math/big/int.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 signed multi-precision integers.

package big

import (
	"fmt"
	"io"
	"math/rand"
	"strings"
)

// An Int represents a signed multi-precision integer.
// The zero value for an Int represents the value 0.
type Int struct {
	neg bool // sign
	abs nat  // absolute value of the integer
}

var intOne = &Int{false, natOne}

// Sign returns:
//
//	-1 if x <  0
//	 0 if x == 0
//	+1 if x >  0
//
func (x *Int) Sign() int {
	if len(x.abs) == 0 {
		return 0
	}
	if x.neg {
		return -1
	}
	return 1
}

// SetInt64 sets z to x and returns z.
func (z *Int) SetInt64(x int64) *Int {
	neg := false
	if x < 0 {
		neg = true
		x = -x
	}
	z.abs = z.abs.setUint64(uint64(x))
	z.neg = neg
	return z
}

// SetUint64 sets z to x and returns z.
func (z *Int) SetUint64(x uint64) *Int {
	z.abs = z.abs.setUint64(x)
	z.neg = false
	return z
}

// NewInt allocates and returns a new Int set to x.
func NewInt(x int64) *Int {
	return new(Int).SetInt64(x)
}

// Set sets z to x and returns z.
func (z *Int) Set(x *Int) *Int {
	if z != x {
		z.abs = z.abs.set(x.abs)
		z.neg = x.neg
	}
	return z
}

// Bits provides raw (unchecked but fast) access to x by returning its
// absolute value as a little-endian Word slice. The result and x share
// the same underlying array.
// Bits is intended to support implementation of missing low-level Int
// functionality outside this package; it should be avoided otherwise.
func (x *Int) Bits() []Word {
	return x.abs
}

// SetBits provides raw (unchecked but fast) access to z by setting its
// value to abs, interpreted as a little-endian Word slice, and returning
// z. The result and abs share the same underlying array.
// SetBits is intended to support implementation of missing low-level Int
// functionality outside this package; it should be avoided otherwise.
func (z *Int) SetBits(abs []Word) *Int {
	z.abs = nat(abs).norm()
	z.neg = false
	return z
}

// Abs sets z to |x| (the absolute value of x) and returns z.
func (z *Int) Abs(x *Int) *Int {
	z.Set(x)
	z.neg = false
	return z
}

// Neg sets z to -x and returns z.
func (z *Int) Neg(x *Int) *Int {
	z.Set(x)
	z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
	return z
}

// Add sets z to the sum x+y and returns z.
func (z *Int) Add(x, y *Int) *Int {
	neg := x.neg
	if x.neg == y.neg {
		// x + y == x + y
		// (-x) + (-y) == -(x + y)
		z.abs = z.abs.add(x.abs, y.abs)
	} else {
		// x + (-y) == x - y == -(y - x)
		// (-x) + y == y - x == -(x - y)
		if x.abs.cmp(y.abs) >= 0 {
			z.abs = z.abs.sub(x.abs, y.abs)
		} else {
			neg = !neg
			z.abs = z.abs.sub(y.abs, x.abs)
		}
	}
	z.neg = len(z.abs) > 0 && neg // 0 has no sign
	return z
}

// Sub sets z to the difference x-y and returns z.
func (z *Int) Sub(x, y *Int) *Int {
	neg := x.neg
	if x.neg != y.neg {
		// x - (-y) == x + y
		// (-x) - y == -(x + y)
		z.abs = z.abs.add(x.abs, y.abs)
	} else {
		// x - y == x - y == -(y - x)
		// (-x) - (-y) == y - x == -(x - y)
		if x.abs.cmp(y.abs) >= 0 {
			z.abs = z.abs.sub(x.abs, y.abs)
		} else {
			neg = !neg
			z.abs = z.abs.sub(y.abs, x.abs)
		}
	}
	z.neg = len(z.abs) > 0 && neg // 0 has no sign
	return z
}

