Imported Upstream version 1.4upstream/1.4

1 files changed, 0 insertions, 1508 deletions

diff --git a/src/pkg/math/big/nat.go b/src/pkg/math/big/nat.go
deleted file mode 100644
index 16a87f5c5..000000000
--- a/src/pkg/math/big/nat.go
+++ /dev/null
@@ -1,1508 +0,0 @@
-// 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
-//
-// Methods are typically of the form:
-//
-//	func (z *Int) Op(x, y *Int) *Int	(similar for *Rat)
-//
-// and implement operations z = x Op y with the result as 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. Methods
-// returning a result other than *Int or *Rat take one of the operands as
-// the receiver.
-//
-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"
-	"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}
-	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 = 40 // 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                    + 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 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<<(_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 {
-		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)
-	// 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()
-}
-
-// 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.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 r2 Word
-		q, r2 = z.divW(u, v[0])
-		r = z2.setWord(r2)
-		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 := int(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 and <= 256.
-func (x nat) string(charset string) string {
-	b := Word(len(charset))
-
-	// special cases
-	switch {
-	case b < 2 || MaxBase > 256:
-		panic("illegal base")
-	case len(x) == 0:
-		return string(charset[0])
-	}
-
-	// allocate buffer for conversion
-	i := int(float64(x.bitLen())/math.Log2(float64(b))) + 1 // off by one at most
-	s := make([]byte, i)
-
-	// convert power of two and non power of two bases separately
-	if b == b&-b {
-		// shift is base-b digit size in bits
-		shift := 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
-		}
-
-	} else {
-		// determine "big base"; i.e., the largest possible value bb
-		// that is a power of base b and still fits into a Word
-		// (as in 10^19 for 19 decimal digits in a 64bit Word)
-		bb := b      // big base is b**ndigits
-		ndigits := 1 // number of base b digits
-		for max := Word(_M / b); bb <= max; bb *= b {
-			ndigits++ // maximize ndigits where bb = b**ndigits, bb <= _M
-		}
-
-		// construct table of successive squares of bb*leafSize to use in subdivisions
-		// result (table != nil) <=> (len(x) > leafSize > 0)
-		table := divisors(len(x), b, ndigits, bb)
-
-		// preserve x, create local copy for use by convertWords
-		q := nat(nil).set(x)
-
-		// convert q to string s in base b
-		q.convertWords(s, charset, b, ndigits, bb, table)
-
-		// strip leading zeros
-		// (x != 0; thus s must contain at least one non-zero digit
-		// and the loop will terminate)
-		i = 0
-		for zero := charset[0]; s[i] == zero; {
-			i++
-		}
-	}
-
-	return string(s[i:])
-}
-
-// Convert words of q to base b digits in s. If q is large, it is recursively "split in half"
-// by nat/nat division using tabulated divisors. Otherwise, it is converted iteratively using
-// repeated nat/Word division.
-//
-// The iterative method processes n Words by n divW() calls, each of which visits every Word in the
-// incrementally shortened q for a total of n + (n-1) + (n-2) ... + 2 + 1, or n(n+1)/2 divW()'s.
-// Recursive conversion divides q by its approximate square root, yielding two parts, each half
-// the size of q. Using the iterative method on both halves means 2 * (n/2)(n/2 + 1)/2 divW()'s
-// plus the expensive long div(). Asymptotically, the ratio is favorable at 1/2 the divW()'s, and
-// is made better by splitting the subblocks recursively. Best is to split blocks until one more
-// split would take longer (because of the nat/nat div()) than the twice as many divW()'s of the
-// iterative approach. This threshold is represented by leafSize. Benchmarking of leafSize in the
-// range 2..64 shows that values of 8 and 16 work well, with a 4x speedup at medium lengths and
-// ~30x for 20000 digits. Use nat_test.go's BenchmarkLeafSize tests to optimize leafSize for
-// specific hardware.
