Parallel Lattice Basis Reduction Using a Multi-threaded Schnorr-Euchner LLL Algorithm

Abstract

Abstract. In this paper, we introduce a new parallel variant of the LLL lattice basis reduction algorithm. Our new, multi-threaded algorithm is the first to provide an efficient, parallel implementation of the Schorr-Euchner algorithm for today's multi-processor, multi-core computer architectures. Experiments with sparse and dense lattice bases show a speed-up factor of about 1.8 for the 2-thread and about factor 3.2 for the 4-thread version of our new parallel lattice basis reduction algorithm in comparison to the traditional non-parallel algorithm.

Three Strategies for Improving Shortest Vector Enumeration Using GPUs

“…ere are much works for that either for approximation algorithms such as LLL and BKZ or for exact algorithms such as Enumeration [16] and Sieving [18]. Backes and Wetzel [24] presented the first parallel implementation of LLL on multicores computer architecture.…”

“…ey have shown an improvement of about 3.2 times more than that in the traditional nonparallel algorithm. Afterward, the same authors introduced numbers of modifications [25] that increase the speedup of the algorithm in [24] by a factor 1.25.…”

“…Enumerate the GPU-points in G on CPU (11) break; (12) end if (13) Copy G to GPU memories (14) Start the GPU stage (15) for each thread in GPU do (16) if the current enumeration corresponding to a start-point p is cut off or reaches level α then (17) Produce a g neighbor-point of the start-point p (18) Calculate l α for the neighbor-point g (19) if l α ≤ A then (20) e enumeration corresponding to g is launched on the current thread (21) else (22) if this start-point p is the last point in G then (23) Stop enumeration on GPU and return to CPU (24) end if (25) Set a flag that indicates that the start-point p terminates its corresponding enumeration (26) Select a new start-point and launch its corresponding enumeration on the current thread (27) end if (28) end if (29) end for each (30) Stop the GPU stage (31) Copy start points that terminate their enumerations in GPU to R (32) G � G − R (33) end while (34) Output: a shorter vector…”

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.…”

ETRUSign: NTRUSign sobre los enteros de Eisenstein

“…Dado que encontrar vectores cortos en un lattice es uno de los problemas más difíciles de resolver computacionalmente, ya que es un problema de tipo NP, algunos autores como C. Schnorr y M. Euchner, en [5], se han adentrado a buscar nuevos algoritmos que resuelvan el problema SV P de manera aproximada considerando la reducción de lattices. Un algoritmo eficiente que resuelve este tipo de problemas es el algoritmo LLL, y que además es de tiempo polinomial determinista, es decir, que tal algoritmo es factible o eficiente para algún ordenador.…”