Adding upstream version 1.20.14.upstream/1.20.14upstream

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

1 files changed, 703 insertions, 0 deletions

diff --git a/src/crypto/internal/bigmod/nat.go b/src/crypto/internal/bigmod/nat.go
new file mode 100644
index 0000000..804316f
--- /dev/null
+++ b/src/crypto/internal/bigmod/nat.go
@@ -0,0 +1,703 @@
+// Copyright 2021 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 bigmod
+
+import (
+	"errors"
+	"math/big"
+	"math/bits"
+)
+
+const (
+	// _W is the number of bits we use for our limbs.
+	_W = bits.UintSize - 1
+	// _MASK selects _W bits from a full machine word.
+	_MASK = (1 << _W) - 1
+)
+
+// choice represents a constant-time boolean. The value of choice is always
+// either 1 or 0. We use an int instead of bool in order to make decisions in
+// constant time by turning it into a mask.
+type choice uint
+
+func not(c choice) choice { return 1 ^ c }
+
+const yes = choice(1)
+const no = choice(0)
+
+// ctSelect returns x if on == 1, and y if on == 0. The execution time of this
+// function does not depend on its inputs. If on is any value besides 1 or 0,
+// the result is undefined.
+func ctSelect(on choice, x, y uint) uint {
+	// When on == 1, mask is 0b111..., otherwise mask is 0b000...
+	mask := -uint(on)
+	// When mask is all zeros, we just have y, otherwise, y cancels with itself.
+	return y ^ (mask & (y ^ x))
+}
+
+// ctEq returns 1 if x == y, and 0 otherwise. The execution time of this
+// function does not depend on its inputs.
+func ctEq(x, y uint) choice {
+	// If x != y, then either x - y or y - x will generate a carry.
+	_, c1 := bits.Sub(x, y, 0)
+	_, c2 := bits.Sub(y, x, 0)
+	return not(choice(c1 | c2))
+}
+
+// ctGeq returns 1 if x >= y, and 0 otherwise. The execution time of this
+// function does not depend on its inputs.
+func ctGeq(x, y uint) choice {
+	// If x < y, then x - y generates a carry.
+	_, carry := bits.Sub(x, y, 0)
+	return not(choice(carry))
+}
+
+// Nat represents an arbitrary natural number
+//
+// Each Nat has an announced length, which is the number of limbs it has stored.
+// Operations on this number are allowed to leak this length, but will not leak
+// any information about the values contained in those limbs.
+type Nat struct {
+	// limbs is a little-endian representation in base 2^W with
+	// W = bits.UintSize - 1. The top bit is always unset between operations.
+	//
+	// The top bit is left unset to optimize Montgomery multiplication, in the
+	// inner loop of exponentiation. Using fully saturated limbs would leave us
+	// working with 129-bit numbers on 64-bit platforms, wasting a lot of space,
+	// and thus time.
+	limbs []uint
+}
+
+// preallocTarget is the size in bits of the numbers used to implement the most
+// common and most performant RSA key size. It's also enough to cover some of
+// the operations of key sizes up to 4096.
+const preallocTarget = 2048
+const preallocLimbs = (preallocTarget + _W - 1) / _W
+
+// NewNat returns a new nat with a size of zero, just like new(Nat), but with
+// the preallocated capacity to hold a number of up to preallocTarget bits.
+// NewNat inlines, so the allocation can live on the stack.
+func NewNat() *Nat {
+	limbs := make([]uint, 0, preallocLimbs)
+	return &Nat{limbs}
+}
+
+// expand expands x to n limbs, leaving its value unchanged.