// Mul sets z to the product x*y and returns z.
func (z *Int) Mul(x, y *Int) *Int {
	// x * y == x * y
	// x * (-y) == -(x * y)
	// (-x) * y == -(x * y)
	// (-x) * (-y) == x * y
	if x == y {
		z.abs = z.abs.sqr(x.abs)
		z.neg = false
		return z
	}
	z.abs = z.abs.mul(x.abs, y.abs)
	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
	return z
}

// MulRange sets z to the product of all integers
// in the range [a, b] inclusively and returns z.
// If a > b (empty range), the result is 1.
func (z *Int) MulRange(a, b int64) *Int {
	switch {
	case a > b:
		return z.SetInt64(1) // empty range
	case a <= 0 && b >= 0:
		return z.SetInt64(0) // range includes 0
	}
	// a <= b && (b < 0 || a > 0)

	neg := false
	if a < 0 {
		neg = (b-a)&1 == 0
		a, b = -b, -a
	}

	z.abs = z.abs.mulRange(uint64(a), uint64(b))
	z.neg = neg
	return z
}

// Binomial sets z to the binomial coefficient of (n, k) and returns z.
func (z *Int) Binomial(n, k int64) *Int {
	// reduce the number of multiplications by reducing k
	if n/2 < k && k <= n {
		k = n - k // Binomial(n, k) == Binomial(n, n-k)
	}
	var a, b Int
	a.MulRange(n-k+1, n)
	b.MulRange(1, k)
	return z.Quo(&a, &b)
}

// Quo sets z to the quotient x/y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Quo implements truncated division (like Go); see QuoRem for more details.
func (z *Int) Quo(x, y *Int) *Int {
	z.abs, _ = z.abs.div(nil, x.abs, y.abs)
	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
	return z
}

// Rem sets z to the remainder x%y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Rem implements truncated modulus (like Go); see QuoRem for more details.
func (z *Int) Rem(x, y *Int) *Int {
	_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
	z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
	return z
}

// QuoRem sets z to the quotient x/y and r to the remainder x%y
// and returns the pair (z, r) for y != 0.
// If y == 0, a division-by-zero run-time panic occurs.
//
// QuoRem implements T-division and modulus (like Go):
//
//	q = x/y      with the result truncated to zero
//	r = x - y*q
//
// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
// See DivMod for Euclidean division and modulus (unlike Go).
//
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
	z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
	z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
	return z, r
}

// Div sets z to the quotient x/y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Div implements Euclidean division (unlike Go); see DivMod for more details.
func (z *Int) Div(x, y *Int) *Int {
	y_neg := y.neg // z may be an alias for y
	var r Int
	z.QuoRem(x, y, &r)
	if r.neg {
		if y_neg {
			z.Add(z, intOne)
		} else {
			z.Sub(z, intOne)
		}
	}
	return z
}

// Mod sets z to the modulus x%y for y != 0 and returns z.
// If y == 0, a division-by-zero run-time panic occurs.
// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
func (z *Int) Mod(x, y *Int) *Int {
	y0 := y // save y
	if z == y || alias(z.abs, y.abs) {
		y0 = new(Int).Set(y)
	}
	var q Int
	q.QuoRem(x, y, z)
	if z.neg {
		if y0.neg {
			z.Sub(z, y0)
		} else {
			z.Add(z, y0)
		}
	}
	return z
}

// DivMod sets z to the quotient x div y and m to the modulus x mod y
// and returns the pair (z, m) for y != 0.
// If y == 0, a division-by-zero run-time panic occurs.
//
// DivMod implements Euclidean division and modulus (unlike Go):
//
//	q = x div y  such that
//	m = x - y*q  with 0 <= m < |y|
//
// (See Raymond T. Boute, ``The Euclidean definition of the functions
// div and mod''. ACM Transactions on Programming Languages and
// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
// ACM press.)
// See QuoRem for T-division and modulus (like Go).
//
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
	y0 := y // save y
	if z == y || alias(z.abs, y.abs) {
		y0 = new(Int).Set(y)
	}
	z.QuoRem(x, y, m)
	if m.neg {
		if y0.neg {
			z.Add(z, intOne)
			m.Sub(m, y0)
		} else {
			z.Sub(z, intOne)
			m.Add(m, y0)
		}
	}
	return z, m
}

