// Copyright 2009 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.
// This file implements unsigned multi-precision integers (natural
// numbers). They are the building blocks for the implementation
// of signed integers, rationals, and floating-point numbers.
//
// Caution: This implementation relies on the function "alias"
// which assumes that (nat) slice capacities are never
// changed (no 3-operand slice expressions). If that
// changes, alias needs to be updated for correctness.
package big
import (
"encoding/binary"
"math/bits"
"math/rand"
"sync"
)
// An unsigned integer x of the form
//
// x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0]
//
// with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n,
// with the digits x[i] as the slice elements.
//
// A number is normalized if the slice contains no leading 0 digits.
// During arithmetic operations, denormalized values may occur but are
// always normalized before returning the final result. The normalized
// representation of 0 is the empty or nil slice (length = 0).
//
type nat []Word
var (
natOne = nat{1}
natTwo = nat{2}
natFive = nat{5}
natTen = nat{10}
)
func (z nat) clear() {
for i := range z {
z[i] = 0
}
}
func (z nat) norm() nat {
i := len(z)
for i > 0 && z[i-1] == 0 {
i--
}
return z[0:i]
}
func (z nat) make(n int) nat {
if n <= cap(z) {
return z[:n] // reuse z
}
if n == 1 {
// Most nats start small and stay that way; don't over-allocate.
return make(nat, 1)
}
// Choosing a good value for e has significant performance impact
// because it increases the chance that a value can be reused.
const e = 4 // extra capacity
return make(nat, n, n+e)
}
func (z nat) setWord(x Word) nat {
if x == 0 {
return z[:0]
}
z = z.make(1)
z[0] = x
return z
}
func (z nat) setUint64(x uint64) nat {
// single-word value
if w := Word(x); uint64(w) == x {
return z.setWord(w)
}
// 2-word value
z = z.make(2)
z[1] = Word(x >> 32)
z[0] = Word(x)
return z
}
func (z nat) set(x nat) nat {
z = z.make(len(x))
copy(z, x)
return z
}
func (z nat) add(x, y nat) nat {
m := len(x)
n := len(y)
switch {
case m < n:
return z.add(y, x)
case m == 0:
// n == 0 because m >= n; result is 0
return z[:0]
case n == 0:
// result is x
return z.set(x)
}
// m > 0
z = z.make(m + 1)
c := addVV(z[0:n], x, y)
if m > n {
c = addVW(z[n:m], x[n:], c)
}
z[m] = c
return z.norm()
}
func (z nat) sub(x, y nat) nat {
m := len(x)
n := len(y)
switch {
case m < n:
panic("underflow")
case m == 0:
// n == 0 because m >= n; result is 0
return z[:0]
case n == 0:
// result is x
return z.set(x)
}
// m > 0
z = z.make(m)
c := subVV(z[0:n], x, y)
if m > n {
c = subVW(z[n:], x[n:], c)
}
if c != 0 {
panic("underflow")
}
return z.norm()
}
func (x nat) cmp(y nat) (r int) {
m := len(x)
n := len(y)
if m != n || m == 0 {
switch {
case m < n:
r = -1
case m > n:
r = 1
}
return
}
i := m - 1
for i > 0 && x[i] == y[i] {
i--
}
switch {
case x[i] < y[i]:
r = -1
case x[i] > y[i]:
r = 1
}
return
}
func (z nat) mulAddWW(x nat, y, r Word) nat {
m := len(x)
if m == 0 || y == 0 {
return z.setWord(r) // result is r
}
// m > 0
z = z.make(m + 1)
z[m] = mulAddVWW(z[0:m], x, y, r)
return z.norm()
}
// 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 nat) {
z[0 : len(x)+len(y)].clear() // initialize z
for i, d := range y {
if d != 0 {
z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
}
}
}
// montgomery computes z mod m = x*y*2**(-n*_W) mod m,
// assuming k = -1/m mod 2**_W.
// z is used for storing the result which is returned;
// z must not alias x, y or m.
// See Gueron, "Efficient Software Implementations of Modular Exponentiation".
// https://eprint.iacr.org/2011/239.pdf
// In the terminology of that paper, this is an "Almost Montgomery Multiplication":
// x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result
// z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m.
func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
// This code assumes x, y, m are all the same length, n.
