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For cryptanalysis in lattice-based schemes, the performance evaluation of lattice basis reduction using high-performance computers is becoming increasingly important for the determination of the security level. We propose a distributed and asynchronous parallel reduction algorithm based on randomization and DeepBKZ, which is an improved variant of the block Korkine-Zolotarev (BKZ) reduction algorithm. Randomized copies of a lattice basis are distributed to up to 103,680 cores and independently reduced in parallel, while some basis vectors are shared asynchronously among all processes via MPI. There is a trade-off between randomization and information sharing; if a substantial amount of information is shared, all processes will work on the same problem, thereby diminishing the benefit of parallelization. To monitor this balance between randomness and sharing, we propose a metric to quantify the variety of lattice bases. We empirically find an optimal parameter of sharing for high-dimensional lattices. We demonstrate the efficacy of our proposed parallel algorithm and implementation with respect to both performance and scalability through our experiments.
Lattice problems are a class of optimization problems that are notably hard. There are no classical or quantum algorithms known to solve these problems efficiently. Their hardness has made lattices a major cryptographic primitive for post-quantum cryptography. Several different approaches have been used for lattice problems with different computational profiles; some suffer from super-exponential time, and others require exponential space. This motivated us to develop a novel lattice problem solver, CMAP-LAP, based on the clever coordination of different algorithms that run massively in parallel. With our flexible framework, heterogeneous modules run asynchronously in parallel on a large-scale distributed system while exchanging information, which drastically boosts the overall performance. We also implement full checkpoint-and-restart functionality, which is vital to high-dimensional lattice problems. Through numerical experiments with up to 103,680 cores, we evaluated the performance and stability of our system and demonstrated its high capability for future massive-scale experiments.