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#136: Start working on database persistence

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This commit is contained in:
TwinProduction
2021-07-12 00:56:30 -04:00
committed by Chris
parent e6335da94f
commit bd1eb7c61b
657 changed files with 2190821 additions and 82 deletions
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27
vendor/github.com/remyoudompheng/bigfft/LICENSE generated vendored Normal file
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Copyright (c) 2012 The Go Authors. All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
* Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
* Redistributions in binary form must reproduce the above
copyright notice, this list of conditions and the following disclaimer
in the documentation and/or other materials provided with the
distribution.
* Neither the name of Google Inc. nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

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vendor/github.com/remyoudompheng/bigfft/README generated vendored Normal file
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Benchmarking math/big vs. bigfft
Number size old ns/op new ns/op delta
1kb 1599 1640 +2.56%
10kb 61533 62170 +1.04%
50kb 833693 831051 -0.32%
100kb 2567995 2693864 +4.90%
1Mb 105237800 28446400 -72.97%
5Mb 1272947000 168554600 -86.76%
10Mb 3834354000 405120200 -89.43%
20Mb 11514488000 845081600 -92.66%
50Mb 49199945000 2893950000 -94.12%
100Mb 147599836000 5921594000 -95.99%
Benchmarking GMP vs bigfft
Number size GMP ns/op Go ns/op delta
1kb 536 1500 +179.85%
10kb 26669 50777 +90.40%
50kb 252270 658534 +161.04%
100kb 686813 2127534 +209.77%
1Mb 12100000 22391830 +85.06%
5Mb 111731843 133550600 +19.53%
10Mb 212314000 318595800 +50.06%
20Mb 490196000 671512800 +36.99%
50Mb 1280000000 2451476000 +91.52%
100Mb 2673000000 5228991000 +95.62%
Benchmarks were run on a Core 2 Quad Q8200 (2.33GHz).
FFT is enabled when input numbers are over 200kbits.
Scanning large decimal number from strings.
(math/big [n^2 complexity] vs bigfft [n^1.6 complexity], Core i5-4590)
Digits old ns/op new ns/op delta
1e3 9995 10876 +8.81%
1e4 175356 243806 +39.03%
1e5 9427422 6780545 -28.08%
1e6 1776707489 144867502 -91.85%
2e6 6865499995 346540778 -94.95%
5e6 42641034189 1069878799 -97.49%
10e6 151975273589 2693328580 -98.23%

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vendor/github.com/remyoudompheng/bigfft/arith_386.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
JMP math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
TEXT ·subVV(SB),NOSPLIT,$0
JMP math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
JMP math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
TEXT ·subVW(SB),NOSPLIT,$0
JMP math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
JMP math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
JMP math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
JMP math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
JMP math∕big·addMulVVW(SB)

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vendor/github.com/remyoudompheng/bigfft/arith_amd64.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
JMP math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
// (same as addVV except for SBBQ instead of ADCQ and label names)
TEXT ·subVV(SB),NOSPLIT,$0
JMP math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
JMP math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
// (same as addVW except for SUBQ/SBBQ instead of ADDQ/ADCQ and label names)
TEXT ·subVW(SB),NOSPLIT,$0
JMP math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
JMP math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
JMP math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
JMP math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
JMP math∕big·addMulVVW(SB)

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vendor/github.com/remyoudompheng/bigfft/arith_arm.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
B math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
TEXT ·subVV(SB),NOSPLIT,$0
B math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
B math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
TEXT ·subVW(SB),NOSPLIT,$0
B math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
B math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
B math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
B math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
B math∕big·addMulVVW(SB)

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vendor/github.com/remyoudompheng/bigfft/arith_arm64.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
B math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
TEXT ·subVV(SB),NOSPLIT,$0
B math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
B math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
TEXT ·subVW(SB),NOSPLIT,$0
B math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
B math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
B math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
B math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
B math∕big·addMulVVW(SB)

