// generateNoncePair deterministically generate a nonce pair for use in
// partial signing of a message. Returns a public key (nonce to disseminate)
// and a private nonce to keep as a secret for the signer.
func generateNoncePair(curve *secp256k1.KoblitzCurve, msg []byte, priv []byte,
nonceFunction func([]byte, []byte, []byte, []byte) []byte, extra []byte,
version []byte) ([]byte, *secp256k1.PublicKey, error) {
k := nonceFunction(priv, msg, extra, version)
bigK := new(big.Int).SetBytes(k)
// k scalar sanity checks.
if bigK.Cmp(bigZero) == 0 {
str := fmt.Sprintf("k scalar is zero")
return nil, nil, schnorrError(ErrBadNonce, str)
}
if bigK.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("k scalar is >= curve.N")
return nil, nil, schnorrError(ErrBadNonce, str)
}
bigK.SetInt64(0)
pubx, puby := curve.ScalarBaseMult(k)
pubnonce := secp256k1.NewPublicKey(curve, pubx, puby)
return k, pubnonce, nil
}
// schnorrSign signs a Schnorr signature using a specified hash function
// and the given nonce, private key, message, and optional public nonce.
// CAVEAT: Lots of variable time algorithms using both the private key and
// k, which can expose the signer to constant time attacks. You have been
// warned! DO NOT use this algorithm where you might have the possibility
// of someone having EM field/cache/etc access.
// Memory management is also kind of sloppy and whether or not your keys
// or nonces can be found in memory later is likely a product of when the
// garbage collector runs.
// TODO Use field elements with constant time algorithms to prevent said
// attacks.
// This is identical to the Schnorr signature function found in libsecp256k1:
// https://github.com/bitcoin/secp256k1/tree/master/src/modules/schnorr
func schnorrSign(curve *secp256k1.KoblitzCurve, msg []byte, ps []byte, k []byte,
pubNonceX *big.Int, pubNonceY *big.Int,
hashFunc func([]byte) []byte) (*Signature, error) {
if len(msg) != scalarSize {
str := fmt.Sprintf("wrong size for message (got %v, want %v)",
len(msg), scalarSize)
return nil, schnorrError(ErrBadInputSize, str)
}
if len(ps) != scalarSize {
str := fmt.Sprintf("wrong size for privkey (got %v, want %v)",
len(ps), scalarSize)
return nil, schnorrError(ErrBadInputSize, str)
}
if len(k) != scalarSize {
str := fmt.Sprintf("wrong size for nonce k (got %v, want %v)",
len(k), scalarSize)
return nil, schnorrError(ErrBadInputSize, str)
}
psBig := new(big.Int).SetBytes(ps)
kBig := new(big.Int).SetBytes(k)
if psBig.Cmp(bigZero) == 0 {
str := fmt.Sprintf("secret scalar is zero")
return nil, schnorrError(ErrInputValue, str)
}
if psBig.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("secret scalar is out of bounds")
return nil, schnorrError(ErrInputValue, str)
}
if kBig.Cmp(bigZero) == 0 {
str := fmt.Sprintf("k scalar is zero")
return nil, schnorrError(ErrInputValue, str)
}
if kBig.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("k scalar is out of bounds")
return nil, schnorrError(ErrInputValue, str)
}
// R = kG
var Rpx, Rpy *big.Int
Rpx, Rpy = curve.ScalarBaseMult(k)
if pubNonceX != nil && pubNonceY != nil {
// Optional: if k' exists then R = R+k'
Rpx, Rpy = curve.Add(Rpx, Rpy, pubNonceX, pubNonceY)
}
// Check if the field element that would be represented by Y is odd.