// Cmp compares x and y and returns:
//
//   -1 if x <  y
//    0 if x == y
//   +1 if x >  y
//
func (x *Int) Cmp(y *Int) (r int) {
	// x cmp y == x cmp y
	// x cmp (-y) == x
	// (-x) cmp y == y
	// (-x) cmp (-y) == -(x cmp y)
	switch {
	case x.neg == y.neg:
		r = x.abs.cmp(y.abs)
		if x.neg {
			r = -r
		}
	case x.neg:
		r = -1
	default:
		r = 1
	}
	return
}

// CmpAbs compares the absolute values of x and y and returns:
//
//   -1 if |x| <  |y|
//    0 if |x| == |y|
//   +1 if |x| >  |y|
//
func (x *Int) CmpAbs(y *Int) int {
	return x.abs.cmp(y.abs)
}

// low32 returns the least significant 32 bits of x.
func low32(x nat) uint32 {
	if len(x) == 0 {
		return 0
	}
	return uint32(x[0])
}

// low64 returns the least significant 64 bits of x.
func low64(x nat) uint64 {
	if len(x) == 0 {
		return 0
	}
	v := uint64(x[0])
	if _W == 32 && len(x) > 1 {
		return uint64(x[1])<<32 | v
	}
	return v
}

// Int64 returns the int64 representation of x.
// If x cannot be represented in an int64, the result is undefined.
func (x *Int) Int64() int64 {
	v := int64(low64(x.abs))
	if x.neg {
		v = -v
	}
	return v
}

// Uint64 returns the uint64 representation of x.
// If x cannot be represented in a uint64, the result is undefined.
func (x *Int) Uint64() uint64 {
	return low64(x.abs)
}

// IsInt64 reports whether x can be represented as an int64.
func (x *Int) IsInt64() bool {
	if len(x.abs) <= 64/_W {
		w := int64(low64(x.abs))
		return w >= 0 || x.neg && w == -w
	}
	return false
}

// IsUint64 reports whether x can be represented as a uint64.
func (x *Int) IsUint64() bool {
	return !x.neg && len(x.abs) <= 64/_W
}

// SetString sets z to the value of s, interpreted in the given base,
// and returns z and a boolean indicating success. The entire string
// (not just a prefix) must be valid for success. If SetString fails,
// the value of z is undefined but the returned value is nil.
//
// The base argument must be 0 or a value between 2 and 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.
//
// For bases <= 36, lower and upper case letters are considered the same:
// The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
// For bases > 36, the upper case letters 'A' to 'Z' represent the digit
// values 36 to 61.
//
func (z *Int) SetString(s string, base int) (*Int, bool) {
	return z.setFromScanner(strings.NewReader(s), base)
}

// setFromScanner implements SetString given an io.BytesScanner.
// For documentation see comments of SetString.
func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
	if _, _, err := z.scan(r, base); err != nil {
		return nil, false
	}
	// entire content must have been consumed
	if _, err := r.ReadByte(); err != io.EOF {
		return nil, false
	}
	return z, true // err == io.EOF => scan consumed all content of r
}

// SetBytes interprets buf as the bytes of a big-endian unsigned
// integer, sets z to that value, and returns z.
func (z *Int) SetBytes(buf []byte) *Int {
	z.abs = z.abs.setBytes(buf)
	z.neg = false
	return z
}

// Bytes returns the absolute value of x as a big-endian byte slice.
func (x *Int) Bytes() []byte {
	buf := make([]byte, len(x.abs)*_S)
	return buf[x.abs.bytes(buf):]
}

// BitLen returns the length of the absolute value of x in bits.
// The bit length of 0 is 0.
func (x *Int) BitLen() int {
	return x.abs.bitLen()
}

// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
// If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y.
//
// Modular exponentation of inputs of a particular size is not a
// cryptographically constant-time operation.
func (z *Int) Exp(x, y, m *Int) *Int {
	// See Knuth, volume 2, section 4.6.3.
	var yWords nat
	if !y.neg {
		yWords = y.abs
	}
	// y >= 0

	var mWords nat
	if m != nil {
		mWords = m.abs // m.abs may be nil for m == 0
	}

	z.abs = z.abs.expNN(x.abs, yWords, mWords)
	z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
	if z.neg && len(mWords) > 0 {
		// make modulus result positive
		z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
		z.neg = false
	}

	return z
}

// GCD sets z to the greatest common divisor of a and b, which both must
// be > 0, and returns z.
// If x or y are not nil, GCD sets their value such that z = a*x + b*y.
// If either a or b is <= 0, GCD sets z = x = y = 0.
func (z *Int) GCD(x, y, a, b *Int) *Int {
	if a.Sign() <= 0 || b.Sign() <= 0 {
		z.SetInt64(0)
		if x != nil {
			x.SetInt64(0)
		}
		if y != nil {
			y.SetInt64(0)
		}
		return z
	}
	if x == nil && y == nil {
		return z.lehmerGCD(a, b)
	}

	A := new(Int).Set(a)
	B := new(Int).Set(b)

	X := new(Int)
	lastX := new(Int).SetInt64(1)

	q := new(Int)
	temp := new(Int)

	r := new(Int)
	for len(B.abs) > 0 {
		q, r = q.QuoRem(A, B, r)

		A, B, r = B, r, A

		temp.Set(X)
		X.Mul(X, q)
		X.Sub(lastX, X)
		lastX.Set(temp)
	}

	if x != nil {
		*x = *lastX
	}

	if y != nil {
		// y = (z - a*x)/b
		y.Mul(a, lastX)
		y.Sub(A, y)
		y.Div(y, b)
	}

	*z = *A
	return z
}

// lehmerGCD sets z to the greatest common divisor of a and b,
// which both must be > 0, and returns z.
// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
// This implementation uses the improved condition by Collins requiring only one
// quotient and avoiding the possibility of single Word overflow.
// See Jebelean, "Improving the multiprecision Euclidean algorithm",
// Design and Implementation of Symbolic Computation Systems, pp 45-58.
func (z *Int) lehmerGCD(a, b *Int) *Int {
	// ensure a >= b
	if a.abs.cmp(b.abs) < 0 {
		a, b = b, a
	}

	// don't destroy incoming values of a and b
	B := new(Int).Set(b) // must be set first in case b is an alias of z
	A := z.Set(a)

	// temp variables for multiprecision update
	t := new(Int)
	r := new(Int)
	s := new(Int)
	w := new(Int)

	// loop invariant A >= B
	for len(B.abs) > 1 {
		// initialize the digits
		var a1, a2, u0, u1, u2, v0, v1, v2 Word

		m := len(B.abs) // m >= 2
		n := len(A.abs) // n >= m >= 2

		// extract the top Word of bits from A and B
		h := nlz(A.abs[n-1])
		a1 = (A.abs[n-1] << h) | (A.abs[n-2] >> (_W - h))
		// B may have implicit zero words in the high bits if the lengths differ
		switch {
		case n == m:
			a2 = (B.abs[n-1] << h) | (B.abs[n-2] >> (_W - h))
		case n == m+1:
			a2 = (B.abs[n-2] >> (_W - h))
		default:
			a2 = 0
		}

		// Since we are calculating with full words to avoid overflow,
		// we use 'even' to track the sign of the cosequences.
		// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
		// For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
		// The first iteration starts with k=1 (odd).
		even := false
		// variables to track the cosequences
		u0, u1, u2 = 0, 1, 0
		v0, v1, v2 = 0, 0, 1

		// Calculate the quotient and cosequences using Collins' stopping condition.
		// Note that overflow of a Word is not possible when computing the remainder
		// sequence and cosequences since the cosequence size is bounded by the input size.
		// See section 4.2 of Jebelean for details.
		for a2 >= v2 && a1-a2 >= v1+v2 {
			q := a1 / a2
			a1, a2 = a2, a1-q*a2
			u0, u1, u2 = u1, u2, u1+q*u2
			v0, v1, v2 = v1, v2, v1+q*v2
			even = !even
		}

