Improved Combinatorial Algorithms for the Inhomogeneous Short Integer Solution Problem

The paper is about algorithms for the inhomogeneous short integer solution problem: given (𝐀 , 𝐬 ) to find a short vector 𝐱 such that 𝐀𝐱 ≡𝐬q . We consider algorithms for this problem due to Camion and Patarin; Wagner; Schroeppel and Shamir; Minder and Sinclair; Howgrave–Graham and Joux (HGJ); Becker, Coron and Joux (BCJ). Our main results include: applying the Hermite normal form (HNF) to get faster algorithms; a heuristic analysis of the HGJ and BCJ algorithms in the case of density greater than one; an improved cryptanalysis of the SWIFFT hash function; a new method that exploits symmetries to speed up algorithms for Ring-SIS in some cases.