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diff --git a/libgo/go/math/big/nat.go b/libgo/go/math/big/nat.go
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+// 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.
+
+// Package big implements multi-precision arithmetic (big numbers).
+// The following numeric types are supported:
+//
+// - Int signed integers
+// - Rat rational numbers
+//
+// All methods on Int take the result as the receiver; if it is one
+// of the operands it may be overwritten (and its memory reused).
+// To enable chaining of operations, the result is also returned.
+//
+package big
+
+// This file contains operations on unsigned multi-precision integers.
+// These are the building blocks for the operations on signed integers
+// and rationals.
+
+import (
+ "errors"
+ "io"
+ "math/rand"
+)
+
+// 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}
+ 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[0:n] // reuse z
+ }
+ // 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.make(0)
+ }
+ z = z.make(1)
+ z[0] = x
+ return z
+}
+
+func (z nat) setUint64(x uint64) nat {
+ // single-digit values
+ if w := Word(x); uint64(w) == x {
+ return z.setWord(w)
+ }
+
+ // compute number of words n required to represent x
+ n := 0
+ for t := x; t > 0; t >>= _W {
+ n++
+ }
+
+ // split x into n words
+ z = z.make(n)
+ for i := range z {
+ z[i] = Word(x & _M)
+ x >>= _W
+ }
+
+ 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.make(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.make(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)
+ }
+ }
+}
+
+// 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 int = 32 // computed by calibrate.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 + z1 + z0
+ // = (x1-x0)*(y0 - y1) + z1 + z0
+ // = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z1 + z0
+ // = x1*y0 - z1 - z0 + x0*y1 + z1 + 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)
+
+ // 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 returns true if x and y share the same base array.
+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*(1<<(_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 <= karatsubaThreshold and an i >= 0. Thus, the
+// result is the largest number that can be divided repeatedly by 2 before
+// becoming about the value of karatsubaThreshold.
+func karatsubaLen(n int) int {
+ i := uint(0)
+ for n > karatsubaThreshold {
+ 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.make(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 || n < 2 {
+ 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 = x1*b + x0
+ // y = y1*b + y0 (and k <= len(y), which implies k <= len(x))
+ // b = 1<<(_W*k) ("base" of digits xi, yi)
+ //
+ k := karatsubaLen(n)
+ // 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, upper portion is garbage
+
+ // If x1 and/or y1 are not 0, add missing terms to z explicitly:
+ //
+ // m+n 2*k 0
+ // z = [ ... | x0*y0 ]
+ // + [ x1*y1 ]
+ // + [ x1*y0 ]
+ // + [ x0*y1 ]
+ //
+ if k < n || m != n {
+ x1 := x[k:] // x1 is normalized because x is
+ y1 := y[k:] // y1 is normalized because y is
+ var t nat
+ t = t.mul(x1, y1)
+ copy(z[2*k:], t)
+ z[2*k+len(t):].clear() // upper portion of z is garbage
+ t = t.mul(x1, y0.norm())
+ addAt(z, t, k)
+ t = t.mul(x0.norm(), y1)
+ addAt(z, t, k)
+ }
+
+ 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{}.setUint64(a), nat{}.setUint64(b))
+ }
+ m := (a + b) / 2
+ return z.mul(nat{}.mulRange(a, m), nat{}.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.make(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.make(0)
+ r = z2.set(u)
+ return
+ }
+
+ if len(v) == 1 {
+ var rprime Word
+ q, rprime = z.divW(u, v[0])
+ if rprime > 0 {
+ r = z2.make(1)
+ r[0] = rprime
+ } else {
+ r = z2.make(0)
+ }
+ return
+ }
+
+ q, r = z.divLarge(z2, u, v)
+ return
+}
+
+// q = (uIn-r)/v, 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(v) >= 2
+// len(uIn) >= len(v)
+func (z nat) divLarge(u, uIn, v nat) (q, r nat) {
+ n := len(v)
+ m := len(uIn) - n
+
+ // determine if z can be reused
+ // TODO(gri) should find a better solution - this if statement
+ // is very costly (see e.g. time pidigits -s -n 10000)
+ if alias(z, uIn) || alias(z, v) {
+ z = nil // z is an alias for uIn or v - cannot reuse
+ }
+ q = z.make(m + 1)
+
+ qhatv := make(nat, n+1)
+ if alias(u, uIn) || alias(u, v) {
+ u = nil // u is an alias for uIn or v - cannot reuse
+ }
+ u = u.make(len(uIn) + 1)
+ u.clear()
+
+ // D1.
