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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 186-4 and SEC 1, Version 2.0.
//
// Signatures generated by this package are not deterministic, but entropy is
// mixed with the private key and the message, achieving the same level of
// security in case of randomness source failure.
package ecdsa
// [FIPS 186-4] references ANSI X9.62-2005 for the bulk of the ECDSA algorithm.
// That standard is not freely available, which is a problem in an open source
// implementation, because not only the implementer, but also any maintainer,
// contributor, reviewer, auditor, and learner needs access to it. Instead, this
// package references and follows the equivalent [SEC 1, Version 2.0].
//
// [FIPS 186-4]: https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
// [SEC 1, Version 2.0]: https://www.secg.org/sec1-v2.pdf
import (
"bytes"
"crypto"
"crypto/aes"
"crypto/cipher"
"crypto/ecdh"
"crypto/elliptic"
"crypto/internal/bigmod"
"crypto/internal/boring"
"crypto/internal/boring/bbig"
"crypto/internal/nistec"
"crypto/internal/randutil"
"crypto/sha512"
"crypto/subtle"
"errors"
"io"
"math/big"
"sync"
"golang.org/x/crypto/cryptobyte"
"golang.org/x/crypto/cryptobyte/asn1"
)
// PublicKey represents an ECDSA public key.
type PublicKey struct {
elliptic.Curve
X, Y *big.Int
}
// Any methods implemented on PublicKey might need to also be implemented on
// PrivateKey, as the latter embeds the former and will expose its methods.
// ECDH returns k as a [ecdh.PublicKey]. It returns an error if the key is
// invalid according to the definition of [ecdh.Curve.NewPublicKey], or if the
// Curve is not supported by crypto/ecdh.
func (k *PublicKey) ECDH() (*ecdh.PublicKey, error) {
c := curveToECDH(k.Curve)
if c == nil {
return nil, errors.New("ecdsa: unsupported curve by crypto/ecdh")
}
if !k.Curve.IsOnCurve(k.X, k.Y) {
return nil, errors.New("ecdsa: invalid public key")
}
return c.NewPublicKey(elliptic.Marshal(k.Curve, k.X, k.Y))
}
// Equal reports whether pub and x have the same value.
//
// Two keys are only considered to have the same value if they have the same Curve value.
// Note that for example elliptic.P256() and elliptic.P256().Params() are different
// values, as the latter is a generic not constant time implementation.
func (pub *PublicKey) Equal(x crypto.PublicKey) bool {
xx, ok := x.(*PublicKey)
if !ok {
return false
}
return bigIntEqual(pub.X, xx.X) && bigIntEqual(pub.Y, xx.Y) &&
// Standard library Curve implementations are singletons, so this check
// will work for those. Other Curves might be equivalent even if not
// singletons, but there is no definitive way to check for that, and
// better to err on the side of safety.
pub.Curve == xx.Curve
}
// PrivateKey represents an ECDSA private key.
type PrivateKey struct {
PublicKey
D *big.Int
}
// ECDH returns k as a [ecdh.PrivateKey]. It returns an error if the key is
// invalid according to the definition of [ecdh.Curve.NewPrivateKey], or if the
// Curve is not supported by crypto/ecdh.
func (k *PrivateKey) ECDH() (*ecdh.PrivateKey, error) {
c := curveToECDH(k.Curve)
if c == nil {
return nil, errors.New("ecdsa: unsupported curve by crypto/ecdh")
}
size := (k.Curve.Params().N.BitLen() + 7) / 8
if k.D.BitLen() > size*8 {
return nil, errors.New("ecdsa: invalid private key")
}
return c.NewPrivateKey(k.D.FillBytes(make([]byte, size)))
}
func curveToECDH(c elliptic.Curve) ecdh.Curve {
switch c {
case elliptic.P256():
return ecdh.P256()
case elliptic.P384():
return ecdh.P384()
case elliptic.P521():
return ecdh.P521()
default:
return nil
}
}
// Public returns the public key corresponding to priv.
func (priv *PrivateKey) Public() crypto.PublicKey {
return &priv.PublicKey
}
// Equal reports whether priv and x have the same value.
