Exemple #1
0
// 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
}
Exemple #2
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// p256FromBig sets out = R*in.
func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
	tmp := new(big.Int).Lsh(in, 257)
	tmp.Mod(tmp, p256.P)

	for i := 0; i < p256Limbs; i++ {
		if bits := tmp.Bits(); len(bits) > 0 {
			out[i] = uint32(bits[0]) & bottom29Bits
		} else {
			out[i] = 0
		}
		tmp.Rsh(tmp, 29)

		i++
		if i == p256Limbs {
			break
		}

		if bits := tmp.Bits(); len(bits) > 0 {
			out[i] = uint32(bits[0]) & bottom28Bits
		} else {
			out[i] = 0
		}
		tmp.Rsh(tmp, 28)
	}
}
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
}
Exemple #4
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// 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
}
Exemple #5
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func TestModAdc(t *testing.T) {
	A := new(big.Int)
	B := new(big.Int)
	C := new(big.Int)
	Carry := new(big.Int)
	Mask := new(big.Int)
	for _, a := range numbers {
		A.SetUint64(a)
		for _, b := range numbers {
			B.SetUint64(b)
			for width := uint8(1); width < 64; width++ {
				carry := b
				c := mod_adc(a, width, &carry)
				C.Add(A, B)
				Carry.Rsh(C, uint(width))
				expectedCarry := Carry.Uint64()
				Mask.SetUint64(uint64(1)<<width - 1)
				C.And(C, Mask)
				expected := C.Uint64()
				if c != expected || expectedCarry != carry {
					t.Fatalf("adc(%d,%d,%d): Expecting %d carry %d but got %d carry %d", a, b, width, expected, expectedCarry, c, carry)
				}
			}
		}
	}
}
Exemple #6
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func bigRsh(z, x, y *big.Int) *big.Int {
	i := y.Int64()
	if i < 0 {
		panic("negative shift")
	}
	return z.Rsh(x, uint(i))
}
Exemple #7
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// 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
}
Exemple #8
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func (self *Encoder) encode_int(number *big.Int, size uint) []byte {
	if size == 1 {
		return []byte{uint8(int8(number.Int64()))}
	} else if size == 2 {
		number_buf := uint16(int16(number.Int64()))
		return []byte{
			uint8(number_buf >> 8),
			uint8(number_buf),
		}
	} else if size == 4 {
		number_buf := uint32(int32(number.Int64()))
		return []byte{
			uint8(number_buf >> 24),
			uint8(number_buf >> 16),
			uint8(number_buf >> 8),
			uint8(number_buf),
		}
	} else if size == 0 {
		if number.Sign() < 0 {
			panic("jksn: number < 0")
		}
		result := []byte{uint8(new(big.Int).And(number, big.NewInt(0x7f)).Uint64())}
		number.Rsh(number, 7)
		for number.Sign() != 0 {
			result = append(result, uint8(new(big.Int).And(number, big.NewInt(0x7f)).Uint64())|0x80)
			number.Rsh(number, 7)
		}
		for i, j := 0, len(result)-1; i < j; i, j = i+1, j-1 {
			result[i], result[j] = result[j], result[i]
		}
		return result
	} else {
		panic("jksn: size not in (1, 2, 4, 0)")
	}
}
Exemple #9
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// pow sets d to x ** y and returns z.
func (z *Big) pow(x *Big, y *big.Int) *Big {
	switch {
	case y.Sign() < 0, (x.ez() || y.Sign() == 0):
		return z.SetMantScale(1, 0)
	case y.Cmp(oneInt) == 0:
		return z.Set(x)
	case x.ez():
		if x.isOdd() {
			return z.Set(x)
		}
		z.form = zero
		return z
	}

	x0 := new(Big).Set(x)
	y0 := new(big.Int).Set(y)
	ret := New(1, 0)
	var odd big.Int
	for y0.Sign() > 0 {
		if odd.And(y0, oneInt).Sign() != 0 {
			ret.Mul(ret, x0)
		}
		y0.Rsh(y0, 1)
		x0.Mul(x0, x0)
	}
	*z = *ret
	return ret
}
Exemple #10
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/*
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
}
Exemple #11
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func (self *Decoder) unsigned_to_signed(x *big.Int, bits uint) *big.Int {
	// return x - ((x >> (bits - 1)) << bits)
	temp := new(big.Int)
	temp.Rsh(x, bits-1)
	temp.Lsh(temp, bits)
	return temp.Sub(x, temp)
}
Exemple #12
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// Shift returns the result of the shift expression x op s
// with op == token.SHL or token.SHR (<< or >>). x must be
// an Int.
//
func Shift(x Value, op token.Token, s uint) Value {
	switch x := x.(type) {
	case unknownVal:
		return x

