Feat: Eisenstein integers

field/eisenstein/doc.go

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+// Package Eisenstein provides Eisenstein integer arithmetic.
+//
+// The Eisenstein integers form a commutative ring of algebraic integers in the
+// algebraic number field Q(ω) – the third cyclotomic field.  These are of the
+// form z = a + bω, where a and b are integers and ω is a primitive third root
+// of unity i.e. ω²+ω+1 = 0.
+package eisenstein

field/eisenstein/eisenstein.go

@@ -0,0 +1,179 @@
+package eisenstein
+
+import (
+	"math/big"
+)
+
+// A ComplexNumber represents an arbitrary-precision Eisenstein integer.
+type ComplexNumber struct {
+	A0, A1 big.Int
+}
+
+// String implements Stringer interface for fancy printing
+func (z *ComplexNumber) String() string {
+	return z.A0.String() + "+(" + z.A1.String() + "*ω)"
+}
+
+// Equal returns true if z equals x, false otherwise
+func (z *ComplexNumber) Equal(x *ComplexNumber) bool {
+	return z.A0.Cmp(&x.A0) == 0 && z.A1.Cmp(&x.A1) == 0
+}
+
+// Set sets z to x, and returns z.
+func (z *ComplexNumber) Set(x *ComplexNumber) *ComplexNumber {
+	z.A0.Set(&x.A0)
+	z.A1.Set(&x.A1)
+	return z
+}
+
+// SetZero sets z to 0, and returns z.
+func (z *ComplexNumber) SetZero() *ComplexNumber {
+	z.A0 = *big.NewInt(0)
+	z.A1 = *big.NewInt(0)
+	return z
+}
+
+// SetOne sets z to 1, and returns z.
+func (z *ComplexNumber) SetOne() *ComplexNumber {
+	z.A0 = *big.NewInt(1)
+	z.A1 = *big.NewInt(0)
+	return z
+}
+
+// Neg sets z to the negative of x, and returns z.
+func (z *ComplexNumber) Neg(x *ComplexNumber) *ComplexNumber {
+	z.A0.Neg(&x.A0)
+	z.A1.Neg(&x.A1)
+	return z
+}
+
+// Conjugate sets z to the conjugate of x, and returns z.
+func (z *ComplexNumber) Conjugate(x *ComplexNumber) *ComplexNumber {
+	z.A0.Sub(&x.A0, &x.A1)
+	z.A1.Neg(&x.A1)
+	return z
+}
+
+// Add sets z to the sum of x and y, and returns z.
+func (z *ComplexNumber) Add(x, y *ComplexNumber) *ComplexNumber {
+	z.A0.Add(&x.A0, &y.A0)
+	z.A1.Add(&x.A1, &y.A1)
+	return z
+}
+
+// Sub sets z to the difference of x and y, and returns z.
+func (z *ComplexNumber) Sub(x, y *ComplexNumber) *ComplexNumber {
+	z.A0.Sub(&x.A0, &y.A0)
+	z.A1.Sub(&x.A1, &y.A1)
+	return z
+}
+
+// Mul sets z to the product of x and y, and returns z.
+//
+// Given that ω²+ω+1=0, the explicit formula is:
+//
+//	(x0+x1ω)(y0+y1ω) = (x0y0-x1y1) + (x0y1+x1y0-x1y1)ω
+func (z *ComplexNumber) Mul(x, y *ComplexNumber) *ComplexNumber {
+	var t [3]big.Int
+	var z0, z1 big.Int
+	t[0].Mul(&x.A0, &y.A0)
+	t[1].Mul(&x.A1, &y.A1)
+	z0.Sub(&t[0], &t[1])
+	t[0].Mul(&x.A0, &y.A1)
+	t[2].Mul(&x.A1, &y.A0)
+	t[0].Add(&t[0], &t[2])
+	z1.Sub(&t[0], &t[1])
+	z.A0.Set(&z0)
+	z.A1.Set(&z1)
+	return z
+}
+
+// Norm returns the norm of z.
+//
+// The explicit formula is:
+//
+//	N(x0+x1ω) = x0² + x1² - x0*x1
+func (z *ComplexNumber) Norm() *big.Int {
+	norm := new(big.Int)
+	temp := new(big.Int)
+	norm.Add(
+		norm.Mul(&z.A0, &z.A0),
+		temp.Mul(&z.A1, &z.A1),
+	)
+	norm.Sub(
+		norm,
+		temp.Mul(&z.A0, &z.A1),
+	)
+	return norm
+}
+
+// Quo sets z to the quotient of x and y, and returns z.
+func (z *ComplexNumber) Quo(x, y *ComplexNumber) *ComplexNumber {
+	norm := y.Norm()
+	if norm.Cmp(big.NewInt(0)) == 0 {
+		panic("division by zero")
+	}
+	z.Conjugate(y)
+	z.Mul(x, z)
+	z.A0.Div(&z.A0, norm)
+	z.A1.Div(&z.A1, norm)
+	return z
+}
+
+// QuoRem sets z to the quotient of x and y, r to the remainder, and returns z and r.
+func (z *ComplexNumber) QuoRem(x, y, r *ComplexNumber) (*ComplexNumber, *ComplexNumber) {
+	norm := y.Norm()
+	if norm.Cmp(big.NewInt(0)) == 0 {
+		panic("division by zero")
+	}
+	z.Conjugate(y)
+	z.Mul(x, z)
+	z.A0.Div(&z.A0, norm)
+	z.A1.Div(&z.A1, norm)
+	var t ComplexNumber
+	r.Sub(x, t.Mul(y, z))
+	return z, r
+}
+
+// Min returns the minimum of z... with respect to the norm.
+func Min(z ...*ComplexNumber) *ComplexNumber {
+	min := z[0]
+	for _, v := range z {
+		if v.Norm().Cmp(min.Norm()) == -1 {
+			min = v
+		}
+	}
+	return min
+}
+
+// HalfGCD returns the rational reconstruction of a, b.
+// This outputs w, v, u s.t. w = a*u + b*v.
+func HalfGCD(a, b *ComplexNumber) [3]*ComplexNumber {
+
+	var check, aRun, bRun, u, v, u_, v_, quotient, remainder, t, t1, t2 ComplexNumber
+
+	aRun.Set(a)
+	bRun.Set(b)
+	u.SetOne()
+	v.SetZero()
+	u_.SetZero()
+	v_.SetOne()
+
+	// Eisenstein integers form an Euclidean domain for the norm
+	aNorm := a.Norm()
+	for check.Mul(&aRun, &aRun).Norm().Cmp(aNorm) == 1 {
+		quotient.QuoRem(&aRun, &bRun, &remainder)
+		t.Mul(&u_, &quotient)
+		t1.Sub(&u, &t)
+		t.Mul(&v_, &quotient)
+		t2.Sub(&v, &t)
+		aRun.Set(&bRun)
+		u.Set(&u_)
+		v.Set(&v_)
+		bRun.Set(&remainder)
+		u_.Set(&t1)
+		v_.Set(&t2)
+	}
+
+	return [3]*ComplexNumber{&aRun, &v, &u}
+}