+func (x *Nat) expand(n int) *Nat {
+	if len(x.limbs) > n {
+		panic("bigmod: internal error: shrinking nat")
+	}
+	if cap(x.limbs) < n {
+		newLimbs := make([]uint, n)
+		copy(newLimbs, x.limbs)
+		x.limbs = newLimbs
+		return x
+	}
+	extraLimbs := x.limbs[len(x.limbs):n]
+	for i := range extraLimbs {
+		extraLimbs[i] = 0
+	}
+	x.limbs = x.limbs[:n]
+	return x
+}
+
+// reset returns a zero nat of n limbs, reusing x's storage if n <= cap(x.limbs).
+func (x *Nat) reset(n int) *Nat {
+	if cap(x.limbs) < n {
+		x.limbs = make([]uint, n)
+		return x
+	}
+	for i := range x.limbs {
+		x.limbs[i] = 0
+	}
+	x.limbs = x.limbs[:n]
+	return x
+}
+
+// set assigns x = y, optionally resizing x to the appropriate size.
+func (x *Nat) set(y *Nat) *Nat {
+	x.reset(len(y.limbs))
+	copy(x.limbs, y.limbs)
+	return x
+}
+
+// setBig assigns x = n, optionally resizing n to the appropriate size.
+//
+// The announced length of x is set based on the actual bit size of the input,
+// ignoring leading zeroes.
+func (x *Nat) setBig(n *big.Int) *Nat {
+	requiredLimbs := (n.BitLen() + _W - 1) / _W
+	x.reset(requiredLimbs)
+
+	outI := 0
+	shift := 0
+	limbs := n.Bits()
+	for i := range limbs {
+		xi := uint(limbs[i])
+		x.limbs[outI] |= (xi << shift) & _MASK
+		outI++
+		if outI == requiredLimbs {
+			return x
+		}
+		x.limbs[outI] = xi >> (_W - shift)
+		shift++ // this assumes bits.UintSize - _W = 1
+		if shift == _W {
+			shift = 0
+			outI++
+		}
+	}
+	return x
+}
+
+// Bytes returns x as a zero-extended big-endian byte slice. The size of the
+// slice will match the size of m.
+//
+// x must have the same size as m and it must be reduced modulo m.
+func (x *Nat) Bytes(m *Modulus) []byte {
+	bytes := make([]byte, m.Size())
+	shift := 0
+	outI := len(bytes) - 1
+	for _, limb := range x.limbs {
+		remainingBits := _W
+		for remainingBits >= 8 {
+			bytes[outI] |= byte(limb) << shift
+			consumed := 8 - shift
+			limb >>= consumed
+			remainingBits -= consumed
+			shift = 0
+			outI--
+			if outI < 0 {
+				return bytes
+			}
+		}
+		bytes[outI] = byte(limb)
+		shift = remainingBits
+	}
+	return bytes
+}
+
+// SetBytes assigns x = b, where b is a slice of big-endian bytes.
+// SetBytes returns an error if b >= m.
+//
+// The output will be resized to the size of m and overwritten.
+func (x *Nat) SetBytes(b []byte, m *Modulus) (*Nat, error) {
+	if err := x.setBytes(b, m); err != nil {
+		return nil, err
+	}
+	if x.cmpGeq(m.nat) == yes {
+		return nil, errors.New("input overflows the modulus")
+	}
+	return x, nil
+}
+
+// SetOverflowingBytes assigns x = b, where b is a slice of big-endian bytes. SetOverflowingBytes
+// returns an error if b has a longer bit length than m, but reduces overflowing
+// values up to 2^⌈log2(m)⌉ - 1.
+//
+// The output will be resized to the size of m and overwritten.