// If it is, just keep k in the group order.
if Rpy.Bit(0) == 1 {
kBig.Mod(kBig, curve.N)
kBig.Sub(curve.N, kBig)
}
// h = Hash(r || m)
Rpxb := BigIntToEncodedBytes(Rpx)
hashInput := make([]byte, 0, scalarSize*2)
hashInput = append(hashInput, Rpxb[:]...)
hashInput = append(hashInput, msg...)
h := hashFunc(hashInput)
hBig := new(big.Int).SetBytes(h)
// If the hash ends up larger than the order of the curve, abort.
if hBig.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("hash of (R || m) too big")
return nil, schnorrError(ErrSchnorrHashValue, str)
}
// s = k - hx
// TODO Speed this up a bunch by using field elements, not
// big ints. That we multiply the private scalar using big
// ints is also probably bad because we can only assume the
// math isn't in constant time, thus opening us up to side
// channel attacks. Using a constant time field element
// implementation will fix this.
sBig := new(big.Int)
sBig.Mul(hBig, psBig)
sBig.Sub(kBig, sBig)
sBig.Mod(sBig, curve.N)
if sBig.Cmp(bigZero) == 0 {
str := fmt.Sprintf("sig s %v is zero")
return nil, schnorrError(ErrZeroSigS, str)
}
// Zero out the private key and nonce when we're done with it.
kBig.SetInt64(0)
zeroSlice(k)
psBig.SetInt64(0)
zeroSlice(ps)
return &Signature{Rpx, sBig}, nil
}
// schnorrRecover recovers a public key using a signature, hash function,
// and message. It also attempts to verify the signature against the
// regenerated public key.
func schnorrRecover(curve *secp256k1.KoblitzCurve, sig, msg []byte,
hashFunc func([]byte) []byte) (*secp256k1.PublicKey, bool, error) {
if len(msg) != scalarSize {
str := fmt.Sprintf("wrong size for message (got %v, want %v)",
len(msg), scalarSize)
return nil, false, schnorrError(ErrBadInputSize, str)
}
if len(sig) != SignatureSize {
str := fmt.Sprintf("wrong size for signature (got %v, want %v)",
len(sig), SignatureSize)
return nil, false, schnorrError(ErrBadInputSize, str)
}
sigR := sig[:32]
sigS := sig[32:]
sigRCopy := make([]byte, scalarSize, scalarSize)
copy(sigRCopy, sigR)
toHash := append(sigRCopy, msg...)
h := hashFunc(toHash)
hBig := new(big.Int).SetBytes(h)
// If the hash ends up larger than the order of the curve, abort.
// Same thing for hash == 0 (as unlikely as that is...).
if hBig.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("hash of (R || m) too big")
return nil, false, schnorrError(ErrSchnorrHashValue, str)
}
if hBig.Cmp(bigZero) == 0 {
str := fmt.Sprintf("hash of (R || m) is zero value")
return nil, false, schnorrError(ErrSchnorrHashValue, str)
}
// Convert s to big int.
sBig := EncodedBytesToBigInt(copyBytes(sigS))
// We also can't have s greater than the order of the curve.
if sBig.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("s value is too big")
return nil, false, schnorrError(ErrInputValue, str)
}
// r can't be larger than the curve prime.
rBig := EncodedBytesToBigInt(copyBytes(sigR))
if rBig.Cmp(curve.P) == 1 {
str := fmt.Sprintf("given R was greater than curve prime")
return nil, false, schnorrError(ErrBadSigRNotOnCurve, str)
}
// Decompress the Y value. We know that the first bit must
// be even. Use the PublicKey struct to make it easier.
compressedPoint := make([]byte, PubKeyBytesLen, PubKeyBytesLen)
compressedPoint[0] = pubkeyCompressed
copy(compressedPoint[1:], sigR)
rPoint, err := secp256k1.ParsePubKey(compressedPoint, curve)
if err != nil {
str := fmt.Sprintf("bad r point")
return nil, false, schnorrError(ErrRegenerateRPoint, str)
}
// Get the inverse of the hash.
hInv := new(big.Int).ModInverse(hBig, curve.N)
hInv.Mod(hInv, curve.N)
// Negate s.
sBig.Sub(curve.N, sBig)
sBig.Mod(sBig, curve.N)
// s' = -s * inverse(h).
sBig.Mul(sBig, hInv)
sBig.Mod(sBig, curve.N)
// Q = h^(-1)R + s'G
lx, ly := curve.ScalarMult(rPoint.GetX(), rPoint.GetY(), hInv.Bytes())
rx, ry := curve.ScalarBaseMult(sBig.Bytes())
pkx, pky := curve.Add(lx, ly, rx, ry)
// Check if the public key is on the curve.
if !curve.IsOnCurve(pkx, pky) {
str := fmt.Sprintf("pubkey not on curve")
return nil, false, schnorrError(ErrPubKeyOffCurve, str)
}
pubkey := secp256k1.NewPublicKey(curve, pkx, pky)
// Verify this signature. Slow, lots of double checks, could be more
// cheaply implemented as
// hQ + sG - R == 0
// which this function checks.
