Adding upstream version 1.20.14.upstream/1.20.14upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 1429 insertions, 0 deletions

diff --git a/src/math/big/nat.go b/src/math/big/nat.go
new file mode 100644
index 0000000..90ce6d1
--- /dev/null
+++ b/src/math/big/nat.go
@@ -0,0 +1,1429 @@
+// 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) String() string {
+	return "0x" + string(z.itoa(false, 16))
+}
+
+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 recurring)
+	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))
+}
+
+// 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)
+	if n > 0 {
+		(*z)[0] = 0xfedcb // break code expecting zero
+	}
+	return z
+}
+
+func putNat(x *nat) {
+	natPool.Put(x)
+}
+
+var natPool sync.Pool
+
+// bitLen returns the length of x in bits.
+// Unlike most methods, it works even if x is not normalized.
+func (x nat) bitLen() int {
+	// This function is used in cryptographic operations. It must not leak
+	// anything but the Int's sign and bit size through side-channels. Any
+	// changes must be reviewed by a security expert.
+	if i := len(x) - 1; i >= 0 {
+		// bits.Len uses a lookup table for the low-order bits on some
+		// architectures. Neutralize any input-dependent behavior by setting all
+		// bits after the first one bit.
+		top := uint(x[i])
+		top |= top >> 1
+		top |= top >> 2
+		top |= top >> 4
+		top |= top >> 8
+		top |= top >> 16
+		top |= top >> 16 >> 16 // ">> 32" doesn't compile on 32-bit architectures
+		return i*_W + bits.Len(top)
+	}
+	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])))
+}
+
+// isPow2 returns i, true when x == 2**i and 0, false otherwise.
+func (x nat) isPow2() (uint, bool) {
+	var i uint
+	for x[i] == 0 {
+		i++
+	}
+	if i == uint(len(x))-1 && x[i]&(x[i]-1) == 0 {
+		return i*_W + uint(bits.TrailingZeros(uint(x[i]))), true
+	}
+	return 0, false
+}
+
+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()
+}
+
+// trunc returns z = x mod 2ⁿ.
+func (z nat) trunc(x nat, n uint) nat {
+	w := (n + _W - 1) / _W
+	if uint(len(x)) < w {
+		return z.set(x)
+	}
+	z = z.make(int(w))
+	copy(z, x)
+	if n%_W != 0 {
+		z[len(z)-1] &= 1<<(n%_W) - 1
+	}
+	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()
+}
+
+// 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, slow bool) 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
+
+	// 0**y = 0
+	if len(x) == 0 {
+		return z.setWord(0)
+	}
+	// x > 0
+
+	// 1**y = 1
+	if len(x) == 1 && x[0] == 1 {
+		return z.setWord(1)
+	}
+	// x > 1
+
+	// x**1 == x
+	if len(y) == 1 && y[0] == 1 {
+		if len(m) != 0 {
+			return z.rem(x, m)
+		}
+		return z.set(x)
+	}
+	// y > 1
+
+	if len(m) != 0 {
+		// We likely end up being as long as the modulus.
+		z = z.make(len(m))
+
+		// If the exponent is large, we use the Montgomery method for odd values,
+		// and a 4-bit, windowed exponentiation for powers of two,
+		// and a CRT-decomposed Montgomery method for the remaining values
+		// (even values times non-trivial odd values, which decompose into one
+		// instance of each of the first two cases).
+		if len(y) > 1 && !slow {
+			if m[0]&1 == 1 {
+				return z.expNNMontgomery(x, y, m)
+			}
+			if logM, ok := m.isPow2(); ok {
+				return z.expNNWindowed(x, y, logM)
+			}
+			return z.expNNMontgomeryEven(x, y, m)
+		}
+	}
+
+	z = z.set(x)
+	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()
+}
+
+// expNNMontgomeryEven calculates x**y mod m where m = m1 × m2 for m1 = 2ⁿ and m2 odd.
