Source file src/crypto/ecdsa/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  //
    10  // SHA2-512(priv.D || entropy || hash)[:32]
    11  //
    12  // The CSPRNG key is indifferentiable from a random oracle as shown in
    13  // [Coron], the AES-CTR stream is indifferentiable from a random oracle
    14  // under standard cryptographic assumptions (see [Larsson] for examples).
    15  //
    16  // References:
    17  //   [Coron]
    18  //     https://cs.nyu.edu/~dodis/ps/merkle.pdf
    19  //   [Larsson]
    20  //     https://www.nada.kth.se/kurser/kth/2D1441/semteo03/lecturenotes/assump.pdf
    21  package ecdsa
    22  
    23  // Further references:
    24  //   [NSA]: Suite B implementer's guide to FIPS 186-3
    25  //     https://apps.nsa.gov/iaarchive/library/ia-guidance/ia-solutions-for-classified/algorithm-guidance/suite-b-implementers-guide-to-fips-186-3-ecdsa.cfm
    26  //   [SECG]: SECG, SEC1
    27  //     http://www.secg.org/sec1-v2.pdf
    28  
    29  import (
    30  	"crypto"
    31  	"crypto/aes"
    32  	"crypto/cipher"
    33  	"crypto/elliptic"
    34  	"crypto/internal/randutil"
    35  	"crypto/sha512"
    36  	"encoding/asn1"
    37  	"errors"
    38  	"io"
    39  	"math/big"
    40  )
    41  
    42  // A invertible implements fast inverse mod Curve.Params().N
    43  type invertible interface {
    44  	// Inverse returns the inverse of k in GF(P)
    45  	Inverse(k *big.Int) *big.Int
    46  }
    47  
    48  // combinedMult implements fast multiplication S1*g + S2*p (g - generator, p - arbitrary point)
    49  type combinedMult interface {
    50  	CombinedMult(bigX, bigY *big.Int, baseScalar, scalar []byte) (x, y *big.Int)
    51  }
    52  
    53  const (
    54  	aesIV = "IV for ECDSA CTR"
    55  )
    56  
    57  // PublicKey represents an ECDSA public key.
    58  type PublicKey struct {
    59  	elliptic.Curve
    60  	X, Y *big.Int
    61  }
    62  
    63  // PrivateKey represents an ECDSA private key.
    64  type PrivateKey struct {
    65  	PublicKey
    66  	D *big.Int
    67  }
    68  
    69  type ecdsaSignature struct {
    70  	R, S *big.Int
    71  }
    72  
    73  // Public returns the public key corresponding to priv.
    74  func (priv *PrivateKey) Public() crypto.PublicKey {
    75  	return &priv.PublicKey
    76  }
    77  
    78  // Sign signs digest with priv, reading randomness from rand. The opts argument
    79  // is not currently used but, in keeping with the crypto.Signer interface,
    80  // should be the hash function used to digest the message.
    81  //
    82  // This method implements crypto.Signer, which is an interface to support keys
    83  // where the private part is kept in, for example, a hardware module. Common
    84  // uses should use the Sign function in this package directly.
    85  func (priv *PrivateKey) Sign(rand io.Reader, digest []byte, opts crypto.SignerOpts) ([]byte, error) {
    86  	r, s, err := Sign(rand, priv, digest)
    87  	if err != nil {
    88  		return nil, err
    89  	}
    90  
    91  	return asn1.Marshal(ecdsaSignature{r, s})
    92  }
    93  
    94  var one = new(big.Int).SetInt64(1)
    95  
    96  // randFieldElement returns a random element of the field underlying the given
    97  // curve using the procedure given in [NSA] A.2.1.