//
// See PublicKey.Equal for details on how Curve is compared.
func (priv *PrivateKey) Equal(x crypto.PrivateKey) bool {
xx, ok := x.(*PrivateKey)
if !ok {
return false
}
return priv.PublicKey.Equal(&xx.PublicKey) && bigIntEqual(priv.D, xx.D)
}
// bigIntEqual reports whether a and b are equal leaking only their bit length
// through timing side-channels.
func bigIntEqual(a, b *big.Int) bool {
return subtle.ConstantTimeCompare(a.Bytes(), b.Bytes()) == 1
}
// 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 can use the SignASN1 function in this package directly.
func (priv *PrivateKey) Sign(rand io.Reader, digest []byte, opts crypto.SignerOpts) ([]byte, error) {
return SignASN1(rand, priv, digest)
}
// GenerateKey generates a new ECDSA private key for the specified curve.
//
// Most applications should use [crypto/rand.Reader] as rand. Note that the
// returned key does not depend deterministically on the bytes read from rand,
// and may change between calls and/or between versions.
func GenerateKey(c elliptic.Curve, rand io.Reader) (*PrivateKey, error) {
randutil.MaybeReadByte(rand)
if boring.Enabled && rand == boring.RandReader {
x, y, d, err := boring.GenerateKeyECDSA(c.Params().Name)
if err != nil {
return nil, err
}
return &PrivateKey{PublicKey: PublicKey{Curve: c, X: bbig.Dec(x), Y: bbig.Dec(y)}, D: bbig.Dec(d)}, nil
}
boring.UnreachableExceptTests()
switch c.Params() {
case elliptic.P224().Params():
return generateNISTEC(p224(), rand)
case elliptic.P256().Params():
return generateNISTEC(p256(), rand)
case elliptic.P384().Params():
return generateNISTEC(p384(), rand)
case elliptic.P521().Params():
return generateNISTEC(p521(), rand)
default:
return generateLegacy(c, rand)
}
}
func generateNISTEC[Point nistPoint[Point]](c *nistCurve[Point], rand io.Reader) (*PrivateKey, error) {
k, Q, err := randomPoint(c, rand)
if err != nil {
return nil, err
}
priv := new(PrivateKey)
priv.PublicKey.Curve = c.curve
priv.D = new(big.Int).SetBytes(k.Bytes(c.N))
priv.PublicKey.X, priv.PublicKey.Y, err = c.pointToAffine(Q)
if err != nil {
return nil, err
}
return priv, nil
}
// randomPoint returns a random scalar and the corresponding point using the
// procedure given in FIPS 186-4, Appendix B.5.2 (rejection sampling).
func randomPoint[Point nistPoint[Point]](c *nistCurve[Point], rand io.Reader) (k *bigmod.Nat, p Point, err error) {
k = bigmod.NewNat()
for {
b := make([]byte, c.N.Size())
if _, err = io.ReadFull(rand, b); err != nil {
return
}
// Mask off any excess bits to increase the chance of hitting a value in
// (0, N). These are the most dangerous lines in the package and maybe in
// the library: a single bit of bias in the selection of nonces would likely
// lead to key recovery, but no tests would fail. Look but DO NOT TOUCH.
if excess := len(b)*8 - c.N.BitLen(); excess > 0 {
// Just to be safe, assert that this only happens for the one curve that
// doesn't have a round number of bits.
if excess != 0 && c.curve.Params().Name != "P-521" {
panic("ecdsa: internal error: unexpectedly masking off bits")
}
b[0] >>= excess
}
// FIPS 186-4 makes us check k <= N - 2 and then add one.
// Checking 0 < k <= N - 1 is strictly equivalent.
// None of this matters anyway because the chance of selecting
// zero is cryptographically negligible.
if _, err = k.SetBytes(b, c.N); err == nil && k.IsZero() == 0 {
break
}
if testingOnlyRejectionSamplingLooped != nil {
testingOnlyRejectionSamplingLooped()
}
}
p, err = c.newPoint().ScalarBaseMult(k.Bytes(c.N))
return
}
// testingOnlyRejectionSamplingLooped is called when rejection sampling in
// randomPoint rejects a candidate for being higher than the modulus.
var testingOnlyRejectionSamplingLooped func()
// errNoAsm is returned by signAsm and verifyAsm when the assembly
// implementation is not available.
var errNoAsm = errors.New("no assembly implementation available")
// SignASN1 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 ASN.1 encoded signature.