+// It uses two recursive calls to expNN for x**y mod m1 and x**y mod m2
+// and then uses the Chinese Remainder Theorem to combine the results.
+// The recursive call using m1 will use expNNWindowed,
+// while the recursive call using m2 will use expNNMontgomery.
+// For more details, see Ç. K. Koç, “Montgomery Reduction with Even Modulus”,
+// IEE Proceedings: Computers and Digital Techniques, 141(5) 314-316, September 1994.
+// http://www.people.vcu.edu/~jwang3/CMSC691/j34monex.pdf
+func (z nat) expNNMontgomeryEven(x, y, m nat) nat {
+	// Split m = m₁ × m₂ where m₁ = 2ⁿ
+	n := m.trailingZeroBits()
+	m1 := nat(nil).shl(natOne, n)
+	m2 := nat(nil).shr(m, n)
+
+	// We want z = x**y mod m.
+	// z₁ = x**y mod m1 = (x**y mod m) mod m1 = z mod m1
+	// z₂ = x**y mod m2 = (x**y mod m) mod m2 = z mod m2
+	// (We are using the math/big convention for names here,
+	// where the computation is z = x**y mod m, so its parts are z1 and z2.
+	// The paper is computing x = a**e mod n; it refers to these as x2 and z1.)
+	z1 := nat(nil).expNN(x, y, m1, false)
+	z2 := nat(nil).expNN(x, y, m2, false)
+
+	// Reconstruct z from z₁, z₂ using CRT, using algorithm from paper,
+	// which uses only a single modInverse (and an easy one at that).
+	//	p = (z₁ - z₂) × m₂⁻¹ (mod m₁)
+	//	z = z₂ + p × m₂
+	// The final addition is in range because:
+	//	z = z₂ + p × m₂
+	//	  ≤ z₂ + (m₁-1) × m₂
+	//	  < m₂ + (m₁-1) × m₂
+	//	  = m₁ × m₂
+	//	  = m.
+	z = z.set(z2)
+
+	// Compute (z₁ - z₂) mod m1 [m1 == 2**n] into z1.
+	z1 = z1.subMod2N(z1, z2, n)
+
+	// Reuse z2 for p = (z₁ - z₂) [in z1] * m2⁻¹ (mod m₁ [= 2ⁿ]).
+	m2inv := nat(nil).modInverse(m2, m1)
+	z2 = z2.mul(z1, m2inv)
+	z2 = z2.trunc(z2, n)
+
+	// Reuse z1 for p * m2.
+	z = z.add(z, z1.mul(z2, m2))
+
+	return z
+}
+
+// expNNWindowed calculates x**y mod m using a fixed, 4-bit window,
+// where m = 2**logM.
+func (z nat) expNNWindowed(x, y nat, logM uint) nat {
+	if len(y) <= 1 {
+		panic("big: misuse of expNNWindowed")
+	}
+	if x[0]&1 == 0 {
+		// len(y) > 1, so y  > logM.
+		// x is even, so x**y is a multiple of 2**y which is a multiple of 2**logM.
+		return z.setWord(0)
+	}
+	if logM == 1 {
+		return z.setWord(1)
+	}
+
+	// zz is used to avoid allocating in mul as otherwise
+	// the arguments would alias.
+	w := int((logM + _W - 1) / _W)
+	zzp := getNat(w)
+	zz := *zzp
+
+	const n = 4
+	// powers[i] contains x^i.
+	var powers [1 << n]*nat
+	for i := range powers {
+		powers[i] = getNat(w)
+	}
+	*powers[0] = powers[0].set(natOne)
+	*powers[1] = powers[1].trunc(x, logM)
+	for i := 2; i < 1<<n; i += 2 {
+		p2, p, p1 := powers[i/2], powers[i], powers[i+1]
+		*p = p.sqr(*p2)
+		*p = p.trunc(*p, logM)
+		*p1 = p1.mul(*p, x)
+		*p1 = p1.trunc(*p1, logM)
+	}
+
+	// Because phi(2**logM) = 2**(logM-1), x**(2**(logM-1)) = 1,
+	// so we can compute x**(y mod 2**(logM-1)) instead of x**y.