    98  func randFieldElement(c elliptic.Curve, rand io.Reader) (k *big.Int, err error) {
    99  	params := c.Params()
   100  	b := make([]byte, params.BitSize/8+8)
   101  	_, err = io.ReadFull(rand, b)
   102  	if err != nil {
   103  		return
   104  	}
   105  
   106  	k = new(big.Int).SetBytes(b)
   107  	n := new(big.Int).Sub(params.N, one)
   108  	k.Mod(k, n)
   109  	k.Add(k, one)
   110  	return
   111  }
   112  
   113  // GenerateKey generates a public and private key pair.
   114  func GenerateKey(c elliptic.Curve, rand io.Reader) (*PrivateKey, error) {
   115  	k, err := randFieldElement(c, rand)
   116  	if err != nil {
   117  		return nil, err
   118  	}
   119  
   120  	priv := new(PrivateKey)
   121  	priv.PublicKey.Curve = c
   122  	priv.D = k
   123  	priv.PublicKey.X, priv.PublicKey.Y = c.ScalarBaseMult(k.Bytes())
   124  	return priv, nil
   125  }
   126  
   127  // hashToInt converts a hash value to an integer. There is some disagreement
   128  // about how this is done. [NSA] suggests that this is done in the obvious
   129  // manner, but [SECG] truncates the hash to the bit-length of the curve order
   130  // first. We follow [SECG] because that's what OpenSSL does. Additionally,
   131  // OpenSSL right shifts excess bits from the number if the hash is too large
   132  // and we mirror that too.
   133  func hashToInt(hash []byte, c elliptic.Curve) *big.Int {
   134  	orderBits := c.Params().N.BitLen()
   135  	orderBytes := (orderBits + 7) / 8
   136  	if len(hash) > orderBytes {
   137  		hash = hash[:orderBytes]
   138  	}
   139  
   140  	ret := new(big.Int).SetBytes(hash)
   141  	excess := len(hash)*8 - orderBits
   142  	if excess > 0 {
   143  		ret.Rsh(ret, uint(excess))
   144  	}
   145  	return ret
   146  }
   147  
   148  // fermatInverse calculates the inverse of k in GF(P) using Fermat's method.
   149  // This has better constant-time properties than Euclid's method (implemented
   150  // in math/big.Int.ModInverse) although math/big itself isn't strictly
   151  // constant-time so it's not perfect.
   152  func fermatInverse(k, N *big.Int) *big.Int {
   153  	two := big.NewInt(2)
   154  	nMinus2 := new(big.Int).Sub(N, two)
   155  	return new(big.Int).Exp(k, nMinus2, N)
   156  }
   157  
   158  var errZeroParam = errors.New("zero parameter")
   159  
   160  // Sign signs a hash (which should be the result of hashing a larger message)
   161  // using the private key, priv. If the hash is longer than the bit-length of the
   162  // private key's curve order, the hash will be truncated to that length.  It
   163  // returns the signature as a pair of integers. The security of the private key
   164  // depends on the entropy of rand.
   165  func Sign(rand io.Reader, priv *PrivateKey, hash []byte) (r, s *big.Int, err error) {
   166  	randutil.MaybeReadByte(rand)
   167  
   168  	// Get min(log2(q) / 2, 256) bits of entropy from rand.
   169  	entropylen := (priv.Curve.Params().BitSize + 7) / 16
   170  	if entropylen > 32 {
   171  		entropylen = 32
   172  	}
   173  	entropy := make([]byte, entropylen)
   174  	_, err = io.ReadFull(rand, entropy)
   175  	if err != nil {
   176  		return
   177  	}
   178  
   179  	// Initialize an SHA-512 hash context; digest ...
   180  	md := sha512.New()
   181  	md.Write(priv.D.Bytes()) // the private key,
   182  	md.Write(entropy)        // the entropy,
   183  	md.Write(hash)           // and the input hash;
   184  	key := md.Sum(nil)[:32]  // and compute ChopMD-256(SHA-512),
   185  	// which is an indifferentiable MAC.
   186  
   187  	// Create an AES-CTR instance to use as a CSPRNG.