+func (x *Nat) SetOverflowingBytes(b []byte, m *Modulus) (*Nat, error) {
+	if err := x.setBytes(b, m); err != nil {
+		return nil, err
+	}
+	leading := _W - bitLen(x.limbs[len(x.limbs)-1])
+	if leading < m.leading {
+		return nil, errors.New("input overflows the modulus")
+	}
+	x.sub(x.cmpGeq(m.nat), m.nat)
+	return x, nil
+}
+
+func (x *Nat) setBytes(b []byte, m *Modulus) error {
+	outI := 0
+	shift := 0
+	x.resetFor(m)
+	for i := len(b) - 1; i >= 0; i-- {
+		bi := b[i]
+		x.limbs[outI] |= uint(bi) << shift
+		shift += 8
+		if shift >= _W {
+			shift -= _W
+			x.limbs[outI] &= _MASK
+			overflow := bi >> (8 - shift)
+			outI++
+			if outI >= len(x.limbs) {
+				if overflow > 0 || i > 0 {
+					return errors.New("input overflows the modulus")
+				}
+				break
+			}
+			x.limbs[outI] = uint(overflow)
+		}
+	}
+	return nil
+}
+
+// Equal returns 1 if x == y, and 0 otherwise.
+//
+// Both operands must have the same announced length.
+func (x *Nat) Equal(y *Nat) choice {
+	// Eliminate bounds checks in the loop.
+	size := len(x.limbs)
+	xLimbs := x.limbs[:size]
+	yLimbs := y.limbs[:size]
+
+	equal := yes
+	for i := 0; i < size; i++ {
+		equal &= ctEq(xLimbs[i], yLimbs[i])
+	}
+	return equal
+}
+
+// IsZero returns 1 if x == 0, and 0 otherwise.
+func (x *Nat) IsZero() choice {
+	// Eliminate bounds checks in the loop.
+	size := len(x.limbs)
+	xLimbs := x.limbs[:size]
+
+	zero := yes
+	for i := 0; i < size; i++ {
+		zero &= ctEq(xLimbs[i], 0)
+	}
+	return zero
+}
+
+// cmpGeq returns 1 if x >= y, and 0 otherwise.
+//
+// Both operands must have the same announced length.
+func (x *Nat) cmpGeq(y *Nat) choice {
+	// Eliminate bounds checks in the loop.
+	size := len(x.limbs)
+	xLimbs := x.limbs[:size]
+	yLimbs := y.limbs[:size]
+
+	var c uint
+	for i := 0; i < size; i++ {
+		c = (xLimbs[i] - yLimbs[i] - c) >> _W
+	}
+	// If there was a carry, then subtracting y underflowed, so
+	// x is not greater than or equal to y.
+	return not(choice(c))
+}
+
+// assign sets x <- y if on == 1, and does nothing otherwise.
+//
+// Both operands must have the same announced length.
+func (x *Nat) assign(on choice, y *Nat) *Nat {
+	// Eliminate bounds checks in the loop.
+	size := len(x.limbs)
+	xLimbs := x.limbs[:size]
+	yLimbs := y.limbs[:size]
+
+	for i := 0; i < size; i++ {
+		xLimbs[i] = ctSelect(on, yLimbs[i], xLimbs[i])
+	}
+	return x
+}
+
+// add computes x += y if on == 1, and does nothing otherwise. It returns the
+// carry of the addition regardless of on.
+//
+// Both operands must have the same announced length.
+func (x *Nat) add(on choice, y *Nat) (c uint) {
+	// Eliminate bounds checks in the loop.
+	size := len(x.limbs)
+	xLimbs := x.limbs[:size]
+	yLimbs := y.limbs[:size]
+
+	for i := 0; i < size; i++ {
+		res := xLimbs[i] + yLimbs[i] + c
+		xLimbs[i] = ctSelect(on, res&_MASK, xLimbs[i])
+		c = res >> _W
+	}
+	return
+}
+
+// sub computes x -= y if on == 1, and does nothing otherwise. It returns the
+// borrow of the subtraction regardless of on.
+//
+// Both operands must have the same announced length.
+func (x *Nat) sub(on choice, y *Nat) (c uint) {
+	// Eliminate bounds checks in the loop.
