1package bls12381 2 3import ( 4 "math/big" 5 6 "github.com/consensys/gnark-crypto/ecc" 7 "github.com/consensys/gnark-crypto/ecc/bls12-381/fp" 8 "github.com/consensys/gnark-crypto/ecc/bls12-381/fr" 9 "github.com/consensys/gnark-crypto/ecc/bls12-381/internal/fptower" 10) 11 12// E: y**2=x**3+4 13// Etwist: y**2 = x**3+4*(u+1) 14// Tower: Fp->Fp2, u**2=-1 -> Fp12, v**6=u+1 15// Generator (BLS12 family): x=-15132376222941642752 16// optimal Ate loop: trace(frob)-1=x 17// trace of pi: x+1 18// Fp: p=4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787 19// Fr: r=52435875175126190479447740508185965837690552500527637822603658699938581184513 (x**4-x**2+1) 20 21// ID bls381 ID 22const ID = ecc.BLS12_381 23 24// bCurveCoeff b coeff of the curve 25var bCurveCoeff fp.Element 26 27// twist 28var twist fptower.E2 29 30// bTwistCurveCoeff b coeff of the twist (defined over Fp2) curve 31var bTwistCurveCoeff fptower.E2 32 33// twoInv 1/2 mod p (needed for DoubleStep in Miller loop) 34var twoInv fp.Element 35 36// generators of the r-torsion group, resp. in ker(pi-id), ker(Tr) 37var g1Gen G1Jac 38var g2Gen G2Jac 39 40var g1GenAff G1Affine 41var g2GenAff G2Affine 42 43// point at infinity 44var g1Infinity G1Jac 45var g2Infinity G2Jac 46 47// optimal Ate loop counter (=trace-1 = x in BLS family) 48var loopCounter [64]int8 49 50// Parameters useful for the GLV scalar multiplication. The third roots define the 51// endomorphisms phi1 and phi2 for <G1Affine> and <G2Affine>. lambda is such that <r, phi-lambda> lies above 52// <r> in the ring Z[phi]. More concretely it's the associated eigenvalue 53// of phi1 (resp phi2) restricted to <G1Affine> (resp <G2Affine>) 54// cf https://www.cosic.esat.kuleuven.be/nessie/reports/phase2/GLV.pdf 55var thirdRootOneG1 fp.Element 56var thirdRootOneG2 fp.Element 57var lambdaGLV big.Int 58 59// glvBasis stores R-linearly independant vectors (a,b), (c,d) 60// in ker((u,v)->u+vlambda[r]), and their determinant 61var glvBasis ecc.Lattice 62 63// psi o pi o psi**-1, where psi:E->E' is the degree 6 iso defined over Fp12 64var endo struct { 65 u fptower.E2 66 v fptower.E2 67} 68 69// generator of the curve 70var xGen big.Int 71 72func init() { 73 74 bCurveCoeff.SetUint64(4) 75 twist.A0.SetUint64(1) 76 twist.A1.SetUint64(1) 77 bTwistCurveCoeff.MulByElement(&twist, &bCurveCoeff) 78 79 twoInv.SetOne().Double(&twoInv).Inverse(&twoInv) 80 81 g1Gen.X.SetString("3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507") 82 g1Gen.Y.SetString("1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569") 83 g1Gen.Z.SetString("1") 84 85 g2Gen.X.SetString("352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160", 86 "3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758") 87 g2Gen.Y.SetString("1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905", 88 "927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582") 89 g2Gen.Z.SetString("1", 90 "0") 91 92 g1GenAff.FromJacobian(&g1Gen) 93 g2GenAff.FromJacobian(&g2Gen) 94 95 g1Infinity.X.SetOne() 96 g1Infinity.Y.SetOne() 97 g2Infinity.X.SetOne() 98 g2Infinity.Y.SetOne() 99 100 thirdRootOneG1.SetString("4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939436") 101 thirdRootOneG2.Square(&thirdRootOneG1) 102 lambdaGLV.SetString("228988810152649578064853576960394133503", 10) //(x**2-1) 103 _r := fr.Modulus() 104 ecc.PrecomputeLattice(_r, &lambdaGLV, &glvBasis) 105 106 endo.u.A0.SetString("0") 107 endo.u.A1.SetString("4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939437") 108 endo.v.A0.SetString("2973677408986561043442465346520108879172042883009249989176415018091420807192182638567116318576472649347015917690530") 109 endo.v.A1.SetString("1028732146235106349975324479215795277384839936929757896155643118032610843298655225875571310552543014690878354869257") 110 111 // binary decomposition of 15132376222941642752 little endian 112 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} 113 114 xGen.SetString("15132376222941642752", 10) 115 116} 117 118// Generators return the generators of the r-torsion group, resp. in ker(pi-id), ker(Tr) 119func Generators() (g1Jac G1Jac, g2Jac G2Jac, g1Aff G1Affine, g2Aff G2Affine) { 120 g1Aff = g1GenAff 121 g2Aff = g2GenAff 122 g1Jac = g1Gen 123 g2Jac = g2Gen 124 return 125} 126