In this paper we further the study of index calculus methods for solving the elliptic curve discrete logarithm problem (ECDLP). We focus on the index calculus for subfield curves, also called Koblitz curves, defined over Fq with ECDLP in Fqn. Instead of accelerating the solution of polynomial systems during index calculus as was predominantly done in previous work, we define factor bases that are invariant under the q-power Frobenius automorphism of the field Fqn, reducing the number of polynomial systems that need to be solved. A reduction by a factor of 1/n is the best one could hope for. We show how to choose factor bases to achieve this, while simultaneously accelerating the linear algebra step of the index calculus method for Koblitz curves by a factor n2. Furthermore, we show how to use the Frobenius endomorphism to improve symmetry breaking for Koblitz curves. We provide constructions of factor bases with the desired properties, and we study their impact on the polynomial system solving costs experimentally.
|Title of host publication||Selected Areas in Cryptography - 27th International Conference, 2020, Revised Selected Papers|
|Editors||Orr Dunkelman, Michael J. Jacobson, Jr., Colin O’Flynn|
|Number of pages||24|
|Publication status||Published - 21 Jul 2021|
|Event||27th International Conference on Selected Areas in Cryptography, SAC 2020 - Virtual, Online|
Duration: 21 Oct 2020 → 23 Oct 2020
|Name||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|Conference||27th International Conference on Selected Areas in Cryptography, SAC 2020|
|Period||21/10/20 → 23/10/20|
Bibliographical noteFunding Information:
Acknowledgements. We thank Jean-Marc Couveignes and Reynald Lercier for their work on Galois invariant smoothness bases  and helpful conversations about the topic. Furthermore, we would like to thank the anonymous reviewers for their helpful comments on the submitted manuscript of this paper. Christophe Petit’s work was supported by EPSRC grant EP/S01361X/1. Simon-Philipp Merz was supported by the EPSRC grant EP/P009301/1.
© 2021, Springer Nature Switzerland AG.
ASJC Scopus subject areas
- Theoretical Computer Science
- Computer Science(all)