// This will sometimes pass even for corrupted signatures, but
// this shouldn't be a concern because whoever is using the
// results should be checking the returned public key against
// some known one anyway. In the case of these Schnorr signatures,
// relatively high numbers of corrupted signatures (50-70%)
// seem to produce valid pubkeys and valid signatures.
_, err = schnorrVerify(curve, sig, pubkey, msg, hashFunc)
if err != nil {
str := fmt.Sprintf("pubkey/sig pair could not be validated")
return nil, false, schnorrError(ErrRegenSig, str)
}
return pubkey, true, nil
}
// schnorrVerify is the internal function for verification of a secp256k1
// Schnorr signature. A secure hash function may be passed for the calculation
// of r.
// This is identical to the Schnorr verification function found in libsecp256k1:
// https://github.com/bitcoin/secp256k1/tree/master/src/modules/schnorr
func schnorrVerify(curve *secp256k1.KoblitzCurve, sig []byte,
pubkey *secp256k1.PublicKey, msg []byte, hashFunc func([]byte) []byte) (bool,
error) {
if len(msg) != scalarSize {
str := fmt.Sprintf("wrong size for message (got %v, want %v)",
len(msg), scalarSize)
return false, schnorrError(ErrBadInputSize, str)
}
if len(sig) != SignatureSize {
str := fmt.Sprintf("wrong size for signature (got %v, want %v)",
len(sig), SignatureSize)
return false, schnorrError(ErrBadInputSize, str)
}
if pubkey == nil {
str := fmt.Sprintf("nil pubkey")
return false, schnorrError(ErrInputValue, str)
}
if !curve.IsOnCurve(pubkey.GetX(), pubkey.GetY()) {
str := fmt.Sprintf("pubkey point is not on curve")
return false, schnorrError(ErrPointNotOnCurve, str)
}
sigR := sig[:32]
sigS := sig[32:]
sigRCopy := make([]byte, scalarSize, scalarSize)
copy(sigRCopy, sigR)
toHash := append(sigRCopy, msg...)
h := hashFunc(toHash)
hBig := new(big.Int).SetBytes(h)
// If the hash ends up larger than the order of the curve, abort.
// Same thing for hash == 0 (as unlikely as that is...).
if hBig.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("hash of (R || m) too big")
return false, schnorrError(ErrSchnorrHashValue, str)
}
if hBig.Cmp(bigZero) == 0 {
str := fmt.Sprintf("hash of (R || m) is zero value")
return false, schnorrError(ErrSchnorrHashValue, str)
}
// Convert s to big int.
sBig := EncodedBytesToBigInt(copyBytes(sigS))
// We also can't have s greater than the order of the curve.
if sBig.Cmp(curve.N) >= 0 {
str := fmt.Sprintf("s value is too big")
return false, schnorrError(ErrInputValue, str)
}
// r can't be larger than the curve prime.
rBig := EncodedBytesToBigInt(copyBytes(sigR))
if rBig.Cmp(curve.P) == 1 {
str := fmt.Sprintf("given R was greater than curve prime")
return false, schnorrError(ErrBadSigRNotOnCurve, str)
}
// r' = hQ + sG
lx, ly := curve.ScalarMult(pubkey.GetX(), pubkey.GetY(), h)
rx, ry := curve.ScalarBaseMult(sigS)
rlx, rly := curve.Add(lx, ly, rx, ry)
if rly.Bit(0) == 1 {
str := fmt.Sprintf("calculated R y-value was odd")
return false, schnorrError(ErrBadSigRYValue, str)
}
if !curve.IsOnCurve(rlx, rly) {
str := fmt.Sprintf("calculated R point was not on curve")
return false, schnorrError(ErrBadSigRNotOnCurve, str)
}
rlxB := BigIntToEncodedBytes(rlx)
// r == r' --> valid signature
if !bytes.Equal(sigR, rlxB[:]) {
str := fmt.Sprintf("calculated R point was not given R")
return false, schnorrError(ErrUnequalRValues, str)
}
return true, nil
}