Further Optimizations of CSIDH: A Systematic Approach to Efficient Strategies, Permutations, and Bound Vectors

Abstract

CSIDH is a recent post-quantum key establishment protocol based on constructing isogenies between supersingular elliptic curves. Several recent works give constant-time implementations of CSIDH along with some optimizations of the ideal class group action evaluation algorithm, including the SIMBA technique of Meyer et al. and the “two-point method” of Onuki et al. A recent work of Cervantes-Vázquez et al. details a number of improvements to the works of Meyer et al. and Onuki et al. Several of these optimizations—in particular, the choice of ordering of the primes, the choice of SIMBA partition and strategies, and the choice of bound vector which defines the secret keyspace—have been made in an ad hoc fashion, and so while they yield performance improvements it has not been clear whether these choices could be improved upon, or how to do so. In this work we present a framework for improving these optimizations using (respectively) linear programming, dynamic programming, and convex programming techniques. Our framework is applicable to any CSIDH security level, to all currently-proposed paradigms for computing the class group action, and to any choice of model for the underlying curves. Using our framework we find improved parameter sets for the two major methods of computing the group action: in the case of the implementation of Meyer et al. we obtain a 13.04% speedup without applying the further optimizations proposed by Cervantes-Vázquez et al., while for that of Cervantes-Vázquez et al. under the two-point method we obtain a speedup of 5.23%, giving the fastest constant-time implementation of CSIDH to date.