   188  	block, err := aes.NewCipher(key)
   189  	if err != nil {
   190  		return nil, nil, err
   191  	}
   192  
   193  	// Create a CSPRNG that xors a stream of zeros with
   194  	// the output of the AES-CTR instance.
   195  	csprng := cipher.StreamReader{
   196  		R: zeroReader,
   197  		S: cipher.NewCTR(block, []byte(aesIV)),
   198  	}
   199  
   200  	// See [NSA] 3.4.1
   201  	c := priv.PublicKey.Curve
   202  	N := c.Params().N
   203  	if N.Sign() == 0 {
   204  		return nil, nil, errZeroParam
   205  	}
   206  	var k, kInv *big.Int
   207  	for {
   208  		for {
   209  			k, err = randFieldElement(c, csprng)
   210  			if err != nil {
   211  				r = nil
   212  				return
   213  			}
   214  
   215  			if in, ok := priv.Curve.(invertible); ok {
   216  				kInv = in.Inverse(k)
   217  			} else {
   218  				kInv = fermatInverse(k, N) // N != 0
   219  			}
   220  
   221  			r, _ = priv.Curve.ScalarBaseMult(k.Bytes())
   222  			r.Mod(r, N)
   223  			if r.Sign() != 0 {
   224  				break
   225  			}
   226  		}
   227  
   228  		e := hashToInt(hash, c)
   229  		s = new(big.Int).Mul(priv.D, r)
   230  		s.Add(s, e)
   231  		s.Mul(s, kInv)
   232  		s.Mod(s, N) // N != 0
   233  		if s.Sign() != 0 {
   234  			break
   235  		}
   236  	}
   237  
   238  	return
   239  }
   240  
   241  // Verify verifies the signature in r, s of hash using the public key, pub. Its
   242  // return value records whether the signature is valid.
   243  func Verify(pub *PublicKey, hash []byte, r, s *big.Int) bool {
   244  	// See [NSA] 3.4.2
   245  	c := pub.Curve
   246  	N := c.Params().N
   247  
   248  	if r.Sign() <= 0 || s.Sign() <= 0 {
   249  		return false
   250  	}
   251  	if r.Cmp(N) >= 0 || s.Cmp(N) >= 0 {
   252  		return false
   253  	}
   254  	e := hashToInt(hash, c)
   255  
   256  	var w *big.Int
   257  	if in, ok := c.(invertible); ok {
   258  		w = in.Inverse(s)
   259  	} else {
   260  		w = new(big.Int).ModInverse(s, N)
   261  	}
   262  
   263  	u1 := e.Mul(e, w)
   264  	u1.Mod(u1, N)
   265  	u2 := w.Mul(r, w)
   266  	u2.Mod(u2, N)
   267  
   268  	// Check if implements S1*g + S2*p
   269  	var x, y *big.Int
   270  	if opt, ok := c.(combinedMult); ok {
   271  		x, y = opt.CombinedMult(pub.X, pub.Y, u1.Bytes(), u2.Bytes())
   272  	} else {
   273  		x1, y1 := c.ScalarBaseMult(u1.Bytes())
   274  		x2, y2 := c.ScalarMult(pub.X, pub.Y, u2.Bytes())
   275  		x, y = c.Add(x1, y1, x2, y2)
   276  	}
   277  
   278  	if x.Sign() == 0 && y.Sign() == 0 {
   279  		return false
   280  	}
   281  	x.Mod(x, N)
   282  	return x.Cmp(r) == 0
   283  }
   284  
   285  type zr struct {
   286  	io.Reader
   287  }
   288  
   289  // Read replaces the contents of dst with zeros.
   290  func (z *zr) Read(dst []byte) (n int, err error) {
   291  	for i := range dst {
   292  		dst[i] = 0
   293  	}
   294  	return len(dst), nil
   295  }
   296  
   297  var zeroReader = &zr{}
   298  

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