// (required by addMulVVW and the for loop).
// It also assumes that x, y are already reduced mod m,
// or else the result will not be properly reduced.
if len(x) != n || len(y) != n || len(m) != n {
panic("math/big: mismatched montgomery number lengths")
}
z = z.make(n * 2)
z.clear()
var c Word
for i := 0; i < n; i++ {
d := y[i]
c2 := addMulVVW(z[i:n+i], x, d)
t := z[i] * k
c3 := addMulVVW(z[i:n+i], m, t)
cx := c + c2
cy := cx + c3
z[n+i] = cy
if cx < c2 || cy < c3 {
c = 1
} else {
c = 0
}
}
if c != 0 {
subVV(z[:n], z[n:], m)
} else {
copy(z[:n], z[n:])
}
return z[:n]
}
// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
// Factored out for readability - do not use outside karatsuba.
func karatsubaAdd(z, x nat, n int) {
if c := addVV(z[0:n], z, x); c != 0 {
addVW(z[n:n+n>>1], z[n:], c)
}
}
// Like karatsubaAdd, but does subtract.
func karatsubaSub(z, x nat, n int) {
if c := subVV(z[0:n], z, x); c != 0 {
subVW(z[n:n+n>>1], z[n:], c)
}
}
// Operands that are shorter than karatsubaThreshold are multiplied using
// "grade school" multiplication; for longer operands the Karatsuba algorithm
// is used.
var karatsubaThreshold = 40 // computed by calibrate_test.go
// karatsuba multiplies x and y and leaves the result in z.
// Both x and y must have the same length n and n must be a
// power of 2. The result vector z must have len(z) >= 6*n.
// The (non-normalized) result is placed in z[0 : 2*n].
func karatsuba(z, x, y nat) {
n := len(y)
// Switch to basic multiplication if numbers are odd or small.
// (n is always even if karatsubaThreshold is even, but be
// conservative)
if n&1 != 0 || n < karatsubaThreshold || n < 2 {
basicMul(z, x, y)
return
}
// n&1 == 0 && n >= karatsubaThreshold && n >= 2
// Karatsuba multiplication is based on the observation that
// for two numbers x and y with:
//
// x = x1*b + x0
// y = y1*b + y0
//
// the product x*y can be obtained with 3 products z2, z1, z0
// instead of 4:
//
// x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
// = z2*b*b + z1*b + z0
//
// with:
//
// xd = x1 - x0
// yd = y0 - y1
//
// z1 = xd*yd + z2 + z0
// = (x1-x0)*(y0 - y1) + z2 + z0
// = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
// = x1*y0 - z2 - z0 + x0*y1 + z2 + z0
// = x1*y0 + x0*y1
// split x, y into "digits"
n2 := n >> 1 // n2 >= 1
x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
// z is used for the result and temporary storage:
//
// 6*n 5*n 4*n 3*n 2*n 1*n 0*n
// z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
//
// For each recursive call of karatsuba, an unused slice of
// z is passed in that has (at least) half the length of the
// caller's z.
// compute z0 and z2 with the result "in place" in z
karatsuba(z, x0, y0) // z0 = x0*y0
karatsuba(z[n:], x1, y1) // z2 = x1*y1
// compute xd (or the negative value if underflow occurs)
s := 1 // sign of product xd*yd
xd := z[2*n : 2*n+n2]
if subVV(xd, x1, x0) != 0 { // x1-x0
s = -s
subVV(xd, x0, x1) // x0-x1
}
// compute yd (or the negative value if underflow occurs)
yd := z[2*n+n2 : 3*n]
if subVV(yd, y0, y1) != 0 { // y0-y1
s = -s
subVV(yd, y1, y0) // y1-y0
}
// p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
// p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
p := z[n*3:]
karatsuba(p, xd, yd)
// save original z2:z0
// (ok to use upper half of z since we're done recursing)
r := z[n*4:]
copy(r, z[:n*2])
// add up all partial products
//
// 2*n n 0
// z = [ z2 | z0 ]
// + [ z0 ]
// + [ z2 ]
// + [ p ]
//
karatsubaAdd(z[n2:], r, n)
karatsubaAdd(z[n2:], r[n:], n)
if s > 0 {
karatsubaAdd(z[n2:], p, n)
} else {
karatsubaSub(z[n2:], p, n)
}
}
// alias reports whether x and y share the same base array.