	case int64Val:
		if s == 0 {
			return x
		}
		switch op {
		case token.SHL:
			z := big.NewInt(int64(x))
			return normInt(z.Lsh(z, s))
		case token.SHR:
			return x >> s
		}

	case intVal:
		if s == 0 {
			return x
		}
		var z big.Int
		switch op {
		case token.SHL:
			return normInt(z.Lsh(x.val, s))
		case token.SHR:
			return normInt(z.Rsh(x.val, s))
		}
	}

	panic(fmt.Sprintf("invalid shift %v %s %d", x, op, s))
}
Exemple #13
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Fichier : graph.go Projet : rsc/tmp
func ratProb(mode int) func(*big.Rat) *big.Rat {
	return func(x *big.Rat) *big.Rat {
		lo := big.NewInt(0)
		hi := new(big.Int).Set(big2p63)
		n := 0
		for lo.Cmp(hi) != 0 {
			m := new(big.Int).Add(lo, hi)
			m = m.Rsh(m, 1)
			if n++; n > 100 {
				fmt.Printf("??? %v %v %v\n", lo, hi, m)
				break
			}
			v := new(big.Rat).SetFrac(m, big2p63)
			f, _ := v.Float64()
			v.SetFloat64(f)
			if v.Cmp(x) < 0 {
				lo.Add(m, bigOne)
			} else {
				hi.Set(m)
			}
		}
		switch mode {
		default: // case 0
			return new(big.Rat).SetFrac(lo, big2p63)
		case 1:
			if lo.Cmp(big.NewInt(cutoff1)) <= 0 {
				lo.Add(lo, big.NewInt(1<<63-cutoff1))
			}
			return new(big.Rat).SetFrac(lo, big2p63)
		case 2:
			return new(big.Rat).SetFrac(lo, big.NewInt(cutoff1))
		}
	}
}
Exemple #14
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// ISqrt returns the greatest number x such that x^2 <= n. n must be
// non-negative.
//
// See https://www.akalin.com/computing-isqrt for an analysis.
func ISqrt(n *big.Int) *big.Int {
	s := n.Sign()
	if s < 0 {
		panic("negative radicand")
	}
	if s == 0 {
		return &big.Int{}
	}

	// x = 2^ceil(Bits(n)/2)
	var x big.Int
	x.Lsh(big.NewInt(1), (uint(n.BitLen())+1)/2)
	for {
		// y = floor((x + floor(n/x))/2)
		var y big.Int
		y.Div(n, &x)
		y.Add(&y, &x)
		y.Rsh(&y, 1)

		if y.Cmp(&x) >= 0 {
			return &x
		}
		x = y
	}
}
Exemple #15
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// 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)
	}
}
Exemple #16
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func ParseId(id *big.Int) (timestamp, workerid, sequence int64) {
	bigS := big.NewInt(0)
	bigW := big.NewInt(0)

	bigS.And(id, big.NewInt((1<<defaultSequenceBits)-1))
	id.Rsh(id, uint(defaultSequenceBits))
	bigW.And(id, big.NewInt((1<<defaultWorkerIdBits)-1))
	id.Rsh(id, uint(defaultWorkerIdBits))

	return id.Int64(), bigW.Int64(), bigS.Int64()
}
func sqrt(n *big.Int) *big.Int {
	a := new(big.Int)
	for b := new(big.Int).Set(n); ; {
		a.Set(b)
		b.Rsh(b.Add(b.Quo(n, a), a), 1)
		if b.Cmp(a) >= 0 {
			return a
		}
	}
	return a.SetInt64(0)
}
Exemple #18
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// StakePoolTicketFee determines the stake pool ticket fee for a given ticket
// from the passed percentage. Pool fee as a percentage is truncated from 0.01%
// to 100.00%. This all must be done with integers, so bear with the big.Int
// usage below.
//
// See the included doc.go of this package for more information about the
// calculation of this fee.
func StakePoolTicketFee(stakeDiff dcrutil.Amount, relayFee dcrutil.Amount,
	height int32, poolFee float64, params *chaincfg.Params) dcrutil.Amount {
	// Shift the decimal two places, e.g. 1.00%
	// to 100. This assumes that the proportion
	// is already multiplied by 100 to give a
	// percentage, thus making the entirety
	// be a multiplication by 10000.
	poolFeeAbs := math.Floor(poolFee * 100.0)
	poolFeeInt := int64(poolFeeAbs)

