A Parallel LLL using POSIX Threads

In this paper we introduce a new parallel variant of the LLL lattice basis reduction algorithm. Lattice theory and in particular lattice basis reduction continues to play an integral role in cryptography. Not only does it provide effective cryptanalysis tools but it is also believed to bring about new cryptographic primitives that exhibit strong security even in the presence of quantum computers. In theory, many aspects of lattices are already well-understood. Yet, many practical aspects, like the performance of lattice basis reduction algorithms, are still under investigation. In this paper, we introduce a new parallel lattice basis reduction algorithm that overcomes shortcomings of previously introduced algorithms. First and foremost, our new algorithm is based on the Schnorr-Euchner algorithm and as such is the first—to the best of our knowledge—to provide a parallel implementation for the Schnorr-Euchner algorithm. Second, using POSIX threads allows us to make effective use of today’s multi-processor, multi-core computer architecture. Developing in a shared memory setting allows us to replace time consuming inter-process communication with synchronization points (barriers) and locks (mutexes). Our implementation of the parallel LLL is optimized for reducing high dimensional lattice bases with big entries that would require a multiprecision floating-point arithmetic to approximate the lattice basis if the original Schnorr-Euchner algorithm was used for the reduction. The reduction of these lattice bases is of great interest, e.g., for cryptanalyzing RSA. In experiments with sparse and dense lattice bases, experiments with our new parallel LLL show (compared to the non-parallel algorithm) a speed-up factor of about 1.75 for the 2-thread and close to factor 3 for the 4-thread version. The overhead of the parallel LLL decreases with increasing dimension of the lattice basis to less than 10% for the 2-thread and less than 15% for the 4-thread version.