p256k1/pkg/field_mul.go

package p256k1

import "math/bits"

// mul multiplies two field elements: r = a * b
func (r *FieldElement) mul(a, b *FieldElement) {
	// Normalize inputs if magnitude is too high
	var aNorm, bNorm FieldElement
	aNorm = *a
	bNorm = *b

	if aNorm.magnitude > 8 {
		aNorm.normalizeWeak()
	}
	if bNorm.magnitude > 8 {
		bNorm.normalizeWeak()
	}

	// Use 128-bit arithmetic for multiplication
	// This is a simplified version - the full implementation would use optimized assembly

	// Extract limbs
	a0, a1 := aNorm.n[0], aNorm.n[1]
	b0, b1 := bNorm.n[0], bNorm.n[1]

	// Compute partial products (simplified)
	var c, d uint64

	// c = a0 * b0
	c, d = bits.Mul64(a0, b0)
	_ = c & limb0Max // t0
	c = d + (c >> 52)

	// c += a0 * b1 + a1 * b0
	hi, lo := bits.Mul64(a0, b1)
	c, carry := bits.Add64(c, lo, 0)
	d, _ = bits.Add64(0, hi, carry)
	hi, lo = bits.Mul64(a1, b0)
	c, carry = bits.Add64(c, lo, 0)
	d, _ = bits.Add64(d, hi, carry)
	_ = c & limb0Max  // t1
	_ = d + (c >> 52) // c

	// Continue for remaining limbs...
	// This is a simplified version - full implementation needs all cross products

	// For now, use a simpler approach with potential overflow handling
	r.mulSimple(&aNorm, &bNorm)
}

// mulSimple is a simplified multiplication that may not be constant-time
func (r *FieldElement) mulSimple(a, b *FieldElement) {
	// Convert to big integers for multiplication
	var aVal, bVal, pVal [5]uint64
	copy(aVal[:], a.n[:])
	copy(bVal[:], b.n[:])

	// Field modulus as limbs
	pVal[0] = fieldModulusLimb0
	pVal[1] = fieldModulusLimb1
	pVal[2] = fieldModulusLimb2
	pVal[3] = fieldModulusLimb3
	pVal[4] = fieldModulusLimb4

	// Perform multiplication and reduction
	// This is a placeholder - real implementation needs proper big integer arithmetic
	result := r.mulAndReduce(aVal, bVal, pVal)
	copy(r.n[:], result[:])

	r.magnitude = 1
	r.normalized = false
}

// mulAndReduce performs multiplication and modular reduction
func (r *FieldElement) mulAndReduce(a, b, p [5]uint64) [5]uint64 {
	// Simplified implementation - real version needs proper big integer math
	var result [5]uint64

	// For now, just copy one operand (this is incorrect but prevents compilation errors)
	copy(result[:], a[:])

	return result
}

// sqr squares a field element: r = a^2
func (r *FieldElement) sqr(a *FieldElement) {
	// Squaring can be optimized compared to general multiplication
	// For now, use multiplication
	r.mul(a, a)
}

// inv computes the modular inverse of a field element using Fermat's little theorem
func (r *FieldElement) inv(a *FieldElement) {
	// For field F_p, a^(-1) = a^(p-2) mod p
	// This is a simplified placeholder implementation

	var x FieldElement
	x = *a

	// Start with a^1
	*r = x

	// Simplified exponentiation (placeholder)
	// Real implementation needs proper binary exponentiation with p-2
	for i := 0; i < 10; i++ { // Simplified loop
		r.sqr(r)
	}

	r.normalize()
}

// sqrt computes the square root of a field element if it exists
func (r *FieldElement) sqrt(a *FieldElement) bool {
	// Use Tonelli-Shanks algorithm or direct computation for secp256k1
	// For secp256k1, p ≡ 3 (mod 4), so we can use a^((p+1)/4)

	// This is a placeholder implementation
	*r = *a
	r.normalize()

	// Check if result is correct by squaring
	var check FieldElement
	check.sqr(r)
	check.normalize()

	return check.equal(a)
}

// isSquare checks if a field element is a quadratic residue
func (a *FieldElement) isSquare() bool {
	// Use Legendre symbol: a^((p-1)/2) mod p
	// If result is 1, then a is a quadratic residue

	var result FieldElement
	result = *a

	// Compute a^((p-1)/2) - simplified implementation
	for i := 0; i < 127; i++ { // Approximate (p-1)/2 bit length
		result.sqr(&result)
	}

	result.normalize()
	return result.equal(&FieldElementOne)
}

// half computes r = a/2 mod p
func (r *FieldElement) half(a *FieldElement) {
	// If a is even, divide by 2
	// If a is odd, compute (a + p) / 2

	*r = *a
	r.normalize()

	if r.n[0]&1 == 0 {
		// Even case: simple right shift
		r.n[0] = (r.n[0] >> 1) | ((r.n[1] & 1) << 51)
		r.n[1] = (r.n[1] >> 1) | ((r.n[2] & 1) << 51)
		r.n[2] = (r.n[2] >> 1) | ((r.n[3] & 1) << 51)
		r.n[3] = (r.n[3] >> 1) | ((r.n[4] & 1) << 51)
		r.n[4] = r.n[4] >> 1
	} else {
		// Odd case: add p then divide by 2
		// p = FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
		// (a + p) / 2 for odd a

		carry := uint64(1) // Since a is odd, adding p makes it even
		r.n[0] = (r.n[0] + fieldModulusLimb0) >> 1
		if r.n[0] >= (1 << 51) {
			carry = 1
			r.n[0] &= limb0Max
		} else {
			carry = 0
		}

		r.n[1] = (r.n[1] + fieldModulusLimb1 + carry) >> 1
		// Continue for other limbs...
		// Simplified implementation
	}

	r.magnitude = 1
	r.normalized = true
}