gnark-plonky2-verifier/field/quadratic_extension.go

package field

import (
	"math/bits"

	"github.com/consensys/gnark-crypto/field/goldilocks"
	"github.com/consensys/gnark/frontend"
)

const D = 2

type QuadraticExtension = [2]F
type QEAlgebra = [D]QuadraticExtension

type QuadraticExtensionAPI struct {
	api      frontend.API
	fieldAPI FieldAPI

	W        F
	DTH_ROOT F

	ONE_QE  QuadraticExtension
	ZERO_QE QuadraticExtension

	ZERO_QE_ALGEBRA QEAlgebra
}

func NewQuadraticExtensionAPI(api frontend.API, fieldAPI FieldAPI) *QuadraticExtensionAPI {
	var ZERO_QE = QuadraticExtension{ZERO_F, ZERO_F}

	var ZERO_QE_ALGEBRA QEAlgebra

	for i := 0; i < D; i++ {
		ZERO_QE_ALGEBRA[i] = ZERO_QE
	}

	return &QuadraticExtensionAPI{
		api:      api,
		fieldAPI: fieldAPI,

		W:        NewFieldConst(7),
		DTH_ROOT: NewFieldConst(18446744069414584320),

		ONE_QE:  QuadraticExtension{ONE_F, ZERO_F},
		ZERO_QE: ZERO_QE,

		ZERO_QE_ALGEBRA: ZERO_QE_ALGEBRA,
	}
}

func (c *QuadraticExtensionAPI) SquareExtension(a QuadraticExtension) QuadraticExtension {
	return c.MulExtension(a, a)
}

func (c *QuadraticExtensionAPI) MulExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
	c_0 := c.fieldAPI.Add(c.fieldAPI.Mul(a[0], b[0]), c.fieldAPI.Mul(c.fieldAPI.Mul(c.W, a[1]), b[1]))
	c_1 := c.fieldAPI.Add(c.fieldAPI.Mul(a[0], b[1]), c.fieldAPI.Mul(a[1], b[0]))
	return QuadraticExtension{c_0, c_1}
}

func (c *QuadraticExtensionAPI) AddExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
	c_0 := c.fieldAPI.Add(a[0], b[0])
	c_1 := c.fieldAPI.Add(a[1], b[1])
	return QuadraticExtension{c_0, c_1}
}

func (c *QuadraticExtensionAPI) SubExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
	c_0 := c.fieldAPI.Sub(a[0], b[0])
	c_1 := c.fieldAPI.Sub(a[1], b[1])
	return QuadraticExtension{c_0, c_1}
}

func (c *QuadraticExtensionAPI) DivExtension(a QuadraticExtension, b QuadraticExtension) QuadraticExtension {
	return c.MulExtension(a, c.InverseExtension(b))
}

func (c *QuadraticExtensionAPI) IsZero(a QuadraticExtension) frontend.Variable {
	return c.api.Mul(IsZero(c.api, c.fieldAPI, a[0]), IsZero(c.api, c.fieldAPI, a[1]))
}

// TODO: Instead of calculating the inverse within the circuit, can witness the
// inverse and assert that a_inverse * a = 1.  Should reduce # of constraints.
func (c *QuadraticExtensionAPI) InverseExtension(a QuadraticExtension) QuadraticExtension {
	// First assert that a doesn't have 0 value coefficients
	a0_is_zero := IsZero(c.api, c.fieldAPI, a[0])
	a1_is_zero := IsZero(c.api, c.fieldAPI, a[1])

	// assert that a0_is_zero OR a1_is_zero == false
	c.api.AssertIsEqual(c.api.Mul(a0_is_zero, a1_is_zero), frontend.Variable(0))

	a_pow_r_minus_1 := QuadraticExtension{a[0], c.fieldAPI.Mul(a[1], c.DTH_ROOT)}
	a_pow_r := c.MulExtension(a_pow_r_minus_1, a)
	return c.ScalarMulExtension(a_pow_r_minus_1, c.fieldAPI.Inverse(a_pow_r[0]))
}

func (c *QuadraticExtensionAPI) ScalarMulExtension(a QuadraticExtension, scalar F) QuadraticExtension {
	return QuadraticExtension{c.fieldAPI.Mul(a[0], scalar), c.fieldAPI.Mul(a[1], scalar)}
}

func (c *QuadraticExtensionAPI) FieldToQE(a F) QuadraticExtension {
	return QuadraticExtension{a, ZERO_F}
}

// / Exponentiate `base` to the power of a known `exponent`.
func (c *QuadraticExtensionAPI) ExpU64Extension(a QuadraticExtension, exponent uint64) QuadraticExtension {
	switch exponent {
	case 0:
		return c.ONE_QE
	case 1:
		return a
	case 2:
		return c.SquareExtension(a)
	default:
	}

	current := a
	product := c.ONE_QE

	for i := 0; i < bits.Len64(exponent); i++ {
		if i != 0 {
			current = c.SquareExtension(current)
		}

		if (exponent >> i & 1) != 0 {
			product = c.MulExtension(product, current)
		}
	}

	return product
}

func (c *QuadraticExtensionAPI) ReduceWithPowers(terms []QuadraticExtension, scalar QuadraticExtension) QuadraticExtension {
	sum := c.ZERO_QE