		// multiprecision step
		if v0 != 0 {
			// simulate the effect of the single precision steps using the cosequences
			// A = u0*A + v0*B
			// B = u1*A + v1*B

			t.abs = t.abs.setWord(u0)
			s.abs = s.abs.setWord(v0)
			t.neg = !even
			s.neg = even

			t.Mul(A, t)
			s.Mul(B, s)

			r.abs = r.abs.setWord(u1)
			w.abs = w.abs.setWord(v1)
			r.neg = even
			w.neg = !even

			r.Mul(A, r)
			w.Mul(B, w)

			A.Add(t, s)
			B.Add(r, w)

		} else {
			// single-digit calculations failed to simluate any quotients
			// do a standard Euclidean step
			t.Rem(A, B)
			A, B, t = B, t, A
		}
	}

	if len(B.abs) > 0 {
		// standard Euclidean algorithm base case for B a single Word
		if len(A.abs) > 1 {
			// A is longer than a single Word
			t.Rem(A, B)
			A, B, t = B, t, A
		}
		if len(B.abs) > 0 {
			// A and B are both a single Word
			a1, a2 := A.abs[0], B.abs[0]
			for a2 != 0 {
				a1, a2 = a2, a1%a2
			}
			A.abs[0] = a1
		}
	}
	*z = *A
	return z
}

// Rand sets z to a pseudo-random number in [0, n) and returns z.
//
// As this uses the math/rand package, it must not be used for
// security-sensitive work. Use crypto/rand.Int instead.
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
	z.neg = false
	if n.neg || len(n.abs) == 0 {
		z.abs = nil
		return z
	}
	z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
	return z
}

// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
// and returns z. If g and n are not relatively prime, the result is undefined.
func (z *Int) ModInverse(g, n *Int) *Int {
	if g.neg {
		// GCD expects parameters a and b to be > 0.
		var g2 Int
		g = g2.Mod(g, n)
	}
	var d Int
	d.GCD(z, nil, g, n)
	// x and y are such that g*x + n*y = d. Since g and n are
	// relatively prime, d = 1. Taking that modulo n results in
	// g*x = 1, therefore x is the inverse element.
	if z.neg {
		z.Add(z, n)
	}
	return z
}

// Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
// The y argument must be an odd integer.
func Jacobi(x, y *Int) int {
	if len(y.abs) == 0 || y.abs[0]&1 == 0 {
		panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
	}

	// We use the formulation described in chapter 2, section 2.4,
	// "The Yacas Book of Algorithms":
	// http://yacas.sourceforge.net/Algo.book.pdf

	var a, b, c Int
	a.Set(x)
	b.Set(y)
	j := 1

	if b.neg {
		if a.neg {
			j = -1
		}
		b.neg = false
	}

	for {
		if b.Cmp(intOne) == 0 {
			return j
		}
		if len(a.abs) == 0 {
			return 0
		}
		a.Mod(&a, &b)
		if len(a.abs) == 0 {
			return 0
		}
		// a > 0

		// handle factors of 2 in 'a'
		s := a.abs.trailingZeroBits()
		if s&1 != 0 {
			bmod8 := b.abs[0] & 7
			if bmod8 == 3 || bmod8 == 5 {
				j = -j
			}
		}
		c.Rsh(&a, s) // a = 2^s*c

		// swap numerator and denominator
		if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
			j = -j
		}
		a.Set(&b)
		b.Set(&c)
	}
}

// modSqrt3Mod4 uses the identity
//      (a^((p+1)/4))^2  mod p
//   == u^(p+1)          mod p
//   == u^2              mod p
// to calculate the square root of any quadratic residue mod p quickly for 3
// mod 4 primes.
func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
	e := new(Int).Add(p, intOne) // e = p + 1
	e.Rsh(e, 2)                  // e = (p + 1) / 4
	z.Exp(x, e, p)               // z = x^e mod p
	return z
}

// modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
// root of a quadratic residue modulo any prime.
func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
	// Break p-1 into s*2^e such that s is odd.
	var s Int
	s.Sub(p, intOne)
	e := s.abs.trailingZeroBits()
	s.Rsh(&s, e)

	// find some non-square n
	var n Int
	n.SetInt64(2)
	for Jacobi(&n, p) != -1 {
		n.Add(&n, intOne)
	}

	// Core of the Tonelli-Shanks algorithm. Follows the description in
	// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
	// Brown:
	// https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
	var y, b, g, t Int
	y.Add(&s, intOne)
	y.Rsh(&y, 1)
	y.Exp(x, &y, p)  // y = x^((s+1)/2)
	b.Exp(x, &s, p)  // b = x^s
	g.Exp(&n, &s, p) // g = n^s
	r := e
	for {
		// find the least m such that ord_p(b) = 2^m
		var m uint
		t.Set(&b)
		for t.Cmp(intOne) != 0 {
			t.Mul(&t, &t).Mod(&t, p)
			m++
		}

		if m == 0 {
			return z.Set(&y)
		}

		t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
		// t = g^(2^(r-m-1)) mod p
		g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
		y.Mul(&y, &t).Mod(&y, p)
		b.Mul(&b, &g).Mod(&b, p)
		r = m
	}
}

// ModSqrt sets z to a square root of x mod p if such a square root exists, and
// returns z. The modulus p must be an odd prime. If x is not a square mod p,
// ModSqrt leaves z unchanged and returns nil. This function panics if p is
// not an odd integer.
func (z *Int) ModSqrt(x, p *Int) *Int {
	switch Jacobi(x, p) {
	case -1:
		return nil // x is not a square mod p
	case 0:
		return z.SetInt64(0) // sqrt(0) mod p = 0
	case 1:
		break
	}
	if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
		x = new(Int).Mod(x, p)
	}

	// Check whether p is 3 mod 4, and if so, use the faster algorithm.
	if len(p.abs) > 0 && p.abs[0]%4 == 3 {
		return z.modSqrt3Mod4Prime(x, p)
	}
	// Otherwise, use Tonelli-Shanks.
	return z.modSqrtTonelliShanks(x, p)
}

// Lsh sets z = x << n and returns z.
func (z *Int) Lsh(x *Int, n uint) *Int {
	z.abs = z.abs.shl(x.abs, n)
	z.neg = x.neg
	return z
}

// Rsh sets z = x >> n and returns z.
func (z *Int) Rsh(x *Int, n uint) *Int {
	if x.neg {
		// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
		t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
		t = t.shr(t, n)
		z.abs = t.add(t, natOne)
		z.neg = true // z cannot be zero if x is negative
		return z
	}

	z.abs = z.abs.shr(x.abs, n)
	z.neg = false
	return z
}

// Bit returns the value of the i'th bit of x. That is, it
// returns (x>>i)&1. The bit index i must be >= 0.
func (x *Int) Bit(i int) uint {
	if i == 0 {
		// optimization for common case: odd/even test of x
		if len(x.abs) > 0 {
			return uint(x.abs[0] & 1) // bit 0 is same for -x
		}
		return 0
	}
	if i < 0 {
		panic("negative bit index")
	}
	if x.neg {
		t := nat(nil).sub(x.abs, natOne)
		return t.bit(uint(i)) ^ 1
	}

	return x.abs.bit(uint(i))
}

// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
// That is, if b is 1 SetBit sets z = x | (1 << i);
// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
// SetBit will panic.
func (z *Int) SetBit(x *Int, i int, b uint) *Int {
	if i < 0 {
		panic("negative bit index")
	}
	if x.neg {
		t := z.abs.sub(x.abs, natOne)
		t = t.setBit(t, uint(i), b^1)
		z.abs = t.add(t, natOne)
		z.neg = len(z.abs) > 0
		return z
	}
	z.abs = z.abs.setBit(x.abs, uint(i), b)
	z.neg = false
	return z
}