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vendor/github.com/remyoudompheng/bigfft/arith_decl.go generated vendored Normal file
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// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package bigfft
import . "math/big"
// implemented in arith_$GOARCH.s
func addVV(z, x, y []Word) (c Word)
func subVV(z, x, y []Word) (c Word)
func addVW(z, x []Word, y Word) (c Word)
func subVW(z, x []Word, y Word) (c Word)
func shlVU(z, x []Word, s uint) (c Word)
func mulAddVWW(z, x []Word, y, r Word) (c Word)
func addMulVVW(z, x []Word, y Word) (c Word)

40
vendor/github.com/remyoudompheng/bigfft/arith_mips64x.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
// +build mips64 mips64le
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
JMP math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
// (same as addVV except for SBBQ instead of ADCQ and label names)
TEXT ·subVV(SB),NOSPLIT,$0
JMP math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
JMP math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
// (same as addVW except for SUBQ/SBBQ instead of ADDQ/ADCQ and label names)
TEXT ·subVW(SB),NOSPLIT,$0
JMP math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
JMP math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
JMP math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
JMP math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
JMP math∕big·addMulVVW(SB)

40
vendor/github.com/remyoudompheng/bigfft/arith_mipsx.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
// +build mips mipsle
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
JMP math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
// (same as addVV except for SBBQ instead of ADCQ and label names)
TEXT ·subVV(SB),NOSPLIT,$0
JMP math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
JMP math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
// (same as addVW except for SUBQ/SBBQ instead of ADDQ/ADCQ and label names)
TEXT ·subVW(SB),NOSPLIT,$0
JMP math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
JMP math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
JMP math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
JMP math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
JMP math∕big·addMulVVW(SB)

38
vendor/github.com/remyoudompheng/bigfft/arith_ppc64x.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
// +build ppc64 ppc64le
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
BR math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
TEXT ·subVV(SB),NOSPLIT,$0
BR math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
BR math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
TEXT ·subVW(SB),NOSPLIT,$0
BR math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
BR math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
BR math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
BR math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
BR math∕big·addMulVVW(SB)

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vendor/github.com/remyoudompheng/bigfft/arith_s390x.s generated vendored Normal file
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// Trampolines to math/big assembly implementations.
#include "textflag.h"
// func addVV(z, x, y []Word) (c Word)
TEXT ·addVV(SB),NOSPLIT,$0
BR math∕big·addVV(SB)
// func subVV(z, x, y []Word) (c Word)
TEXT ·subVV(SB),NOSPLIT,$0
BR math∕big·subVV(SB)
// func addVW(z, x []Word, y Word) (c Word)
TEXT ·addVW(SB),NOSPLIT,$0
BR math∕big·addVW(SB)
// func subVW(z, x []Word, y Word) (c Word)
TEXT ·subVW(SB),NOSPLIT,$0
BR math∕big·subVW(SB)
// func shlVU(z, x []Word, s uint) (c Word)
TEXT ·shlVU(SB),NOSPLIT,$0
BR math∕big·shlVU(SB)
// func shrVU(z, x []Word, s uint) (c Word)
TEXT ·shrVU(SB),NOSPLIT,$0
BR math∕big·shrVU(SB)
// func mulAddVWW(z, x []Word, y, r Word) (c Word)
TEXT ·mulAddVWW(SB),NOSPLIT,$0
BR math∕big·mulAddVWW(SB)
// func addMulVVW(z, x []Word, y Word) (c Word)
TEXT ·addMulVVW(SB),NOSPLIT,$0
BR math∕big·addMulVVW(SB)