+ shift := leadingZeros(v[n-1])
+ if shift > 0 {
+ // do not modify v, it may be used by another goroutine simultaneously
+ v1 := make(nat, n)
+ shlVU(v1, v, shift)
+ v = v1
+ }
+ u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
+
+ // D2.
+ for j := m; j >= 0; j-- {
+ // D3.
+ qhat := Word(_M)
+ if u[j+n] != v[n-1] {
+ var rhat Word
+ qhat, rhat = divWW(u[j+n], u[j+n-1], v[n-1])
+
+ // x1 | x2 = q̂v_{n-2}
+ x1, x2 := mulWW(qhat, v[n-2])
+ // test if q̂v_{n-2} > br̂ + u_{j+n-2}
+ for greaterThan(x1, x2, rhat, u[j+n-2]) {
+ qhat--
+ prevRhat := rhat
+ rhat += v[n-1]
+ // v[n-1] >= 0, so this tests for overflow.
+ if rhat < prevRhat {
+ break
+ }
+ x1, x2 = mulWW(qhat, v[n-2])
+ }
+ }
+
+ // 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
+ }
+
+ 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 + bitLen(x[i])
+ }
+ return 0
+}
+
+// MaxBase is the largest number base accepted for string conversions.
+const MaxBase = 'z' - 'a' + 10 + 1 // = hexValue('z') + 1
+
+func hexValue(ch rune) Word {
+ d := MaxBase + 1 // illegal base
+ switch {
+ case '0' <= ch && ch <= '9':
+ d = int(ch - '0')
+ case 'a' <= ch && ch <= 'z':
+ d = int(ch - 'a' + 10)
+ case 'A' <= ch && ch <= 'Z':
+ d = int(ch - 'A' + 10)
+ }
+ return Word(d)
+}
+
+// scan sets z to the natural number corresponding to the longest possible prefix
+// read from r representing an unsigned integer in a given conversion base.
+// It returns z, the actual conversion base used, and an error, if any. In the
+// error case, the value of z is undefined. The syntax follows the syntax of
+// unsigned integer literals in Go.
+//
+// The base argument must be 0 or a value from 2 through MaxBase. If the base
+// is 0, the string prefix determines the actual conversion base. A prefix of
+// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
+// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
+//
+func (z nat) scan(r io.RuneScanner, base int) (nat, int, error) {
+ // reject illegal bases
+ if base < 0 || base == 1 || MaxBase < base {
+ return z, 0, errors.New("illegal number base")
+ }
+
+ // one char look-ahead
+ ch, _, err := r.ReadRune()
+ if err != nil {
+ return z, 0, err
+ }
+
+ // determine base if necessary
+ b := Word(base)
+ if base == 0 {
+ b = 10
+ if ch == '0' {
+ switch ch, _, err = r.ReadRune(); err {
+ case nil:
+ b = 8
+ switch ch {
+ case 'x', 'X':
+ b = 16
+ case 'b', 'B':
+ b = 2
+ }
+ if b == 2 || b == 16 {
+ if ch, _, err = r.ReadRune(); err != nil {
+ return z, 0, err
+ }
+ }
+ case io.EOF:
+ return z.make(0), 10, nil
+ default:
+ return z, 10, err
+ }
+ }
+ }
+
+ // convert string
+ // - group as many digits d as possible together into a "super-digit" dd with "super-base" bb
+ // - only when bb does not fit into a word anymore, do a full number mulAddWW using bb and dd
+ z = z.make(0)
+ bb := Word(1)
+ dd := Word(0)
+ for max := _M / b; ; {
+ d := hexValue(ch)
+ if d >= b {
+ r.UnreadRune() // ch does not belong to number anymore
+ break
+ }
+
+ if bb <= max {
+ bb *= b
+ dd = dd*b + d
+ } else {
+ // bb * b would overflow
+ z = z.mulAddWW(z, bb, dd)
+ bb = b
+ dd = d
+ }
+
+ if ch, _, err = r.ReadRune(); err != nil {
+ if err != io.EOF {
+ return z, int(b), err
+ }
+ break
+ }
+ }
+
+ switch {
+ case bb > 1:
+ // there was at least one mantissa digit
+ z = z.mulAddWW(z, bb, dd)
+ case base == 0 && b == 8:
+ // there was only the octal prefix 0 (possibly followed by digits > 7);
+ // return base 10, not 8
+ return z, 10, nil
+ case base != 0 || b != 8:
+ // there was neither a mantissa digit nor the octal prefix 0
+ return z, int(b), errors.New("syntax error scanning number")
+ }
+
+ return z.norm(), int(b), nil
+}
+
+// Character sets for string conversion.
