cln16sidh: Add prototype of extension field multiplication

field.go

@@ -1,5 +1,58 @@
 package cln16sidh
 
+// Represents an element of the extension field F_{p^2}.
+type FieldElement struct {
+	A Fp751Element
+	B Fp751Element
+}
+
+func (dest *FieldElement) Mul(lhs, rhs *FieldElement) {
+	// Let (a,b,c,d) = (lhs.A,lhs.B,rhs.A,rhs.B).
+
+	a := &lhs.A
+	b := &lhs.B
+	c := &rhs.A
+	d := &rhs.B
+
+	// We want to compute
+	//
+	// (a + bi)*(c + di) = (a*c - b*d) + (a*d + b*c)i
+	//
+	// Use Karatsuba's trick: note that
+	//
+	// (b - a)*(c - d) = (b*c + a*d) - a*c - b*d
+	//
+	// so (a*d + b*c) = (b-a)*(c-d) + a*c + b*d.
+
+	var ac, bd Fp751X2
+	Fp751Mul(&ac, a, c)
+	Fp751Mul(&bd, b, d)
+
+	var b_minus_a, c_minus_d Fp751Element
+	Fp751SubReduced(&b_minus_a, b, a)
+	Fp751SubReduced(&c_minus_d, c, d)
+
+	var ad_plus_bc Fp751X2
+	Fp751Mul(&ad_plus_bc, &b_minus_a, &c_minus_d)
+	Fp751X2AddLazy(&ad_plus_bc, &ad_plus_bc, &ac)
+	Fp751X2AddLazy(&ad_plus_bc, &ad_plus_bc, &bd)
+
+	Fp751Reduce(&dest.B, &ad_plus_bc)
+
+	Fp751X2AddLazy(&ac, &ac, &bd)
+	Fp751Reduce(&dest.A, &ac)
+}
+
+func (dest *FieldElement) Add(lhs, rhs *FieldElement) {
+	Fp751AddReduced(&dest.A, &lhs.A, &rhs.A)
+	Fp751AddReduced(&dest.B, &lhs.B, &rhs.B)
+}
+
+func (dest *FieldElement) Sub(lhs, rhs *FieldElement) {
+	Fp751SubReduced(&dest.A, &lhs.A, &rhs.A)
+	Fp751SubReduced(&dest.B, &lhs.B, &rhs.B)
+}
+
 const Fp751NumWords = 12
 
 // Represents an element of the base field F_p, in Montgomery form.

field_test.go

@@ -236,3 +236,16 @@ func BenchmarkFp751SubReduced(b *testing.B) {
 		Fp751SubReduced(&benchmarkFp751Element, &x, &y)
 	}
 }
+
+func BenchmarkFieldElementMultiply(b *testing.B) {
+	x := Fp751Element{17026702066521327207, 5108203422050077993, 10225396685796065916, 11153620995215874678, 6531160855165088358, 15302925148404145445, 1248821577836769963, 9789766903037985294, 7493111552032041328, 10838999828319306046, 18103257655515297935, 27403304611634}
+
+	y := Fp751Element{4227467157325093378, 10699492810770426363, 13500940151395637365, 12966403950118934952, 16517692605450415877, 13647111148905630666, 14223628886152717087, 7167843152346903316, 15855377759596736571, 4300673881383687338, 6635288001920617779, 30486099554235}
+
+	z := &FieldElement{A: x, B: y}
+	w := new(FieldElement)
+
+	for n := 0; n < b.N; n++ {
+		w.Mul(z, z)
+	}
+}