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vendor/github.com/remyoudompheng/bigfft/fermat.go generated vendored Normal file
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package bigfft
import (
"math/big"
)
// Arithmetic modulo 2^n+1.
// A fermat of length w+1 represents a number modulo 2^(w*_W) + 1. The last
// word is zero or one. A number has at most two representatives satisfying the
// 0-1 last word constraint.
type fermat nat
func (n fermat) String() string { return nat(n).String() }
func (z fermat) norm() {
n := len(z) - 1
c := z[n]
if c == 0 {
return
}
if z[0] >= c {
z[n] = 0
z[0] -= c
return
}
// z[0] < z[n].
subVW(z, z, c) // Substract c
if c > 1 {
z[n] -= c - 1
c = 1
}
// Add back c.
if z[n] == 1 {
z[n] = 0
return
} else {
addVW(z, z, 1)
}
}
// Shift computes (x << k) mod (2^n+1).
func (z fermat) Shift(x fermat, k int) {
if len(z) != len(x) {
panic("len(z) != len(x) in Shift")
}
n := len(x) - 1
// Shift by n*_W is taking the opposite.
k %= 2 * n * _W
if k < 0 {
k += 2 * n * _W
}
neg := false
if k >= n*_W {
k -= n * _W
neg = true
}
kw, kb := k/_W, k%_W
z[n] = 1 // Add (-1)
if !neg {
for i := 0; i < kw; i++ {
z[i] = 0
}
// Shift left by kw words.
// x = a·2^(n-k) + b
// x<<k = (b<<k) - a
copy(z[kw:], x[:n-kw])
b := subVV(z[:kw+1], z[:kw+1], x[n-kw:])
if z[kw+1] > 0 {
z[kw+1] -= b
} else {
subVW(z[kw+1:], z[kw+1:], b)
}
} else {
for i := kw + 1; i < n; i++ {
z[i] = 0
}
// Shift left and negate, by kw words.
copy(z[:kw+1], x[n-kw:n+1]) // z_low = x_high
b := subVV(z[kw:n], z[kw:n], x[:n-kw]) // z_high -= x_low
z[n] -= b
}
// Add back 1.
if z[n] > 0 {
z[n]--
} else if z[0] < ^big.Word(0) {
z[0]++
} else {
addVW(z, z, 1)
}
// Shift left by kb bits
shlVU(z, z, uint(kb))
z.norm()
}
// ShiftHalf shifts x by k/2 bits the left. Shifting by 1/2 bit
// is multiplication by sqrt(2) mod 2^n+1 which is 2^(3n/4) - 2^(n/4).
// A temporary buffer must be provided in tmp.
func (z fermat) ShiftHalf(x fermat, k int, tmp fermat) {
n := len(z) - 1
if k%2 == 0 {
z.Shift(x, k/2)
return
}
u := (k - 1) / 2
a := u + (3*_W/4)*n
b := u + (_W/4)*n
z.Shift(x, a)
tmp.Shift(x, b)
z.Sub(z, tmp)
}
// Add computes addition mod 2^n+1.
func (z fermat) Add(x, y fermat) fermat {
if len(z) != len(x) {
panic("Add: len(z) != len(x)")
}
addVV(z, x, y) // there cannot be a carry here.
z.norm()
return z
}
// Sub computes substraction mod 2^n+1.
func (z fermat) Sub(x, y fermat) fermat {
if len(z) != len(x) {
panic("Add: len(z) != len(x)")
}
n := len(y) - 1
b := subVV(z[:n], x[:n], y[:n])
b += y[n]
// If b > 0, we need to subtract b<<n, which is the same as adding b.
z[n] = x[n]
if z[0] <= ^big.Word(0)-b {
z[0] += b
} else {
addVW(z, z, b)
}
z.norm()
return z
}
func (z fermat) Mul(x, y fermat) fermat {
if len(x) != len(y) {
panic("Mul: len(x) != len(y)")
}
n := len(x) - 1
if n < 30 {
z = z[:2*n+2]
basicMul(z, x, y)
z = z[:2*n+1]
} else {
var xi, yi, zi big.Int
xi.SetBits(x)
yi.SetBits(y)
zi.SetBits(z)
zb := zi.Mul(&xi, &yi).Bits()
if len(zb) <= n {
// Short product.
copy(z, zb)
for i := len(zb); i < len(z); i++ {
z[i] = 0
}
return z
}
z = zb
}
// len(z) is at most 2n+1.
if len(z) > 2*n+1 {
panic("len(z) > 2n+1")
}
// We now have
// z = z[:n] + 1<<(n*W) * z[n:2n+1]
// which normalizes to:
// z = z[:n] - z[n:2n] + z[2n]
c1 := big.Word(0)
if len(z) > 2*n {
c1 = addVW(z[:n], z[:n], z[2*n])
}
c2 := big.Word(0)
if len(z) >= 2*n {
c2 = subVV(z[:n], z[:n], z[n:2*n])
} else {
m := len(z) - n
c2 = subVV(z[:m], z[:m], z[n:])
c2 = subVW(z[m:n], z[m:n], c2)
}
// Restore carries.
// Substracting z[n] -= c2 is the same
// as z[0] += c2
z = z[:n+1]
z[n] = c1
c := addVW(z, z, c2)
if c != 0 {
panic("impossible")
}
z.norm()
return z
}
// copied from math/big
//
// basicMul multiplies x and y and leaves the result in z.
// The (non-normalized) result is placed in z[0 : len(x) + len(y)].
func basicMul(z, x, y fermat) {
// initialize z
for i := 0; i < len(z); i++ {
z[i] = 0
}
for i, d := range y {
if d != 0 {
z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
}
}
}

