úÎ!|t„      !"#$% & ' ( )*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|} ~ None "#,.7=>?@AHMSVX_g^ pairing3Pairing-friendly field multiplicative target group U_r.pairing,Pairing-friendly elliptic curve right group E(Fq').pairing+Pairing-friendly elliptic curve left group E(Fq).pairing9Pairings of a family of pairing-friendly elliptic curves.Let E(Fq)) be an elliptic curve over a prime field Fq , and let Fq < Fq' < Fq'' < Fq'''Q be a tower of simple field extensions defined by irreducible monic polynomials u, v, and w.*Then the pairing is defined to be of type E(Fq) x E(Fq') -> U_r , where U_r is the r.-th roots of unity multiplicative subgroup of Fq''', and r is the order of E(Fq) and the order of a prime field Fr. pairing)Pairings of general cryptographic groups.Let G1 and G2* be additive cyclic groups of prime order r, and GT1 be a multiplicative cyclic group of prime order r.*Then the pairing is defined to be of type  G1 x G2 -> GT@, and satisfies bilinearity, non-degeneracy, and computability.!pairing Left group G1."pairing Right group G2.#pairing Target group GT.$pairing'Computable non-degenerate bilinear map.  #!"$ #!"$ None "#,.7=>?@AHMSVX_g+u%pairing $https://eprint.iacr.org/2016/130.pdf8Miller algorithm for Barreto-Lynn-Scott degree 12 curves.&pairing $https://eprint.iacr.org/2010/354.pdf+Miller algorithm for Barreto-Naehrig curves.'pairing $https://eprint.iacr.org/2016/130.pdf<Final exponentiation for Barreto-Lynn-Scott degree 12 curves.(pairing $https://eprint.iacr.org/2010/354.pdf/Final exponentiation for Barreto-Naehrig curves.pairingLine function evaluation  Line(T, Q, P).-Compute the line function between two points T and Q in G2), evaluate the line function at a point P in G1-, and embed the line function evaluation in GT.€pairingTwisted Frobenius endomorphism Frob(P)..Compute the Frobenius endomorphism on a point P given a twist xi.pairingUnitary exponentiation ^.$Exponentiation of a unitary element x to an arbitrary integer n% in a specified cyclotomic subgroup.pairingPoint P.pairingPoint T.pairingPoint Q.pairingPoints T + Q and  Line(T, Q, P).€pairingTwist xi.pairingPoint P.pairingPoint Frob(P).pairingElement x in cyclotomic subgroup.pairingInteger n.pairingElement x ^ n.  #!"$%&'(%'&(None "#,.7=>?@AHMSVX_g3»)pairingFq12 multiplicative target group GT.*pairingBLS12381 curve right group  G2 = E'(Fq2).+pairingBLS12381 curve left group  G1 = E(Fq).,pairing'Field of points of BLS12381 curve over Fq12.-pairing'Field of points of BLS12381 curve over Fq6..pairingBLS12381 curve parameter s = t in signed binary./pairingBLS12381 curve parameter t in hexadecimal.0pairingBPrecompute primitive roots of unity for binary powers that divide r - 1. #!"$%&'()*+,-./0 ./-,+*)0None "#,.7=>?@AHMSVX_g<5pairingFq12 multiplicative target group GT.6pairingBN254 curve right group  G2 = E'(Fq2).7pairingBN254 curve left group  G1 = E(Fq).8pairing$Field of points of BN254 curve over Fq12.9pairing$Field of points of BN254 curve over Fq6.:pairingBN254 curve parameter  s = 6t + 2 in signed binary.;pairingBN254 curve parameter t in hexadecimal.<pairingBPrecompute primitive roots of unity for binary powers that divide r - 1. #!"$%&'(56789:;< :;98765<None "#,.7=>?@AHMSVX_gD—ApairingFq12 multiplicative target group GT.BpairingBN254A curve right group  G2 = E'(Fq2).CpairingBN254A curve left group  G1 = E(Fq).Dpairing%Field of points of BN254A curve over Fq12.Epairing%Field of points of BN254A curve over Fq6.FpairingBN254A curve parameter  s = 6t + 2 in signed binary.GpairingBN254A curve parameter t in hexadecimal.HpairingBPrecompute primitive roots of unity for binary powers that divide r - 1. #!"$%&'(ABCDEFGH FGEDCBAHNone "#,.7=>?