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package main
import (
"fmt"
"math/big"
)
// secp256k1 curve: y² = x³ + 7
// The 'a' coefficient is 0, 'b' is 7
var curveB = NewFieldElementFromInt64(7)
// Generator point G for secp256k1
var (
Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16)
Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16)
G = &Point{
x: NewFieldElement(Gx),
y: NewFieldElement(Gy),
infinity: false,
}
)
// Curve order (number of points on the curve)
var N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16)
// Point represents a point on the secp256k1 curve
type Point struct {
x, y *FieldElement
infinity bool // true if this is the point at infinity (identity)
}
// NewPoint creates a point from x, y coordinates
// Returns error if the point is not on the curve
func NewPoint(x, y *FieldElement) (*Point, error) {
p := &Point{x: x, y: y, infinity: false}
if !p.IsOnCurve() {
return nil, fmt.Errorf("point (%s, %s) is not on the curve", x.String(), y.String())
}
return p, nil
}
// Infinity returns the point at infinity (identity element)
func Infinity() *Point {
return &Point{infinity: true}
}
// IsInfinity returns true if this is the point at infinity
func (p *Point) IsInfinity() bool {
return p.infinity
}
// IsOnCurve checks if the point satisfies y² = x³ + 7
func (p *Point) IsOnCurve() bool {
if p.infinity {
return true
}
// y² = x³ + 7
left := p.y.Square() // y²
right := p.x.Square().Mul(p.x).Add(curveB) // x³ + 7
return left.Equal(right)
}
// Equal checks if two points are the same
func (p *Point) Equal(q *Point) bool {
if p.infinity && q.infinity {
return true
}
if p.infinity || q.infinity {
return false
}
return p.x.Equal(q.x) && p.y.Equal(q.y)
}
// Add returns p + q using the elliptic curve addition formulas
func (p *Point) Add(q *Point) *Point {
// Handle infinity (identity element)
if p.infinity {
return q
}
if q.infinity {
return p
}
// If points are inverses (same x, opposite y), return infinity
if p.x.Equal(q.x) && !p.y.Equal(q.y) {
return Infinity()
}
// If points are the same, use doubling formula
if p.Equal(q) {
return p.Double()
}
// Standard addition formula for P ≠ Q:
// slope = (y2 - y1) / (x2 - x1)
// x3 = slope² - x1 - x2
// y3 = slope * (x1 - x3) - y1
slope := q.y.Sub(p.y).Div(q.x.Sub(p.x))
x3 := slope.Square().Sub(p.x).Sub(q.x)
y3 := slope.Mul(p.x.Sub(x3)).Sub(p.y)
return &Point{x: x3, y: y3, infinity: false}
}
// Double returns 2P (point added to itself)
func (p *Point) Double() *Point {
if p.infinity {
return Infinity()
}
// If y = 0, tangent is vertical, return infinity
if p.y.IsZero() {
return Infinity()
}
// Doubling formula:
// slope = (3x² + a) / (2y) -- for secp256k1, a = 0
// x3 = slope² - 2x
// y3 = slope * (x - x3) - y
three := NewFieldElementFromInt64(3)
two := NewFieldElementFromInt64(2)
slope := three.Mul(p.x.Square()).Div(two.Mul(p.y))
x3 := slope.Square().Sub(two.Mul(p.x))
y3 := slope.Mul(p.x.Sub(x3)).Sub(p.y)
return &Point{x: x3, y: y3, infinity: false}
}
// ScalarMul returns k * P (point multiplied by scalar)
// Uses double-and-add algorithm
func (p *Point) ScalarMul(k *big.Int) *Point {
result := Infinity()
addend := p
// Clone k so we don't modify the original
scalar := new(big.Int).Set(k)
for scalar.Sign() > 0 {
// If lowest bit is 1, add current addend
if scalar.Bit(0) == 1 {
result = result.Add(addend)
}
// Double the addend
addend = addend.Double()
// Shift scalar right by 1
scalar.Rsh(scalar, 1)
}
return result
}
// Negate returns -P (same x, negated y)
func (p *Point) Negate() *Point {
if p.infinity {
return Infinity()
}
// -y mod P
negY := NewFieldElement(new(big.Int).Sub(P, p.y.value))
return &Point{x: p.x.Clone(), y: negY, infinity: false}
}
// String returns a readable representation
func (p *Point) String() string {
if p.infinity {
return "Point(infinity)"
}
return fmt.Sprintf("Point(%s, %s)", p.x.String()[:8]+"...", p.y.String()[:8]+"...")
}
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