370
vendor/github.com/remyoudompheng/bigfft/fft.go generated vendored Normal file
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// 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
}

3
vendor/github.com/remyoudompheng/bigfft/go.mod generated vendored Normal file
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module github.com/remyoudompheng/bigfft
go 1.12

70
vendor/github.com/remyoudompheng/bigfft/scan.go generated vendored Normal file
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package bigfft
import (
"math/big"
)
// FromDecimalString converts the base 10 string
// representation of a natural (non-negative) number
// into a *big.Int.
// Its asymptotic complexity is less than quadratic.
func FromDecimalString(s string) *big.Int {
var sc scanner
z := new(big.Int)
sc.scan(z, s)
return z
}
type scanner struct {
// powers[i] is 10^(2^i * quadraticScanThreshold).
powers []*big.Int
}
func (s *scanner) chunkSize(size int) (int, *big.Int) {
if size <= quadraticScanThreshold {
panic("size < quadraticScanThreshold")
}
pow := uint(0)
for n := size; n > quadraticScanThreshold; n /= 2 {
pow++
}
// threshold * 2^(pow-1) <= size < threshold * 2^pow
return quadraticScanThreshold << (pow - 1), s.power(pow - 1)
}
func (s *scanner) power(k uint) *big.Int {
for i := len(s.powers); i <= int(k); i++ {
z := new(big.Int)
if i == 0 {
if quadraticScanThreshold%14 != 0 {
panic("quadraticScanThreshold % 14 != 0")
}
z.Exp(big.NewInt(1e14), big.NewInt(quadraticScanThreshold/14), nil)
} else {
z.Mul(s.powers[i-1], s.powers[i-1])
}
s.powers = append(s.powers, z)
}
return s.powers[k]
}
func (s *scanner) scan(z *big.Int, str string) {
if len(str) <= quadraticScanThreshold {
z.SetString(str, 10)
return
}
sz, pow := s.chunkSize(len(str))
// Scan the left half.
s.scan(z, str[:len(str)-sz])
// FIXME: reuse temporaries.
left := Mul(z, pow)
// Scan the right half
s.scan(z, str[len(str)-sz:])
z.Add(z, left)
}
// quadraticScanThreshold is the number of digits
// below which big.Int.SetString is more efficient
// than subquadratic algorithms.
// 1232 digits fit in 4096 bits.
const quadraticScanThreshold = 1232