ecdsa.go

Documentation: crypto/ecdsa

     1  // Copyright 2011 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  // Package ecdsa implements the Elliptic Curve Digital Signature Algorithm, as
     6  // defined in FIPS 186-3.
     7  //
     8  // This implementation  derives the nonce from an AES-CTR CSPRNG keyed by
     9  // ChopMD(256, SHA2-512(priv.D || entropy || hash)). The CSPRNG key is IRO by
    10  // a result of Coron; the AES-CTR stream is IRO under standard assumptions.
    11  package ecdsa
    12  
    13  // References:
    14  //   [NSA]: Suite B implementer's guide to FIPS 186-3,
    15  //     https://apps.nsa.gov/iaarchive/library/ia-guidance/ia-solutions-for-classified/algorithm-guidance/suite-b-implementers-guide-to-fips-186-3-ecdsa.cfm
    16  //   [SECG]: SECG, SEC1
    17  //     http://www.secg.org/sec1-v2.pdf
    18  
    19  import (
    20  	"crypto"
    21  	"crypto/aes"
    22  	"crypto/cipher"
    23  	"crypto/elliptic"
    24  	"crypto/sha512"
    25  	"encoding/asn1"
    26  	"errors"
    27  	"io"
    28  	"math/big"
    29  
    30  	"crypto/internal/randutil"
    31  )
    32  
    33  // A invertible implements fast inverse mod Curve.Params().N
    34  type invertible interface {
    35  	// Inverse returns the inverse of k in GF(P)
    36  	Inverse(k *big.Int) *big.Int
    37  }
    38  
    39  // combinedMult implements fast multiplication S1*g + S2*p (g - generator, p - arbitrary point)
    40  type combinedMult interface {
    41  	CombinedMult(bigX, bigY *big.Int, baseScalar, scalar []byte) (x, y *big.Int)
    42  }
    43  
    44  const (
    45  	aesIV = "IV for ECDSA CTR"
    46  )
    47  
    48  // PublicKey represents an ECDSA public key.
    49  type PublicKey struct {
    50  	elliptic.Curve
    51  	X, Y *big.Int
    52  }
    53  
    54  // PrivateKey represents an ECDSA private key.
    55  type PrivateKey struct {
    56  	PublicKey
    57  	D *big.Int
    58  }
    59  
    60  type ecdsaSignature struct {
    61  	R, S *big.Int
    62  }
    63  
    64  // Public returns the public key corresponding to priv.
    65  func (priv *PrivateKey) Public() crypto.PublicKey {
    66  	return &priv.PublicKey
    67  }
    68  
    69  // Sign signs digest with priv, reading randomness from rand. The opts argument
    70  // is not currently used but, in keeping with the crypto.Signer interface,
    71  // should be the hash function used to digest the message.
    72  //
    73  // This method implements crypto.Signer, which is an interface to support keys
    74  // where the private part is kept in, for example, a hardware module. Common
    75  // uses should use the Sign function in this package directly.
    76  func (priv *PrivateKey) Sign(rand io.Reader, digest []byte, opts crypto.SignerOpts) ([]byte, error) {
    77  	r, s, err := Sign(rand, priv, digest)
    78  	if err != nil {
    79  		return nil, err
    80  	}
    81  
    82  	return asn1.Marshal(ecdsaSignature{r, s})
    83  }
    84  
    85  var one = new(big.Int).SetInt64(1)
    86  
    87  // randFieldElement returns a random element of the field underlying the given
    88  // curve using the procedure given in [NSA] A.2.1.
    89  func randFieldElement(c elliptic.Curve, rand io.Reader) (k *big.Int, err error) {
    90  	params := c.Params()
    91  	b := make([]byte, params.BitSize/8+8)
    92  	_, err = io.ReadFull(rand, b)
    93  	if err != nil {
    94  		return
    95  	}
    96  
    97  	k = new(big.Int).SetBytes(b)
    98  	n := new(big.Int).Sub(params.N, one)
    99  	k.Mod(k, n)
   100  	k.Add(k, one)
   101  	return
   102  }
   103  
   104  // GenerateKey generates a public and private key pair.
   105  func GenerateKey(c elliptic.Curve, rand io.Reader) (*PrivateKey, error) {
   106  	k, err := randFieldElement(c, rand)
   107  	if err != nil {
   108  		return nil, err
   109  	}
   110  
   111  	priv := new(PrivateKey)
   112  	priv.PublicKey.Curve = c
   113  	priv.D = k
   114  	priv.PublicKey.X, priv.PublicKey.Y = c.ScalarBaseMult(k.Bytes())
   115  	return priv, nil
   116  }
   117  
   118  // hashToInt converts a hash value to an integer. There is some disagreement
   119  // about how this is done. [NSA] suggests that this is done in the obvious
   120  // manner, but [SECG] truncates the hash to the bit-length of the curve order
   121  // first. We follow [SECG] because that's what OpenSSL does. Additionally,
   122  // OpenSSL right shifts excess bits from the number if the hash is too large
   123  // and we mirror that too.
   124  func hashToInt(hash []byte, c elliptic.Curve) *big.Int {
   125  	orderBits := c.Params().N.BitLen()
   126  	orderBytes := (orderBits + 7) / 8
   127  	if len(hash) > orderBytes {
   128  		hash = hash[:orderBytes]
   129  	}
   130  
   131  	ret := new(big.Int).SetBytes(hash)
   132  	excess := len(hash)*8 - orderBits
   133  	if excess > 0 {
   134  		ret.Rsh(ret, uint(excess))
   135  	}
   136  	return ret
   137  }
   138  
   139  // fermatInverse calculates the inverse of k in GF(P) using Fermat's method.
