// ProbablyPrimeBigInt returns true if n is prime or n is a pseudoprime to base
// a. It implements the Miller-Rabin primality test for one specific value of
// 'a' and k == 1. See also ProbablyPrimeUint32.
func ProbablyPrimeBigInt(n, a *big.Int) bool {
var d big.Int
d.Set(n)
d.Sub(&d, _1) // d <- n-1
s := 0
for ; d.Bit(s) == 0; s++ {
}
nMinus1 := big.NewInt(0).Set(&d)
d.Rsh(&d, uint(s))
x := ModPowBigInt(a, &d, n)
if x.Cmp(_1) == 0 || x.Cmp(nMinus1) == 0 {
return true
}
for ; s > 1; s-- {
if x = x.Mod(x.Mul(x, x), n); x.Cmp(_1) == 0 {
return false
}
if x.Cmp(nMinus1) == 0 {
return true
}
}
return false
}
/*
FromFactorBigInt returns n such that d | Mn if n <= max and d is odd. In other
cases zero is returned.
It is conjectured that every odd d ∊ N divides infinitely many Mersenne numbers.
The returned n should be the exponent of smallest such Mn.
NOTE: The computation of n from a given d performs roughly in O(n). It is
thus highly recomended to use the 'max' argument to limit the "searched"
exponent upper bound as appropriate. Otherwise the computation can take a long
time as a large factor can be a divisor of a Mn with exponent above the uint32
limits.
The FromFactorBigInt function is a modification of the original Will
Edgington's "reverse method", discussed here:
http://tech.groups.yahoo.com/group/primenumbers/message/15061
*/
func FromFactorBigInt(d *big.Int, max uint32) (n uint32) {
if d.Bit(0) == 0 {
return
}
var m big.Int
for n < max {
m.Add(&m, d)
i := 0
for ; m.Bit(i) == 1; i++ {
if n == math.MaxUint32 {
return 0
}
n++
}
m.Rsh(&m, uint(i))
if m.Sign() == 0 {
if n > max {
n = 0
}
return
}
}
return 0
}
// polyPowMod computes ``f**n`` in ``GF(p)[x]/(g)`` using repeated squaring.
// Given polynomials ``f`` and ``g`` in ``GF(p)[x]`` and a non-negative
// integer ``n``, efficiently computes ``f**n (mod g)`` i.e. the remainder
// of ``f**n`` from division by ``g``, using the repeated squaring algorithm.
// This function was ported from sympy.polys.galoistools.
func polyPowMod(f *Poly, n *big.Int, g *Poly) (h *Poly, err error) {
zero := big.NewInt(int64(0))
one := big.NewInt(int64(1))
n = big.NewInt(int64(0)).Set(n)
if n.BitLen() < 3 {
// Small values of n not useful for recon
err = powModSmallN
return
}
h = NewPoly(Zi(f.p, 1))
for {
if n.Bit(0) > 0 {
h = NewPoly().Mul(h, f)
h, err = PolyMod(h, g)
if err != nil {
return
}
n.Sub(n, one)
}
n.Rsh(n, 1)
if n.Cmp(zero) == 0 {
break
}
f = NewPoly().Mul(f, f)
f, err = PolyMod(f, g)
if err != nil {
return
}
}
return
}
// StrongMillerRabin checks if N is a
// strong Miller-Rabin pseudoprime in base a.
// That is, it checks if a is a witness
// for compositeness of N or if N is a strong
// pseudoprime base a.
//
// Use builtin ProbablyPrime if you want to do a lot
// of random tests, this is for one specific
// base value.