+	size := len(x.limbs)
+	xLimbs := x.limbs[:size]
+	yLimbs := y.limbs[:size]
+
+	for i := 0; i < size; i++ {
+		res := xLimbs[i] - yLimbs[i] - c
+		xLimbs[i] = ctSelect(on, res&_MASK, xLimbs[i])
+		c = res >> _W
+	}
+	return
+}
+
+// Modulus is used for modular arithmetic, precomputing relevant constants.
+//
+// Moduli are assumed to be odd numbers. Moduli can also leak the exact
+// number of bits needed to store their value, and are stored without padding.
+//
+// Their actual value is still kept secret.
+type Modulus struct {
+	// The underlying natural number for this modulus.
+	//
+	// This will be stored without any padding, and shouldn't alias with any
+	// other natural number being used.
+	nat     *Nat
+	leading int  // number of leading zeros in the modulus
+	m0inv   uint // -nat.limbs[0]⁻¹ mod _W
+	rr      *Nat // R*R for montgomeryRepresentation
+}
+
+// rr returns R*R with R = 2^(_W * n) and n = len(m.nat.limbs).
+func rr(m *Modulus) *Nat {
+	rr := NewNat().ExpandFor(m)
+	// R*R is 2^(2 * _W * n). We can safely get 2^(_W * (n - 1)) by setting the
+	// most significant limb to 1. We then get to R*R by shifting left by _W
+	// n + 1 times.
+	n := len(rr.limbs)
+	rr.limbs[n-1] = 1
+	for i := n - 1; i < 2*n; i++ {
+		rr.shiftIn(0, m) // x = x * 2^_W mod m
+	}
+	return rr
+}
+
+// minusInverseModW computes -x⁻¹ mod _W with x odd.
+//
+// This operation is used to precompute a constant involved in Montgomery
+// multiplication.
+func minusInverseModW(x uint) uint {
+	// Every iteration of this loop doubles the least-significant bits of
+	// correct inverse in y. The first three bits are already correct (1⁻¹ = 1,
+	// 3⁻¹ = 3, 5⁻¹ = 5, and 7⁻¹ = 7 mod 8), so doubling five times is enough
+	// for 61 bits (and wastes only one iteration for 31 bits).
+	//
+	// See https://crypto.stackexchange.com/a/47496.
+	y := x
+	for i := 0; i < 5; i++ {
+		y = y * (2 - x*y)
+	}
+	return (1 << _W) - (y & _MASK)
+}
+
+// NewModulusFromBig creates a new Modulus from a [big.Int].
+//
+// The Int must be odd. The number of significant bits must be leakable.
+func NewModulusFromBig(n *big.Int) *Modulus {
+	m := &Modulus{}
+	m.nat = NewNat().setBig(n)
+	m.leading = _W - bitLen(m.nat.limbs[len(m.nat.limbs)-1])
+	m.m0inv = minusInverseModW(m.nat.limbs[0])
+	m.rr = rr(m)
+	return m
+}
+
+// bitLen is a version of bits.Len that only leaks the bit length of n, but not
+// its value. bits.Len and bits.LeadingZeros use a lookup table for the
+// low-order bits on some architectures.
+func bitLen(n uint) int {
+	var len int
+	// We assume, here and elsewhere, that comparison to zero is constant time
+	// with respect to different non-zero values.
+	for n != 0 {
+		len++
+		n >>= 1
+	}
+	return len
+}
+
+// Size returns the size of m in bytes.
+func (m *Modulus) Size() int {
+	return (m.BitLen() + 7) / 8
+}
+
+// BitLen returns the size of m in bits.
+func (m *Modulus) BitLen() int {
+	return len(m.nat.limbs)*_W - int(m.leading)
+}
+
+// Nat returns m as a Nat. The return value must not be written to.
+func (m *Modulus) Nat() *Nat {
+	return m.nat
+}
+
+// shiftIn calculates x = x << _W + y mod m.
+//
+// This assumes that x is already reduced mod m, and that y < 2^_W.
+func (x *Nat) shiftIn(y uint, m *Modulus) *Nat {
+	d := NewNat().resetFor(m)
+
+	// Eliminate bounds checks in the loop.