@AHMSVX_gMMpairingFq12 multiplicative target group GT.NpairingBN254B curve right group  G2 = E'(Fq2).OpairingBN254B curve left group  G1 = E(Fq).Ppairing%Field of points of BN254B curve over Fq12.Qpairing%Field of points of BN254B curve over Fq6.RpairingBN254B curve parameter  s = 6t + 2 in signed binary.SpairingBN254B curve parameter t in hexadecimal.TpairingBPrecompute primitive roots of unity for binary powers that divide r - 1.  #!"$%&'(MNOPQRST RS QPONMTNone "#,.7=>?@AHMSVX_gU‹YpairingFq12 multiplicative target group GT.ZpairingBN254C curve right group  G2 = E'(Fq2).[pairingBN254C curve left group  G1 = E(Fq).\pairing%Field of points of BN254C curve over Fq12.]pairing%Field of points of BN254C curve over Fq6.^pairingBN254C curve parameter  s = 6t + 2 in signed binary._pairingBN254C curve parameter t in hexadecimal.`pairingBPrecompute primitive roots of unity for binary powers that divide r - 1.  #!"$%&'(YZ[\]^_` ^_ ]\ [ZY`None "#,.7=>?@AHMSVX_g^epairingFq12 multiplicative target group GT.fpairingBN254D curve right group  G2 = E'(Fq2).gpairingBN254D curve left group  G1 = E(Fq).hpairing%Field of points of BN254D curve over Fq12.ipairing%Field of points of BN254D curve over Fq6.jpairingBN254D curve parameter  s = 6t + 2 in signed binary.kpairingBN254D curve parameter t in hexadecimal.lpairingBPrecompute primitive roots of unity for binary powers that divide r - 1. #!"$%&'(efghijkl jkihgfelNone "#,.7=>?@AHMSVX_gfgqpairingFq12 multiplicative target group GT.rpairingBN462 curve right group  G2 = E'(Fq2).spairingBN462 curve left group  G1 = E(Fq).tpairing$Field of points of BN462 curve over Fq12.upairing$Field of points of BN462 curve over Fq6.vpairingBN462 curve parameter  s = 6t + 2 in signed binary.wpairingBN462 curve parameter t in hexadecimal.xpairingBPrecompute primitive roots of unity for binary powers that divide r - 1. #!"$%&'(qrstuvwx vwutsrqx None "#,.7=>?@AHMSVX_gs5}pairing:Encodes a given byte string to a point on the BLS12 curve.lThe implementation uses the Shallue-van de Woestijne encoding to BLS12 curves as specified in Section 3 of  $https://eprint.iacr.org/2019/403.pdfFFast and simple constant-time hashing to the BLS12-381 elliptic curve.%This function is not implemented yet.~pairing7Encodes a given byte string to a point on the BN curve.iThe implementation uses the Shallue-van de Woestijne encoding to BN curves as specified in Section 6 of  2https://www.di.ens.fr/~fouque/pub/latincrypt12.pdf2Indifferentiable Hashing to Barreto-Naehrig Curves.}This function evaluates an empty bytestring or one that contains NUL to zero and is sent to an arbitrary point on the curve.  #!"$}~}~ None "#,.7=>?@AHMSVX_gt_‚ƒ„…†‡ˆ‰Š                      !   " #$ # #%&'()*+,- . / 0 123456789:;<=23456789:;<>23456789:;<?23456789:;<@23456789:;<A23456789:;<B23456789:;<C D E F G H I J K L M N O PQ$pairing-1.1.0-9efc2eUPjVxB721NHtl3W2Data.Pairing.BN462Data.Pairing.BN254Data.Pairing.BN254DData.Pairing.BN254CData.Pairing.BN254BData.Pairing.BN254AData.Pairing.BLS12381 Data.PairingData.Pairing.AteData.Pairing.Hash Paths_pairing+elliptic-curve-0.3.0-Er01DllX3dI4K8HP3nyHUjData.Curve.Weierstrass.BN462TFq2Data.Curve.Weierstrass.BN462BN462FqFrData.Curve.Weierstrass.BN254TData.Curve.Weierstrass.BN254DTData.Curve.Weierstrass.BN254DBN254DData.Curve.Weierstrass.BN254CTData.Curve.Weierstrass.BN254CBN254CData.Curve.Weierstrass.BN254BTData.Curve.Weierstrass.BN254BBN254BData.Curve.Weierstrass.BN254ATData.Curve.Weierstrass.BN254ABN254AData.Curve.Weierstrass.BN254BN254 Data.Curve.Weierstrass.BLS12381TData.Curve.Weierstrass.BLS12381BLS12381 ECPairingGT ECPairingG2 ECPairingG1 ECPairingPairingG1G2GTpairingmillerAlgorithmBLS12millerAlgorithmBNfinalExponentiationBLS12finalExponentiationBNGT'G2'G1'Fq12Fq6 parameterBin parameterHexgetRootOfUnity$fIrreducibleMonicVExtension$fIrreducibleMonicWExtension$fCyclicSubgroupRootsOfUnity$fPairingBLS12381$fPairingBN254$fPairingBN254A$fPairingBN254B$fPairingBN254C$fPairingBN254D$fPairingBN462 swEncBLS12swEncBN lineFunction frobTwisted powUnitaryversion getBinDir getLibDir getDynLibDir getDataDir getLibexecDir getSysconfDirgetDataFileName