// Verify verifies the signature in r, s of hash using the public key, pub. It
// returns true iff the signature is valid.
func Verify(pub *PublicKey, hash []byte, r, s *big.Int) bool {
// See [NSA] 3.4.2
c := pub.Curve
if r.Sign() == 0 || s.Sign() == 0 {
return false
}
if r.Cmp(c.N) >= 0 || s.Cmp(c.N) >= 0 {
return false
}
e := hashToInt(hash, c)
w := new(big.Int).ModInverse(s, c.N)
u1 := e.Mul(e, w)
u2 := w.Mul(r, w)
x1, y1 := c.ScalarBaseMult(u1.Bytes())
x2, y2 := c.ScalarMult(pub.X, pub.Y, u2.Bytes())
if x1.Cmp(x2) == 0 {
return false
}
x, _ := c.Add(x1, y1, x2, y2)
x.Mod(x, c.N)
return x.Cmp(r) == 0
}
// Verify verifies the signature in r, s of hash using the public key, pub. It
// returns true iff the signature is valid.
func Verify(pub *PublicKey, hash []byte, r, s *big.Int) bool {
// FIPS 186-3, section 4.7
if r.Sign() < 1 || r.Cmp(pub.Q) >= 0 {
return false
}
if s.Sign() < 1 || s.Cmp(pub.Q) >= 0 {
return false
}
w := new(big.Int).ModInverse(s, pub.Q)
n := pub.Q.BitLen()
if n&7 != 0 {
return false
}
n >>= 3
if n > len(hash) {
n = len(hash)
}
z := new(big.Int).SetBytes(hash[:n])
u1 := new(big.Int).Mul(z, w)
u1.Mod(u1, pub.Q)
u2 := w.Mul(r, w)
u2.Mod(u2, pub.Q)
v := u1.Exp(pub.G, u1, pub.P)
u2.Exp(pub.Y, u2, pub.P)
v.Mul(v, u2)
v.Mod(v, pub.P)
v.Mod(v, pub.Q)
return v.Cmp(r) == 0
}
// rawValueForBig returns an asn1.RawValue which represents the given integer.
func rawValueForBig(n *big.Int) asn1.RawValue {
b := n.Bytes()
if n.Sign() >= 0 && len(b) > 0 && b[0]&0x80 != 0 {
// This positive number would be interpreted as a negative
// number in ASN.1 because the MSB is set.
padded := make([]byte, len(b)+1)
copy(padded[1:], b)
b = padded
}
return asn1.RawValue{Tag: 2, Bytes: b}
}
// Sign signs an arbitrary length hash (which should be the result of hashing a
// larger message) using the private key, priv. 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 os.Error) {
// FIPS 186-3, section 4.6
n := priv.Q.BitLen()
if n&7 != 0 {
err = InvalidPublicKeyError
return
}
n >>= 3
for {
k := new(big.Int)
buf := make([]byte, n)
for {
_, err = io.ReadFull(rand, buf)
if err != nil {
return
}
k.SetBytes(buf)
if k.Sign() > 0 && k.Cmp(priv.Q) < 0 {
break
}
}
kInv := new(big.Int).ModInverse(k, priv.Q)
r = new(big.Int).Exp(priv.G, k, priv.P)
r.Mod(r, priv.Q)
if r.Sign() == 0 {
continue
}
if n > len(hash) {
n = len(hash)
}
z := k.SetBytes(hash[:n])
s = new(big.Int).Mul(priv.X, r)
s.Add(s, z)
s.Mod(s, priv.Q)
s.Mul(s, kInv)
s.Mod(s, priv.Q)
if s.Sign() != 0 {
break
}
}
return
}
// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
// returns its double, also in Jacobian form.