+const (
+ lowercaseDigits = "0123456789abcdefghijklmnopqrstuvwxyz"
+ uppercaseDigits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+)
+
+// decimalString returns a decimal representation of x.
+// It calls x.string with the charset "0123456789".
+func (x nat) decimalString() string {
+ return x.string(lowercaseDigits[0:10])
+}
+
+// string converts x to a string using digits from a charset; a digit with
+// value d is represented by charset[d]. The conversion base is determined
+// by len(charset), which must be >= 2.
+func (x nat) string(charset string) string {
+ b := Word(len(charset))
+
+ // special cases
+ switch {
+ case b < 2 || b > 256:
+ panic("illegal base")
+ case len(x) == 0:
+ return string(charset[0])
+ }
+
+ // allocate buffer for conversion
+ i := x.bitLen()/log2(b) + 1 // +1: round up
+ s := make([]byte, i)
+
+ // special case: power of two bases can avoid divisions completely
+ if b == b&-b {
+ // shift is base-b digit size in bits
+ shift := uint(trailingZeroBits(b)) // shift > 0 because b >= 2
+ mask := Word(1)<<shift - 1
+ w := x[0]
+ nbits := uint(_W) // number of unprocessed bits in w
+
+ // convert less-significant words
+ for k := 1; k < len(x); k++ {
+ // convert full digits
+ for nbits >= shift {
+ i--
+ s[i] = charset[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ // convert any partial leading digit and advance to next word
+ if nbits == 0 {
+ // no partial digit remaining, just advance
+ w = x[k]
+ nbits = _W
+ } else {
+ // partial digit in current (k-1) and next (k) word
+ w |= x[k] << nbits
+ i--
+ s[i] = charset[w&mask]
+
+ // advance
+ w = x[k] >> (shift - nbits)
+ nbits = _W - (shift - nbits)
+ }
+ }
+
+ // convert digits of most-significant word (omit leading zeros)
+ for nbits >= 0 && w != 0 {
+ i--
+ s[i] = charset[w&mask]
+ w >>= shift
+ nbits -= shift
+ }
+
+ return string(s[i:])
+ }
+
+ // general case: extract groups of digits by multiprecision division
+
+ // maximize ndigits where b**ndigits < 2^_W; bb (big base) is b**ndigits
+ bb := Word(1)
+ ndigits := 0
+ for max := Word(_M / b); bb <= max; bb *= b {
+ ndigits++
+ }
+
+ // preserve x, create local copy for use in repeated divisions
+ q := nat{}.set(x)
+ var r Word
+
+ // convert
+ if b == 10 { // hard-coding for 10 here speeds this up by 1.25x
+ for len(q) > 0 {
+ // extract least significant, base bb "digit"
+ q, r = q.divW(q, bb) // N.B. >82% of time is here. Optimize divW
+ if len(q) == 0 {
+ // skip leading zeros in most-significant group of digits
+ for j := 0; j < ndigits && r != 0; j++ {
+ i--
+ s[i] = charset[r%10]
+ r /= 10
+ }
+ } else {
+ for j := 0; j < ndigits; j++ {
+ i--
+ s[i] = charset[r%10]
+ r /= 10
+ }
+ }
+ }
+ } else {
+ for len(q) > 0 {
+ // extract least significant group of digits
+ q, r = q.divW(q, bb) // N.B. >82% of time is here. Optimize divW
+ if len(q) == 0 {
+ // skip leading zeros in most-significant group of digits
+ for j := 0; j < ndigits && r != 0; j++ {
+ i--
+ s[i] = charset[r%b]
+ r /= b
+ }
+ } else {
+ for j := 0; j < ndigits; j++ {
+ i--
+ s[i] = charset[r%b]
+ r /= b
+ }
+ }
+ }
+ }
+
+ return string(s[i:])
+}
+
+const deBruijn32 = 0x077CB531
+
