1 files changed, 370 insertions, 0 deletions
diff --git a/vendor/github.com/remyoudompheng/bigfft/fft.go b/vendor/github.com/remyoudompheng/bigfft/fft.go
new file mode 100644
index 0000000..2d4c1e7
--- /dev/null
+++ b/vendor/github.com/remyoudompheng/bigfft/fft.go
@@ -0,0 +1,370 @@
+// Package bigfft implements multiplication of big.Int using FFT.
+//
+// The implementation is based on the Schönhage-Strassen method
+// using integer FFT modulo 2^n+1.
+package bigfft
+
+import (
+	"math/big"
+	"unsafe"
+)
+
+const _W = int(unsafe.Sizeof(big.Word(0)) * 8)
+
+type nat []big.Word
+
+func (n nat) String() string {
+	v := new(big.Int)
+	v.SetBits(n)
+	return v.String()
+}
+
+// fftThreshold is the size (in words) above which FFT is used over
+// Karatsuba from math/big.
+//
+// TestCalibrate seems to indicate a threshold of 60kbits on 32-bit
+// arches and 110kbits on 64-bit arches.
+var fftThreshold = 1800
+
+// Mul computes the product x*y and returns z.
+// It can be used instead of the Mul method of
+// *big.Int from math/big package.
+func Mul(x, y *big.Int) *big.Int {
+	xwords := len(x.Bits())
+	ywords := len(y.Bits())
+	if xwords > fftThreshold && ywords > fftThreshold {
+		return mulFFT(x, y)
+	}
+	return new(big.Int).Mul(x, y)
+}
+
+func mulFFT(x, y *big.Int) *big.Int {
+	var xb, yb nat = x.Bits(), y.Bits()
+	zb := fftmul(xb, yb)
+	z := new(big.Int)
+	z.SetBits(zb)
+	if x.Sign()*y.Sign() < 0 {
+		z.Neg(z)
+	}
+	return z
+}
+
+// A FFT size of K=1<<k is adequate when K is about 2*sqrt(N) where
+// N = x.Bitlen() + y.Bitlen().
+
+func fftmul(x, y nat) nat {
+	k, m := fftSize(x, y)
+	xp := polyFromNat(x, k, m)
+	yp := polyFromNat(y, k, m)
+	rp := xp.Mul(&yp)
+	return rp.Int()
+}
+
+// fftSizeThreshold[i] is the maximal size (in bits) where we should use
+// fft size i.
+var fftSizeThreshold = [...]int64{0, 0, 0,
+	4 << 10, 8 << 10, 16 << 10, // 5 
+	32 << 10, 64 << 10, 1 << 18, 1 << 20, 3 << 20, // 10
+	8 << 20, 30 << 20, 100 << 20, 300 << 20, 600 << 20,
+}
+
+// returns the FFT length k, m the number of words per chunk
+// such that m << k is larger than the number of words
+// in x*y.
+func fftSize(x, y nat) (k uint, m int) {
+	words := len(x) + len(y)
+	bits := int64(words) * int64(_W)
+	k = uint(len(fftSizeThreshold))
+	for i := range fftSizeThreshold {
+		if fftSizeThreshold[i] > bits {
+			k = uint(i)
+			break
+		}
+	}
+	// The 1<<k chunks of m words must have N bits so that
+	// 2^N-1 is larger than x*y. That is, m<<k > words
+	m = words>>k + 1
+	return
+}
+
+// valueSize returns the length (in words) to use for polynomial
+// coefficients, to compute a correct product of polynomials P*Q
+// where deg(P*Q) < K (== 1<<k) and where coefficients of P and Q are
+// less than b^m (== 1 << (m*_W)).
+// The chosen length (in bits) must be a multiple of 1 << (k-extra).
+func valueSize(k uint, m int, extra uint) int {
+	// The coefficients of P*Q are less than b^(2m)*K
+	// so we need W * valueSize >= 2*m*W+K
+	n := 2*m*_W + int(k) // necessary bits
+	K := 1 << (k - extra)
+	if K < _W {
+		K = _W
+	}
+	n = ((n / K) + 1) * K // round to a multiple of K
+	return n / _W
+}
+
+// poly represents an integer via a polynomial in Z[x]/(x^K+1)
+// where K is the FFT length and b^m is the computation basis 1<<(m*_W).
+// If P = a[0] + a[1] x + ... a[n] x^(K-1), the associated natural number
+// is P(b^m).
+type poly struct {
+	k uint  // k is such that K = 1<<k.
+	m int   // the m such that P(b^m) is the original number.
+	a []nat // a slice of at most K m-word coefficients.
+}
+
+// polyFromNat slices the number x into a polynomial
+// with 1<<k coefficients made of m words.