func (curve *Curve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
delta := new(big.Int).Mul(z, z)
delta.Mod(delta, curve.P)
gamma := new(big.Int).Mul(y, y)
gamma.Mod(gamma, curve.P)
alpha := new(big.Int).Sub(x, delta)
if alpha.Sign() == -1 {
alpha.Add(alpha, curve.P)
}
alpha2 := new(big.Int).Add(x, delta)
alpha.Mul(alpha, alpha2)
alpha2.Set(alpha)
alpha.Lsh(alpha, 1)
alpha.Add(alpha, alpha2)
beta := alpha2.Mul(x, gamma)
x3 := new(big.Int).Mul(alpha, alpha)
beta8 := new(big.Int).Lsh(beta, 3)
x3.Sub(x3, beta8)
for x3.Sign() == -1 {
x3.Add(x3, curve.P)
}
x3.Mod(x3, curve.P)
z3 := new(big.Int).Add(y, z)
z3.Mul(z3, z3)
z3.Sub(z3, gamma)
if z3.Sign() == -1 {
z3.Add(z3, curve.P)
}
z3.Sub(z3, delta)
if z3.Sign() == -1 {
z3.Add(z3, curve.P)
}
z3.Mod(z3, curve.P)
beta.Lsh(beta, 2)
beta.Sub(beta, x3)
if beta.Sign() == -1 {
beta.Add(beta, curve.P)
}
y3 := alpha.Mul(alpha, beta)
gamma.Mul(gamma, gamma)
gamma.Lsh(gamma, 3)
gamma.Mod(gamma, curve.P)
y3.Sub(y3, gamma)
if y3.Sign() == -1 {
y3.Add(y3, curve.P)
}
y3.Mod(y3, curve.P)
return x3, y3, z3
}
func marshalInt(to []byte, n *big.Int) []byte {
lengthBytes := to
to = to[4:]
length := 0
if n.Sign() < 0 {
// A negative number has to be converted to two's-complement
// form. So we'll subtract 1 and invert. If the
// most-significant-bit isn't set then we'll need to pad the
// beginning with 0xff in order to keep the number negative.
nMinus1 := new(big.Int).Neg(n)
nMinus1.Sub(nMinus1, bigOne)
bytes := nMinus1.Bytes()
for i := range bytes {
bytes[i] ^= 0xff
}
if len(bytes) == 0 || bytes[0]&0x80 == 0 {
to[0] = 0xff
to = to[1:]
length++
}
nBytes := copy(to, bytes)
to = to[nBytes:]
length += nBytes
} else if n.Sign() == 0 {
// A zero is the zero length string
} else {
bytes := n.Bytes()
if len(bytes) > 0 && bytes[0]&0x80 != 0 {
// We'll have to pad this with a 0x00 in order to
// stop it looking like a negative number.
to[0] = 0
to = to[1:]
length++
}
nBytes := copy(to, bytes)
to = to[nBytes:]
length += nBytes
}
lengthBytes[0] = byte(length >> 24)
lengthBytes[1] = byte(length >> 16)
lengthBytes[2] = byte(length >> 8)
lengthBytes[3] = byte(length)
return to
}
func testParameterGeneration(t *testing.T, sizes ParameterSizes, L, N int) {
var priv PrivateKey
params := &priv.Parameters
err := GenerateParameters(params, rand.Reader, sizes)
if err != nil {
t.Errorf("%d: %s", int(sizes), err)
return
}
if params.P.BitLen() != L {
t.Errorf("%d: params.BitLen got:%d want:%d", int(sizes), params.P.BitLen(), L)
}
if params.Q.BitLen() != N {
t.Errorf("%d: q.BitLen got:%d want:%d", int(sizes), params.Q.BitLen(), L)
}
one := new(big.Int)
one.SetInt64(1)
pm1 := new(big.Int).Sub(params.P, one)
quo, rem := new(big.Int).DivMod(pm1, params.Q, new(big.Int))
if rem.Sign() != 0 {
t.Errorf("%d: p-1 mod q != 0", int(sizes))
}
x := new(big.Int).Exp(params.G, quo, params.P)
if x.Cmp(one) == 0 {
t.Errorf("%d: invalid generator", int(sizes))
}
err = GenerateKey(&priv, rand.Reader)
if err != nil {
t.Errorf("error generating key: %s", err)
return
}
testSignAndVerify(t, int(sizes), &priv)
}
// GenerateKey generates a public&private key pair. The Parameters of the
// PrivateKey must already be valid (see GenerateParameters).
func GenerateKey(priv *PrivateKey, rand io.Reader) os.Error {
if priv.P == nil || priv.Q == nil || priv.G == nil {
return os.ErrorString("crypto/dsa: parameters not set up before generating key")
}
x := new(big.Int)
xBytes := make([]byte, priv.Q.BitLen()/8)
for {
_, err := io.ReadFull(rand, xBytes)
if err != nil {
return err
}
x.SetBytes(xBytes)
if x.Sign() != 0 && x.Cmp(priv.Q) < 0 {
break
}
}
priv.X = x
priv.Y = new(big.Int)
priv.Y.Exp(priv.G, x, priv.P)
return nil
}
func marshalBigInt(out *forkableWriter, n *big.Int) (err os.Error) {
if n.Sign() < 0 {
// A negative number has to be converted to two's-complement
// form. So we'll subtract 1 and invert. If the
// most-significant-bit isn't set then we'll need to pad the
// beginning with 0xff in order to keep the number negative.
