Parallel Enumeration of Shortest Lattice Vectors

Abstract

Abstract. Lattice basis reduction is the problem of finding short vectors in lattices. The security of lattice based cryptosystems is based on the hardness of lattice reduction. Furthermore, lattice reduction is used to attack well-known cryptosystems like RSA. One of the algorithms used in lattice reduction is the enumeration algorithm (ENUM), that provably finds a shortest vector of a lattice. We present a parallel version of the lattice enumeration algorithm. Using multi-core CPU systems with up to 16 cores, our implementation gains a speed-up of up to factor 14. Compared to the currently best public implementation, our parallel algorithm saves more than 90% of runtime.

Another 80-dimensional extremal lattice

“…However, after finding a sufficiently good basis for the lattice using block Korkine-Zolotareff (BKZ) reduction [32] (with a dimension parameter of about 30 -this is usually all that is useful, and takes less than 10 minutes), it only takes only a couple of cpu-months to do an exhaustive search, and parallel code for this is now available from Pujol [29] (described in [10], and see also [9]). Using 12 cpus and Pujol's code, it took about 4 days to show that our lattice has no vectors of norm 2 or 4.…”

Parallel Improved Schnorr-Euchner Enumeration SE++ for the CVP and SVP

“…Results: Our results show that enumeration-based CVP-solvers, whose scalability was never studied, can be parallelized at least as efficiently as enumerationbased SVP-solvers, based on a comparison of the CVP and SVP versions of our algorithm and the state of the art SVP implementation described in [10]. In particular, our parallel version of this algorithm achieves superlinear speedups in some instances on up to 8 cores and a speedup factor of 14.8x for 16 cores when solving the CVP on a 50-dimensional lattice, on a dual-socket machine with 16 physical cores.…”

“…On one hand, we study the practicability of the CVP, to which end we implement and assess the performance of an enhanced version of the Schnorr-Euchner enumeration routine, described in [12], a CVP-solver that can easily be modified to solve the SVP, from here on referred to as SE++. In particular, we propose a parallel version of this algorithm for sharedmemory CPU systems, implemented with OpenMP, and we analyze its performance on a 16-core CPU system against the parallel SVP-solver proposed in [10].…”

“…In particular, the computational practicability of the CVP has received little attention. While several SVP-solvers have been implemented on various computer architectures [10], [15], few open implementations of CVP-solvers are available. In particular, the scalability of CVP-solvers was never studied.…”

Random Sampling for Short Lattice Vectors on Graphics Cards

“…There are approaches of parallelizing LLL in the SIMD model, e.g. [30,2] and also for enumeration [9,15,10]. The combination of both however has not yet been tried.…”