+func polyFromNat(x nat, k uint, m int) poly {
+	p := poly{k: k, m: m}
+	length := len(x)/m + 1
+	p.a = make([]nat, length)
+	for i := range p.a {
+		if len(x) < m {
+			p.a[i] = make(nat, m)
+			copy(p.a[i], x)
+			break
+		}
+		p.a[i] = x[:m]
+		x = x[m:]
+	}
+	return p
+}
+
+// Int evaluates back a poly to its integer value.
+func (p *poly) Int() nat {
+	length := len(p.a)*p.m + 1
+	if na := len(p.a); na > 0 {
+		length += len(p.a[na-1])
+	}
+	n := make(nat, length)
+	m := p.m
+	np := n
+	for i := range p.a {
+		l := len(p.a[i])
+		c := addVV(np[:l], np[:l], p.a[i])
+		if np[l] < ^big.Word(0) {
+			np[l] += c
+		} else {
+			addVW(np[l:], np[l:], c)
+		}
+		np = np[m:]
+	}
+	n = trim(n)
+	return n
+}
+
+func trim(n nat) nat {
+	for i := range n {
+		if n[len(n)-1-i] != 0 {
+			return n[:len(n)-i]
+		}
+	}
+	return nil
+}
+
+// Mul multiplies p and q modulo X^K-1, where K = 1<<p.k.
+// The product is done via a Fourier transform.
+func (p *poly) Mul(q *poly) poly {
+	// extra=2 because:
+	// * some power of 2 is a K-th root of unity when n is a multiple of K/2.
+	// * 2 itself is a square (see fermat.ShiftHalf)
+	n := valueSize(p.k, p.m, 2)
+
+	pv, qv := p.Transform(n), q.Transform(n)
+	rv := pv.Mul(&qv)
+	r := rv.InvTransform()
+	r.m = p.m
+	return r
+}
+
+// A polValues represents the value of a poly at the powers of a
+// K-th root of unity θ=2^(l/2) in Z/(b^n+1)Z, where b^n = 2^(K/4*l).
+type polValues struct {
+	k      uint     // k is such that K = 1<<k.
+	n      int      // the length of coefficients, n*_W a multiple of K/4.
+	values []fermat // a slice of K (n+1)-word values
+}
+
+// Transform evaluates p at θ^i for i = 0...K-1, where
+// θ is a K-th primitive root of unity in Z/(b^n+1)Z.
+func (p *poly) Transform(n int) polValues {
+	k := p.k
+	inputbits := make([]big.Word, (n+1)<<k)
+	input := make([]fermat, 1<<k)
+	// Now computed q(ω^i) for i = 0 ... K-1
+	valbits := make([]big.Word, (n+1)<<k)
+	values := make([]fermat, 1<<k)
+	for i := range values {
+		input[i] = inputbits[i*(n+1) : (i+1)*(n+1)]
+		if i < len(p.a) {
+			copy(input[i], p.a[i])
+		}
+		values[i] = fermat(valbits[i*(n+1) : (i+1)*(n+1)])
+	}
+	fourier(values, input, false, n, k)
+	return polValues{k, n, values}
+}
+
+// InvTransform reconstructs p (modulo X^K - 1) from its
+// values at θ^i for i = 0..K-1.
+func (v *polValues) InvTransform() poly {
+	k, n := v.k, v.n
+
+	// Perform an inverse Fourier transform to recover p.
+	pbits := make([]big.Word, (n+1)<<k)
+	p := make([]fermat, 1<<k)
+	for i := range p {
+		p[i] = fermat(pbits[i*(n+1) : (i+1)*(n+1)])
+	}
+	fourier(p, v.values, true, n, k)
+	// Divide by K, and untwist q to recover p.
+	u := make(fermat, n+1)
+	a := make([]nat, 1<<k)
+	for i := range p {
+		u.Shift(p[i], -int(k))
+		copy(p[i], u)
+		a[i] = nat(p[i])
+	}
+	return poly{k: k, m: 0, a: a}
+}
+
+// NTransform evaluates p at θω^i for i = 0...K-1, where
+// θ is a (2K)-th primitive root of unity in Z/(b^n+1)Z
+// and ω = θ².
+func (p *poly) NTransform(n int) polValues {
+	k := p.k
+	if len(p.a) >= 1<<k {
+		panic("Transform: len(p.a) >= 1<<k")
+	}
+	// θ is represented as a shift.
+	θshift := (n * _W) >> k
+	// p(x) = a_0 + a_1 x + ... + a_{K-1} x^(K-1)
+	// p(θx) = q(x) where
+	// q(x) = a_0 + θa_1 x + ... + θ^(K-1) a_{K-1} x^(K-1)
+	//
+	// Twist p by θ to obtain q.
