Algorithm For Shortest Vector Problem Research Articles

Improving genetic algorithms for solving the SVP: focusing on low memory consumption and high reproducibility

The shortest vector problem (SVP) is absolutely essential in lattice-based cryptography. In this paper, we significantly improve genetic algorithms (GAs) for solving the SVP. GAs, which are simple and powerful optimization techniques, that have the potential to eliminate the limitations of the existing fast SVP algorithms. We improve the entire phase of the GA construction. Our proposed method is based on the concept of low memory consumption and high reproducibility, and this is the main and significant difference between our algorithm and the other SVP algorithms. Our contributions are twofold. First, we developed a new GA for solving the SVP and achieved a considerable improvement in the running time performance. Second, we interpreted certain genetic operations, such as mutation and crossover, in the context of lattices, which has not been done in previous studies. The general result of this paper is that we showed the potential of GAs in the field of lattices.

A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

We give deterministic $\tilde{O}(2^{2n})$-time $\tilde{O}(2^n)$-space algorithms to solve all the most important computational problems on point lattices in NP, including the shortest vector problem (SVP), closest vector problem (CVP), and shortest independent vectors problem (SIVP). This improves the $n^{O(n)}$ running time of the best previously known algorithms for CVP [R. Kannan, Math. Oper. Res., 12 (1987), pp. 415--440] and SIVP [D. Micciancio, Proceedings of the $19$th Annual ACM-SIAM Symposium on Discrete Algorithms, 2008, pp. 84--93] and gives a deterministic and asymptotically faster alternative to the $2^{O(n)}$-time (and space) randomized algorithm for SVP of Ajtai, Kumar, and Sivakumar [Proceedings of the $33$rd Annual ACM Symposium on Theory of Computing, 2001, pp. 266--275]. The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preproces...

Sieve algorithms for the shortest vector problem are practical

The most famous lattice problem is the Shortest Vector Problem (SVP), which has many applications in cryptology. The best approximation algorithms known for SVP in high dimension rely on a subroutine for exact SVP in low dimension. In this paper, we assess the practicality of the best (theoretical) algorithm known for exact SVP in low dimension: the sieve algorithm proposed by Ajtai, Kumar and Sivakumar (AKS) in 2001. AKS is a randomized algorithm of time and space complexity 2 O ( n ) , which is theoretically much lower than the super-exponential complexity of all alternative SVP algorithms. Surprisingly, no implementation and no practical analysis of AKS has ever been reported. It was in fact widely believed that AKS was impractical: for instance, Schnorr claimed in 2003 that the constant hidden in the 2 O ( n ) complexity was at least 30. In this paper, we show that AKS can actually be made practical: we present a heuristic variant of AKS whose running time is polynomial-time operations, and whose space requirement is polynomially many bits. Our implementation can experimentally find shortest lattice vectors up to dimension 50, but is slower than classical alternative SVP algorithms in these dimensions.