func StrongMillerRabin(N *big.Int, a int64) int {
// Step 0: parse input
if N.Sign() < 0 || N.Bit(0) == 0 || a < 2 {
panic("MR is for positive odd integers with a >= 2")
}
A := big.NewInt(a)
if (a == 2 && N.Bit(0) == 0) || new(big.Int).GCD(nil, nil, N, A).Cmp(one) != 0 {
return IsComposite
}
// Step 1: find d,s, so that n - 1 = d*2^s
// with d odd
d := new(big.Int).Sub(N, one)
s := trailingZeroBits(d)
d.Rsh(d, s)
// Step 2: compute powers a^d
// and then a^(d*2^r) for 0<r<s
nm1 := new(big.Int).Sub(N, one)
Ad := new(big.Int).Exp(A, d, N)
if Ad.Cmp(one) == 0 || Ad.Cmp(nm1) == 0 {
return Undetermined
}
for r := uint(1); r < s; r++ {
Ad.Exp(Ad, two, N)
if Ad.Cmp(nm1) == 0 {
return Undetermined
}
}
// Step 3: a is a witness for compositeness
return IsComposite
}
func EncodePublicKey(x, y *big.Int, compressed bool) ([]byte, error) {
var pubkey []byte
if compressed {
pubkey = make([]byte, 33)
pubkey[0] = 2 + byte(y.Bit(0))
} else {
pubkey = make([]byte, 65)
pubkey[0] = 4
}
// Right-align x coordinate
bytes := x.Bytes()
if len(bytes) > 32 {
return nil, fmt.Errorf("Value of x has > 32 bytes")
}
copy(pubkey[1+(32-len(bytes)):33], bytes)
if !compressed {
// Right-align y coordinate
bytes = y.Bytes()
if len(bytes) > 32 {
return nil, fmt.Errorf("Value of y has > 32 bytes")
}
copy(pubkey[33+(32-len(bytes)):65], bytes)
}
return pubkey, nil
}
// ScalarMult computes Q = k * P on EllipticCurve ec.
func (ec *EllipticCurve) ScalarMult(k *big.Int, P Point) (Q Point) {
/* Note: this function is not constant time, due to the branching nature of
* the underlying point Add() function. */
/* Montgomery Ladder Point Multiplication
*
* Implementation based on pseudocode here:
* See https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Montgomery_ladder */
var R0 Point
var R1 Point
R0.X = nil
R0.Y = nil
R1.X = new(big.Int).Set(P.X)
R1.Y = new(big.Int).Set(P.Y)
for i := ec.N.BitLen() - 1; i >= 0; i-- {
if k.Bit(i) == 0 {
R1 = ec.Add(R0, R1)
R0 = ec.Add(R0, R0)
} else {
R0 = ec.Add(R0, R1)
R1 = ec.Add(R1, R1)
}
}
return R0
}
func factor2(n *big.Int) int {
// could be improved for large factors
f := 0
for ; n.Bit(f) == 0; f++ {
}
return f
}
func factor(n *big.Int) (pf []pExp) {
var e int64
for ; n.Bit(int(e)) == 0; e++ {
}
if e > 0 {
n.Rsh(n, uint(e))
pf = []pExp{{big.NewInt(2), e}}
}
s := sqrt(n)
q, r := new(big.Int), new(big.Int)
for d := big.NewInt(3); n.Cmp(one) > 0; d.Add(d, two) {
if d.Cmp(s) > 0 {
d.Set(n)
}
for e = 0; ; e++ {
q.QuoRem(n, d, r)
if r.BitLen() > 0 {
break
}
n.Set(q)
}
if e > 0 {
pf = append(pf, pExp{new(big.Int).Set(d), e})
s = sqrt(n)
}
}
return
}
func ScanLeftInt(x *big.Int) int {
for k := 0; k < x.BitLen(); k += 1 {
if x.Bit(k) != 0 {
return k
}
}
return -1
}
// compute the position of the leading one of a bit vector
func rank(hv *big.Int) int {
for i := 0; i < hv.BitLen(); i++ {
if hv.Bit(i) > 0 {
return i + 1
}
}
return MAX_LEN
}
func findSetBit(s *big.Int, start int) int {
for i := start; i < s.BitLen(); i++ {
if s.Bit(i) == 1 {
return i
}
}
return -1
}
func FirstBitSet(v *big.Int) int {
for i := 0; i < v.BitLen(); i++ {
if v.Bit(i) > 0 {
return i
}
}
return v.BitLen()
}
// JacobiSymbol returns the jacobi symbol ( N / D ) of
// N (numerator) over D (denominator).