// And sets z = x & y and returns z.
func (z *Int) And(x, y *Int) *Int {
	if x.neg == y.neg {
		if x.neg {
			// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
			x1 := nat(nil).sub(x.abs, natOne)
			y1 := nat(nil).sub(y.abs, natOne)
			z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
			z.neg = true // z cannot be zero if x and y are negative
			return z
		}

		// x & y == x & y
		z.abs = z.abs.and(x.abs, y.abs)
		z.neg = false
		return z
	}

	// x.neg != y.neg
	if x.neg {
		x, y = y, x // & is symmetric
	}

	// x & (-y) == x & ^(y-1) == x &^ (y-1)
	y1 := nat(nil).sub(y.abs, natOne)
	z.abs = z.abs.andNot(x.abs, y1)
	z.neg = false
	return z
}

// AndNot sets z = x &^ y and returns z.
func (z *Int) AndNot(x, y *Int) *Int {
	if x.neg == y.neg {
		if x.neg {
			// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
			x1 := nat(nil).sub(x.abs, natOne)
			y1 := nat(nil).sub(y.abs, natOne)
			z.abs = z.abs.andNot(y1, x1)
			z.neg = false
			return z
		}

		// x &^ y == x &^ y
		z.abs = z.abs.andNot(x.abs, y.abs)
		z.neg = false
		return z
	}

	if x.neg {
		// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
		x1 := nat(nil).sub(x.abs, natOne)
		z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
		z.neg = true // z cannot be zero if x is negative and y is positive
		return z
	}

	// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
	y1 := nat(nil).sub(y.abs, natOne)
	z.abs = z.abs.and(x.abs, y1)
	z.neg = false
	return z
}

// Or sets z = x | y and returns z.
func (z *Int) Or(x, y *Int) *Int {
	if x.neg == y.neg {
		if x.neg {
			// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
			x1 := nat(nil).sub(x.abs, natOne)
			y1 := nat(nil).sub(y.abs, natOne)
			z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
			z.neg = true // z cannot be zero if x and y are negative
			return z
		}

		// x | y == x | y
		z.abs = z.abs.or(x.abs, y.abs)
		z.neg = false
		return z
	}

	// x.neg != y.neg
	if x.neg {
		x, y = y, x // | is symmetric
	}

	// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
	y1 := nat(nil).sub(y.abs, natOne)
	z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
	z.neg = true // z cannot be zero if one of x or y is negative
	return z
}

// Xor sets z = x ^ y and returns z.
func (z *Int) Xor(x, y *Int) *Int {
	if x.neg == y.neg {
		if x.neg {
			// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
			x1 := nat(nil).sub(x.abs, natOne)
			y1 := nat(nil).sub(y.abs, natOne)
			z.abs = z.abs.xor(x1, y1)
			z.neg = false
			return z
		}

		// x ^ y == x ^ y
		z.abs = z.abs.xor(x.abs, y.abs)
		z.neg = false
		return z
	}

	// x.neg != y.neg
	if x.neg {
		x, y = y, x // ^ is symmetric
	}

	// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
	y1 := nat(nil).sub(y.abs, natOne)
	z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
	z.neg = true // z cannot be zero if only one of x or y is negative
	return z
}

// Not sets z = ^x and returns z.
func (z *Int) Not(x *Int) *Int {
	if x.neg {
		// ^(-x) == ^(^(x-1)) == x-1
		z.abs = z.abs.sub(x.abs, natOne)
		z.neg = false
		return z
	}

	// ^x == -x-1 == -(x+1)
	z.abs = z.abs.add(x.abs, natOne)
	z.neg = true // z cannot be zero if x is positive
	return z
}

// Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
// It panics if x is negative.
func (z *Int) Sqrt(x *Int) *Int {
	if x.neg {
		panic("square root of negative number")
	}
	z.neg = false
	z.abs = z.abs.sqrt(x.abs)
	return z
}