aboutsummaryrefslogtreecommitdiffstats
path: root/vendor/github.com/remyoudompheng/bigfft/fft.go
blob: 2d4c1e7a98fd249dbe3a4a7862e76a330799f0f8 (plain) (blame)
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
367
368
369
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
}