+	size := len(m.nat.limbs)
+	xLimbs := x.limbs[:size]
+	dLimbs := d.limbs[:size]
+	mLimbs := m.nat.limbs[:size]
+
+	// Each iteration of this loop computes x = 2x + b mod m, where b is a bit
+	// from y. Effectively, it left-shifts x and adds y one bit at a time,
+	// reducing it every time.
+	//
+	// To do the reduction, each iteration computes both 2x + b and 2x + b - m.
+	// The next iteration (and finally the return line) will use either result
+	// based on whether the subtraction underflowed.
+	needSubtraction := no
+	for i := _W - 1; i >= 0; i-- {
+		carry := (y >> i) & 1
+		var borrow uint
+		for i := 0; i < size; i++ {
+			l := ctSelect(needSubtraction, dLimbs[i], xLimbs[i])
+
+			res := l<<1 + carry
+			xLimbs[i] = res & _MASK
+			carry = res >> _W
+
+			res = xLimbs[i] - mLimbs[i] - borrow
+			dLimbs[i] = res & _MASK
+			borrow = res >> _W
+		}
+		// See Add for how carry (aka overflow), borrow (aka underflow), and
+		// needSubtraction relate.
+		needSubtraction = ctEq(carry, borrow)
+	}
+	return x.assign(needSubtraction, d)
+}
+
+// Mod calculates out = x mod m.
+//
+// This works regardless how large the value of x is.
+//
+// The output will be resized to the size of m and overwritten.
+func (out *Nat) Mod(x *Nat, m *Modulus) *Nat {
+	out.resetFor(m)
+	// Working our way from the most significant to the least significant limb,
+	// we can insert each limb at the least significant position, shifting all
+	// previous limbs left by _W. This way each limb will get shifted by the
+	// correct number of bits. We can insert at least N - 1 limbs without
+	// overflowing m. After that, we need to reduce every time we shift.
+	i := len(x.limbs) - 1
+	// For the first N - 1 limbs we can skip the actual shifting and position
+	// them at the shifted position, which starts at min(N - 2, i).
+	start := len(m.nat.limbs) - 2
+	if i < start {
+		start = i
+	}
+	for j := start; j >= 0; j-- {
+		out.limbs[j] = x.limbs[i]
+		i--
+	}
+	// We shift in the remaining limbs, reducing modulo m each time.
+	for i >= 0 {
+		out.shiftIn(x.limbs[i], m)
+		i--
+	}
+	return out
+}
+
+// ExpandFor ensures out has the right size to work with operations modulo m.
+//
+// The announced size of out must be smaller than or equal to that of m.
+func (out *Nat) ExpandFor(m *Modulus) *Nat {
+	return out.expand(len(m.nat.limbs))
+}
+
+// resetFor ensures out has the right size to work with operations modulo m.
+//
+// out is zeroed and may start at any size.
+func (out *Nat) resetFor(m *Modulus) *Nat {
+	return out.reset(len(m.nat.limbs))
+}
+
+// Sub computes x = x - y mod m.
+//
+// The length of both operands must be the same as the modulus. Both operands
+// must already be reduced modulo m.
+func (x *Nat) Sub(y *Nat, m *Modulus) *Nat {
+	underflow := x.sub(yes, y)
+	// If the subtraction underflowed, add m.
+	x.add(choice(underflow), m.nat)
+	return x
+}
+
+// Add computes x = x + y mod m.
+//
+// The length of both operands must be the same as the modulus. Both operands
+// must already be reduced modulo m.
+func (x *Nat) Add(y *Nat, m *Modulus) *Nat {
+	overflow := x.add(yes, y)
+	underflow := not(x.cmpGeq(m.nat)) // x < m
+
+	// Three cases are possible:
+	//
+	//   - overflow = 0, underflow = 0
+	//
+	// In this case, addition fits in our limbs, but we can still subtract away
+	// m without an underflow, so we need to perform the subtraction to reduce
+	// our result.
