On Parallel Computation of Large Smooth-Degree Isogeny

Abstract

The computation of large smooth-degree isogenies is considered to be the most time-consuming task in isogeny-based cryptosystems and, to this end, recently several proposals have been made to speed it up. For implementation in software using a single core, De Feo et al. presented an optimal way to compute such isogenies. The multi-core setting is however far more intricate but offers various ways to reduce the computation time and is an active area of research. This thesis presents a study of speeding-up large smooth-degree isogeny computation with various forms of parallelism and consists of three contributions.

The first contribution of this thesis is two novel theoretical techniques for speeding-up the computation with parallelism. Our proposed technique, called precedence-constrained scheduling (PCS), transforms the isogeny computation into a task scheduling problem with precedence constraints and utilizes several task scheduling algorithms to tackle the problem. Another proposed technique of ours is to formulate the isogeny computation as an integer linear program. Combining both techniques, we are able to reduce the theoretical cost of the isogeny computation by up to 13.02% from the state-of-the-art.

The second contribution of this thesis is two software implementations of the isogeny computation based on our PCS technique. We consider two execution environments for the implementations: one relies only on the parallelism provided by multi-core processors, and the other utilizes multi-core processors supporting the Intel's Advanced Vector eXtensions (AVX) technology. To our best knowledge, we are the first to utilize both parallelization technologies for the isogeny computation. Also, to achieve effective implementations, we modify PCS for each execution environments and equip both implementations with a synchronization handling technique. The implementation results show up to 14.36% speed-up for the first implementation and up to 34.05% speed-up for the second implementation.

The third contribution of this thesis is two applications of using learning-based optimizations to speed-up the parallel isogeny computation. We consider the genetic algorithm and the reinforcement learning algorithm and detail our design rationale when instantiating both algorithms for our problem. From experimental results, the genetic algorithm is able to find a better approach for the isogeny computation. The approach found is nontrivial and is up to 9.95% faster than human's heuristic. On the other hand, the reinforcement learning lags PCS by as small as 2.73%. We use the experimental results of the reinforcement learning to argue that PCS may be nearly or even optimal for the computation.