+var deBruijn32Lookup = []byte{
+ 0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8,
+ 31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9,
+}
+
+const deBruijn64 = 0x03f79d71b4ca8b09
+
+var deBruijn64Lookup = []byte{
+ 0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4,
+ 62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5,
+ 63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11,
+ 54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
+}
+
+// trailingZeroBits returns the number of consecutive zero bits on the right
+// side of the given Word.
+// See Knuth, volume 4, section 7.3.1
+func trailingZeroBits(x Word) int {
+ // x & -x leaves only the right-most bit set in the word. Let k be the
+ // index of that bit. Since only a single bit is set, the value is two
+ // to the power of k. Multiplying by a power of two is equivalent to
+ // left shifting, in this case by k bits. The de Bruijn constant is
+ // such that all six bit, consecutive substrings are distinct.
+ // Therefore, if we have a left shifted version of this constant we can
+ // find by how many bits it was shifted by looking at which six bit
+ // substring ended up at the top of the word.
+ switch _W {
+ case 32:
+ return int(deBruijn32Lookup[((x&-x)*deBruijn32)>>27])
+ case 64:
+ return int(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58])
+ default:
+ panic("Unknown word size")
+ }
+
+ return 0
+}
+
+// z = x << s
+func (z nat) shl(x nat, s uint) nat {
+ m := len(x)
+ if m == 0 {
+ return z.make(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 {
+ m := len(x)
+ n := m - int(s/_W)
+ if n <= 0 {
+ return z.make(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 {
+ n = j + 1
+ }
+ z = z.make(n)
+ copy(z, x)
+ z[j] |= m
+ // no need to normalize
+ return z
+ }
+ panic("set bit is not 0 or 1")
+}
+
+func (z nat) bit(i uint) uint {
+ j := int(i / _W)
+ if j >= len(z) {
+ return 0
+ }
+ return uint(z[j] >> (i % _W) & 1)
+}
+
+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 returns true iff (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)
+}
+
+// powersOfTwoDecompose finds q and k with x = q * 1<<k and q is odd, or q and k are 0.
+func (x nat) powersOfTwoDecompose() (q nat, k int) {
+ if len(x) == 0 {
+ return x, 0
+ }
+
+ // One of the words must be non-zero by definition,
+ // so this loop will terminate with i < len(x), and
+ // i is the number of 0 words.
+ i := 0
+ for x[i] == 0 {
+ i++
+ }
+ n := trailingZeroBits(x[i]) // x[i] != 0
+
+ q = make(nat, len(x)-i)
+ shrVU(q, x[i:], uint(n))
+
+ q = q.norm()
+ k = i*_W + n
+ return
+}
+
+// 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 {
+ bitLengthOfMSW := uint(n % _W)
+ if bitLengthOfMSW == 0 {
+ bitLengthOfMSW = _W
+ }
+ mask := Word((1 << bitLengthOfMSW) - 1)
+ z = z.make(len(limit))
+
+ for {
+ for i := range z {
+ switch _W {
+ case 32:
+ z[i] = Word(rand.Uint32())
+ case 64:
+ z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
+ }
+ }
+
+ z[len(limit)-1] &= mask
+
+ if z.cmp(limit) < 0 {
+ break
+ }
+ }
+
+ return z.norm()
+}
+
+// If m != nil, expNN calculates x**y mod m. Otherwise it calculates x**y. It
+// reuses the storage of z if possible.
