úÎ!]ÆTþ“      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrs t u vwxyz{|}~€‚ƒ„…†‡ˆ‰Š‹ Œ  Ž   ‘ ’ Safe"#=?_ 3“pairingISum all the elements of some container according to its group structure.”•“Safe"#=?_( pairingElliptic curve coefficentpairingElliptic curve coefficentpairingEmbedding degree–pairing&BN parameter that determines the primepairing0Characteristic of the finite fields we work with pairingxOrder of elliptic curve E(Fq) G1, and therefore also the characteristic of the prime field we choose our exponents from pairingNParameter used to define the twisted curve over Fq, with xi = xi_a + xi_b * i pairingNParameter used to define the twisted curve over Fq, with xi = xi_a + xi_b * i pairingQuadratic nonresidue in Fq   None "#7=?M_lmÛ pairing Prime field with characteristic _qpairingUse new instead of this constructorpairingTurn an integer into an Fq( number, should be used instead of the Fq constructor.pairingQuadratic non-residuepairingMultiplicative inversepairingAdditive identitypairingMultiplicative identitypairing=Euclidean algorithm to compute inverse in an integral domain a  None "#=?M_lm épairing Prime field with characteristic _r pairingUse new instead of this constructor!pairingTurn an integer into an Fr( number, should be used instead of the Fr constructor.&pairingÁCompute primitive roots of unity for 2^0, 2^1, ..., 2^28. (2^28 is the largest power of two that divides _r - 1, therefore there are no primitive roots of unity for higher powers of 2 in Fr.)&pairingCexponent of 2 for which we want to get the primitive root of unity  !"#$%&'  !"#$%&'None "#79=?_lm(m 1pairingQuadratic extension of Fq defined as  Fq[u]/x^2 + 12pairingUse new instead of this contructor5pairingnew x y creates a value representing  x + y * u 6pairingCubic non-residue in Fq27pairingMultiplicative identity8pairingAdditive identity9pairingMultiplication by a scalar in Fq:pairing Multiply by xi;pairing Divide by xi<pairingSquaring operation=pairingMultiplicative inverse>pairing Conjugation123456789:;<=>?123459=78><:;6?None"#=?_lm.­Fpairing/Field extension defined as Fq2[v]/v^3 - (9 + u)KpairingCreate a new value in Fq6 , should be used instead of the Fq6 constructor.LpairingAdditive identityMpairingMultiplicative identityNpairingSquaring operationOpairing Multiply by xi (cubic nonresidue in Fq2) and reorder coefficientsPpairingMultiplicative inverse FGHIJKLMNOPQ FGHIJKPMLNOQNone"#=?_lm6ã Vpairing)Field extension defined as Fq6[w]/w^2 - vWpairingUse new instead of this constructorZpairingCreate a new value in Fq120 by providing a list of twelve coefficients in Fq , should be used instead of the Fq12 constructor.[pairingDeconstruct a value in Fq12' into a list of twelve coefficients in Fq.\pairingMultiplicative identity]pairingAdditive identity^pairingMultiplicative inverse_pairing Conjugation`pairingIterated Frobenius automorphism VWXYZ[\]^_`a VWXYZ[^\]_`aNone "#479=?_=ĶfpairingPoints on a curve over a field aE represented as either affine coordinates or as a point at infinity.gpairing Affine pointhpairingPoint at infinityipairinghPoint addition, provides a group structure on an elliptic curve with the point at infinity as its unit.jpairingPoint doublingkpairing#Negation (flipping the y component)lpairingMultiplication by a scalarfghijklfghjikl None"#=?