//
// The signature is randomized. Most applications should use [crypto/rand.Reader]
// as rand. Note that the returned signature does not depend deterministically on
// the bytes read from rand, and may change between calls and/or between versions.
func SignASN1(rand io.Reader, priv *PrivateKey, hash []byte) ([]byte, error) {
randutil.MaybeReadByte(rand)
if boring.Enabled && rand == boring.RandReader {
b, err := boringPrivateKey(priv)
if err != nil {
return nil, err
}
return boring.SignMarshalECDSA(b, hash)
}
boring.UnreachableExceptTests()
csprng, err := mixedCSPRNG(rand, priv, hash)
if err != nil {
return nil, err
}
if sig, err := signAsm(priv, csprng, hash); err != errNoAsm {
return sig, err
}
switch priv.Curve.Params() {
case elliptic.P224().Params():
return signNISTEC(p224(), priv, csprng, hash)
case elliptic.P256().Params():
return signNISTEC(p256(), priv, csprng, hash)
case elliptic.P384().Params():
return signNISTEC(p384(), priv, csprng, hash)
case elliptic.P521().Params():
return signNISTEC(p521(), priv, csprng, hash)
default:
return signLegacy(priv, csprng, hash)
}
}
func signNISTEC[Point nistPoint[Point]](c *nistCurve[Point], priv *PrivateKey, csprng io.Reader, hash []byte) (sig []byte, err error) {
// SEC 1, Version 2.0, Section 4.1.3
k, R, err := randomPoint(c, csprng)
if err != nil {
return nil, err
}
// kInv = k⁻¹
kInv := bigmod.NewNat()
inverse(c, kInv, k)
Rx, err := R.BytesX()
if err != nil {
return nil, err
}
r, err := bigmod.NewNat().SetOverflowingBytes(Rx, c.N)
if err != nil {
return nil, err
}
// The spec wants us to retry here, but the chance of hitting this condition
// on a large prime-order group like the NIST curves we support is
// cryptographically negligible. If we hit it, something is awfully wrong.
if r.IsZero() == 1 {
return nil, errors.New("ecdsa: internal error: r is zero")
}
e := bigmod.NewNat()
hashToNat(c, e, hash)
s, err := bigmod.NewNat().SetBytes(priv.D.Bytes(), c.N)
if err != nil {
return nil, err
}
s.Mul(r, c.N)
s.Add(e, c.N)
s.Mul(kInv, c.N)
// Again, the chance of this happening is cryptographically negligible.
if s.IsZero() == 1 {
return nil, errors.New("ecdsa: internal error: s is zero")
}
return encodeSignature(r.Bytes(c.N), s.Bytes(c.N))
}
func encodeSignature(r, s []byte) ([]byte, error) {
var b cryptobyte.Builder
b.AddASN1(asn1.SEQUENCE, func(b *cryptobyte.Builder) {
addASN1IntBytes(b, r)
addASN1IntBytes(b, s)
})
return b.Bytes()
}
// addASN1IntBytes encodes in ASN.1 a positive integer represented as
// a big-endian byte slice with zero or more leading zeroes.
func addASN1IntBytes(b *cryptobyte.Builder, bytes []byte) {
for len(bytes) > 0 && bytes[0] == 0 {
bytes = bytes[1:]
}
if len(bytes) == 0 {
b.SetError(errors.New("invalid integer"))
return
}
b.AddASN1(asn1.INTEGER, func(c *cryptobyte.Builder) {
if bytes[0]&0x80 != 0 {
c.AddUint8(0)
}
c.AddBytes(bytes)
})
}
// inverse sets kInv to the inverse of k modulo the order of the curve.
func inverse[Point nistPoint[Point]](c *nistCurve[Point], kInv, k *bigmod.Nat) {
if c.curve.Params().Name == "P-256" {
kBytes, err := nistec.P256OrdInverse(k.Bytes(c.N))
// Some platforms don't implement P256OrdInverse, and always return an error.
if err == nil {
_, err := kInv.SetBytes(kBytes, c.N)
if err != nil {
panic("ecdsa: internal error: P256OrdInverse produced an invalid value")
}
return
}
}
// Calculate the inverse of s in GF(N) using Fermat's method
// (exponentiation modulo P - 2, per Euler's theorem)
kInv.Exp(k, c.nMinus2, c.N)
}
// hashToNat sets e to the left-most bits of hash, according to
// SEC 1, Section 4.1.3, point 5 and Section 4.1.4, point 3.
func hashToNat[Point nistPoint[Point]](c *nistCurve[Point], e *bigmod.Nat, hash []byte) {
// ECDSA asks us to take the left-most log2(N) bits of hash, and use them as
// an integer modulo N. This is the absolute worst of all worlds: we still
// have to reduce, because the result might still overflow N, but to take
// the left-most bits for P-521 we have to do a right shift.