+	//
+	//   - overflow = 0, underflow = 1
+	//
+	// The addition fits in our limbs, but we can't subtract m without
+	// underflowing. The result is already reduced.
+	//
+	//   - overflow = 1, underflow = 1
+	//
+	// The addition does not fit in our limbs, and the subtraction's borrow
+	// would cancel out with the addition's carry. We need to subtract m to
+	// reduce our result.
+	//
+	// The overflow = 1, underflow = 0 case is not possible, because y is at
+	// most m - 1, and if adding m - 1 overflows, then subtracting m must
+	// necessarily underflow.
+	needSubtraction := ctEq(overflow, uint(underflow))
+
+	x.sub(needSubtraction, m.nat)
+	return x
+}
+
+// montgomeryRepresentation calculates x = x * R mod m, with R = 2^(_W * n) and
+// n = len(m.nat.limbs).
+//
+// Faster Montgomery multiplication replaces standard modular multiplication for
+// numbers in this representation.
+//
+// This assumes that x is already reduced mod m.
+func (x *Nat) montgomeryRepresentation(m *Modulus) *Nat {
+	// A Montgomery multiplication (which computes a * b / R) by R * R works out
+	// to a multiplication by R, which takes the value out of the Montgomery domain.
+	return x.montgomeryMul(NewNat().set(x), m.rr, m)
+}
+
+// montgomeryReduction calculates x = x / R mod m, with R = 2^(_W * n) and
+// n = len(m.nat.limbs).
+//
+// This assumes that x is already reduced mod m.
+func (x *Nat) montgomeryReduction(m *Modulus) *Nat {
+	// By Montgomery multiplying with 1 not in Montgomery representation, we
+	// convert out back from Montgomery representation, because it works out to
+	// dividing by R.
+	t0 := NewNat().set(x)
+	t1 := NewNat().ExpandFor(m)
+	t1.limbs[0] = 1
+	return x.montgomeryMul(t0, t1, m)
+}
+
+// montgomeryMul calculates d = a * b / R mod m, with R = 2^(_W * n) and
+// n = len(m.nat.limbs), using the Montgomery Multiplication technique.
+//
+// All inputs should be the same length, not aliasing d, and already
+// reduced modulo m. d will be resized to the size of m and overwritten.
+func (d *Nat) montgomeryMul(a *Nat, b *Nat, m *Modulus) *Nat {
+	d.resetFor(m)
+	if len(a.limbs) != len(m.nat.limbs) || len(b.limbs) != len(m.nat.limbs) {
+		panic("bigmod: invalid montgomeryMul input")
+	}
+
+	// See https://bearssl.org/bigint.html#montgomery-reduction-and-multiplication
+	// for a description of the algorithm implemented mostly in montgomeryLoop.
+	// See Add for how overflow, underflow, and needSubtraction relate.
+	overflow := montgomeryLoop(d.limbs, a.limbs, b.limbs, m.nat.limbs, m.m0inv)
+	underflow := not(d.cmpGeq(m.nat)) // d < m
+	needSubtraction := ctEq(overflow, uint(underflow))
+	d.sub(needSubtraction, m.nat)
+
+	return d
+}
+
+func montgomeryLoopGeneric(d, a, b, m []uint, m0inv uint) (overflow uint) {
+	// Eliminate bounds checks in the loop.
+	size := len(d)
+	a = a[:size]
+	b = b[:size]
+	m = m[:size]
+
+	for _, ai := range a {
+		// This is an unrolled iteration of the loop below with j = 0.
