Faster Rotation-Based Gauss Sieve for Solving the SVP on General Ideal Lattices

Abstract

The hardness in solving the shortest vector problem (SVP) is a fundamental assumption for the security of lattice-based cryptographic algorithms. In 2010, Micciancio and Voulgaris proposed an algorithm named the Gauss Sieve, which is a fast and heuristic algorithm for solving the SVP. Schneider presented another algorithm named the Ideal Gauss Sieve in 2011, which is applicable to a special class of lattices, called ideal lattices. The Ideal Gauss Sieve speeds up the Gauss Sieve by using some properties of the ideal lattices. However, the algorithm is applicable only if the dimension of the ideal lattice n is a power of two or n + 1 is a prime. Ishiguro et al. proposed an extension to the Ideal Gauss Sieve algorithm in 2014, which is applicable only if the prime factor of n is 2 or 3. In this paper, we first generalize the dimensions that can be applied to the ideal lattice properties to when the prime factor of n is derived from 2, p or q for two primes p and q. To the best of our knowledge, no algorithm using ideal lattice properties has been proposed so far with dimensions such as: 20, 44, 80, 84, and 92. Then we present an algorithm that speeds up the Gauss Sieve for these dimensions. Our experiments show that our proposed algorithm is 10 times faster than the original Gauss Sieve in solving an 80dimensional SVP problem. Moreover, we propose a rotation-based Gauss Sieve that is approximately 1.5 times faster than the Ideal Gauss Sieve. key words: shortest vector problem, Gauss Sieve, ideal lattice, generalization