nMinus1 := new(big.Int).Neg(n)
nMinus1.Sub(nMinus1, bigOne)
bytes := nMinus1.Bytes()
for i := range bytes {
bytes[i] ^= 0xff
}
if len(bytes) == 0 || bytes[0]&0x80 == 0 {
err = out.WriteByte(0xff)
if err != nil {
return
}
}
_, err = out.Write(bytes)
} else if n.Sign() == 0 {
// Zero is written as a single 0 zero rather than no bytes.
err = out.WriteByte(0x00)
} else {
bytes := n.Bytes()
if len(bytes) > 0 && bytes[0]&0x80 != 0 {
// We'll have to pad this with 0x00 in order to stop it
// looking like a negative number.
err = out.WriteByte(0)
if err != nil {
return
}
}
_, err = out.Write(bytes)
}
return
}
func intLength(n *big.Int) int {
length := 4 /* length bytes */
if n.Sign() < 0 {
nMinus1 := new(big.Int).Neg(n)
nMinus1.Sub(nMinus1, bigOne)
bitLen := nMinus1.BitLen()
if bitLen%8 == 0 {
// The number will need 0xff padding
length++
}
length += (bitLen + 7) / 8
} else if n.Sign() == 0 {
// A zero is the zero length string
} else {
bitLen := n.BitLen()
if bitLen%8 == 0 {
// The number will need 0x00 padding
length++
}
length += (bitLen + 7) / 8
}
return length
}
// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
// (x2, y2, z2) and returns their sum, also in Jacobian form.
func (curve *Curve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
// See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
z1z1 := new(big.Int).Mul(z1, z1)
z1z1.Mod(z1z1, curve.P)
z2z2 := new(big.Int).Mul(z2, z2)
z2z2.Mod(z2z2, curve.P)
u1 := new(big.Int).Mul(x1, z2z2)
u1.Mod(u1, curve.P)
u2 := new(big.Int).Mul(x2, z1z1)
u2.Mod(u2, curve.P)
h := new(big.Int).Sub(u2, u1)
if h.Sign() == -1 {
h.Add(h, curve.P)
}
i := new(big.Int).Lsh(h, 1)
i.Mul(i, i)
j := new(big.Int).Mul(h, i)
s1 := new(big.Int).Mul(y1, z2)
s1.Mul(s1, z2z2)
s1.Mod(s1, curve.P)
s2 := new(big.Int).Mul(y2, z1)
s2.Mul(s2, z1z1)
s2.Mod(s2, curve.P)
r := new(big.Int).Sub(s2, s1)
if r.Sign() == -1 {
r.Add(r, curve.P)
}
r.Lsh(r, 1)
v := new(big.Int).Mul(u1, i)
x3 := new(big.Int).Set(r)
x3.Mul(x3, x3)
x3.Sub(x3, j)
x3.Sub(x3, v)
x3.Sub(x3, v)
x3.Mod(x3, curve.P)
y3 := new(big.Int).Set(r)
v.Sub(v, x3)
y3.Mul(y3, v)
s1.Mul(s1, j)
s1.Lsh(s1, 1)
y3.Sub(y3, s1)
y3.Mod(y3, curve.P)
z3 := new(big.Int).Add(z1, z2)
z3.Mul(z3, z3)
z3.Sub(z3, z1z1)
if z3.Sign() == -1 {
z3.Add(z3, curve.P)
}
z3.Sub(z3, z2z2)
if z3.Sign() == -1 {
z3.Add(z3, curve.P)
}
z3.Mul(z3, h)
z3.Mod(z3, curve.P)
return x3, y3, z3
}
// decrypt performs an RSA decryption, resulting in a plaintext integer. If a
// random source is given, RSA blinding is used.
func decrypt(random io.Reader, priv *PrivateKey, c *big.Int) (m *big.Int, err os.Error) {
// TODO(agl): can we get away with reusing blinds?
if c.Cmp(priv.N) > 0 {
err = DecryptionError{}
return
}
var ir *big.Int
if random != nil {
// Blinding enabled. Blinding involves multiplying c by r^e.