// Note: alias assumes that the capacity of underlying arrays
// is never changed for nat values; i.e. that there are
// no 3-operand slice expressions in this code (or worse,
// reflect-based operations to the same effect).
func alias(x, y nat) bool {
return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
}
// addAt implements z += x<<(_W*i); z must be long enough.
// (we don't use nat.add because we need z to stay the same
// slice, and we don't need to normalize z after each addition)
func addAt(z, x nat, i int) {
if n := len(x); n > 0 {
if c := addVV(z[i:i+n], z[i:], x); c != 0 {
j := i + n
if j < len(z) {
addVW(z[j:], z[j:], c)
}
}
}
}
func max(x, y int) int {
if x > y {
return x
}
return y
}
// karatsubaLen computes an approximation to the maximum k <= n such that
// k = p<<i for a number p <= threshold and an i >= 0. Thus, the
// result is the largest number that can be divided repeatedly by 2 before
// becoming about the value of threshold.
func karatsubaLen(n, threshold int) int {
i := uint(0)
for n > threshold {
n >>= 1
i++
}
return n << i
}
func (z nat) mul(x, y nat) nat {
m := len(x)
n := len(y)
switch {
case m < n:
return z.mul(y, x)
case m == 0 || n == 0:
return z[:0]
case n == 1:
return z.mulAddWW(x, y[0], 0)
}
// m >= n > 1
// determine if z can be reused
if alias(z, x) || alias(z, y) {
z = nil // z is an alias for x or y - cannot reuse
}
// use basic multiplication if the numbers are small
if n < karatsubaThreshold {
z = z.make(m + n)
basicMul(z, x, y)
return z.norm()
}
// m >= n && n >= karatsubaThreshold && n >= 2
// determine Karatsuba length k such that
//
// x = xh*b + x0 (0 <= x0 < b)
// y = yh*b + y0 (0 <= y0 < b)
// b = 1<<(_W*k) ("base" of digits xi, yi)
//
k := karatsubaLen(n, karatsubaThreshold)
// k <= n
// multiply x0 and y0 via Karatsuba
x0 := x[0:k] // x0 is not normalized
y0 := y[0:k] // y0 is not normalized
z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
karatsuba(z, x0, y0)
z = z[0 : m+n] // z has final length but may be incomplete
z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
// If xh != 0 or yh != 0, add the missing terms to z. For
//
// xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
// yh = y1*b (0 <= y1 < b)
//
// the missing terms are
//
// x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
//
// since all the yi for i > 1 are 0 by choice of k: If any of them
// were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
// be a larger valid threshold contradicting the assumption about k.
//
if k < n || m != n {
var t nat
// add x0*y1*b
x0 := x0.norm()
y1 := y[k:] // y1 is normalized because y is
t = t.mul(x0, y1) // update t so we don't lose t's underlying array
addAt(z, t, k)
// add xi*y0<<i, xi*y1*b<<(i+k)
y0 := y0.norm()
for i := k; i < len(x); i += k {
xi := x[i:]
if len(xi) > k {
xi = xi[:k]
}
xi = xi.norm()
t = t.mul(xi, y0)
addAt(z, t, i)
t = t.mul(xi, y1)
addAt(z, t, i+k)
}
}
return z.norm()
}
// basicSqr sets z = x*x and is asymptotically faster than basicMul
// by about a factor of 2, but slower for small arguments due to overhead.
// Requirements: len(x) > 0, len(z) == 2*len(x)
// The (non-normalized) result is placed in z.
func basicSqr(z, x nat) {
n := len(x)
t := make(nat, 2*n) // temporary variable to hold the products
z[1], z[0] = mulWW(x[0], x[0]) // the initial square
for i := 1; i < n; i++ {
d := x[i]
// z collects the squares x[i] * x[i]
z[2*i+1], z[2*i] = mulWW(d, d)
// t collects the products x[i] * x[j] where j < i
t[2*i] = addMulVVW(t[i:2*i], x[0:i], d)
}
t[2*n-1] = shlVU(t[1:2*n-1], t[1:2*n-1], 1) // double the j < i products
addVV(z, z, t) // combine the result
}
// karatsubaSqr squares x and leaves the result in z.
