/usr/share/gocode/src/github.com/remyoudompheng/bigfft/fft.go is in golang-github-remyoudompheng-bigfft-dev 0.0~git20130913.0.a8e77dd-1.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 | // 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 smallest multiple of 1<<k greater than
// 2*m*_W + k, that is also a multiple of _W. If extra > 0, the
// returned value is only required to be a multiple of 1<<(k-extra)
func valueSize(k uint, m int, extra uint) int {
n := 2*m*_W + int(k)
K := 1 << (k - extra)
if K < _W {
K = _W
}
n = ((n / K) + 1) * 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 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 odd powers of a
// (2K)-th root of unity θ=2^l in Z/(b^n+1)Z, where b^n = 2^Kl.
type polValues struct {
k uint // k is such that K = 1<<k.
n int // the length of coefficients, n*_W a multiple of 1<<k.
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
}
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