// Then the decryption operation performs (m^e * r^e)^d mod n
// which equals mr mod n. The factor of r can then be removed
// by multiplying by the multiplicative inverse of r.
var r *big.Int
for {
r, err = rand.Int(random, priv.N)
if err != nil {
return
}
if r.Cmp(bigZero) == 0 {
r = bigOne
}
var ok bool
ir, ok = modInverse(r, priv.N)
if ok {
break
}
}
bigE := big.NewInt(int64(priv.E))
rpowe := new(big.Int).Exp(r, bigE, priv.N)
cCopy := new(big.Int).Set(c)
cCopy.Mul(cCopy, rpowe)
cCopy.Mod(cCopy, priv.N)
c = cCopy
}
if priv.Precomputed.Dp == nil {
m = new(big.Int).Exp(c, priv.D, priv.N)
} else {
// We have the precalculated values needed for the CRT.
m = new(big.Int).Exp(c, priv.Precomputed.Dp, priv.Primes[0])
m2 := new(big.Int).Exp(c, priv.Precomputed.Dq, priv.Primes[1])
m.Sub(m, m2)
if m.Sign() < 0 {
m.Add(m, priv.Primes[0])
}
m.Mul(m, priv.Precomputed.Qinv)
m.Mod(m, priv.Primes[0])
m.Mul(m, priv.Primes[1])
m.Add(m, m2)
for i, values := range priv.Precomputed.CRTValues {
prime := priv.Primes[2+i]
m2.Exp(c, values.Exp, prime)
m2.Sub(m2, m)
m2.Mul(m2, values.Coeff)
m2.Mod(m2, prime)
if m2.Sign() < 0 {
m2.Add(m2, prime)
}
m2.Mul(m2, values.R)
m.Add(m, m2)
}
}
if ir != nil {
// Unblind.
m.Mul(m, ir)
m.Mod(m, priv.N)
}
return
}
// decrypt performs an RSA decryption, resulting in a plaintext integer. If a
// random source is given, RSA blinding is used.
func decrypt(rand io.Reader, priv *PrivateKey, c *big.Int) (m *big.Int, err os.Error) {
// TODO(agl): can we get away with reusing blinds?
if c.Cmp(priv.N) > 0 {
err = DecryptionError{}
return
}
var ir *big.Int
if rand != nil {
// Blinding enabled. Blinding involves multiplying c by r^e.
// Then the decryption operation performs (m^e * r^e)^d mod n
// which equals mr mod n. The factor of r can then be removed
// by multipling by the multiplicative inverse of r.
var r *big.Int
for {
r, err = randomNumber(rand, priv.N)
if err != nil {
return
}
if r.Cmp(bigZero) == 0 {
r = bigOne
}
var ok bool
ir, ok = modInverse(r, priv.N)
if ok {
break
}
}
bigE := big.NewInt(int64(priv.E))
rpowe := new(big.Int).Exp(r, bigE, priv.N)
c.Mul(c, rpowe)
c.Mod(c, priv.N)
}
priv.rwMutex.RLock()
if priv.dP == nil && priv.P != nil {
priv.rwMutex.RUnlock()
priv.rwMutex.Lock()
if priv.dP == nil && priv.P != nil {
priv.precompute()
}
priv.rwMutex.Unlock()
priv.rwMutex.RLock()
}
if priv.dP == nil {
m = new(big.Int).Exp(c, priv.D, priv.N)
} else {
// We have the precalculated values needed for the CRT.
m = new(big.Int).Exp(c, priv.dP, priv.P)
m2 := new(big.Int).Exp(c, priv.dQ, priv.Q)
m.Sub(m, m2)
if m.Sign() < 0 {
m.Add(m, priv.P)
}
m.Mul(m, priv.qInv)
m.Mod(m, priv.P)
m.Mul(m, priv.Q)
m.Add(m, m2)
if priv.dR != nil {
// 3-prime CRT.
m2.Exp(c, priv.dR, priv.R)
m2.Sub(m2, m)
m2.Mul(m2, priv.tr)
m2.Mod(m2, priv.R)
if m2.Sign() < 0 {
m2.Add(m2, priv.R)
}
m2.Mul(m2, priv.pq)
m.Add(m, m2)
}
}
priv.rwMutex.RUnlock()
if ir != nil {
// Unblind.
m.Mul(m, ir)
m.Mod(m, priv.N)
}
return
}