   140  // This has better constant-time properties than Euclid's method (implemented
   141  // in math/big.Int.ModInverse) although math/big itself isn't strictly
   142  // constant-time so it's not perfect.
   143  func fermatInverse(k, N *big.Int) *big.Int {
   144  	two := big.NewInt(2)
   145  	nMinus2 := new(big.Int).Sub(N, two)
   146  	return new(big.Int).Exp(k, nMinus2, N)
   147  }
   148  
   149  var errZeroParam = errors.New("zero parameter")
   150  
   151  // Sign signs a hash (which should be the result of hashing a larger message)
   152  // using the private key, priv. If the hash is longer than the bit-length of the
   153  // private key's curve order, the hash will be truncated to that length.  It
   154  // returns the signature as a pair of integers. The security of the private key
   155  // depends on the entropy of rand.
   156  func Sign(rand io.Reader, priv *PrivateKey, hash []byte) (r, s *big.Int, err error) {
   157  	randutil.MaybeReadByte(rand)
   158  
   159  	// Get min(log2(q) / 2, 256) bits of entropy from rand.
   160  	entropylen := (priv.Curve.Params().BitSize + 7) / 16
   161  	if entropylen > 32 {
   162  		entropylen = 32
   163  	}
   164  	entropy := make([]byte, entropylen)
   165  	_, err = io.ReadFull(rand, entropy)
   166  	if err != nil {
   167  		return
   168  	}
   169  
   170  	// Initialize an SHA-512 hash context; digest ...
   171  	md := sha512.New()
   172  	md.Write(priv.D.Bytes()) // the private key,
   173  	md.Write(entropy)        // the entropy,
   174  	md.Write(hash)           // and the input hash;
   175  	key := md.Sum(nil)[:32]  // and compute ChopMD-256(SHA-512),
   176  	// which is an indifferentiable MAC.
   177  
   178  	// Create an AES-CTR instance to use as a CSPRNG.
   179  	block, err := aes.NewCipher(key)
   180  	if err != nil {
   181  		return nil, nil, err
   182  	}
   183  
   184  	// Create a CSPRNG that xors a stream of zeros with
   185  	// the output of the AES-CTR instance.
   186  	csprng := cipher.StreamReader{
   187  		R: zeroReader,
   188  		S: cipher.NewCTR(block, []byte(aesIV)),
   189  	}
   190  
   191  	// See [NSA] 3.4.1
   192  	c := priv.PublicKey.Curve
   193  	N := c.Params().N
   194  	if N.Sign() == 0 {
   195  		return nil, nil, errZeroParam
   196  	}
   197  	var k, kInv *big.Int
   198  	for {
   199  		for {
   200  			k, err = randFieldElement(c, csprng)
   201  			if err != nil {
   202  				r = nil
   203  				return
   204  			}
   205  
   206  			if in, ok := priv.Curve.(invertible); ok {
   207  				kInv = in.Inverse(k)
   208  			} else {
   209  				kInv = fermatInverse(k, N) // N != 0
   210  			}
   211  
   212  			r, _ = priv.Curve.ScalarBaseMult(k.Bytes())
   213  			r.Mod(r, N)
   214  			if r.Sign() != 0 {
   215  				break
   216  			}
   217  		}
   218  
   219  		e := hashToInt(hash, c)
   220  		s = new(big.Int).Mul(priv.D, r)
   221  		s.Add(s, e)
   222  		s.Mul(s, kInv)
   223  		s.Mod(s, N) // N != 0
   224  		if s.Sign() != 0 {
   225  			break
   226  		}
   227  	}
   228  
   229  	return
   230  }
   231  
   232  // Verify verifies the signature in r, s of hash using the public key, pub. Its
   233  // return value records whether the signature is valid.
   234  func Verify(pub *PublicKey, hash []byte, r, s *big.Int) bool {
   235  	// See [NSA] 3.4.2
   236  	c := pub.Curve
   237  	N := c.Params().N
   238  
   239  	if r.Sign() <= 0 || s.Sign() <= 0 {
   240  		return false
   241  	}
   242  	if r.Cmp(N) >= 0 || s.Cmp(N) >= 0 {
   243  		return false
   244  	}
   245  	e := hashToInt(hash, c)
   246  
   247  	var w *big.Int
   248  	if in, ok := c.(invertible); ok {
   249  		w = in.Inverse(s)
   250  	} else {
   251  		w = new(big.Int).ModInverse(s, N)
   252  	}
   253  
   254  	u1 := e.Mul(e, w)
   255  	u1.Mod(u1, N)
   256  	u2 := w.Mul(r, w)
   257  	u2.Mod(u2, N)
   258  
   259  	// Check if implements S1*g + S2*p
   260  	var x, y *big.Int
   261  	if opt, ok := c.(combinedMult); ok {
   262  		x, y = opt.CombinedMult(pub.X, pub.Y, u1.Bytes(), u2.Bytes())
   263  	} else {
   264  		x1, y1 := c.ScalarBaseMult(u1.Bytes())
   265  		x2, y2 := c.ScalarMult(pub.X, pub.Y, u2.Bytes())
   266  		x, y = c.Add(x1, y1, x2, y2)
   267  	}
   268  
   269  	if x.Sign() == 0 && y.Sign() == 0 {
   270  		return false
   271  	}
   272  	x.Mod(x, N)
   273  	return x.Cmp(r) == 0
   274  }
   275  
   276  type zr struct {
   277  	io.Reader
   278  }
   279  
   280  // Read replaces the contents of dst with zeros.
   281  func (z *zr) Read(dst []byte) (n int, err error) {
   282  	for i := range dst {
   283  		dst[i] = 0
   284  	}
   285  	return len(dst), nil
   286  }
   287  
   288  var zeroReader = &zr{}
   289
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