// len(x) must be a power of 2 and len(z) >= 6*len(x).
// The (non-normalized) result is placed in z[0 : 2*len(x)].
//
// The algorithm and the layout of z are the same as for karatsuba.
func karatsubaSqr(z, x nat) {
n := len(x)
if n&1 != 0 || n < karatsubaSqrThreshold || n < 2 {
basicSqr(z[:2*n], x)
return
}
n2 := n >> 1
x1, x0 := x[n2:], x[0:n2]
karatsubaSqr(z, x0)
karatsubaSqr(z[n:], x1)
// s = sign(xd*yd) == -1 for xd != 0; s == 1 for xd == 0
xd := z[2*n : 2*n+n2]
if subVV(xd, x1, x0) != 0 {
subVV(xd, x0, x1)
}
p := z[n*3:]
karatsubaSqr(p, xd)
r := z[n*4:]
copy(r, z[:n*2])
karatsubaAdd(z[n2:], r, n)
karatsubaAdd(z[n2:], r[n:], n)
karatsubaSub(z[n2:], p, n) // s == -1 for p != 0; s == 1 for p == 0
}
// Operands that are shorter than basicSqrThreshold are squared using
// "grade school" multiplication; for operands longer than karatsubaSqrThreshold
// we use the Karatsuba algorithm optimized for x == y.
var basicSqrThreshold = 20 // computed by calibrate_test.go
var karatsubaSqrThreshold = 260 // computed by calibrate_test.go
// z = x*x
func (z nat) sqr(x nat) nat {
n := len(x)
switch {
case n == 0:
return z[:0]
case n == 1:
d := x[0]
z = z.make(2)
z[1], z[0] = mulWW(d, d)
return z.norm()
}
if alias(z, x) {
z = nil // z is an alias for x - cannot reuse
}
if n < basicSqrThreshold {
z = z.make(2 * n)
basicMul(z, x, x)
return z.norm()
}
if n < karatsubaSqrThreshold {
z = z.make(2 * n)
basicSqr(z, x)
return z.norm()
}
// Use Karatsuba multiplication optimized for x == y.
// The algorithm and layout of z are the same as for mul.
// z = (x1*b + x0)^2 = x1^2*b^2 + 2*x1*x0*b + x0^2
k := karatsubaLen(n, karatsubaSqrThreshold)
x0 := x[0:k]
z = z.make(max(6*k, 2*n))
karatsubaSqr(z, x0) // z = x0^2
z = z[0 : 2*n]
z[2*k:].clear()
if k < n {
var t nat
x0 := x0.norm()
x1 := x[k:]
t = t.mul(x0, x1)
addAt(z, t, k)
addAt(z, t, k) // z = 2*x1*x0*b + x0^2
t = t.sqr(x1)
addAt(z, t, 2*k) // z = x1^2*b^2 + 2*x1*x0*b + x0^2
}
return z.norm()
}
// mulRange computes the product of all the unsigned integers in the
// range [a, b] inclusively. If a > b (empty range), the result is 1.
func (z nat) mulRange(a, b uint64) nat {
switch {
case a == 0:
// cut long ranges short (optimization)
return z.setUint64(0)
case a > b:
return z.setUint64(1)
case a == b:
return z.setUint64(a)
case a+1 == b:
return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
}
m := (a + b) / 2
return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
}
// q = (x-r)/y, with 0 <= r < y
func (z nat) divW(x nat, y Word) (q nat, r Word) {
m := len(x)
switch {
case y == 0:
panic("division by zero")
case y == 1:
q = z.set(x) // result is x
return
case m == 0:
q = z[:0] // result is 0
return
}
// m > 0
z = z.make(m)
r = divWVW(z, 0, x, y)
q = z.norm()
return
}
func (z nat) div(z2, u, v nat) (q, r nat) {
if len(v) == 0 {
panic("division by zero")
}
if u.cmp(v) < 0 {
q = z[:0]
r = z2.set(u)
return
}
if len(v) == 1 {
var r2 Word
q, r2 = z.divW(u, v[0])
r = z2.setWord(r2)
return
}
q, r = z.divLarge(z2, u, v)
return
}
// getNat returns a *nat of len n. The contents may not be zero.