_AaspairingCJacobian coordinates for points on an elliptic curve over a field a.tpairing2Convert affine coordinates to Jacobian coordinatesupairing2Convert Jacobian coordinates to affine coordinatesstustuNone"#=?_J vpairingKGT is subgroup of _r-th roots of unity of the multiplicative group of Fq12wpairing+G2 is E'(Fq2) defined by y^2 = x^3 + b / xixpairing$G1 is E(Fq) defined by y^2 = x^3 + bypairingGenerator for G1zpairingGenerator for G2{pairingFTest whether a value in G1 satisfies the corresponding curve equation|pairingFTest whether a value in G2 satisfies the corresponding curve equation}pairing.Test whether a value is an _r-th root of unity~pairingParameter for curve on Fqpairing$Parameter for twisted curve over Fq2vwxyz{|}~xwv{|}yz~ None"#=?_T;‹pairing9Optimal Ate pairing (including final exponentiation step)Œpairing9Optimal Ate pairing without the final exponentiation steppairingXBinary expansion (missing the most-significant bit) representing the number 6 * _t + 2. Z29793968203157093288 = 0b11001110101111001011100000011100110111110011101100011101110101000—pairing*Miller loop with precomputed values for G2ŽpairingQIterated frobenius morphisms on fields of characteristic _q, implemented naivelypairing5Naive implementation of the final exponentiation steppairing8A faster way of performing the final exponentiation step‹ŒŽ‹ŒŽ Safe"#=?_TŲ˜™š›œžŸ       !"#$%&'()*++,!-./0123456789::;<=>?@ABCDE!FGHIJKLLMNOPQRAS!TUVWXXYZ[\]^_`!abcdeefghijklmnop q r stuvwxyz{|}~€‚ƒ„…†‡ˆ ‰ Š ‹ Œ  Ž   ‘ ’ “” • – — ˜ ™ š › œ ž$pairing-0.1.2-3MH45KpXALjL5nqfvowhc4 Pairing.GroupPairing.Params Pairing.Fq Pairing.Fr Pairing.Fq2 Pairing.Fq6 Pairing.Fq12 Pairing.PointPairing.JacobianPairing.PairingPairing.CyclicGroup Paths_pairing CyclicGroup generatororderexpninverse_b_a_k_q_r_xiA_xiB_nqrFqnewfqNqrfqInvfqZerofqOne euclideanrandom$fFractionalFq$fNumFq $fAsIntegerFq$fShowFq$fEqFq$fBitsFq $fGenericFq $fNFDataFq$fOrdFqFrfrInv isRootOfUnityisPrimitiveRootOfUnityprimitiveRootOfUnityprecompRootOfUnity $fPrettyFr$fFractionalFr$fNumFr $fAsIntegerFr$fShowFr$fEqFr$fOrdFr$fBitsFr $fNFDataFrFq2fq2xfq2yxifq2onefq2zero fq2scalarMulmulXidivXifq2sqrfq2invfq2conj$fFractionalFq2$fNumFq2$fEqFq2 $fShowFq2 $fGenericFq2 $fNFDataFq2Fq6fq6xfq6yfq6zfq6zerofq6onefq6sqrfq6inv$fFractionalFq6$fNumFq6$fEqFq6 $fShowFq6Fq12fq12xfq12y deconstructfq12onefq12zerofq12invfq12conj fq12frobenius$fFractionalFq12 $fNumFq12$fEqFq12 $fShowFq12PointInfinitygAddgDoublegNeggMul $fEqPoint $fOrdPoint $fShowPoint$fFunctorPoint$fGenericPoint $fNFDataPointJPoint toJacobian fromJacobianGTG2G1g1g2 isOnCurveG1 isOnCurveG2isInGTb1b2$fArbitraryPoint$fArbitraryPoint0$fCyclicGroupPoint $fMonoidPoint$fSemigroupPoint$fCyclicGroupPoint0$fMonoidPoint0$fSemigroupPoint0$fCyclicGroupFq12 $fMonoidFq12$fSemigroupFq12reducedPairing atePairingateLoopCountBinaryfrobeniusNaivefinalExponentiationNaivefinalExponentiation$fShowEllCoeffs $fEqEllCoeffssumG AsInteger asInteger_t ateMillerLoopversion getBinDir getLibDir getDynLibDir getDataDir getLibexecDir getSysconfDirgetDataFileName