	// Subsidy is fetched from the blockchain package, then
	// pushed forward a number of adjustment periods for
	// compensation in gradual subsidy decay. Recall that
	// the average time to claiming 50% of the tickets as
	// votes is the approximately the same as the ticket
	// pool size (params.TicketPoolSize), so take the
	// ceiling of the ticket pool size divided by the
	// reduction interval.
	adjs := int(math.Ceil(float64(params.TicketPoolSize) /
		float64(params.ReductionInterval)))
	initSubsidyCacheOnce.Do(func() {
		subsidyCache = blockchain.NewSubsidyCache(int64(height), params)
	})
	subsidy := blockchain.CalcStakeVoteSubsidy(subsidyCache, int64(height),
		params)
	for i := 0; i < adjs; i++ {
		subsidy *= 100
		subsidy /= 101
	}

	// The numerator is (p*10000*s*(v+z)) << 64.
	shift := uint(64)
	s := new(big.Int).SetInt64(subsidy)
	v := new(big.Int).SetInt64(int64(stakeDiff))
	z := new(big.Int).SetInt64(int64(relayFee))
	num := new(big.Int).SetInt64(poolFeeInt)
	num.Mul(num, s)
	vPlusZ := new(big.Int).Add(v, z)
	num.Mul(num, vPlusZ)
	num.Lsh(num, shift)

	// The denominator is 10000*(s+v).
	// The extra 10000 above cancels out.
	den := new(big.Int).Set(s)
	den.Add(den, v)
	den.Mul(den, new(big.Int).SetInt64(10000))

	// Divide and shift back.
	num.Div(num, den)
	num.Rsh(num, shift)

	return dcrutil.Amount(num.Int64())
}
Exemple #19
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func legendre(a, p *big.Int) int {
	var r big.Int
	r.Rsh(p, 1)
	r.Exp(a, &r, p)
	switch r.BitLen() {
	case 0:
		return 0
	case 1:
		return 1
	}
	return -1
}
Exemple #20
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//intToBase64 makes string from int.
func intToBase64(n *big.Int) string {
	var result string
	and := big.NewInt(0x3f)
	var tmp, nn big.Int
	nn.Set(n)

	for nn.Cmp(big.NewInt(0)) > 0 {
		bit := tmp.And(&nn, and).Uint64()
		result += string(base64en[bit])
		nn.Rsh(&nn, 6)
	}
	return result + string(base64en[0]*byte(86-len(result)))
}
Exemple #21
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// convert hash value to integer
// [http://www.secg.org/download/aid-780/sec1-v2.pdf]
func convertHash(hash []byte) *big.Int {

	// trim hash value (if required)
	maxSize := (curveN.BitLen() + 7) / 8
	if len(hash) > maxSize {
		hash = hash[:maxSize]
	}

	// convert to integer
	val := new(big.Int).SetBytes(hash)
	val.Rsh(val, uint(maxSize*8-curveN.BitLen()))
	return val
}
Exemple #22
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func basedSolovayStrassen(N, a *big.Int) int {
	// we assume N is odd
	x := JacobiSymbol(a, N)
	if x == 0 {
		return IsComposite
	}
	z := new(big.Int)
	z.Exp(a, z.Rsh(z.Sub(N, one), 1), N) // this step is expensive
	if (x == 1 && z.Cmp(one) == 0) || (x == -1 && z.Sub(N, z).Cmp(one) == 0) {
		return Undetermined
	}
	return IsComposite
}
Exemple #23
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// hashToInt converts a hash value to an integer. There is some disagreement
// about how this is done. [NSA] suggests that this is done in the obvious
// manner, but [SECG] truncates the hash to the bit-length of the curve order
// first. We follow [SECG] because that's what OpenSSL does.
func hashToInt(hash []byte, c *ec256k1.BitCurve) *big.Int {
	orderBits := c.N.BitLen()
	orderBytes := (orderBits + 7) / 8
	if len(hash) > orderBytes {
		hash = hash[:orderBytes]
	}