	for i := len(terms) - 1; i >= 0; i-- {
		sum = c.AddExtension(
			c.MulExtension(
				sum,
				scalar,
			),
			terms[i],
		)
	}

	return sum
}

func (c *QuadraticExtensionAPI) Select(b frontend.Variable, qe0, qe1 QuadraticExtension) QuadraticExtension {
	var retQE QuadraticExtension

	for i := 0; i < 2; i++ {
		retQE[i] = c.fieldAPI.Select(b, qe0[i], qe1[i])
	}

	return retQE
}

func (c *QuadraticExtensionAPI) Lookup2(b0 frontend.Variable, b1 frontend.Variable, qe0, qe1, qe2, qe3 QuadraticExtension) QuadraticExtension {
	var retQE QuadraticExtension

	for i := 0; i < 2; i++ {
		retQE[i] = c.fieldAPI.Lookup2(b0, b1, qe0[i], qe1[i], qe2[i], qe3[i])
	}

	return retQE
}

func (c *QuadraticExtensionAPI) AssertIsEqual(a, b QuadraticExtension) {
	for i := 0; i < 2; i++ {
		c.fieldAPI.AssertIsEqual(a[i], b[i])
	}
}

func (c *QuadraticExtensionAPI) InnerProductExtension(constant F, startingAcc QuadraticExtension, pairs [][2]QuadraticExtension) QuadraticExtension {
	acc := startingAcc

	for i := 0; i < len(pairs); i++ {
		a := pairs[i][0]
		b := pairs[i][1]
		mul := c.ScalarMulExtension(a, constant)
		mul = c.MulExtension(mul, b)
		acc = c.AddExtension(acc, mul)
	}

	return acc
}

/*
func (c *QuadraticExtensionAPI) Println(a QuadraticExtension) {
	fmt.Print("Degree 0 coefficient")
	c.fieldAPI.Println(a[0])

	fmt.Print("Degree 1 coefficient")
	c.fieldAPI.Println(a[1])
}
*/

func (c *QuadraticExtensionAPI) MulExtensionAlgebra(a, b QEAlgebra) QEAlgebra {
	var inner [D][][2]QuadraticExtension
	var inner_w [D][][2]QuadraticExtension
	for i := 0; i < D; i++ {
		for j := 0; j < D-i; j++ {
			idx := (i + j) % D
			inner[idx] = append(inner[idx], [2]QuadraticExtension{a[i], b[j]})
		}
		for j := D - i; j < D; j++ {
			idx := (i + j) % D
			inner_w[idx] = append(inner_w[idx], [2]QuadraticExtension{a[i], b[j]})
		}
	}

	var product QEAlgebra
	for i := 0; i < D; i++ {
		acc := c.InnerProductExtension(c.W, c.ZERO_QE, inner_w[i])
		product[i] = c.InnerProductExtension(ONE_F, acc, inner[i])
	}

	return product
}

func (c *QuadraticExtensionAPI) ScalarMulExtensionAlgebra(a QuadraticExtension, b QEAlgebra) QEAlgebra {
	var product QEAlgebra
	for i := 0; i < D; i++ {
		product[i] = c.MulExtension(a, b[i])
	}

	return product
}

func (c *QuadraticExtensionAPI) AddExtensionAlgebra(a, b QEAlgebra) QEAlgebra {
	var sum QEAlgebra
	for i := 0; i < D; i++ {
		sum[i] = c.AddExtension(a[i], b[i])
	}

	return sum
}

func (c *QuadraticExtensionAPI) SubExtensionAlgebra(a, b QEAlgebra) QEAlgebra {
	var diff QEAlgebra
	for i := 0; i < D; i++ {
		diff[i] = c.SubExtension(a[i], b[i])
	}

	return diff
}

func (c *QuadraticExtensionAPI) PartialInterpolateExtAlgebra(
	domain []goldilocks.Element,
	values []QEAlgebra,
	barycentricWeights []goldilocks.Element,
	point QEAlgebra,
	initialEval QEAlgebra,
	initialPartialProd QEAlgebra,
) (QEAlgebra, QEAlgebra) {
	n := len(values)
	if n == 0 {
		panic("Cannot interpolate with no values")
	}
	if n != len(domain) {
		panic("Domain and values must have the same length")
	}
	if n != len(barycentricWeights) {
		panic("Domain and barycentric weights must have the same length")
	}

	newEval := initialEval
	newPartialProd := initialPartialProd
	for i := 0; i < n; i++ {
		val := values[i]
		x := domain[i]
		xField := NewFieldConst(x.Uint64())
		xQE := QuadraticExtension{xField, ZERO_F}
		xQEAlgebra := QEAlgebra{xQE, c.ZERO_QE}
		weight := QuadraticExtension{NewFieldConst(barycentricWeights[i].Uint64()), ZERO_F}
		term := c.SubExtensionAlgebra(point, xQEAlgebra)
		weightedVal := c.ScalarMulExtensionAlgebra(weight, val)
		newEval = c.MulExtensionAlgebra(newEval, term)
		tmp := c.MulExtensionAlgebra(weightedVal, newPartialProd)
		newEval = c.AddExtensionAlgebra(newEval, tmp)
		newPartialProd = c.MulExtensionAlgebra(newPartialProd, term)
	}

	return newEval, newPartialProd
}