if size := c.N.Size(); len(hash) >= size {
hash = hash[:size]
if excess := len(hash)*8 - c.N.BitLen(); excess > 0 {
hash = bytes.Clone(hash)
for i := len(hash) - 1; i >= 0; i-- {
hash[i] >>= excess
if i > 0 {
hash[i] |= hash[i-1] << (8 - excess)
}
}
}
}
_, err := e.SetOverflowingBytes(hash, c.N)
if err != nil {
panic("ecdsa: internal error: truncated hash is too long")
}
}
// mixedCSPRNG returns a CSPRNG that mixes entropy from rand with the message
// and the private key, to protect the key in case rand fails. This is
// equivalent in security to RFC 6979 deterministic nonce generation, but still
// produces randomized signatures.
func mixedCSPRNG(rand io.Reader, priv *PrivateKey, hash []byte) (io.Reader, error) {
// This implementation derives the nonce from an AES-CTR CSPRNG keyed by:
//
// SHA2-512(priv.D || entropy || hash)[:32]
//
// The CSPRNG key is indifferentiable from a random oracle as shown in
// [Coron], the AES-CTR stream is indifferentiable from a random oracle
// under standard cryptographic assumptions (see [Larsson] for examples).
//
// [Coron]: https://cs.nyu.edu/~dodis/ps/merkle.pdf
// [Larsson]: https://web.archive.org/web/20040719170906/https://www.nada.kth.se/kurser/kth/2D1441/semteo03/lecturenotes/assump.pdf
// Get 256 bits of entropy from rand.
entropy := make([]byte, 32)
if _, err := io.ReadFull(rand, entropy); err != nil {
return nil, err
}
// 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, err
}
// Create a CSPRNG that xors a stream of zeros with
// the output of the AES-CTR instance.
const aesIV = "IV for ECDSA CTR"
return &cipher.StreamReader{
R: zeroReader,
S: cipher.NewCTR(block, []byte(aesIV)),
}, nil
}
type zr struct{}
var zeroReader = zr{}
// Read replaces the contents of dst with zeros. It is safe for concurrent use.
func (zr) Read(dst []byte) (n int, err error) {
for i := range dst {
dst[i] = 0
}
return len(dst), nil
}
// VerifyASN1 verifies the ASN.1 encoded signature, sig, of hash using the
// public key, pub. Its return value records whether the signature is valid.
func VerifyASN1(pub *PublicKey, hash, sig []byte) bool {
if boring.Enabled {
key, err := boringPublicKey(pub)
if err != nil {
return false
}
return boring.VerifyECDSA(key, hash, sig)
}
boring.UnreachableExceptTests()
if err := verifyAsm(pub, hash, sig); err != errNoAsm {
return err == nil
}
switch pub.Curve.Params() {
case elliptic.P224().Params():
return verifyNISTEC(p224(), pub, hash, sig)
case elliptic.P256().Params():
return verifyNISTEC(p256(), pub, hash, sig)
case elliptic.P384().Params():
return verifyNISTEC(p384(), pub, hash, sig)
case elliptic.P521().Params():
return verifyNISTEC(p521(), pub, hash, sig)
default:
return verifyLegacy(pub, hash, sig)
}
}
func verifyNISTEC[Point nistPoint[Point]](c *nistCurve[Point], pub *PublicKey, hash, sig []byte) bool {
rBytes, sBytes, err := parseSignature(sig)
if err != nil {
return false
}
Q, err := c.pointFromAffine(pub.X, pub.Y)
if err != nil {
return false
}
// SEC 1, Version 2.0, Section 4.1.4
r, err := bigmod.NewNat().SetBytes(rBytes, c.N)
if err != nil || r.IsZero() == 1 {
return false
}
s, err := bigmod.NewNat().SetBytes(sBytes, c.N)
if err != nil || s.IsZero() == 1 {
return false
}
e := bigmod.NewNat()
hashToNat(c, e, hash)
// w = s⁻¹
w := bigmod.NewNat()
inverse(c, w, s)
// p₁ = [e * s⁻¹]G
p1, err := c.newPoint().ScalarBaseMult(e.Mul(w, c.N).Bytes(c.N))
if err != nil {
return false
}
// p₂ = [r * s⁻¹]Q
p2, err := Q.ScalarMult(Q, w.Mul(r, c.N).Bytes(c.N))
if err != nil {
return false
}
// BytesX returns an error for the point at infinity.