+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
+ }
+
+ if len(y) == 0 {
+ z = z.make(1)
+ z[0] = 1
+ return z
+ }
+
+ if m != nil {
+ // We likely end up being as long as the modulus.
+ z = z.make(len(m))
+ }
+ z = z.set(x)
+ v := y[len(y)-1]
+ // It's invalid for the most significant word to be zero, therefore we
+ // will find a one bit.
+ shift := leadingZeros(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)
+ for j := 0; j < w; j++ {
+ z = z.mul(z, z)
+
+ if v&mask != 0 {
+ z = z.mul(z, x)
+ }
+
+ if m != nil {
+ q, z = q.div(z, z, m)
+ }
+
+ v <<= 1
+ }
+
+ for i := len(y) - 2; i >= 0; i-- {
+ v = y[i]
+
+ for j := 0; j < _W; j++ {
+ z = z.mul(z, z)
+
+ if v&mask != 0 {
+ z = z.mul(z, x)
+ }
+
+ if m != nil {
+ q, z = q.div(z, z, m)
+ }
+
+ v <<= 1
+ }
+ }
+
+ return z
+}
+
+// probablyPrime performs reps Miller-Rabin tests to check whether n is prime.
+// If it returns true, n is prime with probability 1 - 1/4^reps.
+// If it returns false, n is not prime.
+func (n nat) probablyPrime(reps int) bool {
+ if len(n) == 0 {
+ return false
+ }
+
+ if len(n) == 1 {
+ if n[0] < 2 {
+ return false
+ }
+
+ if n[0]%2 == 0 {
+ return n[0] == 2
+ }
+
+ // We have to exclude these cases because we reject all
+ // multiples of these numbers below.
+ switch n[0] {
+ case 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53:
+ return true
+ }
+ }
+
+ const primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29}
+ const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53}
+
+ var r Word
+ switch _W {
+ case 32:
+ r = n.modW(primesProduct32)
+ case 64:
+ r = n.modW(primesProduct64 & _M)
+ default:
+ panic("Unknown word size")
+ }
+
+ if r%3 == 0 || r%5 == 0 || r%7 == 0 || r%11 == 0 ||
+ r%13 == 0 || r%17 == 0 || r%19 == 0 || r%23 == 0 || r%29 == 0 {
+ return false
+ }
+
+ if _W == 64 && (r%31 == 0 || r%37 == 0 || r%41 == 0 ||
+ r%43 == 0 || r%47 == 0 || r%53 == 0) {
+ return false
+ }
+
+ nm1 := nat{}.sub(n, natOne)
+ // 1<<k * q = nm1;
+ q, k := nm1.powersOfTwoDecompose()
+
+ nm3 := nat{}.sub(nm1, natTwo)
+ rand := rand.New(rand.NewSource(int64(n[0])))
+
+ var x, y, quotient nat
+ nm3Len := nm3.bitLen()
+
+NextRandom:
+ for i := 0; i < reps; i++ {
+ x = x.random(rand, nm3, nm3Len)
+ x = x.add(x, natTwo)
+ y = y.expNN(x, q, n)
+ if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 {
+ continue
+ }
+ for j := 1; j < k; j++ {
+ y = y.mul(y, y)
+ quotient, y = quotient.div(y, y, n)
+ if y.cmp(nm1) == 0 {
+ continue NextRandom
+ }
+ if y.cmp(natOne) == 0 {
+ return false
+ }
+ }
+ return false
+ }
+
+ return true
+}
+
+// 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
+}
+
+// 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)
+
+ k := 0
+ s := uint(0)
+ var d Word
+ for i := len(buf); i > 0; i-- {
+ d |= Word(buf[i-1]) << s
+ if s += 8; s == _S*8 {
+ z[k] = d
+ k++
+ s = 0
+ d = 0
+ }
+ }
+ if k < len(z) {
+ z[k] = d
+ }
+
+ return z.norm()
+}
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