// See http://en.wikipedia.org/wiki/Jacobi_symbol
func JacobiSymbol(N *big.Int, D *big.Int) int {
//Step 0: parse input / easy cases
if D.Sign() <= 0 || D.Bit(0) == 0 {
// we will assume D is positive
// wolfram is ok with negative denominator
// im not sure what is standard though
panic("JacobiSymbol defined for positive odd denominator only")
}
var n, d, tmp big.Int
n.Set(N)
d.Set(D)
j := 1
for {
// Step 1: Reduce the numerator mod the denominator
n.Mod(&n, &d)
if n.Sign() == 0 {
// if n,d not relatively prime
return 0
}
if len(n.Bits()) >= len(d.Bits())-1 {
// n > d/2 so swap n with d-n
// and multiply j by JacobiSymbol(-1 / d)
n.Sub(&d, &n)
if d.Bits()[0]&3 == 3 {
// if d = 3 mod 4
j = -1 * j
}
}
// Step 2: extract factors of 2
s := trailingZeroBits(&n)
n.Rsh(&n, s)
if s&1 == 1 {
switch d.Bits()[0] & 7 {
case 3, 5: // d = 3,5 mod 8
j = -1 * j
}
}
// Step 3: check numerator
if len(n.Bits()) == 1 && n.Bits()[0] == 1 {
// if n = 1 were done
return j
}
// Step 4: flip and go back to step 1
if n.Bits()[0]&3 != 1 { // n = 3 mod 4
if d.Bits()[0]&3 != 1 { // d = 3 mod 4
j = -1 * j
}
}
tmp.Set(&n)
n.Set(&d)
d.Set(&tmp)
}
}
// Set z to one of the square roots of a modulo p if a square root exists.
// The modulus p must be an odd prime.
// Returns true on success, false if input a is not a square modulo p.
func Sqrt(z *big.Int, a *big.Int, p *big.Int) bool {
if a.Sign() == 0 {
z.SetInt64(0) // sqrt(0) = 0
return true
}
if Jacobi(a, p) != 1 {
return false // a is not a square mod M
}
// Break p-1 into s*2^e such that s is odd.
var s big.Int
var e int
s.Sub(p, one)
for s.Bit(0) == 0 {
s.Div(&s, two)
e++
}
// Find some non-square n
var n big.Int
n.SetInt64(2)
for Jacobi(&n, p) != -1 {
n.Add(&n, one)
}
// Heart of the Tonelli-Shanks algorithm.
// Follows the description in
// "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra Brown.
var x, b, g, t big.Int
x.Add(&s, one).Div(&x, two).Exp(a, &x, p)
b.Exp(a, &s, p)
g.Exp(&n, &s, p)
r := e
for {
// Find the least m such that ord_p(b) = 2^m
var m int
t.Set(&b)
for t.Cmp(one) != 0 {
t.Exp(&t, two, p)
m++
}
if m == 0 {
z.Set(&x)
return true
}
t.SetInt64(0).SetBit(&t, r-m-1, 1).Exp(&g, &t, p)
// t = g^(2^(r-m-1)) mod p
g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
x.Mul(&x, &t).Mod(&x, p)
b.Mul(&b, &g).Mod(&b, p)
r = m
}
}
func modPowBigInt(b, e, m *big.Int) (r *big.Int) {
r = big.NewInt(1)
for i, n := 0, e.BitLen(); i < n; i++ {
if e.Bit(i) != 0 {
r.Mod(r.Mul(r, b), m)
}
b.Mod(b.Mul(b, b), m)
}
return
}
func testBig(s string) {
b, _ := new(big.Int).SetString(s, 10)
fmt.Printf("Testing big integer %v: ", b)
// the Bit function is the only sensible test for big ints.