Rx, err := p1.Add(p1, p2).BytesX()
if err != nil {
return false
}
v, err := bigmod.NewNat().SetOverflowingBytes(Rx, c.N)
if err != nil {
return false
}
return v.Equal(r) == 1
}
func parseSignature(sig []byte) (r, s []byte, err error) {
var inner cryptobyte.String
input := cryptobyte.String(sig)
if !input.ReadASN1(&inner, asn1.SEQUENCE) ||
!input.Empty() ||
!inner.ReadASN1Integer(&r) ||
!inner.ReadASN1Integer(&s) ||
!inner.Empty() {
return nil, nil, errors.New("invalid ASN.1")
}
return r, s, nil
}
type nistCurve[Point nistPoint[Point]] struct {
newPoint func() Point
curve elliptic.Curve
N *bigmod.Modulus
nMinus2 []byte
}
// nistPoint is a generic constraint for the nistec Point types.
type nistPoint[T any] interface {
Bytes() []byte
BytesX() ([]byte, error)
SetBytes([]byte) (T, error)
Add(T, T) T
ScalarMult(T, []byte) (T, error)
ScalarBaseMult([]byte) (T, error)
}
// pointFromAffine is used to convert the PublicKey to a nistec Point.
func (curve *nistCurve[Point]) pointFromAffine(x, y *big.Int) (p Point, err error) {
bitSize := curve.curve.Params().BitSize
// Reject values that would not get correctly encoded.
if x.Sign() < 0 || y.Sign() < 0 {
return p, errors.New("negative coordinate")
}
if x.BitLen() > bitSize || y.BitLen() > bitSize {
return p, errors.New("overflowing coordinate")
}
// Encode the coordinates and let SetBytes reject invalid points.
byteLen := (bitSize + 7) / 8
buf := make([]byte, 1+2*byteLen)
buf[0] = 4 // uncompressed point
x.FillBytes(buf[1 : 1+byteLen])
y.FillBytes(buf[1+byteLen : 1+2*byteLen])
return curve.newPoint().SetBytes(buf)
}
// pointToAffine is used to convert a nistec Point to a PublicKey.
func (curve *nistCurve[Point]) pointToAffine(p Point) (x, y *big.Int, err error) {
out := p.Bytes()
if len(out) == 1 && out[0] == 0 {
// This is the encoding of the point at infinity.
return nil, nil, errors.New("ecdsa: public key point is the infinity")
}
byteLen := (curve.curve.Params().BitSize + 7) / 8
x = new(big.Int).SetBytes(out[1 : 1+byteLen])
y = new(big.Int).SetBytes(out[1+byteLen:])
return x, y, nil
}
var p224Once sync.Once
var _p224 *nistCurve[*nistec.P224Point]
func p224() *nistCurve[*nistec.P224Point] {
p224Once.Do(func() {
_p224 = &nistCurve[*nistec.P224Point]{
newPoint: func() *nistec.P224Point { return nistec.NewP224Point() },
}
precomputeParams(_p224, elliptic.P224())
})
return _p224
}
var p256Once sync.Once
var _p256 *nistCurve[*nistec.P256Point]
func p256() *nistCurve[*nistec.P256Point] {
p256Once.Do(func() {
_p256 = &nistCurve[*nistec.P256Point]{
newPoint: func() *nistec.P256Point { return nistec.NewP256Point() },
}
precomputeParams(_p256, elliptic.P256())
})
return _p256
}
var p384Once sync.Once
var _p384 *nistCurve[*nistec.P384Point]
func p384() *nistCurve[*nistec.P384Point] {
p384Once.Do(func() {
_p384 = &nistCurve[*nistec.P384Point]{
newPoint: func() *nistec.P384Point { return nistec.NewP384Point() },
}
precomputeParams(_p384, elliptic.P384())
})
return _p384
}
var p521Once sync.Once
var _p521 *nistCurve[*nistec.P521Point]
func p521() *nistCurve[*nistec.P521Point] {
p521Once.Do(func() {
_p521 = &nistCurve[*nistec.P521Point]{
newPoint: func() *nistec.P521Point { return nistec.NewP521Point() },
}
precomputeParams(_p521, elliptic.P521())
})
return _p521
}
func precomputeParams[Point nistPoint[Point]](c *nistCurve[Point], curve elliptic.Curve) {
params := curve.Params()
c.curve = curve
var err error
c.N, err = bigmod.NewModulusFromBig(params.N)
if err != nil {
panic(err)
}
c.nMinus2 = new(big.Int).Sub(params.N, big.NewInt(2)).Bytes()
}