Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2. (English) Zbl 1101.14036
Summary: We study the computation of the number of isomorphism classes of hyperelliptic curves of genus 2 over finite fields \(\mathbb{F}_ q\) with \(q\) even. We show the formula of the number of isomorphism classes, that is, for \(q = 2m\), if \(4 \nmid m\), then the formula is \(2q^3 + q^2 - q\); if \(4 \mid m\), then the formula is \(2q^3 + q^2 - q + 8\). These results can be used in the classification problems and the hyperelliptic curve cryptosystems.
MSC:
| 14G50 | Applications to coding theory and cryptography of arithmetic geometry |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 14G15 | Finite ground fields in algebraic geometry |
References:
| [1] | Miller, V., Uses of elliptic curves in cryptography, in Advances in Cryptology-CRYPTO’85 (ed. Williams, H.C.), Springer Lecture Notes in Computer Science, 1986, 218: 417-426. |
| [2] | Koblitz, N., Elliptic curve cryptosystems, Math. Comp., 1987, 48: 203–209. · Zbl 0622.94015 · doi:10.1090/S0025-5718-1987-0866109-5 |
| [3] | Koblitz, N., Hyperelliptic cryptosystems, J.Cryptology, 1989, 1: 139–150. · Zbl 0674.94010 · doi:10.1007/BF02252872 |
| [4] | Cantor, D.G., Computing in the Jacobian of a hyperelliptic curve, Math. Comp., 1987, 48: 95–101. · Zbl 0613.14022 · doi:10.1090/S0025-5718-1987-0866101-0 |
| [5] | Adleman, L.M., DeMarrais, J., Huang, M. D. A., A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields, in Algorithmic Number Theory (eds. Adleman, L. M., Huang M. D. A.), Springer Lecture Notes in Computer Science, 1994, 877: 28-40. · Zbl 0829.11068 |
| [6] | Enge, A., Computing discrete logarithms in high-genus hyperelliptic jacobians in provably subexponential time, Math. Comp., 2002, 71: 729–742. · Zbl 0988.68069 · doi:10.1090/S0025-5718-01-01363-1 |
| [7] | Gaudry, P., An algorithm for solving the discrete log problem on hyperelliptic curves, in Advances in Cryptology-Eurocrypt 2000 (ed. Preneel, B.), Springer Lecture Notes in Computer Science, 2000, 1807: 19-34. · Zbl 1082.94517 |
| [8] | Encinas, L. H., Menezes, A. J., Masqué, J. M., Isomorphism classes of genus-2 hyperelliptic curves over finite fields, Appl. Algebra Engrg. Comm. Comput., 2002, 13: 57–65. · Zbl 1066.14026 · doi:10.1007/s002000100092 |
| [9] | Encinas, L. H., Masqué, J. M., Isomorphism classes of hyperelliptic curves of genus 2 in characteristic 5. Tech. Rep. CORR2002-07. Centre For Applied Cryptographic Research, The University of Waterloo, Canada, 2002, Available at http://www.cacr.math.uwaterloo.ca/techreports/2002/corr2002-07.ps |
| [10] | Choie, Y., Yun, D., Isomorphism classes of hyperelliptic curves of genus 2 over F q, in Information Security and Privacy (eds. Batten, L.M., Seberry, J.), Springer Lecture Notes in Computer Science, 2002, 2384: 190-202. · Zbl 1022.11029 |
| [11] | Choie, Y., Jeong, E., Isomorphism classes of hyperelliptic curves of genus 2 over F 2 n, Cryptology ePrint Archive 2003/213. |
| [12] | Cardona, G., On the number of curves of genus 2 over a finite field, Finite Fields Appl., 2003, 9: 505–526. · Zbl 1091.11023 · doi:10.1016/S1071-5797(03)00039-X |
| [13] | Cardona, G., Nart, E., Pujolàs, J., Curves of genus two over fields of even characteristic, Available at http:// arxiv.org/abs/math.NT/0210105 · Zbl 1097.11033 |
| [14] | Menezes, A. J., Wu, Y. H., Zuccherato, R. J., An elementary introduction to hyperelliptic curves, as an appendix in Algebraic aspects of cryptography (by Koblit Z, N.), Springer Algorithms and Computations, 1998, 3: 155-178. |
| [15] | Lockhart, P., On the discriminant of a hyperelliptic curve, Trans. Amer. Math. Soc., 1994, 342: 729–752. · Zbl 0815.11031 · doi:10.2307/2154650 |
| [16] | Dixon, J. D., Mortimer, B., Permutation Groups, GTM 163, Berlin: Springer, 1996. · Zbl 0951.20001 |
| [17] | Lidl, R., Niederreiter, H., Introduction to Finite Fields and Their Applications, Cambridge: Cambridge University Press, 1994. · Zbl 0820.11072 |
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