Feat/adding gnark support for extension (#522)

* initial implementaiton of the wrapper

* started updating the gnark interpolation

prover/maths/fft/fastpolyext/gnark_interpolation.go

@@ -0,0 +1,91 @@
+package fastpolyext
+
+import (
+	"github.com/consensys/gnark/frontend"
+	"github.com/consensys/linea-monorepo/prover/maths/fft"
+	"github.com/consensys/linea-monorepo/prover/maths/field"
+	"github.com/consensys/linea-monorepo/prover/maths/field/fext/gnarkfext"
+	"github.com/consensys/linea-monorepo/prover/utils"
+	"github.com/consensys/linea-monorepo/prover/utils/gnarkutil"
+)
+
+// Evaluate a polynomial in lagrange basis on a gnark circuit
+func InterpolateGnark(api gnarkfext.API, poly []gnarkfext.E2, x gnarkfext.E2) frontend.Variable {
+
+	if !utils.IsPowerOfTwo(len(poly)) {
+		utils.Panic("only support powers of two but poly has length %v", len(poly))
+	}
+
+	// When the poly is of length 1 it means it is a constant polynomial and its
+	// evaluation is trivial.
+	if len(poly) == 1 {
+		return poly[0]
+	}
+
+	n := len(poly)
+	domain := fft.NewDomain(n)
+	one := field.One()
+
+	// Test that x is not a root of unity. In the other case, we would
+	// have to divide by zero. In practice this constraint is not necessary
+	// (because the division constraint would be non-satisfiable anyway)
+	// But doing an explicit check clarifies the need.
+	xN := gnarkutil.Exp(api, x, n)
+	api.AssertIsDifferent(xN, 1)
+
+	// Compute the term-wise summand of the interpolation formula.
+	// This will allow the gnark solver to process the expensive
+	// inverses in parallel.
+	terms := make([]gnarkfext.E2, n)
+
+	// omegaMinI carries the domain's inverse root of unity generator raised to
+	// the power I in the following loop. It is initialized with omega**0 = 1.
+	omegaI := frontend.Variable(1)
+
+	for i := 0; i < n; i++ {
+
+		if i > 0 {
+			omegaI = api.Inner.Mul(omegaI, domain.GeneratorInv)
+		}
+
+		xOmegaN := api.MulByBase(x, omegaI)
+		terms[i] = api.Sub(xOmegaN, gnarkfext.One())
+		// No point doing a batch inverse in a circuit
+		terms[i] = api.Inverse(terms[i])
+		terms[i] = api.Mul(terms[i], poly[i])
+	}
+
+	nonNilTerms := make([]frontend.Variable, 0, len(terms))
+	for i := range terms {
+		if terms[i] == nil {
+			continue
+		}
+
+		nonNilTerms = append(nonNilTerms, terms[i])
+	}
+
+	// Then sum all the terms
+	var res frontend.Variable
+
+	switch {
+	case len(nonNilTerms) == 0:
+		res = 0
+	case len(nonNilTerms) == 1:
+		res = nonNilTerms[0]
+	case len(nonNilTerms) == 2:
+		res = api.Add(nonNilTerms[0], nonNilTerms[1])
+	default:
+		res = api.Add(nonNilTerms[0], nonNilTerms[1], nonNilTerms[2:]...)
+	}
+
+	/*
+		Then multiply the res by a factor \frac{g^{1 - n}X^n -g}{n}
+	*/
+	factor := xN
+	factor = api.Sub(factor, one)
+	factor = api.Mul(factor, domain.CardinalityInv)
+	res = api.Mul(res, factor)
+
+	return res
+
+}