implement binary EEA inversion

crypto/bn256/cloudflare/bn256_test.go

@@ -92,6 +92,64 @@ func TestTripartiteDiffieHellman(t *testing.T) {
 	}
 }
 
+func TestBinaryEAA(t *testing.T) {
+	for i := 0; i < 10000; i++ {
+		_, Ga, err := RandomG1(rand.Reader)
+		if err != nil {
+			t.Fatal(err)
+		}
+		tmpLittleFermat := &gfP{}
+		tmpLittleFermat.Invert(&Ga.p.x)
+
+		tmpBinaryEAA := &gfP{}
+		tmpBinaryEAA.EaaInvert(&Ga.p.x)
+
+		eq := equals(tmpLittleFermat, tmpBinaryEAA)
+		if eq == false {
+			t.Fatalf("results of different inversion do not agree")
+		}
+	}
+}
+
+func BenchmarkLittleFermatInversion(b *testing.B) {
+	el := gfP{0x0, 0x97816a916871ca8d, 0xb85045b68181585d, 0x30644e72e131a029}
+
+	b.ResetTimer()
+
+	tmp := &gfP{}
+	for i := 0; i < b.N; i++ {
+		tmp.Invert(&el)
+	}
+}
+
+func BenchmarkBinaryEEAInversion(b *testing.B) {
+	el := gfP{0x0, 0x97816a916871ca8d, 0xb85045b68181585d, 0x30644e72e131a029}
+
+	b.ResetTimer()
+
+	tmp := &gfP{}
+	for i := 0; i < b.N; i++ {
+		tmp.EaaInvert(&el)
+	}
+}
+
+func BenchmarkG1AddAndMakeAffine(b *testing.B) {
+	_, Ga, err := RandomG1(rand.Reader)
+	if err != nil {
+		b.Fatal(err)
+	}
+	_, Gb, err := RandomG1(rand.Reader)
+	if err != nil {
+		b.Fatal(err)
+	}
+	b.ResetTimer()
+
+	for i := 0; i < b.N; i++ {
+		e := new(G1).Add(Ga, Gb)
+		e.p.MakeAffine()
+	}
+}
+
 func BenchmarkG1(b *testing.B) {
 	x, _ := rand.Int(rand.Reader, Order)
 	b.ResetTimer()

crypto/bn256/cloudflare/curve.go

@@ -217,7 +217,8 @@ func (c *curvePoint) MakeAffine() {
 	}
 
 	zInv := &gfP{}
-	zInv.Invert(&c.z)
+	zInv.EaaInvert(&c.z)
+	// zInv.Invert(&c.z)
 
 	t, zInv2 := &gfP{}, &gfP{}
 	gfpMul(t, &c.y, zInv)

crypto/bn256/cloudflare/gfp.go

@@ -3,6 +3,7 @@ package bn256
 import (
 	"errors"
 	"fmt"
+	"math/bits"
 )
 
 type gfP [4]uint64
@@ -79,3 +80,130 @@ func (e *gfP) Unmarshal(in []byte) error {
 
 func montEncode(c, a *gfP) { gfpMul(c, a, r2) }
 func montDecode(c, a *gfP) { gfpMul(c, a, &gfP{1}) }
+
+func isZero(a *gfP) bool {
+	return a[0] == 0 && a[1] == 0 && a[2] == 0 && a[3] == 0
+}
+
+func isEven(a *gfP) bool {
+	return bits.TrailingZeros64((a[0])) > 0
+}
+
+func div2(a *gfP) {
+	a[0] = a[0]>>1 | a[1]<<63
+	a[1] = a[1]>>1 | a[2]<<63
+	a[2] = a[2]>>1 | a[3]<<63
+	a[3] = a[3] >> 1
+}
+
+func (e *gfP) addNocarry(f *gfP) {
+	carry := uint64(0)
+	e[0], carry = bits.Add64(e[0], f[0], carry)
+	e[1], carry = bits.Add64(e[1], f[1], carry)
+	e[2], carry = bits.Add64(e[2], f[2], carry)
+	e[3], _ = bits.Add64(e[3], f[3], carry)
+}
+
+func (e *gfP) subNoborrow(f *gfP) {
+	borrow := uint64(0)
+	e[0], borrow = bits.Sub64(e[0], f[0], borrow)
+	e[1], borrow = bits.Sub64(e[1], f[1], borrow)
+	e[2], borrow = bits.Sub64(e[2], f[2], borrow)
+	e[3], _ = bits.Sub64(e[3], f[3], borrow)
+}
+
+func gte(a, b *gfP) bool {
+	// subtract b from a. If no borrow occures then a >= b
+	borrow := uint64(0)
+	_, borrow = bits.Sub64(a[0], b[0], borrow)
+	_, borrow = bits.Sub64(a[1], b[1], borrow)
+	_, borrow = bits.Sub64(a[2], b[2], borrow)
+	_, borrow = bits.Sub64(a[3], b[3], borrow)
+
+	return borrow == 0
+}
+
+func equals(a, b *gfP) bool {
+	return a[0] == b[0] && a[1] == b[1] && a[2] == b[2] && a[3] == b[3]
+}
+
+// Performs inversion of the field element using binary EEA.
+// If element is zero (no inverse exists) then set `e` to zero
+func (e *gfP) EaaInvert(f *gfP) {
+	if isZero(f) {
+		e.Set(&gfP{0, 0, 0, 0})
+		return
+	}
+
+	// Guajardo Kumar Paar Pelzl
+	// Efficient Software-Implementation of Finite Fields with Applications to Cryptography
+	// Algorithm 16 (BEA for Inversion in Fp)
+
+	one := gfP{1, 0, 0, 0}
+
+	u, b := gfP{}, gfP{}
+	u.Set(f)
+	b.Set(r2)
+
+	v := gfP{p2[0], p2[1], p2[2], p2[3]}
+	c := gfP{0, 0, 0, 0}
+	modulus := gfP{p2[0], p2[1], p2[2], p2[3]}
+
+	for {
+		if equals(&u, &one) || equals(&v, &one) {
+			break
+		}
+
+		// while u is even
+		for {
+			if !isEven(&u) {
+				break
+			}
+
+			div2(&u)
+			if isEven(&b) {
+				div2(&b)
+			} else {
+				// we will not overflow a modulus here,
+				// so we can use specialized function
+				// do perform addition without reduction
+				b.addNocarry(&modulus)
+				div2(&b)
+			}
+		}
+
+		// while v is even
+		for {
+			if !isEven(&v) {
+				break
+			}
+
+			div2(&v)
+			if isEven(&c) {
+				div2(&c)
+			} else {
+				// we will not overflow a modulus here,
+				// so we can use specialized function
+				// do perform addition without reduction
+				c.addNocarry(&modulus)
+				div2(&c)
+			}
+		}
+
+		if gte(&v, &u) {
+			// v >= u
+			v.subNoborrow(&u)
+			gfpSub(&c, &c, &b)
+		} else {
+			// if v < u
+			u.subNoborrow(&v)
+			gfpSub(&b, &b, &c)
+		}
+	}
+
+	if equals(&u, &one) {
+		e.Set(&b)
+	} else {
+		e.Set(&c)
+	}
+}