-//
-func (q nat) convertWords(s []byte, charset string, b Word, ndigits int, bb Word, table []divisor) {
-	// split larger blocks recursively
-	if table != nil {
-		// len(q) > leafSize > 0
-		var r nat
-		index := len(table) - 1
-		for len(q) > leafSize {
-			// find divisor close to sqrt(q) if possible, but in any case < q
-			maxLength := q.bitLen()     // ~= log2 q, or at of least largest possible q of this bit length
-			minLength := maxLength >> 1 // ~= log2 sqrt(q)
-			for index > 0 && table[index-1].nbits > minLength {
-				index-- // desired
-			}
-			if table[index].nbits >= maxLength && table[index].bbb.cmp(q) >= 0 {
-				index--
-				if index < 0 {
-					panic("internal inconsistency")
-				}
-			}
-
-			// split q into the two digit number (q'*bbb + r) to form independent subblocks
-			q, r = q.div(r, q, table[index].bbb)
-
-			// convert subblocks and collect results in s[:h] and s[h:]
-			h := len(s) - table[index].ndigits
-			r.convertWords(s[h:], charset, b, ndigits, bb, table[0:index])
-			s = s[:h] // == q.convertWords(s, charset, b, ndigits, bb, table[0:index+1])
-		}
-	}
-
-	// having split any large blocks now process the remaining (small) block iteratively
-	i := len(s)
-	var r Word
-	if b == 10 {
-		// hard-coding for 10 here speeds this up by 1.25x (allows for / and % by constants)
-		for len(q) > 0 {
-			// extract least significant, base bb "digit"
-			q, r = q.divW(q, bb)
-			for j := 0; j < ndigits && i > 0; j++ {
-				i--
-				// avoid % computation since r%10 == r - int(r/10)*10;
-				// this appears to be faster for BenchmarkString10000Base10
-				// and smaller strings (but a bit slower for larger ones)
-				t := r / 10
-				s[i] = charset[r-t<<3-t-t] // TODO(gri) replace w/ t*10 once compiler produces better code
-				r = t
-			}
-		}
-	} else {
-		for len(q) > 0 {
-			// extract least significant, base bb "digit"
-			q, r = q.divW(q, bb)
-			for j := 0; j < ndigits && i > 0; j++ {
-				i--
-				s[i] = charset[r%b]
-				r /= b
-			}
-		}
-	}
-
-	// prepend high-order zeroes
-	zero := charset[0]
-	for i > 0 { // while need more leading zeroes
-		i--
-		s[i] = zero
-	}
-}
-
-// Split blocks greater than leafSize Words (or set to 0 to disable recursive conversion)
-// Benchmark and configure leafSize using: go test -bench="Leaf"
-//   8 and 16 effective on 3.0 GHz Xeon "Clovertown" CPU (128 byte cache lines)
-//   8 and 16 effective on 2.66 GHz Core 2 Duo "Penryn" CPU
-var leafSize int = 8 // number of Word-size binary values treat as a monolithic block
-
-type divisor struct {
-	bbb     nat // divisor
-	nbits   int // bit length of divisor (discounting leading zeroes) ~= log2(bbb)
-	ndigits int // digit length of divisor in terms of output base digits
-}
-
-var cacheBase10 struct {
-	sync.Mutex
-	table [64]divisor // cached divisors for base 10
-}
-
-// expWW computes x**y
-func (z nat) expWW(x, y Word) nat {
-	return z.expNN(nat(nil).setWord(x), nat(nil).setWord(y), nil)
-}
-
-// construct table of powers of bb*leafSize to use in subdivisions
-func divisors(m int, b Word, ndigits int, bb Word) []divisor {
-	// only compute table when recursive conversion is enabled and x is large
-	if leafSize == 0 || m <= leafSize {
-		return nil
-	}
-
-	// determine k where (bb**leafSize)**(2**k) >= sqrt(x)
-	k := 1
-	for words := leafSize; words < m>>1 && k < len(cacheBase10.table); words <<= 1 {
-		k++
-	}
-
-	// reuse and extend existing table of divisors or create new table as appropriate
-	var table []divisor // for b == 10, table overlaps with cacheBase10.table
-	if b == 10 {
-		cacheBase10.Lock()
-		table = cacheBase10.table[0:k] // reuse old table for this conversion
-	} else {
-		table = make([]divisor, k) // create new table for this conversion
-	}
-
-	// extend table
-	if table[k-1].ndigits == 0 {
-		// add new entries as needed
-		var larger nat
-		for i := 0; i < k; i++ {
-			if table[i].ndigits == 0 {
-				if i == 0 {
-					table[0].bbb = nat(nil).expWW(bb, Word(leafSize))
-					table[0].ndigits = ndigits * leafSize
-				} else {
-					table[i].bbb = nat(nil).mul(table[i-1].bbb, table[i-1].bbb)
-					table[i].ndigits = 2 * table[i-1].ndigits
-				}
-
-				// optimization: exploit aggregated extra bits in macro blocks
-				larger = nat(nil).set(table[i].bbb)
-				for mulAddVWW(larger, larger, b, 0) == 0 {
-					table[i].bbb = table[i].bbb.set(larger)
-					table[i].ndigits++
-				}
-
-				table[i].nbits = table[i].bbb.bitLen()
-			}
-		}
-	}
-
-	if b == 10 {
-		cacheBase10.Unlock()
-	}
-
-	return table
-}
-
-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 least significant zero
-// bits of x.
-func trailingZeroBits(x Word) uint {
-	// 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.
-	// (Knuth, volume 4, section 7.3.1)
-	switch _W {
-	case 32:
-		return uint(deBruijn32Lookup[((x&-x)*deBruijn32)>>27])
-	case 64:
-		return uint(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58])
-	default:
-		panic("unknown word size")
-	}
-}
-
-// 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 + trailingZeroBits(x[i])
-}
-
-// 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 {
-			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")
-}
-
-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)
-}
-
-// 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
-
-	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.
-	if len(x) > 1 && len(y) > 1 && len(m) > 0 {
-		return z.expNNWindowed(x, y, m)
-	}
-
-	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
-	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)
-	// 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.mul(z, 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.mul(z, 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.mul(*p2, *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.mul(z, z)
-				zz, z = z, zz
-				zz, r = zz.div(r, z, m)
-				z, r = r, z
-
-				zz = zz.mul(z, z)
-				zz, z = z, zz
-				zz, r = zz.div(r, z, m)
-				z, r = r, z
-
-				zz = zz.mul(z, z)
-				zz, z = z, zz
-				zz, r = zz.div(r, z, m)
-				z, r = r, z
-
-				zz = zz.mul(z, 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()
-}
-
-// 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(nil).sub(n, natOne)
-	// determine q, k such that nm1 = q << k
-	k := nm1.trailingZeroBits()
-	q := nat(nil).shr(nm1, k)
-
-	nm3 := nat(nil).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 := uint(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()
-}