+	tbits := make([]big.Word, (n+1)<<k)
+	twisted := make([]fermat, 1<<k)
+	src := make(fermat, n+1)
+	for i := range twisted {
+		twisted[i] = fermat(tbits[i*(n+1) : (i+1)*(n+1)])
+		if i < len(p.a) {
+			for i := range src {
+				src[i] = 0
+			}
+			copy(src, p.a[i])
+			twisted[i].Shift(src, θshift*i)
+		}
+	}
+
+	// Now computed q(ω^i) for i = 0 ... K-1
+	valbits := make([]big.Word, (n+1)<<k)
+	values := make([]fermat, 1<<k)
+	for i := range values {
+		values[i] = fermat(valbits[i*(n+1) : (i+1)*(n+1)])
+	}
+	fourier(values, twisted, false, n, k)
+	return polValues{k, n, values}
+}
+
+// InvTransform reconstructs a polynomial from its values at
+// roots of x^K+1. The m field of the returned polynomial
+// is unspecified.
+func (v *polValues) InvNTransform() poly {
+	k := v.k
+	n := v.n
+	θshift := (n * _W) >> k
+
+	// Perform an inverse Fourier transform to recover q.
+	qbits := make([]big.Word, (n+1)<<k)
+	q := make([]fermat, 1<<k)
+	for i := range q {
+		q[i] = fermat(qbits[i*(n+1) : (i+1)*(n+1)])
+	}
+	fourier(q, v.values, true, n, k)
+
+	// Divide by K, and untwist q to recover p.
+	u := make(fermat, n+1)
+	a := make([]nat, 1<<k)
+	for i := range q {
+		u.Shift(q[i], -int(k)-i*θshift)
+		copy(q[i], u)
+		a[i] = nat(q[i])
+	}
+	return poly{k: k, m: 0, a: a}
+}
+
+// fourier performs an unnormalized Fourier transform
+// of src, a length 1<<k vector of numbers modulo b^n+1
+// where b = 1<<_W.
+func fourier(dst []fermat, src []fermat, backward bool, n int, k uint) {
+	var rec func(dst, src []fermat, size uint)
+	tmp := make(fermat, n+1)  // pre-allocate temporary variables.
+	tmp2 := make(fermat, n+1) // pre-allocate temporary variables.
+
+	// The recursion function of the FFT.
+	// The root of unity used in the transform is ω=1<<(ω2shift/2).
+	// The source array may use shifted indices (i.e. the i-th
+	// element is src[i << idxShift]).
+	rec = func(dst, src []fermat, size uint) {
+		idxShift := k - size
+		ω2shift := (4 * n * _W) >> size
+		if backward {
+			ω2shift = -ω2shift
+		}
+
+		// Easy cases.
+		if len(src[0]) != n+1 || len(dst[0]) != n+1 {
+			panic("len(src[0]) != n+1 || len(dst[0]) != n+1")
+		}
+		switch size {
+		case 0:
+			copy(dst[0], src[0])
+			return
+		case 1:
+			dst[0].Add(src[0], src[1<<idxShift]) // dst[0] = src[0] + src[1]
+			dst[1].Sub(src[0], src[1<<idxShift]) // dst[1] = src[0] - src[1]
+			return
+		}
+
+		// Let P(x) = src[0] + src[1<<idxShift] * x + ... + src[K-1 << idxShift] * x^(K-1)
+		// The P(x) = Q1(x²) + x*Q2(x²)
+		// where Q1's coefficients are src with indices shifted by 1
+		// where Q2's coefficients are src[1<<idxShift:] with indices shifted by 1
+
+		// Split destination vectors in halves.
+		dst1 := dst[:1<<(size-1)]
+		dst2 := dst[1<<(size-1):]
+		// Transform Q1 and Q2 in the halves.
+		rec(dst1, src, size-1)
+		rec(dst2, src[1<<idxShift:], size-1)
+
+		// Reconstruct P's transform from transforms of Q1 and Q2.
+		// dst[i]            is dst1[i] + ω^i * dst2[i]
+		// dst[i + 1<<(k-1)] is dst1[i] + ω^(i+K/2) * dst2[i]
+		//
+		for i := range dst1 {
+			tmp.ShiftHalf(dst2[i], i*ω2shift, tmp2) // ω^i * dst2[i]
+			dst2[i].Sub(dst1[i], tmp)
+			dst1[i].Add(dst1[i], tmp)
+		}
+	}
+	rec(dst, src, k)
+}
+
+// Mul returns the pointwise product of p and q.
+func (p *polValues) Mul(q *polValues) (r polValues) {
+	n := p.n
+	r.k, r.n = p.k, p.n
+	r.values = make([]fermat, len(p.values))
+	bits := make([]big.Word, len(p.values)*(n+1))
+	buf := make(fermat, 8*n)
+	for i := range r.values {
+		r.values[i] = bits[i*(n+1) : (i+1)*(n+1)]
+		z := buf.Mul(p.values[i], q.values[i])
+		copy(r.values[i], z)
+	}
+	return
+}