Add(k, one)
return
}
// GenerateKey generates a public and private key pair.
func GenerateKey(c elliptic.Curve, rand io.Reader) (*PrivateKey, error) {
k, err := randFieldElement(c, rand)
if err != nil {
return nil, err
}
priv := new(PrivateKey)
priv.PublicKey.Curve = c
priv.D = k
priv.PublicKey.X, priv.PublicKey.Y = c.ScalarBaseMult(k.Bytes())
return priv, nil
}
// hashToInt converts a hash value to an integer. There is some disagreement
// about how this is done. [NSA] suggests that this is done in the obvious
// manner, but [SECG] truncates the hash to the bit-length of the curve order
// first. We follow [SECG] because that's what OpenSSL does. Additionally,
// OpenSSL right shifts excess bits from the number if the hash is too large
// and we mirror that too.
func hashToInt(hash []byte, c elliptic.Curve) *big.Int {
orderBits := c.Params().N.BitLen()
orderBytes := (orderBits + 7) / 8
if len(hash) > orderBytes {
hash = hash[:orderBytes]
}
ret := new(big.Int).SetBytes(hash)
excess := len(hash)*8 - orderBits
if excess > 0 {
ret.Rsh(ret, uint(excess))
}
return ret
}
// fermatInverse calculates the inverse of k in GF(P) using Fermat's method.
// This has better constant-time properties than Euclid's method (implemented
// in math/big.Int.ModInverse) although math/big itself isn't strictly
// constant-time so it's not perfect.
func fermatInverse(k, N *big.Int) *big.Int {
two := big.NewInt(2)
nMinus2 := new(big.Int).Sub(N, two)
return new(big.Int).Exp(k, nMinus2, N)
}
var errZeroParam = errors.New("zero parameter")
// Sign signs a hash (which should be the result of hashing a larger message)
// using the private key, priv. If the hash is longer than the bit-length of the
// private key's curve order, the hash will be truncated to that length. It
// returns the signature as a pair of integers. The security of the private key
// depends on the entropy of rand.
func Sign(rand io.Reader, priv *PrivateKey, hash []byte) (r, s *big.Int, err error) {
randutil.MaybeReadByte(rand)
// Get min(log2(q) / 2, 256) bits of entropy from rand.
entropylen := (priv.Curve.Params().BitSize + 7) / 16
if entropylen > 32 {
entropylen = 32
}
entropy := make([]byte, entropylen)
_, err = io.ReadFull(rand, entropy)
if err != nil {
return
}
// Initialize an SHA-512 hash context; digest ...
md := sha512.New()
md.Write(priv.D.Bytes()) // the private key,
md.Write(entropy) // the entropy,
md.Write(hash) // and the input hash;
key := md.Sum(nil)[:32] // and compute ChopMD-256(SHA-512),
// which is an indifferentiable MAC.
// Create an AES-CTR instance to use as a CSPRNG.
block, err := aes.NewCipher(key)
if err != nil {
return nil, nil, err
}
// Create a CSPRNG that xors a stream of zeros with
// the output of the AES-CTR instance.
csprng := cipher.StreamReader{
R: zeroReader,
S: cipher.NewCTR(block, []byte(aesIV)),
}
// See [NSA] 3.4.1
c := priv.PublicKey.Curve
N := c.Params().N
if N.Sign() == 0 {
return nil, nil, errZeroParam
}
var k, kInv *big.Int
for {
for {
k, err = randFieldElement(c, csprng)
if err != nil {
r = nil
return
}
if in, ok := priv.Curve.(invertible); ok {
kInv = in.Inverse(k)
} else {
kInv = fermatInverse(k, N) // N != 0
}
r, _ = priv.Curve.ScalarBaseMult(k.Bytes())
r.Mod(r, N)
if r.Sign() != 0 {
break
}
}
e := hashToInt(hash, c)
s = new(big.Int).Mul(priv.D, r)
s.Add(s, e)
s.Mul(s, kInv)
s.Mod(s, N) // N != 0
if s.Sign() != 0 {
break
}
}
return
}
// Verify verifies the signature in r, s of hash using the public key, pub. Its
// return value records whether the signature is valid.
func Verify(pub *PublicKey, hash []byte, r, s *big.Int) bool {
// See [NSA] 3.4.2
c := pub.Curve
N := c.Params().N
if r.Sign() <= 0 || s.Sign() <= 0 {
return false
}
if r.Cmp(N) >= 0 || s.Cmp(N) >= 0 {
return false
}
e := hashToInt(hash, c)
var w *big.Int
if in, ok := c.(invertible); ok {
w = in.Inverse(s)
} else {
w = new(big.Int).ModInverse(s, N)
}
u1 := e.Mul(e, w)
u1.Mod(u1, N)
u2 := w.Mul(r, w)
u2.Mod(u2, N)
// Check if implements S1*g + S2*p
var x, y *big.Int
if opt, ok := c.(combinedMult); ok {
x, y = opt.CombinedMult(pub.X, pub.Y, u1.Bytes(), u2.Bytes())
} else {
x1, y1 := c.ScalarBaseMult(u1.Bytes())
x2, y2 := c.ScalarMult(pub.X, pub.Y, u2.Bytes())
x, y = c.Add(x1, y1, x2, y2)
}
if x.Sign() == 0 && y.Sign() == 0 {
return false
}
x.Mod(x, N)
return x.Cmp(r) == 0
}
type zr struct {
io.Reader
}
// Read replaces the contents of dst with zeros.
func (z *zr) Read(dst []byte) (n int, err error) {
for i := range dst {
dst[i] = 0
}
return len(dst), nil
}
var zeroReader = &zr{}