	ret := new(big.Int).SetBytes(hash)
	excess := orderBytes*8 - orderBits
	if excess > 0 {
		ret.Rsh(ret, uint(excess))
	}
	return ret
}
Exemple #24
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// IsSquare returns true if N = m^2
// for some positive integer m.
// It uses newtons method and other checks.
func IsSquare(N *big.Int) bool {
	// Step -1: check inputs
	if N.Sign() <= 0 {
		// 0 is a square
		if N.Sign() == 0 {
			return true
		}
		// negative numbers are not
		return false
	}

	// Step 0: Easy case
	if N.BitLen() < 62 { // need padding, 63 is too close to limit
		n := N.Int64()
		a := int64(math.Sqrt(float64(n)))
		if a*a == n {
			return true
		}
		return false
	}

	// Step 1.1: check if it is a square mod small power of 2
	if _, ok := squaresMod128[uint8(N.Uint64())]; !ok {
		return false
	}

	// Setp 1.2: check if it is a square mod a small number
	_z := uint16(new(big.Int).Mod(N, smallSquareMod).Uint64())
	if _, ok := smallSquares[_z]; !ok {
		return false
	}

	// Step 2: run newtons method, see
	// Cohen's book computational alg. number theory
	// Ch. 1, algorithm 1.7.1
	z := new(big.Int)
	x := new(big.Int).Lsh(one, uint(N.BitLen()+2)>>1)
	y := new(big.Int)
	for {
		// Set y = [(x + [N/x])/2]
		y := y.Rsh(z.Add(x, z.Div(N, x)), 1)
		// if y < x, set x to y
		// else return x
		if y.Cmp(x) == -1 {
			x.Set(y)
		} else {
			return z.Mul(x, x).Cmp(N) == 0
		}
	}
}
Exemple #25
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// hashToInt converts a hash value to an integer. There is some disagreement
// about how this is done. [NSA] suggests that this is done in the obvious
// manner, but [SECG] truncates the hash to the bit-length of the curve order
// first. We follow [SECG] because that's what OpenSSL does. Additionally,
// OpenSSL right shifts excess bits from the number if the hash is too large
// and we mirror that too.
func hashToInt(hash []byte, c elliptic.Curve) *big.Int {
	orderBits := c.Params().N.BitLen()
	orderBytes := (orderBits + 7) / 8
	if len(hash) > orderBytes {
		hash = hash[:orderBytes]
	}

