precomp: use normalized extended points (#59)

---------

Signed-off-by: Ignacio Hagopian <[email protected]>

bandersnatch/bandersnatch.go

@@ -5,20 +5,24 @@ import (
 	"io"
 
 	gnarkbandersnatch "github.com/consensys/gnark-crypto/ecc/bls12-381/bandersnatch"
+	gnarkfr "github.com/consensys/gnark-crypto/ecc/bls12-381/fr"
 	"github.com/crate-crypto/go-ipa/bandersnatch/fp"
 )
 
 var CurveParams = gnarkbandersnatch.GetEdwardsCurve()
 
 type PointAffine = gnarkbandersnatch.PointAffine
 type PointProj = gnarkbandersnatch.PointProj
+type PointExtended = gnarkbandersnatch.PointExtended
 
 var Identity = PointProj{
 	X: fp.Zero(),
 	Y: fp.One(),
 	Z: fp.One(),
 }
 
+var IdentityExt = PointExtendedFromProj(&Identity)
+
 // Reads an uncompressed affine point
 // Point is not guaranteed to be in the prime subgroup
 func ReadUncompressedPoint(r io.Reader) (PointAffine, error) {
@@ -92,3 +96,62 @@ func computeY(x *fp.Element, choose_largest bool) *fp.Element {
 		return sqrtY.Neg(sqrtY)
 	}
 }
+
+// PointExtendedFromProj converts a point in projective coordinates to extended coordinates.
+func PointExtendedFromProj(p *PointProj) PointExtended {
+	var pzinv fp.Element
+	pzinv.Inverse(&p.Z)
+	var z fp.Element
+	z.Mul(&p.X, &p.Y).Mul(&z, &pzinv)
+	return PointExtended{
+		X: p.X,
+		Y: p.Y,
+		Z: p.Z,
+		T: z,
+	}
+}
+
+// PointExtendedNormalized is an extended point which is normalized.
+// i.e: Z=1. We store it this way to save 32 bytes per point in memory.
+type PointExtendedNormalized struct {
+	X, Y, T gnarkfr.Element
+}
+
+// Neg computes p = -p1
+func (p *PointExtendedNormalized) Neg(p1 *PointExtendedNormalized) *PointExtendedNormalized {
+	p.X.Neg(&p1.X)
+	p.Y = p1.Y
+	p.T.Neg(&p1.T)
+	return p
+}
+
+// ExtendedAddNormalized computes p = p1 + p2.
+// https://hyperelliptic.org/EFD/g1p/auto-twisted-extended.html#addition-madd-2008-hwcd
+func ExtendedAddNormalized(p, p1 *PointExtended, p2 *PointExtendedNormalized) *gnarkbandersnatch.PointExtended {
+	var A, B, C, D, E, F, G, H, tmp gnarkfr.Element
+	A.Mul(&p1.X, &p2.X)
+	B.Mul(&p1.Y, &p2.Y)
+	C.Mul(&p1.T, &p2.T).Mul(&C, &CurveParams.D)
+	D.Set(&p1.Z)
+	tmp.Add(&p1.X, &p1.Y)
+	E.Add(&p2.X, &p2.Y).
+		Mul(&E, &tmp).
+		Sub(&E, &A).
+		Sub(&E, &B)
+	F.Sub(&D, &C)
+	G.Add(&D, &C)
+	H.Set(&A)
+
+	// mulBy5(&H)
+	H.Neg(&H)
+	gnarkfr.MulBy5(&H)
+
+	H.Sub(&B, &H)
+
+	p.X.Mul(&E, &F)
+	p.Y.Mul(&G, &H)
+	p.T.Mul(&E, &H)
+	p.Z.Mul(&F, &G)
+
+	return p
+}

banderwagon/precomp.go

@@ -61,20 +61,24 @@ func NewPrecompMSM(points []Element) (MSMPrecomp, error) {
 // MSM calculates the 256-MSM of the given scalars on the fixed basis.
 // It automatically detects how many non-zero scalars there are and parallelizes the computation.
 func (msm *MSMPrecomp) MSM(scalars []fr.Element) Element {
-	result := Identity.inner
+	result := bandersnatch.IdentityExt
 
 	for i := range scalars {
 		if !scalars[i].IsZero() {
 			msm.precompPoints[i].ScalarMul(scalars[i], &result)
 		}
 	}
-	return Element{inner: result}
+	return Element{inner: bandersnatch.PointProj{
+		X: result.X,
+		Y: result.Y,
+		Z: result.Z,
+	}}
 }
 
 // PrecompPoint is a precomputed table for a single point.
 type PrecompPoint struct {
 	windowSize int
-	windows    [][]bandersnatch.PointAffine
+	windows    [][]bandersnatch.PointExtendedNormalized
 }
 
 // NewPrecompPoint creates a new PrecompPoint for the given point and window size.
@@ -88,27 +92,23 @@ func NewPrecompPoint(point Element, windowSize int) (PrecompPoint, error) {
 
 	res := PrecompPoint{
 		windowSize: windowSize,
-		windows:    make([][]bandersnatch.PointAffine, 256/windowSize),
+		windows:    make([][]bandersnatch.PointExtendedNormalized, 256/windowSize),
 	}
 