// The pool holds *nat to avoid allocation when converting to interface{}.
func getNat(n int) *nat {
var z *nat
if v := natPool.Get(); v != nil {
z = v.(*nat)
}
if z == nil {
z = new(nat)
}
*z = z.make(n)
return z
}
func putNat(x *nat) {
natPool.Put(x)
}
var natPool sync.Pool
// q = (uIn-r)/vIn, with 0 <= r < y
// Uses z as storage for q, and u as storage for r if possible.
// See Knuth, Volume 2, section 4.3.1, Algorithm D.
// Preconditions:
// len(vIn) >= 2
// len(uIn) >= len(vIn)
// u must not alias z
func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
n := len(vIn)
m := len(uIn) - n
// D1.
shift := nlz(vIn[n-1])
// do not modify vIn, it may be used by another goroutine simultaneously
vp := getNat(n)
v := *vp
shlVU(v, vIn, shift)
// u may safely alias uIn or vIn, the value of uIn is used to set u and vIn was already used
u = u.make(len(uIn) + 1)
u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
// z may safely alias uIn or vIn, both values were used already
if alias(z, u) {
z = nil // z is an alias for u - cannot reuse
}
q = z.make(m + 1)
qhatvp := getNat(n + 1)
qhatv := *qhatvp
// D2.
vn1 := v[n-1]
for j := m; j >= 0; j-- {
// D3.
qhat := Word(_M)
if ujn := u[j+n]; ujn != vn1 {
var rhat Word
qhat, rhat = divWW(ujn, u[j+n-1], vn1)
// x1 | x2 = q̂v_{n-2}
vn2 := v[n-2]
x1, x2 := mulWW(qhat, vn2)
// test if q̂v_{n-2} > br̂ + u_{j+n-2}
ujn2 := u[j+n-2]
for greaterThan(x1, x2, rhat, ujn2) {
qhat--
prevRhat := rhat
rhat += vn1
// v[n-1] >= 0, so this tests for overflow.
if rhat < prevRhat {
break
}
x1, x2 = mulWW(qhat, vn2)
}
}
// D4.
qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
if c != 0 {
c := addVV(u[j:j+n], u[j:], v)
u[j+n] += c
qhat--
}
q[j] = qhat
}
putNat(vp)
putNat(qhatvp)
q = q.norm()
shrVU(u, u, shift)
r = u.norm()
return q, r
}
// Length of x in bits. x must be normalized.
func (x nat) bitLen() int {
if i := len(x) - 1; i >= 0 {
return i*_W + bits.Len(uint(x[i]))
}
return 0
}
// trailingZeroBits returns the number of consecutive least significant zero
// bits of x.
func (x nat) trailingZeroBits() uint {
if len(x) == 0 {
return 0
}
var i uint
for x[i] == 0 {
i++
}
// x[i] != 0
return i*_W + uint(bits.TrailingZeros(uint(x[i])))
}
func same(x, y nat) bool {
return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
}
// z = x << s
func (z nat) shl(x nat, s uint) nat {
if s == 0 {
if same(z, x) {
return z
}
if !alias(z, x) {
return z.set(x)
}
}
m := len(x)
if m == 0 {
return z[:0]
}
// m > 0
n := m + int(s/_W)
z = z.make(n + 1)
z[n] = shlVU(z[n-m:n], x, s%_W)
z[0 : n-m].clear()
return z.norm()
}
// z = x >> s
func (z nat) shr(x nat, s uint) nat {
if s == 0 {
if same(z, x) {
return z
}
if !alias(z, x) {
return z.set(x)
}
}
m := len(x)
n := m - int(s/_W)
if n <= 0 {
return z[:0]
}
// n > 0
z = z.make(n)
shrVU(z, x[m-n:], s%_W)
return z.norm()
}
func (z nat) setBit(x nat, i uint, b uint) nat {
j := int(i / _W)
m := Word(1) << (i % _W)
n := len(x)
switch b {
case 0:
z = z.make(n)
copy(z, x)
if j >= n {
// no need to grow
return z
}
z[j] &^= m
return z.norm()
case 1:
if j >= n {
z = z.make(j + 1)
z[n:].clear()
} else {
z = z.make(n)
}
copy(z, x)
z[j] |= m
// no need to normalize
return z
}
panic("set bit is not 0 or 1")
}
// bit returns the value of the i'th bit, with lsb == bit 0.