if b.Bit(0) == 0 {
fmt.Println("even")
} else {
fmt.Println("odd")
}
}
// contiguousScanStrategy tries to allocate starting at 0 and filling in any gaps
func contiguousScanStrategy(allocated *big.Int, max, count int) (int, bool) {
if count >= max {
return 0, false
}
for i := 0; i < max; i++ {
if allocated.Bit(i) == 0 {
return i, true
}
}
return 0, false
}
// counts the number of zeros at the end of the
// binary expansion. So 2=10 ---> 1, 4=100 ---> 2
// 3=111 ---> 0, see test for more examples
// also 0 ---> 0 and 1 ---> 0
func trailingZeroBits(x *big.Int) (i uint) {
if x.Sign() < 0 {
panic("unknown bits of negative")
}
if x.Sign() == 0 || x.Bit(0) == 1 {
return 0
}
for i = 1; i < uint(x.BitLen()) && x.Bit(int(i)) != 1; i++ {
}
return
}
// Marshal encodes a ECC Point into it's compressed representation
func Marshal(curve elliptic.Curve, x, y *big.Int) []byte {
byteLen := (curve.Params().BitSize + 7) >> 3
ret := make([]byte, 1+byteLen)
ret[0] = 2 + byte(y.Bit(0))
xBytes := x.Bytes()
copy(ret[1+byteLen-len(xBytes):], xBytes)
return ret
}
func bitSetToString(n *big.Int) string {
var b bytes.Buffer
b.WriteRune('{')
for i, bitsLen := 0, n.BitLen(); i < bitsLen; i++ {
if n.Bit(i) == 0 {
continue
}
fmt.Fprintf(&b, "%v, ", i)
}
b.WriteRune('}')
return b.String()
}
//primize makes x a prime number.
//Result will be bigger than x.
func primize(x *big.Int) *big.Int {
var tmp big.Int
if x.Bit(0) == 0 {
x.Add(x, tmp.SetInt64(1))
}
for {
if x.ProbablyPrime(sprpTestCount) {
return x
}
x.Add(x, tmp.SetInt64(2))
}
}
func (d PublicKeyDigest) Log2int() int {
var b big.Int
b.SetBytes(d[:])
b.Add(&b, big.NewInt(1))
l := len(d)*8 - 1
for i := len(d) * 8; i >= 0 && b.Bit(i) == 0; i-- {
l--
}
if l < 0 {
return 0
}
return l
}
// mulMod computes z = (x * y) % p.
func mulMod(x *big.Int, y *big.Int, p *big.Int) (z *big.Int) {
n := new(big.Int).Set(x)
z = big.NewInt(0)
for i := 0; i < y.BitLen(); i++ {
if y.Bit(i) == 1 {
z = addMod(z, n, p)
}
n = addMod(n, n, p)
}
return z
}
// randomScanStrategy chooses a random address from the provided big.Int, and then
// scans forward looking for the next available address (it will wrap the range if
// necessary).
func randomScanStrategy(allocated *big.Int, max, count int) (int, bool) {
if count >= max {
return 0, false
}
offset := rand.Intn(max)
for i := 0; i < max; i++ {
at := (offset + i) % max
if allocated.Bit(at) == 0 {
return at, true
}
}
return 0, false
}
// trunc truncates a value to the range of the given type.
func (t *_type) trunc(x *big.Int) *big.Int {
r := new(big.Int)
m := new(big.Int)
m.Lsh(one, t.bits)
m.Sub(m, one)
r.And(x, m)
if t.signed && r.Bit(int(t.bits)-1) == 1 {
m.Neg(one)
m.Lsh(m, t.bits)
r.Or(r, m)
}
return r
}
// ScalarMult returns k*(Bx,By) where k is a number in big-endian form. This
// uses the repeated doubling method, which is variable time.
// TODO use a constant time method to prevent side channel attacks.
func (curve *TwistedEdwardsCurve) ScalarMult(x1, y1 *big.Int,
k []byte) (x, y *big.Int) {
// Convert the scalar to a big int.
s := new(big.Int).SetBytes(k)
// Get a new group element to do cached doubling
// calculations in.
dEGE := new(edwards25519.ExtendedGroupElement)
dEGE.Zero()
// Use the doubling method for the multiplication.