-	windows := make([][]bandersnatch.PointProj, 256/windowSize)
+	windows := make([][]bandersnatch.PointExtended, 256/windowSize)
 	group, _ := errgroup.WithContext(context.Background())
 	group.SetLimit(runtime.NumCPU())
 	for i := 0; i < len(res.windows); i++ {
 		i := i
-		base := point.inner
+		base := bandersnatch.PointExtendedFromProj(&point.inner)
 		group.Go(func() error {
-			windows[i] = make([]bandersnatch.PointProj, 1<<(windowSize-1))
+			windows[i] = make([]bandersnatch.PointExtended, 1<<(windowSize-1))
 			curr := base
 			for j := 0; j < len(windows[i]); j++ {
 				windows[i][j] = curr
 				curr.Add(&curr, &base)
 			}
-			batchProjToAffine(windows[i])
-			res.windows[i] = make([]bandersnatch.PointAffine, 1<<(windowSize-1))
-			for j := range windows[i] {
-				res.windows[i][j].FromProj(&windows[i][j])
-			}
+			res.windows[i] = batchToExtendedPointNormalized(windows[i])
 			return nil
 		})
 		point.ScalarMul(&point, &specialWindow)
@@ -121,12 +121,12 @@ func NewPrecompPoint(point Element, windowSize int) (PrecompPoint, error) {
 // ScalarMul multiplies the point by the given scalar using the precomputed points.
 // It applies a trick to push a carry between windows since our precomputed tables
 // avoid storing point inverses.
-func (pp *PrecompPoint) ScalarMul(scalar fr.Element, res *bandersnatch.PointProj) {
+func (pp *PrecompPoint) ScalarMul(scalar fr.Element, res *bandersnatch.PointExtended) {
 	numWindowsInLimb := 64 / pp.windowSize
 
 	scalar.FromMont()
 	var carry uint64
-	var pNeg bandersnatch.PointAffine
+	var pNeg bandersnatch.PointExtendedNormalized
 	for l := 0; l < fr.Limbs; l++ {
 		for w := 0; w < numWindowsInLimb; w++ {
 			windowValue := (scalar[l]>>(pp.windowSize*w))&((1<<pp.windowSize)-1) + carry
@@ -139,11 +139,11 @@ func (pp *PrecompPoint) ScalarMul(scalar fr.Element, res *bandersnatch.PointProj
 				windowValue = (1 << pp.windowSize) - windowValue
 				if windowValue != 0 {
 					pNeg.Neg(&pp.windows[l*numWindowsInLimb+w][windowValue-1])
-					res.MixedAdd(res, &pNeg)
+					bandersnatch.ExtendedAddNormalized(res, res, &pNeg)
 				}
 				carry = 1
 			} else {
-				res.MixedAdd(res, &pp.windows[l*numWindowsInLimb+w][windowValue-1])
+				bandersnatch.ExtendedAddNormalized(res, res, &pp.windows[l*numWindowsInLimb+w][windowValue-1])
 			}
 		}
 	}
@@ -195,3 +195,49 @@ func batchProjToAffine(points []bandersnatch.PointProj) []bandersnatch.PointAffi
 
 	return result
 }
+
+func batchToExtendedPointNormalized(points []bandersnatch.PointExtended) []bandersnatch.PointExtendedNormalized {
+	result := make([]bandersnatch.PointExtendedNormalized, len(points))
+	zeroes := make([]bool, len(points))
+	accumulator := fp.One()
+
+	// batch invert all points[].Z coordinates with Montgomery batch inversion trick
+	// (stores points[].Z^-1 in result[i].X to avoid allocating a slice of fr.Elements)
+	for i := 0; i < len(points); i++ {
+		if points[i].Z.IsZero() {
+			zeroes[i] = true
+			continue
+		}
+		result[i].X = accumulator
+		accumulator.Mul(&accumulator, &points[i].Z)
+	}
+
+	var accInverse fp.Element
+	accInverse.Inverse(&accumulator)
+
+	for i := len(points) - 1; i >= 0; i-- {
+		if zeroes[i] {
+			// do nothing, (X=0, Y=0) is infinity point in affine
+			continue
+		}
+		result[i].X.Mul(&result[i].X, &accInverse)
+		accInverse.Mul(&accInverse, &points[i].Z)
+	}
+
+	// batch convert to affine.
+	parallel.Execute(len(points), func(start, end int) {
+		for i := start; i < end; i++ {
+			if zeroes[i] {
+				// do nothing, (X=0, Y=0) is infinity point in affine
+				continue
+			}
+
+			a := result[i].X
+			result[i].X.Mul(&points[i].X, &a)
+			result[i].Y.Mul(&points[i].Y, &a)
+			result[i].T.Mul(&result[i].X, &result[i].Y)
+		}
+	})
+
+	return result
+}