Abstract
In this paper, we revisit the recent small characteristic discrete logarithm algorithms. We show that a simplified description of the algorithm, together with some additional ideas, permits to obtain an improved complexity for the polynomial time precomputation that arises during the discrete logarithm computation. With our new improvements, this is reduced to O(q 6), where q is the cardinality of the basefield we are considering. This should be compared to the best currently documented complexity for this part, namely O(q 7). With our simplified setting, the complexity of the precomputation in the general case becomes similar to the complexity known for Kummer (or twisted Kummer) extensions.
Chapter PDF
Similar content being viewed by others
References
Adj, G., Menezes, A., Oliveira, T., Rodríguez-Henríquez, F.: Computing discrete logarithms in \(\mathbb{F}_{3^{6\cdot 137}}\) and \(\mathbb{F}_{3^{6\cdot 163}}\) using Magma. Cryptology ePrint Archive, Report 2014/057 (2014)
Barbulescu, R., Bouvier, C., Detrey, J., Gaudry, P., Jeljeli, H., Thomé, E., Videau, M., Zimmermann, P.: Discrete logarithm in \(\mathbb{F}_{2^809}\) with ffs. Cryptology ePrint Archive, Report 2013/197 (2013)
Barbulescu, R., Gaudry, P., Joux, A., Thomé, E.: A heuristic quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 1–16. Springer, Heidelberg (2014)
Blake, I.F., Mullin, R.C., Vanstone, S.A.: Computing logarithms in \(\mathbb{F}_{2^n}\). In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 73–82. Springer, Heidelberg (1985)
Göloğlu, F., Granger, R., McGuire, G., Zumbrägel, J.: On the function field sieve and the impact of higher splitting probabilities - application to discrete logarithms in \(\mathbb{F}_{2^{1971}}\) and \(\mathbb{F}_{2^{3164}}\). In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 109–128. Springer, Heidelberg (2013)
Göloglu, F., Granger, R., McGuire, G., Zumbrägel, J.: On the function field sieve and the impact of higher splitting probabilities: Application to discrete logarithms in \(\mathbb{F}_{2^{1971}}\). Cryptology ePrint Archive, Report 2013/074 (2013)
Granger, R., Kleinjung, T., Zumbrägel, J.: Breaking ‘128-bit secure’ supersingular binary curves (or how to solve discrete logarithms in \(\mathbb{F}_{2^{4 \cdot 1223}}\) and \(\mathbb{F}_{2^{12 \cdot 367}}\)). Cryptology ePrint Archive, Report 2014/119 (2014)
Granger, R., Kleinjung, T., Zumbrägel, J.: On the powers of 2. Cryptology ePrint Archive, Report 2014/300 (2014)
Joux, A.: Faster index calculus for the medium prime case application to 1175-bit and 1425-bit finite fields. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 177–193. Springer, Heidelberg (2013)
Joux, A.: A new index calculus algorithm with complexity L(1/4 + o(1)) in very small characteristic. Cryptology ePrint Archive, Report 2013/095 (2013)
Joux, A.: A new index calculus algorithm with complexity L(1/4 + o(1)) in small characteristic. In: Lange, T., Lauter, K., Lisoněk, P. (eds.) SAC 2013. LNCS, vol. 8282, pp. 355–380. Springer, Heidelberg (2014)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 International Association for Cryptologic Research
About this paper
Cite this paper
Joux, A., Pierrot, C. (2014). Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms. In: Sarkar, P., Iwata, T. (eds) Advances in Cryptology – ASIACRYPT 2014. ASIACRYPT 2014. Lecture Notes in Computer Science, vol 8873. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45611-8_20
Download citation
DOI: https://doi.org/10.1007/978-3-662-45611-8_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-45610-1
Online ISBN: 978-3-662-45611-8
eBook Packages: Computer ScienceComputer Science (R0)