math/big/nat.go

// 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 {
		tp := getNat(3 * k)
		t := *tp

		// 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)
		}

		putNat(tp)
	}

	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)
	tp := getNat(2 * n)
	t := *tp // temporary variable to hold the products
	t.clear()
	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
	putNat(tp)
}

// 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 {
		tp := getNat(2 * k)
		t := *tp
		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
		putNat(tp)
	}

	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 < vIn
// 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)

	if n < divRecursiveThreshold {
		q.divBasic(u, v)
	} else {
		q.divRecursive(u, v)
	}
	putNat(vp)

	q = q.norm()
	shrVU(u, u, shift)
	r = u.norm()

	return q, r
}

// divBasic performs word-by-word division of u by v.
// The quotient is written in pre-allocated q.
// The remainder overwrites input u.
//
// Precondition:
// - q is large enough to hold the quotient u / v
//   which has a maximum length of len(u)-len(v)+1.
func (q nat) divBasic(u, v nat) {
	n := len(v)
	m := len(u) - n

	qhatvp := getNat(n + 1)
	qhatv := *qhatvp

	// D2.
	vn1 := v[n-1]
	for j := m; j >= 0; j-- {
		// D3.
		qhat := Word(_M)
		var ujn Word
		if j+n < len(u) {
			ujn = u[j+n]
		}
		if 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.
		// Compute the remainder u - (q̂*v) << (_W*j).
		// The subtraction may overflow if q̂ estimate was off by one.
		qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
		qhl := len(qhatv)
		if j+qhl > len(u) && qhatv[n] == 0 {
			qhl--
		}
		c := subVV(u[j:j+qhl], u[j:], qhatv)
		if c != 0 {
			c := addVV(u[j:j+n], u[j:], v)
			// If n == qhl, the carry from subVV and the carry from addVV
			// cancel out and don't affect u[j+n].
			if n < qhl {
				u[j+n] += c
			}
			qhat--
		}

		if j == m && m == len(q) && qhat == 0 {
			continue
		}
		q[j] = qhat
	}

	putNat(qhatvp)
}

const divRecursiveThreshold = 100

// divRecursive performs word-by-word division of u by v.
// The quotient is written in pre-allocated z.
// The remainder overwrites input u.
//
// Precondition:
// - len(z) >= len(u)-len(v)
//
// See Burnikel, Ziegler, "Fast Recursive Division", Algorithm 1 and 2.
func (z nat) divRecursive(u, v nat) {
	// Recursion depth is less than 2 log2(len(v))
	// Allocate a slice of temporaries to be reused across recursion.
	recDepth := 2 * bits.Len(uint(len(v)))
	// large enough to perform Karatsuba on operands as large as v
	tmp := getNat(3 * len(v))
	temps := make([]*nat, recDepth)
	z.clear()
	z.divRecursiveStep(u, v, 0, tmp, temps)
	for _, n := range temps {
		if n != nil {
			putNat(n)
		}
	}
	putNat(tmp)
}

// divRecursiveStep computes the division of u by v.
// - z must be large enough to hold the quotient
// - the quotient will overwrite z
// - the remainder will overwrite u
func (z nat) divRecursiveStep(u, v nat, depth int, tmp *nat, temps []*nat) {
	u = u.norm()
	v = v.norm()

	if len(u) == 0 {
		z.clear()
		return
	}
	n := len(v)
	if n < divRecursiveThreshold {
		z.divBasic(u, v)
		return
	}
	m := len(u) - n
	if m < 0 {
		return
	}

	// Produce the quotient by blocks of B words.
	// Division by v (length n) is done using a length n/2 division
	// and a length n/2 multiplication for each block. The final
	// complexity is driven by multiplication complexity.
	B := n / 2

	// Allocate a nat for qhat below.
	if temps[depth] == nil {
		temps[depth] = getNat(n)
	} else {
		*temps[depth] = temps[depth].make(B + 1)
	}

	j := m
	for j > B {
		// Divide u[j-B:j+n] by vIn. Keep remainder in u
		// for next block.
		//
		// The following property will be used (Lemma 2):
		// if u = u1 << s + u0
		//    v = v1 << s + v0
		// then floor(u1/v1) >= floor(u/v)
		//
		// Moreover, the difference is at most 2 if len(v1) >= len(u/v)
		// We choose s = B-1 since len(v)-B >= B+1 >= len(u/v)
		s := (B - 1)
		// Except for the first step, the top bits are always
		// a division remainder, so the quotient length is <= n.
		uu := u[j-B:]

		qhat := *temps[depth]
		qhat.clear()
		qhat.divRecursiveStep(uu[s:B+n], v[s:], depth+1, tmp, temps)
		qhat = qhat.norm()
		// Adjust the quotient:
		//    u = u_h << s + u_l
		//    v = v_h << s + v_l
		//  u_h = q̂ v_h + rh
		//    u = q̂ (v - v_l) + rh << s + u_l
		// After the above step, u contains a remainder:
		//    u = rh << s + u_l
		// and we need to subtract q̂ v_l
		//
		// But it may be a bit too large, in which case q̂ needs to be smaller.
		qhatv := tmp.make(3 * n)
		qhatv.clear()
		qhatv = qhatv.mul(qhat, v[:s])
		for i := 0; i < 2; i++ {
			e := qhatv.cmp(uu.norm())
			if e <= 0 {
				break
			}
			subVW(qhat, qhat, 1)
			c := subVV(qhatv[:s], qhatv[:s], v[:s])
			if len(qhatv) > s {
				subVW(qhatv[s:], qhatv[s:], c)
			}
			addAt(uu[s:], v[s:], 0)
		}
		if qhatv.cmp(uu.norm()) > 0 {
			panic("impossible")
		}
		c := subVV(uu[:len(qhatv)], uu[:len(qhatv)], qhatv)
		if c > 0 {
			subVW(uu[len(qhatv):], uu[len(qhatv):], c)
		}
		addAt(z, qhat, j-B)
		j -= B
	}

	// Now u < (v<<B), compute lower bits in the same way.
	// Choose shift = B-1 again.
	s := B
	qhat := *temps[depth]
	qhat.clear()
	qhat.divRecursiveStep(u[s:].norm(), v[s:], depth+1, tmp, temps)
	qhat = qhat.norm()
	qhatv := tmp.make(3 * n)
	qhatv.clear()
	qhatv = qhatv.mul(qhat, v[:s])
	// Set the correct remainder as before.
	for i := 0; i < 2; i++ {
		if e := qhatv.cmp(u.norm()); e > 0 {
			subVW(qhat, qhat, 1)
			c := subVV(qhatv[:s], qhatv[:s], v[:s])
			if len(qhatv) > s {
				subVW(qhatv[s:], qhatv[s:], c)
			}
			addAt(u[s:], v[s:], 0)
		}
	}
	if qhatv.cmp(u.norm()) > 0 {
		panic("impossible")
	}
	c := subVV(u[0:len(qhatv)], u[0:len(qhatv)], qhatv)
	if c > 0 {
		c = subVW(u[len(qhatv):], u[len(qhatv):], c)
	}
	if c > 0 {
		panic("impossible")
	}

	// Done!
	addAt(z, qhat.norm(), 0)
}

// 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
	}
}