func (x nat) bit(i uint) uint {
j := i / _W
if j >= uint(len(x)) {
return 0
}
// 0 <= j < len(x)
return uint(x[j] >> (i % _W) & 1)
}
// sticky returns 1 if there's a 1 bit within the
// i least significant bits, otherwise it returns 0.
func (x nat) sticky(i uint) uint {
j := i / _W
if j >= uint(len(x)) {
if len(x) == 0 {
return 0
}
return 1
}
// 0 <= j < len(x)
for _, x := range x[:j] {
if x != 0 {
return 1
}
}
if x[j]<<(_W-i%_W) != 0 {
return 1
}
return 0
}
func (z nat) and(x, y nat) nat {
m := len(x)
n := len(y)
if m > n {
m = n
}
// m <= n
z = z.make(m)
for i := 0; i < m; i++ {
z[i] = x[i] & y[i]
}
return z.norm()
}
func (z nat) andNot(x, y nat) nat {
m := len(x)
n := len(y)
if n > m {
n = m
}
// m >= n
z = z.make(m)
for i := 0; i < n; i++ {
z[i] = x[i] &^ y[i]
}
copy(z[n:m], x[n:m])
return z.norm()
}
func (z nat) or(x, y nat) nat {
m := len(x)
n := len(y)
s := x
if m < n {
n, m = m, n
s = y
}
// m >= n
z = z.make(m)
for i := 0; i < n; i++ {
z[i] = x[i] | y[i]
}
copy(z[n:m], s[n:m])
return z.norm()
}
func (z nat) xor(x, y nat) nat {
m := len(x)
n := len(y)
s := x
if m < n {
n, m = m, n
s = y
}
// m >= n
z = z.make(m)
for i := 0; i < n; i++ {
z[i] = x[i] ^ y[i]
}
copy(z[n:m], s[n:m])
return z.norm()
}
// greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
func greaterThan(x1, x2, y1, y2 Word) bool {
return x1 > y1 || x1 == y1 && x2 > y2
}
// modW returns x % d.
func (x nat) modW(d Word) (r Word) {
// TODO(agl): we don't actually need to store the q value.
var q nat
q = q.make(len(x))
return divWVW(q, 0, x, d)
}
// random creates a random integer in [0..limit), using the space in z if
// possible. n is the bit length of limit.
func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
if alias(z, limit) {
z = nil // z is an alias for limit - cannot reuse
}
z = z.make(len(limit))
bitLengthOfMSW := uint(n % _W)
if bitLengthOfMSW == 0 {
bitLengthOfMSW = _W
}
mask := Word((1 << bitLengthOfMSW) - 1)
for {
switch _W {
case 32:
for i := range z {
z[i] = Word(rand.Uint32())
}
case 64:
for i := range z {
z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
}
default:
panic("unknown word size")
}
z[len(limit)-1] &= mask
if z.cmp(limit) < 0 {
break
}
}
return z.norm()
}
// If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
// otherwise it sets z to x**y. The result is the value of z.
func (z nat) expNN(x, y, m nat) nat {
if alias(z, x) || alias(z, y) {
// We cannot allow in-place modification of x or y.
z = nil
}
// x**y mod 1 == 0
if len(m) == 1 && m[0] == 1 {
return z.setWord(0)
}
// m == 0 || m > 1
// x**0 == 1
if len(y) == 0 {
return z.setWord(1)
}
// y > 0
// x**1 mod m == x mod m
if len(y) == 1 && y[0] == 1 && len(m) != 0 {
_, z = nat(nil).div(z, x, m)
return z
}
// y > 1
if len(m) != 0 {
// We likely end up being as long as the modulus.