	ret := new(big.Int).SetBytes(hash)
	excess := len(hash)*8 - orderBits
	if excess > 0 {
		ret.Rsh(ret, uint(excess))
	}
	return ret
}
Exemple #26
0
func binaryIntOp(x *big.Int, op token.Token, y *big.Int) interface{} {
	var z big.Int
	switch op {
	case token.ADD:
		return z.Add(x, y)
	case token.SUB:
		return z.Sub(x, y)
	case token.MUL:
		return z.Mul(x, y)
	case token.QUO:
		return z.Quo(x, y)
	case token.REM:
		return z.Rem(x, y)
	case token.AND:
		return z.And(x, y)
	case token.OR:
		return z.Or(x, y)
	case token.XOR:
		return z.Xor(x, y)
	case token.AND_NOT:
		return z.AndNot(x, y)
	case token.SHL:
		// The shift length must be uint, or untyped int and
		// convertible to uint.
		// TODO 32/64bit
		if y.BitLen() > 32 {
			panic("Excessive shift length")
		}
		return z.Lsh(x, uint(y.Int64()))
	case token.SHR:
		if y.BitLen() > 32 {
			panic("Excessive shift length")
		}
		return z.Rsh(x, uint(y.Int64()))
	case token.EQL:
		return x.Cmp(y) == 0
	case token.NEQ:
		return x.Cmp(y) != 0
	case token.LSS:
		return x.Cmp(y) < 0
	case token.LEQ:
		return x.Cmp(y) <= 0
	case token.GTR:
		return x.Cmp(y) > 0
	case token.GEQ:
		return x.Cmp(y) >= 0
	}
	panic("unreachable")
}
Exemple #27
0
// BinomialS computes the binomial coefficient C(n,k) using prime number
// sieve p.  BinomialS returns nil if p is too small.  Otherwise it leaves
// the result in z, replacing the existing value of z, and returning z.
func BinomialS(z *big.Int, p *sieve.Sieve, n, k uint) *big.Int {
	if uint64(n) > p.Lim {
		return nil
	}
	if k > n {
		return z.SetInt64(0)
	}
	if k > n/2 {
		k = n - k
	}
	if k < 3 {
		switch k {
		case 0:
			return z.SetInt64(1)
		case 1:
			return z.SetInt64(int64(n))
		case 2:
			var n1 big.Int
			return z.Rsh(z.Mul(z.SetInt64(int64(n)), n1.SetInt64(int64(n-1))), 1)
		}
	}
	rootN := uint64(xmath.FloorSqrt(n))
	var factors []uint64
	p.Iterate(2, rootN, func(p uint64) (terminate bool) {
		var r, nn, kk uint64 = 0, uint64(n), uint64(k)
		for nn > 0 {
			if nn%p < kk%p+r {
				r = 1
				factors = append(factors, p)
			} else {
				r = 0
			}
			nn /= p
			kk /= p
		}
		return
	})
	p.Iterate(rootN+1, uint64(n/2), func(p uint64) (terminate bool) {
		if uint64(n)%p < uint64(k)%p {
			factors = append(factors, p)
		}
		return
	})
	p.Iterate(uint64(n-k+1), uint64(n), func(p uint64) (terminate bool) {
		factors = append(factors, p)
		return
	})
	return xmath.Product(z, factors)
}
Exemple #28
0
// sqrtMod computes z = sqrt(x) % p.
func sqrtMod(x *big.Int, p *big.Int) (z *big.Int) {
	/* assert that p % 4 == 3 */
	if new(big.Int).Mod(p, big.NewInt(4)).Cmp(big.NewInt(3)) != 0 {
		panic("p is not equal to 3 mod 4!")
	}

	/* z = sqrt(x) % p = x^((p+1)/4) % p */

	/* e = (p+1)/4 */
	e := new(big.Int).Add(p, big.NewInt(1))
	e = e.Rsh(e, 2)

	z = expMod(x, e, p)
	return z
}
Exemple #29
0
func (enc *Encoding) decodeBlock(dst []byte, src []byte, baseOffset int) (int, int, error) {
	si := 0 // source index
	numGoodChars := 0
	res := new(big.Int)

	for i, b := range src {
		v := enc.decodeMap[b]
		si++

		if v == invalidByte {
			return 0, 0, CorruptInputError(i + baseOffset)
		}
		if v == skipByte {
			continue
		}

		numGoodChars++
		res.Mul(res, enc.baseBig)
		res.Add(res, big.NewInt(int64(v)))

		if numGoodChars == enc.baseXBlockLen {
			break
		}
	}

	if !enc.IsValidEncodingLength(numGoodChars) {
		return 0, 0, ErrInvalidEncodingLength
	}

	paddedLen := enc.DecodedLen(numGoodChars)

	// Compute the corresponding right shift, to move the number
	// over to its natural base.
	res = res.Rsh(res, enc.extraBits(paddedLen, numGoodChars))

	// Use big-endian representation (the default with Go's library)
	raw := res.Bytes()
	p := 0
	if len(raw) < paddedLen {
		p = paddedLen - len(raw)
		copy(dst, bytes.Repeat([]byte{0}, p))
	}
	copy(dst[p:paddedLen], raw)
	return paddedLen, si, nil
}
Exemple #30
0
// Mod sets mod to n % Mexp and returns mod. It panics for exp == 0 || exp >=
// math.MaxInt32 || n < 0.
func Mod(mod, n *big.Int, exp uint32) *big.Int {
	if exp == 0 || exp >= math.MaxInt32 || n.Sign() < 0 {
		panic(0)
	}

	m := New(exp)
	mod.Set(n)
	var x big.Int
	for mod.BitLen() > int(exp) {
		x.Set(mod)
		x.Rsh(&x, uint(exp))
		mod.And(mod, m)
		mod.Add(mod, &x)
	}
	if mod.BitLen() == int(exp) && mod.Cmp(m) == 0 {
		mod.SetInt64(0)
	}
	return mod
}