Abstract
The discrete logarithm problem in Jacobians of curves of high genus g over finite fields \(\mathbb {F}_q\) is known to be computable with subexponential complexity \(L_{q^g}(1/2, O(1))\). We present an algorithm for a family of plane curves whose degrees in X and Y are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of \(L_{q^g}(1/3, O(1))\), and a discrete logarithm computation takes subexponential time of \(L_{q^g}(1/3+ \varepsilon, o(1))\) for any positive ε. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.
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Adleman, L.M., DeMarrais, J., Huang, M.-D.: A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 28–40. Springer, Heidelberg (1994)
Bender, R.L., Pomerance, C.: Rigorous discrete logarithm computations in finite fields via smooth polynomials. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Perspectives on Number Theory: Proceedings of a Conference in Honor of A.O.L. Atkin. Studies in Advanced Mathematics, vol. 7, pp. 221–232. American Mathematical Society, Providence (1998)
Couveignes, J.-M.: Algebraic groups and discrete logarithm. In: Public-key cryptography and computational number theory, pp. 17–27. Walter de Gruyter, Berlin (2001)
Diem, C.: An index calculus algorithm for plane curves of small degree. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 543–557. Springer, Heidelberg (2006)
Enge, A.: Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time. Math. Comp. 71, 729–742 (2002)
Enge, A., Gaudry, P.: A general framework for subexponential discrete logarithm algorithms. Acta Arith. 102, 83–103 (2002)
Enge, A., Stein, A.: Smooth ideals in hyperelliptic function fields. Math. Comp. 71, 1219–1230 (2002)
Gaudry, P.: An algorithm for solving the discrete log problem on hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19–34. Springer, Heidelberg (2000)
Gaudry, P., Thomé, E., Thériault, N., Diem, C.: A double large prime variation for small genus hyperelliptic index calculus. Math. Comp. 76, 475–492 (2007)
Haché, G.: Construction effective de codes géométriques. PhD thesis, Université de Paris VI (1996)
Haffner, J.L., McCurley, K.S.: A rigorous subexponential algorithm for computation of class groups. J. Amer. Math. Soc. 2(4), 837–850 (1989)
Heß, F.: Computing Riemann-Roch spaces in algebraic function fields and related topics. J. Symbolic Comput. 33, 425–445 (2002)
Heß, F.: Computing relations in divisor class groups of algebraic curves over finite fields. Preprint (2004)
Manstavičius, E.: Semigroup elements free of large prime factors. In: Schweiger, F., Manstavičius, E. (eds.) New Trends in Probability and Statistic, pp. 135–153 (1992)
Müller, V., Stein, A., Thiel, C.: Computing discrete logarithms in real quadratic congruence function fields of large genus. Math. Comp. 68(226), 807–822 (1999)
Storjohann, A.: Algorithms for Matrix Canonical Forms. PhD thesis, Eidgenössische Technische Hochschule Zürich (2000)
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Enge, A., Gaudry, P. (2007). An L (1/3 + ε) Algorithm for the Discrete Logarithm Problem for Low Degree Curves. In: Naor, M. (eds) Advances in Cryptology - EUROCRYPT 2007. EUROCRYPT 2007. Lecture Notes in Computer Science, vol 4515. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72540-4_22
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DOI: https://doi.org/10.1007/978-3-540-72540-4_22
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