z = z.make(len(m))
}
z = z.set(x)
// If the base is non-trivial and the exponent is large, we use
// 4-bit, windowed exponentiation. This involves precomputing 14 values
// (x^2...x^15) but then reduces the number of multiply-reduces by a
// third. Even for a 32-bit exponent, this reduces the number of
// operations. Uses Montgomery method for odd moduli.
if x.cmp(natOne) > 0 && len(y) > 1 && len(m) > 0 {
if m[0]&1 == 1 {
return z.expNNMontgomery(x, y, m)
}
return z.expNNWindowed(x, y, m)
}
v := y[len(y)-1] // v > 0 because y is normalized and y > 0
shift := nlz(v) + 1
v <<= shift
var q nat
const mask = 1 << (_W - 1)
// We walk through the bits of the exponent one by one. Each time we
// see a bit, we square, thus doubling the power. If the bit is a one,
// we also multiply by x, thus adding one to the power.
w := _W - int(shift)
// zz and r are used to avoid allocating in mul and div as
// otherwise the arguments would alias.
var zz, r nat
for j := 0; j < w; j++ {
zz = zz.sqr(z)
zz, z = z, zz
if v&mask != 0 {
zz = zz.mul(z, x)
zz, z = z, zz
}
if len(m) != 0 {
zz, r = zz.div(r, z, m)
zz, r, q, z = q, z, zz, r
}
v <<= 1
}
for i := len(y) - 2; i >= 0; i-- {
v = y[i]
for j := 0; j < _W; j++ {
zz = zz.sqr(z)
zz, z = z, zz
if v&mask != 0 {
zz = zz.mul(z, x)
zz, z = z, zz
}
if len(m) != 0 {
zz, r = zz.div(r, z, m)
zz, r, q, z = q, z, zz, r
}
v <<= 1
}
}
return z.norm()
}
// expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
func (z nat) expNNWindowed(x, y, m nat) nat {
// zz and r are used to avoid allocating in mul and div as otherwise
// the arguments would alias.
var zz, r nat
const n = 4
// powers[i] contains x^i.
var powers [1 << n]nat
powers[0] = natOne
powers[1] = x
for i := 2; i < 1<<n; i += 2 {
p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
*p = p.sqr(*p2)
zz, r = zz.div(r, *p, m)
*p, r = r, *p
*p1 = p1.mul(*p, x)
zz, r = zz.div(r, *p1, m)
*p1, r = r, *p1
}
z = z.setWord(1)
for i := len(y) - 1; i >= 0; i-- {
yi := y[i]
for j := 0; j < _W; j += n {
if i != len(y)-1 || j != 0 {
// Unrolled loop for significant performance
// gain. Use go test -bench=".*" in crypto/rsa
// to check performance before making changes.
zz = zz.sqr(z)
zz, z = z, zz
zz, r = zz.div(r, z, m)
z, r = r, z
zz = zz.sqr(z)
zz, z = z, zz
zz, r = zz.div(r, z, m)
z, r = r, z
zz = zz.sqr(z)
zz, z = z, zz
zz, r = zz.div(r, z, m)
z, r = r, z
zz = zz.sqr(z)
zz, z = z, zz
zz, r = zz.div(r, z, m)
z, r = r, z
}
zz = zz.mul(z, powers[yi>>(_W-n)])
zz, z = z, zz
zz, r = zz.div(r, z, m)
z, r = r, z
yi <<= n
}
}
return z.norm()
}
// expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
// Uses Montgomery representation.
func (z nat) expNNMontgomery(x, y, m nat) nat {
numWords := len(m)
// We want the lengths of x and m to be equal.
// It is OK if x >= m as long as len(x) == len(m).
if len(x) > numWords {
_, x = nat(nil).div(nil, x, m)
// Note: now len(x) <= numWords, not guaranteed ==.
