src/crypto/ecdsa/ecdsa.go - go - Git at Google

 // 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 (
 	"crypto"
 	"crypto/aes"
 	"crypto/cipher"
 	"crypto/elliptic"
 	"crypto/internal/randutil"
 	"crypto/sha512"
 	"errors"
 	"io"
 	"math/big"

 	"golang.org/x/crypto/cryptobyte"
 	"golang.org/x/crypto/cryptobyte/asn1"
 )

 // A invertible implements fast inverse in GF(N).
 type invertible interface {
 	// Inverse returns the inverse of k mod Params().N.
 	Inverse(k *big.Int) *big.Int
 }

 // A combinedMult implements fast combined multiplication for verification.
 type combinedMult interface {
 	// CombinedMult returns [s1]G + [s2]P where G is the generator.
 	CombinedMult(Px, Py *big.Int, s1, s2 []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
 }

 // Any methods implemented on PublicKey might need to also be implemented on
 // PrivateKey, as the latter embeds the former and will expose its methods.

 // 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 pub.X.Cmp(xx.X) == 0 && pub.Y.Cmp(xx.Y) == 0 &&
 		// 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
 }

 // 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) && priv.D.Cmp(xx.D) == 0
 }

 // 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) {
 	r, s, err := Sign(rand, priv, digest)
 	if err != nil {
 		return nil, err
 	}

 	var b cryptobyte.Builder
 	b.AddASN1(asn1.SEQUENCE, func(b *cryptobyte.Builder) {
 		b.AddASN1BigInt(r)
 		b.AddASN1BigInt(s)
 	})
 	return b.Bytes()
 }

 var one = new(big.Int).SetInt64(1)

 // randFieldElement returns a random element of the order of the given
 // curve using the procedure given in FIPS 186-4, Appendix B.5.1.
 func randFieldElement(c elliptic.Curve, rand io.Reader) (k *big.Int, err error) {
 	params := c.Params()
 	// Note that for P-521 this will actually be 63 bits more than the order, as
 	// division rounds down, but the extra bit is inconsequential.
 	b := make([]byte, params.BitSize/8+8) // TODO: use params.N.BitLen()
 	_, 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. Per FIPS 186-4, Section 6.4,
 // we use the left-most bits of the hash to match the bit-length of the order of
 // the curve. This also performs Step 5 of SEC 1, Version 2.0, Section 4.1.3.
 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
 // (exponentiation modulo P - 2, per Euler's theorem). This has better
 // constant-time properties than Euclid's method (implemented in
 // math/big.Int.ModInverse and FIPS 186-4, Appendix C.1) 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. Most applications should use
 // SignASN1 instead of dealing directly with r, s.
 func Sign(rand io.Reader, priv *PrivateKey, hash []byte) (r, s *big.Int, err error) {
 	randutil.MaybeReadByte(rand)

 	// 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)
 	_, 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)),
 	}

 	c := priv.PublicKey.Curve
 	return sign(priv, &csprng, c, hash)
 }

 func signGeneric(priv *PrivateKey, csprng *cipher.StreamReader, c elliptic.Curve, hash []byte) (r, s *big.Int, err error) {
 	// SEC 1, Version 2.0, Section 4.1.3
 	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
 }

 // 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.
 func SignASN1(rand io.Reader, priv *PrivateKey, hash []byte) ([]byte, error) {
 	return priv.Sign(rand, hash, nil)
 }

 // Verify verifies the signature in r, s of hash using the public key, pub. Its
 // return value records whether the signature is valid. Most applications should
 // use VerifyASN1 instead of dealing directly with r, s.
 func Verify(pub *PublicKey, hash []byte, r, s *big.Int) bool {
 	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
 	}
 	return verify(pub, c, hash, r, s)
 }

 func verifyGeneric(pub *PublicKey, c elliptic.Curve, hash []byte, r, s *big.Int) bool {
 	// SEC 1, Version 2.0, Section 4.1.4
 	e := hashToInt(hash, c)
 	var w *big.Int
 	N := c.Params().N
 	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
 }

 // 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 {
 	var (
 		r, s  = &big.Int{}, &big.Int{}
 		inner cryptobyte.String
 	)
 	input := cryptobyte.String(sig)
 	if !input.ReadASN1(&inner, asn1.SEQUENCE) ||
 		!input.Empty() ||
 		!inner.ReadASN1Integer(r) ||
 		!inner.ReadASN1Integer(s) ||
 		!inner.Empty() {
 		return false
 	}
 	return Verify(pub, hash, r, s)
 }

