Golang Int.Bit方法代碼示例

// 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
}