}
if len(x) < numWords {
rr := make(nat, numWords)
copy(rr, x)
x = rr
}
// Ideally the precomputations would be performed outside, and reused
// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
// Iteration for Multiplicative Inverses Modulo Prime Powers".
k0 := 2 - m[0]
t := m[0] - 1
for i := 1; i < _W; i <<= 1 {
t *= t
k0 *= (t + 1)
}
k0 = -k0
// RR = 2**(2*_W*len(m)) mod m
RR := nat(nil).setWord(1)
zz := nat(nil).shl(RR, uint(2*numWords*_W))
_, RR = nat(nil).div(RR, zz, m)
if len(RR) < numWords {
zz = zz.make(numWords)
copy(zz, RR)
RR = zz
}
// one = 1, with equal length to that of m
one := make(nat, numWords)
one[0] = 1
const n = 4
// powers[i] contains x^i
var powers [1 << n]nat
powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
for i := 2; i < 1<<n; i++ {
powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
}
// initialize z = 1 (Montgomery 1)
z = z.make(numWords)
copy(z, powers[0])
zz = zz.make(numWords)
// same windowed exponent, but with Montgomery multiplications
for i := len(y) - 1; i >= 0; i-- {
yi := y[i]
for j := 0; j < _W; j += n {
if i != len(y)-1 || j != 0 {
zz = zz.montgomery(z, z, m, k0, numWords)
z = z.montgomery(zz, zz, m, k0, numWords)
zz = zz.montgomery(z, z, m, k0, numWords)
z = z.montgomery(zz, zz, m, k0, numWords)
}
zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
z, zz = zz, z
yi <<= n
}
}
// convert to regular number
zz = zz.montgomery(z, one, m, k0, numWords)
// One last reduction, just in case.
// See golang.org/issue/13907.
if zz.cmp(m) >= 0 {
// Common case is m has high bit set; in that case,
// since zz is the same length as m, there can be just
// one multiple of m to remove. Just subtract.
// We think that the subtract should be sufficient in general,
// so do that unconditionally, but double-check,
// in case our beliefs are wrong.
// The div is not expected to be reached.
zz = zz.sub(zz, m)
if zz.cmp(m) >= 0 {
_, zz = nat(nil).div(nil, zz, m)
}
}
return zz.norm()
}
// bytes writes the value of z into buf using big-endian encoding.
// len(buf) must be >= len(z)*_S. The value of z is encoded in the
// slice buf[i:]. The number i of unused bytes at the beginning of
// buf is returned as result.
func (z nat) bytes(buf []byte) (i int) {
i = len(buf)
for _, d := range z {
for j := 0; j < _S; j++ {
i--
buf[i] = byte(d)
d >>= 8
}
}
for i < len(buf) && buf[i] == 0 {
i++
}
return
}
// bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
func bigEndianWord(buf []byte) Word {
if _W == 64 {
return Word(binary.BigEndian.Uint64(buf))
}
return Word(binary.BigEndian.Uint32(buf))
}
// setBytes interprets buf as the bytes of a big-endian unsigned
// integer, sets z to that value, and returns z.
func (z nat) setBytes(buf []byte) nat {
z = z.make((len(buf) + _S - 1) / _S)
i := len(buf)
for k := 0; i >= _S; k++ {
z[k] = bigEndianWord(buf[i-_S : i])
i -= _S
}
if i > 0 {
var d Word
for s := uint(0); i > 0; s += 8 {
d |= Word(buf[i-1]) << s
i--
}
z[len(z)-1] = d
}
return z.norm()
}
// sqrt sets z = ⌊√x⌋
func (z nat) sqrt(x nat) nat {
if x.cmp(natOne) <= 0 {
return z.set(x)
}
if alias(z, x) {
z = nil
}
// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
// See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
// https://members.loria.fr/PZimmermann/mca/pub226.html
// If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
// otherwise it converges to the correct z and stays there.
var z1, z2 nat
z1 = z
z1 = z1.setUint64(1)
z1 = z1.shl(z1, uint(x.bitLen()+1)/2) // must be ≥ √x
for n := 0; ; n++ {
z2, _ = z2.div(nil, x, z1)
z2 = z2.add(z2, z1)
z2 = z2.shr(z2, 1)
if z2.cmp(z1) >= 0 {
// z1 is answer.
// Figure out whether z1 or z2 is currently aliased to z by looking at loop count.
if n&1 == 0 {
return z1
}
return z.set(z1)
}
z1, z2 = z2, z1
}
}
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