 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{}
	// 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 (
	"crypto"
	"crypto/aes"
	"crypto/cipher"
	"crypto/elliptic"
	"crypto/internal/randutil"
	"crypto/sha512"
	"errors"
	"io"
	"math/big"

	"golang.org/x/crypto/cryptobyte"
	"golang.org/x/crypto/cryptobyte/asn1"
	)

	// A invertible implements fast inverse in GF(N).
	type invertible interface {
	// Inverse returns the inverse of k mod Params().N.
	Inverse(k big.Int) big.Int
	}

	// A combinedMult implements fast combined multiplication for verification.
	type combinedMult interface {
	// CombinedMult returns [s1]G + [s2]P where G is the generator.
	CombinedMult(Px, Py big.Int, s1, s2 []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
	}

	// Any methods implemented on PublicKey might need to also be implemented on
	// PrivateKey, as the latter embeds the former and will expose its methods.

	// 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 pub.X.Cmp(xx.X) == 0 && pub.Y.Cmp(xx.Y) == 0 &&
	// 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
	}

	// 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) && priv.D.Cmp(xx.D) == 0
	}

	// 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) {
	r, s, err := Sign(rand, priv, digest)
	if err != nil {
	return nil, err
	}

	var b cryptobyte.Builder
	b.AddASN1(asn1.SEQUENCE, func(b *cryptobyte.Builder) {
	b.AddASN1BigInt(r)
	b.AddASN1BigInt(s)
	})
	return b.Bytes()
	}

	var one = new(big.Int).SetInt64(1)

	// randFieldElement returns a random element of the order of the given
	// curve using the procedure given in FIPS 186-4, Appendix B.5.1.
	func randFieldElement(c elliptic.Curve, rand io.Reader) (k *big.Int, err error) {
	params := c.Params()
	// Note that for P-521 this will actually be 63 bits more than the order, as
	// division rounds down, but the extra bit is inconsequential.
	b := make([]byte, params.BitSize/8+8) // TODO: use params.N.BitLen()
	_, 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. Per FIPS 186-4, Section 6.4,
	// we use the left-most bits of the hash to match the bit-length of the order of
	// the curve. This also performs Step 5 of SEC 1, Version 2.0, Section 4.1.3.
	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
	// (exponentiation modulo P - 2, per Euler's theorem). This has better
	// constant-time properties than Euclid's method (implemented in
	// math/big.Int.ModInverse and FIPS 186-4, Appendix C.1) 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. Most applications should use
	// SignASN1 instead of dealing directly with r, s.
	func Sign(rand io.Reader, priv PrivateKey, hash []byte) (r, s big.Int, err error) {
	randutil.MaybeReadByte(rand)

	// 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)
	_, 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)),
	}

	c := priv.PublicKey.Curve
	return sign(priv, &csprng, c, hash)
	}

	func signGeneric(priv PrivateKey, csprng cipher.StreamReader, c elliptic.Curve, hash []byte) (r, s *big.Int, err error) {
	// SEC 1, Version 2.0, Section 4.1.3
	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
	}

	// 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.
	func SignASN1(rand io.Reader, priv *PrivateKey, hash []byte) ([]byte, error) {
	return priv.Sign(rand, hash, nil)
	}

	// Verify verifies the signature in r, s of hash using the public key, pub. Its
	// return value records whether the signature is valid. Most applications should
	// use VerifyASN1 instead of dealing directly with r, s.
	func Verify(pub PublicKey, hash []byte, r, s big.Int) bool {
	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
	}
	return verify(pub, c, hash, r, s)
	}

	func verifyGeneric(pub PublicKey, c elliptic.Curve, hash []byte, r, s big.Int) bool {
	// SEC 1, Version 2.0, Section 4.1.4
	e := hashToInt(hash, c)
	var w *big.Int
	N := c.Params().N
	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 S1g + S2p
	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
	}

	// 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 {
	var (
	r, s = &big.Int{}, &big.Int{}
	inner cryptobyte.String
	)
	input := cryptobyte.String(sig)
	if !input.ReadASN1(&inner, asn1.SEQUENCE) \|\|
	!input.Empty() \|\|
	!inner.ReadASN1Integer(r) \|\|
	!inner.ReadASN1Integer(s) \|\|
	!inner.Empty() {
	return false
	}
	return Verify(pub, hash, r, s)
	}

	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{}