// p := given point
// q := point(zero)
// for each bit in the scalar, descending:
// double(q)
// if bit == 1:
// add(q, p)
// return q
//
// Note that the addition is skipped for zero bits,
// making this variable time and thus vulnerable to
// side channel attack vectors.
for i := s.BitLen() - 1; i >= 0; i-- {
dCGE := new(edwards25519.CompletedGroupElement)
dEGE.Double(dCGE)
dCGE.ToExtended(dEGE)
if s.Bit(i) == 1 {
ss := new([32]byte)
dEGE.ToBytes(ss)
var err error
xi, yi, err := curve.EncodedBytesToBigIntPoint(ss)
if err != nil {
return nil, nil
}
xAdd, yAdd := curve.Add(xi, yi, x1, y1)
dTempBytes := BigIntPointToEncodedBytes(xAdd, yAdd)
dEGE.FromBytes(dTempBytes)
}
}
finalBytes := new([32]byte)
dEGE.ToBytes(finalBytes)
var err error
x, y, err = curve.EncodedBytesToBigIntPoint(finalBytes)
if err != nil {
return nil, nil
}
return
}
// findComplements returns the indices of two character arrays a and b
// in charTab such that (a & maskIncl) == (~b & maskIncl),
// a & maskExcl == maskExcl, and b & maskExcl == maskExcl.
func findComplements(charTab CharTable, maskIncl, maskExcl *big.Int) (s, t int) {
// w := maxInt(maskIncl.BitLen(), maskExcl.BitLen())
// fmt.Printf("findComplements\n.incl=%0*b\n.excl=%0*b\n", w, maskIncl, w, maskExcl)
if maskIncl.BitLen() == 0 {
panic("Empty mask")
}
var minInclBit int
for minInclBit = 0; minInclBit < maskIncl.BitLen(); minInclBit++ {
if maskIncl.Bit(minInclBit) != 0 {
break
}
}
arraysByNorm := map[string][]int{}
for i, a := range charTab {
b := &big.Int{}
b.And(a, maskExcl)
if b.Cmp(maskExcl) != 0 {
continue
}
if a.Bit(minInclBit) == 0 {
b.AndNot(maskIncl, a)
} else {
b.And(maskIncl, a)
}
key := fmt.Sprintf("%x", b)
arraysByNorm[key] = append(arraysByNorm[key], i)
}
for _, cands := range arraysByNorm {
if len(cands) < 2 {
continue
}
for i := 0; i < len(cands)-1; i++ {
a := charTab[cands[i]]
for j := i + 1; j < len(cands); j++ {
b := charTab[cands[j]]
c := &big.Int{}
c.Xor(a, b)
c.And(c, maskIncl)
if c.Cmp(maskIncl) == 0 {
return cands[i], cands[j]
}
}
}
}
return -1, -1
}
// Scan a bit-field within x starting from bit i,
// either upward or downward to but not including bit j,
// for the first bit with value b.
// Returns the position of the first b-bit found,
// or -1 if every bit in the bit-field is set to 1-b.
func BitScan(x *big.Int, i, j int, b uint) int {
// XXX could be made a lot more efficient using x.Words()
inc := 1
if i > j {
inc = -1
}
for ; i != j; i += inc {
if x.Bit(i) == b {
return i
}
}
return -1
}
func compressPublicKey(x *big.Int, y *big.Int) []byte {
var key bytes.Buffer
// Write header; 0x2 for even y value; 0x3 for odd
key.WriteByte(byte(0x2) + byte(y.Bit(0)))
// Write X coord; Pad the key so x is aligned with the LSB. Pad size is key length - header size (1) - xBytes size
xBytes := x.Bytes()
for i := 0; i < (PublicKeyCompressedLength - 1 - len(xBytes)); i++ {
key.WriteByte(0x0)
}
key.Write(xBytes)
return key.Bytes()
}
// SolovayStrassen chooses k random numbers in [2,...,N]
// and checks that there was no
// "Euler liar". That is, every number a we chose
// satisfied a^((n-1)/2) = Jacobi(a/N) mod N.
// See https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test.
// Probability it passes and is not prime is 2^(-k).
func SolovayStrassen(N *big.Int, k int) int {
if N.Bit(0) == 0 && N.BitLen() > 1 {
return IsComposite
}
a := new(big.Int)
b := make([]byte, N.BitLen())
for i := 0; i < k; i++ {
rand.Read(b)
a.SetBytes(b)
if basedSolovayStrassen(N, a) == IsComposite {
return IsComposite
}
}
return Undetermined
}
