bls12-381.go

// Package bls12381 efficient elliptic curve, pairing and hash to curve implementation for bls12-381.
//
// bls12-381: A Barreto--Lynn--Scott curve
// 		embedding degree k=12
// 		seed x₀=-15132376222941642752
// 		𝔽r: r=52435875175126190479447740508185965837690552500527637822603658699938581184513 (x₀⁴-x₀²+1)
// 		𝔽p: p=4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787 ((x₀-1)² ⋅ r(x₀)/3+x₀)
// 		(E/𝔽p): Y²=X³+4
// 		(Eₜ/𝔽p²): Y² = X³+4(u+1) (M-type twist)
// 		r ∣ #E(Fp) and r ∣ #Eₜ(𝔽p²)
// Extension fields tower:
//     𝔽p²[u] = 𝔽p/u²+1
//     𝔽p⁶[v] = 𝔽p²/v³-1-u
//     𝔽p¹²[w] = 𝔽p⁶/w²-v
// optimal Ate loop size:
//		x₀
// Security: estimated 126-bit level following [https://eprint.iacr.org/2019/885.pdf]
// (r is 255 bits and p¹² is 4569 bits)
//
// Warning
//
// This code has been partially audited and is provided as-is. In particular, there is no security guarantees such as constant time implementation or side-channel attack resistance.
package bls12381

import (
	"math/big"

	"github.com/liyue201/gnark-crypto/ecc"
	"github.com/liyue201/gnark-crypto/ecc/bls12-381/fp"
	"github.com/liyue201/gnark-crypto/ecc/bls12-381/fr"
	"github.com/liyue201/gnark-crypto/ecc/bls12-381/pkg/fptower"
)

// ID bls381 ID
const ID = ecc.BLS12_381

// bCurveCoeff b coeff of the curve Y²=X³+b
var bCurveCoeff fp.Element

// twist
var twist fptower.E2

// bTwistCurveCoeff b coeff of the twist (defined over 𝔽p²) curve
var bTwistCurveCoeff fptower.E2

// generators of the r-torsion group, resp. in ker(pi-id), ker(Tr)
var g1Gen G1Jac
var g2Gen G2Jac

var g1GenAff G1Affine
var g2GenAff G2Affine

// point at infinity
var g1Infinity G1Jac
var g2Infinity G2Jac

// optimal Ate loop counter
var loopCounter [64]int8

// Parameters useful for the GLV scalar multiplication. The third roots define the
// endomorphisms ϕ₁ and ϕ₂ for <G1Affine> and <G2Affine>. lambda is such that <r, ϕ-λ> lies above
// <r> in the ring Z[ϕ]. More concretely it's the associated eigenvalue
// of ϕ₁ (resp ϕ₂) restricted to <G1Affine> (resp <G2Affine>)
// see https://www.cosic.esat.kuleuven.be/nessie/reports/phase2/GLV.pdf
var thirdRootOneG1 fp.Element
var thirdRootOneG2 fp.Element
var lambdaGLV big.Int

// glvBasis stores R-linearly independent vectors (a,b), (c,d)
// in ker((u,v) → u+vλ[r]), and their determinant
var glvBasis ecc.Lattice

// ψ o π o ψ^{-1}, where ψ:E → E' is the degree 6 iso defined over 𝔽p¹²
var endo struct {
	u fptower.E2
	v fptower.E2
}

// seed x₀ of the curve
var xGen big.Int

func init() {

	bCurveCoeff.SetUint64(4)
	// M-twist
	twist.A0.SetUint64(1)
	twist.A1.SetUint64(1)
	bTwistCurveCoeff.MulByElement(&twist, &bCurveCoeff)

	g1Gen.X.SetString("3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507")
	g1Gen.Y.SetString("1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569")
	g1Gen.Z.SetOne()

	g2Gen.X.SetString("352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160",
		"3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758")
	g2Gen.Y.SetString("1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905",
		"927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582")
	g2Gen.Z.SetString("1",
		"0")

	g1GenAff.FromJacobian(&g1Gen)
	g2GenAff.FromJacobian(&g2Gen)

	// (X,Y,Z) = (1,1,0)
	g1Infinity.X.SetOne()
	g1Infinity.Y.SetOne()
	g2Infinity.X.SetOne()
	g2Infinity.Y.SetOne()

	thirdRootOneG1.SetString("4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939436")
	thirdRootOneG2.Square(&thirdRootOneG1)
	lambdaGLV.SetString("228988810152649578064853576960394133503", 10) //(x₀²-1)
	_r := fr.Modulus()
	ecc.PrecomputeLattice(_r, &lambdaGLV, &glvBasis)

	endo.u.A0.SetString("0")
	endo.u.A1.SetString("4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939437")
	endo.v.A0.SetString("2973677408986561043442465346520108879172042883009249989176415018091420807192182638567116318576472649347015917690530")
	endo.v.A1.SetString("1028732146235106349975324479215795277384839936929757896155643118032610843298655225875571310552543014690878354869257")

	// binary decomposition of -x₀ little endian
	loopCounter = [64]int8{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1}

	// -x₀
	xGen.SetString("15132376222941642752", 10)

}

// Generators return the generators of the r-torsion group, resp. in ker(pi-id), ker(Tr)
func Generators() (g1Jac G1Jac, g2Jac G2Jac, g1Aff G1Affine, g2Aff G2Affine) {
	g1Aff = g1GenAff
	g2Aff = g2GenAff
	g1Jac = g1Gen
	g2Jac = g2Gen
	return
}