+	// That is, we can throw away all but the bottom logM-1 bits of y.
+	// Instead of allocating a new y, we start reading y at the right word
+	// and truncate it appropriately at the start of the loop.
+	i := len(y) - 1
+	mtop := int((logM - 2) / _W) // -2 because the top word of N bits is the (N-1)/W'th word.
+	mmask := ^Word(0)
+	if mbits := (logM - 1) & (_W - 1); mbits != 0 {
+		mmask = (1 << mbits) - 1
+	}
+	if i > mtop {
+		i = mtop
+	}
+	advance := false
+	z = z.setWord(1)
+	for ; i >= 0; i-- {
+		yi := y[i]
+		if i == mtop {
+			yi &= mmask
+		}
+		for j := 0; j < _W; j += n {
+			if advance {
+				// Account for use of 4 bits in previous iteration.
+				// 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
+				z = z.trunc(z, logM)
+
+				zz = zz.sqr(z)
+				zz, z = z, zz
+				z = z.trunc(z, logM)
+
+				zz = zz.sqr(z)
+				zz, z = z, zz
+				z = z.trunc(z, logM)
+
+				zz = zz.sqr(z)
+				zz, z = z, zz
+				z = z.trunc(z, logM)
+			}
+
+			zz = zz.mul(z, *powers[yi>>(_W-n)])
+			zz, z = z, zz
+			z = z.trunc(z, logM)
+
+			yi <<= n
+			advance = true
+		}
+	}
+
+	*zzp = zz
+	putNat(zzp)
+	for i := range powers {
+		putNat(powers[i])
+	}
+
+	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.
+// The value of z is encoded in the slice buf[i:]. If the value of z
+// cannot be represented in buf, bytes panics. The number i of unused
+// bytes at the beginning of buf is returned as result.
+func (z nat) bytes(buf []byte) (i int) {
+	// This function is used in cryptographic operations. It must not leak
+	// anything but the Int's sign and bit size through side-channels. Any
+	// changes must be reviewed by a security expert.
+	i = len(buf)
+	for _, d := range z {
+		for j := 0; j < _S; j++ {
+			i--
+			if i >= 0 {
+				buf[i] = byte(d)
+			} else if byte(d) != 0 {
+				panic("math/big: buffer too small to fit value")
+			}
+			d >>= 8
+		}
+	}
+
+	if i < 0 {
+		i = 0
+	}
+	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
+	}
+}
+
+// subMod2N returns z = (x - y) mod 2ⁿ.
+func (z nat) subMod2N(x, y nat, n uint) nat {
+	if uint(x.bitLen()) > n {
+		if alias(z, x) {
+			// ok to overwrite x in place
+			x = x.trunc(x, n)
+		} else {
+			x = nat(nil).trunc(x, n)
+		}
+	}
+	if uint(y.bitLen()) > n {
+		if alias(z, y) {
+			// ok to overwrite y in place
+			y = y.trunc(y, n)
+		} else {
+			y = nat(nil).trunc(y, n)
+		}
+	}
+	if x.cmp(y) >= 0 {
+		return z.sub(x, y)
+	}
+	// x - y < 0; x - y mod 2ⁿ = x - y + 2ⁿ = 2ⁿ - (y - x) = 1 + 2ⁿ-1 - (y - x) = 1 + ^(y - x).
+	z = z.sub(y, x)
+	for uint(len(z))*_W < n {
+		z = append(z, 0)
+	}
+	for i := range z {
+		z[i] = ^z[i]
+	}
+	z = z.trunc(z, n)
+	return z.add(z, natOne)
+}