github.com/cloudflare/circl@v1.5.0/math/fp25519/fp.go

// Package fp25519 provides prime field arithmetic over GF(2^255-19).
package fp25519

import (
	"errors"

	"github.com/cloudflare/circl/internal/conv"
)

// Size in bytes of an element.
const Size = 32

// Elt is a prime field element.
type Elt [Size]byte

func (e Elt) String() string { return conv.BytesLe2Hex(e[:]) }

// p is the prime modulus 2^255-19.
var p = Elt{
	0xed, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
	0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
	0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
	0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x7f,
}

// P returns the prime modulus 2^255-19.
func P() Elt { return p }

// ToBytes stores in b the little-endian byte representation of x.
func ToBytes(b []byte, x *Elt) error {
	if len(b) != Size {
		return errors.New("wrong size")
	}
	Modp(x)
	copy(b, x[:])
	return nil
}

// IsZero returns true if x is equal to 0.
func IsZero(x *Elt) bool { Modp(x); return *x == Elt{} }

// SetOne assigns x=1.
func SetOne(x *Elt) { *x = Elt{}; x[0] = 1 }

// Neg calculates z = -x.
func Neg(z, x *Elt) { Sub(z, &p, x) }

// InvSqrt calculates z = sqrt(x/y) iff x/y is a quadratic-residue, which is
// indicated by returning isQR = true. Otherwise, when x/y is a quadratic
// non-residue, z will have an undetermined value and isQR = false.
func InvSqrt(z, x, y *Elt) (isQR bool) {
	sqrtMinusOne := &Elt{
		0xb0, 0xa0, 0x0e, 0x4a, 0x27, 0x1b, 0xee, 0xc4,
		0x78, 0xe4, 0x2f, 0xad, 0x06, 0x18, 0x43, 0x2f,
		0xa7, 0xd7, 0xfb, 0x3d, 0x99, 0x00, 0x4d, 0x2b,
		0x0b, 0xdf, 0xc1, 0x4f, 0x80, 0x24, 0x83, 0x2b,
	}
	t0, t1, t2, t3 := &Elt{}, &Elt{}, &Elt{}, &Elt{}

	Mul(t0, x, y)   // t0 = u*v
	Sqr(t1, y)      // t1 = v^2
	Mul(t2, t0, t1) // t2 = u*v^3
	Sqr(t0, t1)     // t0 = v^4
	Mul(t1, t0, t2) // t1 = u*v^7

	var Tab [4]*Elt
	Tab[0] = &Elt{}
	Tab[1] = &Elt{}
	Tab[2] = t3
	Tab[3] = t1

	*Tab[0] = *t1
	Sqr(Tab[0], Tab[0])
	Sqr(Tab[1], Tab[0])
	Sqr(Tab[1], Tab[1])
	Mul(Tab[1], Tab[1], Tab[3])
	Mul(Tab[0], Tab[0], Tab[1])
	Sqr(Tab[0], Tab[0])
	Mul(Tab[0], Tab[0], Tab[1])
	Sqr(Tab[1], Tab[0])
	for i := 0; i < 4; i++ {
		Sqr(Tab[1], Tab[1])
	}
	Mul(Tab[1], Tab[1], Tab[0])
	Sqr(Tab[2], Tab[1])
	for i := 0; i < 4; i++ {
		Sqr(Tab[2], Tab[2])
	}
	Mul(Tab[2], Tab[2], Tab[0])
	Sqr(Tab[1], Tab[2])
	for i := 0; i < 14; i++ {
		Sqr(Tab[1], Tab[1])
	}
	Mul(Tab[1], Tab[1], Tab[2])
	Sqr(Tab[2], Tab[1])
	for i := 0; i < 29; i++ {
		Sqr(Tab[2], Tab[2])
	}
	Mul(Tab[2], Tab[2], Tab[1])
	Sqr(Tab[1], Tab[2])
	for i := 0; i < 59; i++ {
		Sqr(Tab[1], Tab[1])
	}
	Mul(Tab[1], Tab[1], Tab[2])
	for i := 0; i < 5; i++ {
		Sqr(Tab[1], Tab[1])
	}
	Mul(Tab[1], Tab[1], Tab[0])
	Sqr(Tab[2], Tab[1])
	for i := 0; i < 124; i++ {
		Sqr(Tab[2], Tab[2])
	}
	Mul(Tab[2], Tab[2], Tab[1])
	Sqr(Tab[2], Tab[2])
	Sqr(Tab[2], Tab[2])
	Mul(Tab[2], Tab[2], Tab[3])

	Mul(z, t3, t2) // z = xy^(p+3)/8 = xy^3*(xy^7)^(p-5)/8
	// Checking whether y z^2 == x
	Sqr(t0, z)     // t0 = z^2
	Mul(t0, t0, y) // t0 = yz^2
	Sub(t1, t0, x) // t1 = t0-u
	Add(t2, t0, x) // t2 = t0+u
	if IsZero(t1) {
		return true
	} else if IsZero(t2) {
		Mul(z, z, sqrtMinusOne) // z = z*sqrt(-1)
		return true
	} else {
		return false
	}
}

// Inv calculates z = 1/x mod p.
func Inv(z, x *Elt) {
	x0, x1, x2 := &Elt{}, &Elt{}, &Elt{}
	Sqr(x1, x)
	Sqr(x0, x1)
	Sqr(x0, x0)
	Mul(x0, x0, x)
	Mul(z, x0, x1)
	Sqr(x1, z)
	Mul(x0, x0, x1)
	Sqr(x1, x0)
	for i := 0; i < 4; i++ {
		Sqr(x1, x1)
	}
	Mul(x0, x0, x1)
	Sqr(x1, x0)
	for i := 0; i < 9; i++ {
		Sqr(x1, x1)
	}
	Mul(x1, x1, x0)
	Sqr(x2, x1)
	for i := 0; i < 19; i++ {
		Sqr(x2, x2)
	}
	Mul(x2, x2, x1)
	for i := 0; i < 10; i++ {
		Sqr(x2, x2)
	}
	Mul(x2, x2, x0)
	Sqr(x0, x2)
	for i := 0; i < 49; i++ {
		Sqr(x0, x0)
	}
	Mul(x0, x0, x2)
	Sqr(x1, x0)
	for i := 0; i < 99; i++ {
		Sqr(x1, x1)
	}
	Mul(x1, x1, x0)
	for i := 0; i < 50; i++ {
		Sqr(x1, x1)
	}
	Mul(x1, x1, x2)
	for i := 0; i < 5; i++ {
		Sqr(x1, x1)
	}
	Mul(z, z, x1)
}

// Cmov assigns y to x if n is 1.
func Cmov(x, y *Elt, n uint) { cmov(x, y, n) }

// Cswap interchanges x and y if n is 1.
func Cswap(x, y *Elt, n uint) { cswap(x, y, n) }

// Add calculates z = x+y mod p.
func Add(z, x, y *Elt) { add(z, x, y) }

// Sub calculates z = x-y mod p.
func Sub(z, x, y *Elt) { sub(z, x, y) }

// AddSub calculates (x,y) = (x+y mod p, x-y mod p).
func AddSub(x, y *Elt) { addsub(x, y) }

// Mul calculates z = x*y mod p.
func Mul(z, x, y *Elt) { mul(z, x, y) }

// Sqr calculates z = x^2 mod p.
func Sqr(z, x *Elt) { sqr(z, x) }

// Modp ensures that z is between [0,p-1].
func Modp(z *Elt) { modp(z) }