+		hi, lo := bits.Mul(ai, b[0])
+		z_lo, c := bits.Add(d[0], lo, 0)
+		f := (z_lo * m0inv) & _MASK // (d[0] + a[i] * b[0]) * m0inv
+		z_hi, _ := bits.Add(0, hi, c)
+		hi, lo = bits.Mul(f, m[0])
+		z_lo, c = bits.Add(z_lo, lo, 0)
+		z_hi, _ = bits.Add(z_hi, hi, c)
+		carry := z_hi<<1 | z_lo>>_W
+
+		for j := 1; j < size; j++ {
+			// z = d[j] + a[i] * b[j] + f * m[j] + carry <= 2^(2W+1) - 2^(W+1) + 2^W
+			hi, lo := bits.Mul(ai, b[j])
+			z_lo, c := bits.Add(d[j], lo, 0)
+			z_hi, _ := bits.Add(0, hi, c)
+			hi, lo = bits.Mul(f, m[j])
+			z_lo, c = bits.Add(z_lo, lo, 0)
+			z_hi, _ = bits.Add(z_hi, hi, c)
+			z_lo, c = bits.Add(z_lo, carry, 0)
+			z_hi, _ = bits.Add(z_hi, 0, c)
+			d[j-1] = z_lo & _MASK
+			carry = z_hi<<1 | z_lo>>_W // carry <= 2^(W+1) - 2
+		}
+
+		z := overflow + carry // z <= 2^(W+1) - 1
+		d[size-1] = z & _MASK
+		overflow = z >> _W // overflow <= 1
+	}
+	return
+}
+
+// Mul calculates x *= y mod m.
+//
+// x and y must already be reduced modulo m, they must share its announced
+// length, and they may not alias.
+func (x *Nat) Mul(y *Nat, m *Modulus) *Nat {
+	// A Montgomery multiplication by a value out of the Montgomery domain
+	// takes the result out of Montgomery representation.
+	xR := NewNat().set(x).montgomeryRepresentation(m) // xR = x * R mod m
+	return x.montgomeryMul(xR, y, m)                  // x = xR * y / R mod m
+}
+
+// Exp calculates out = x^e mod m.
+//
+// The exponent e is represented in big-endian order. The output will be resized
+// to the size of m and overwritten. x must already be reduced modulo m.
+func (out *Nat) Exp(x *Nat, e []byte, m *Modulus) *Nat {
+	// We use a 4 bit window. For our RSA workload, 4 bit windows are faster
+	// than 2 bit windows, but use an extra 12 nats worth of scratch space.
+	// Using bit sizes that don't divide 8 are more complex to implement.
+
+	table := [(1 << 4) - 1]*Nat{ // table[i] = x ^ (i+1)
+		// newNat calls are unrolled so they are allocated on the stack.
+		NewNat(), NewNat(), NewNat(), NewNat(), NewNat(),
+		NewNat(), NewNat(), NewNat(), NewNat(), NewNat(),
+		NewNat(), NewNat(), NewNat(), NewNat(), NewNat(),
+	}
+	table[0].set(x).montgomeryRepresentation(m)
+	for i := 1; i < len(table); i++ {
+		table[i].montgomeryMul(table[i-1], table[0], m)
+	}
+
+	out.resetFor(m)
+	out.limbs[0] = 1
+	out.montgomeryRepresentation(m)
+	t0 := NewNat().ExpandFor(m)
+	t1 := NewNat().ExpandFor(m)
+	for _, b := range e {
+		for _, j := range []int{4, 0} {
+			// Square four times.
+			t1.montgomeryMul(out, out, m)
+			out.montgomeryMul(t1, t1, m)
+			t1.montgomeryMul(out, out, m)
+			out.montgomeryMul(t1, t1, m)
+
+			// Select x^k in constant time from the table.
+			k := uint((b >> j) & 0b1111)
+			for i := range table {
+				t0.assign(ctEq(k, uint(i+1)), table[i])
+			}
+
+			// Multiply by x^k, discarding the result if k = 0.
+			t1.montgomeryMul(out, t0, m)
+			out.assign(not(ctEq(k, 0)), t1)
+		}
+	}
+
+	return out.montgomeryReduction(m)
+}