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arithmetic on Hessian curves Public Key Cryptography – PKC 2010 Paris 2010 Lecture Notes in Comput. Sci. 6056 Springer Berlin 2010 243 260 9 D. Freeman, M. Scott and E. Teske, A taxonomy of pairing-friendly elliptic curves, J. Cryptology 23 (2010), 2, 224–280. Freeman D. Scott M. Teske E. A taxonomy of pairing-friendly elliptic curves J. Cryptology 23 2010 2 224 280 10 G. Frey, M. Müller and H. Rück, The tate pairing and the discrete logarithm applied to elliptic curve cryptosystems, IEEE Trans. Inform. Theory 45 (1999), 5, 1717–1719. Frey G. Müller M. Rück H. The tate

Abstract

This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y2=dX4+Z4. We improve the doubling and addition steps in Miller's algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best result to date among curves with quartic twists. For the doubling and addition steps in Miller's algorithm for the computation of the Tate pairing, we obtain a theoretical gain up to 27% and 39%, depending on the embedding degree and the extension field arithmetic, with respect to Weierstrass curves and previous results on Jacobi quartic curves. Furthermore and for the first time, we compute and implement Ate, twisted Ate and optimal pairings on the Jacobi quartic curves. Our results are up to 27% more efficient compared to the case of Weierstrass curves with quartic twists.

this pairing in cryptographic applications. Keywords. Tate pairing, Weil pairing, self-pairing, pairing based cryptography. 2010 Mathematics Subject Classification. 14G50, 11T71, 11G20, 14Q05. 1 Introduction A pairing is a non-degenerate bilinear map e W G1 G2 7! GT where G1;G2;GT are cyclic groups of prime order r (the first two are usually written additively, and the third multiplicatively). Such groups are found from elliptic or hyperelliptic curves and the pairing is usually the Tate–Lichtenbaum pairing or one of its variants. Pairings have found many

. Scott M. Faster squaring in the cyclotomic subgroup of sixth degree extensions Public Key Cryptography PKC 2010 Lecture Notes in Comput. Sci. 6056 Springer Berlin 2010 209 223 11 L. Hu, J. Dong and D. Pei, Implementation of cryptosystems based on Tate pairing, J. Comput. Sci. Tech. 20 (2005), 2, 264–269. Hu L. Dong J. Pei D. Implementation of cryptosystems based on Tate pairing J. Comput. Sci. Tech. 20 2005 2 264 269 12 M. Joye and J. J. Quisquater, Efficient computation of full Lucas sequences, Electron. Lett. 36 (1996), 6, 537–538. Joye M. Quisquater J. J

-adic representation, 9 Perrin-Riou's "logarithme elargi", 165 pseudo-isomorphism of lwasawa mod- ules, 40 pseudo-null lwasawa module, 40 representation, p-adic, 9 rigidity, 176 Selmer group, 21 Selmer group of an abelian variety, 27 Selmer sequence, 132 semilocal Galois cohomology, 93, 202 singular cohomology classes, 14 Stickelberger element, 56 symmetric square of an elliptic curve, 73, 170 Tate pairing, 18 Teichmiiller character, 54 Thaine, Francisco, 3 twist (by arbitrary characters), 123 twist (by characters of finite order), 44 twisting homomorphism, 119

projective, 1 Néron, see height, see theorem of Néron, Tate Néron–Tate pairing, 124 Nagell, see theorem of Nagell, Lutz, and Cassels, see theorem of Nagell, Lutz non-split multiplicative reduction, see reduction nonsingular, 4 O, 93 one-way function, 82 ordinary, 65 Index 367 parametrization, 41 period, 33, 52, 54 period parallelogram, 33 plane algebraic curve, 2 affine, 2 projective, 2 Pohlig-Hellmann reduction of DLP, 83 point at infinity, 2, 3 Pollard, 84 primality test, Goldwasser, Kilian, 27 projective n-space, see n-space public key, see cryptosystem purely

.37), respectively, are related by OhD.P / D 3 Oh.P /. A very important property of the canonical height is that, by means of it, a positive- definite quadratic form is defined as follows: First, one defines the so-called Néron– Tate (or Weil) pairing by hP ,Qi D Oh.P CQ/ Oh.P / Oh.Q/. The following important properties for the canonical height and the Néron–Tate pair- ing hold (see [45, Theorem 9.3]): The Néron–Tate pairing is bilinear. For any P 2 E with PE 2 E.Q/ and anym 2 Z, Oh.mP / D m2 Oh.P /; in particular, Oh.P / D Oh.P /. Oh.P / 0 and Oh.P / D 0 if and only if PE

application to the arithmetic theta correspondence, which is modeled on [6], we need Howe’s abstract formulation. The additional facts we need, some of which are special to this particular dual pair, can be found in [10]. 2This follows from invariance under HB(Af ) of the hermitian form on MW(MB) de- termined by the Néron-Tate pairing. CENTRAL DERIVATIVES OF L-FUNCTIONS 355 where δp(σp, ψ−p ) is the local dichotomy sign defined in (8.2.8). Note that the occurrence here of the additive ψ−p , where ψ − p (x) = ψp(−x), is ex- plained in Corollary 8.2.6. Since B is indefinite

(K)→ R 〈P,Q〉 = ĥ(P +Q)− ĥ(P )− ĥ(Q) is symmetric and bilinear. This pairing is called Néron–Tate pairing. (Sometimes a factor 12 is placed in front of the right-hand side. This has the advantage that then ĥ(P ) = 〈P, P 〉.) Proof. a) As we have seen in Theorem 5.13, one has 2h(P )+ 2h(Q)− c1 ≤ h(P +Q)+ h(P −Q) ≤ 2h(P )+ 2h(Q)+ c2 for all P,Q ∈ E(K) with constants c1 and c2. Replacing P and Q by 2nP and 2nQ, and dividing the equation by 22n, one gets the equation 2 h(2nP ) 22n + 2h(2 nQ) 22n − c1 22n ≤ h(2 n(P +Q)) 22n + h(2 n(P −Q)) 22n ≤ 2h(2 nP ) 22n + 2h(2 nQ) 22