Improved Combinatorial Algorithms for the Inhomogeneous Short Integer Solution Problem

The paper is about algorithms for the inhomogeneous short integer solution problem: given $$(\mathbf A , \mathbf s )$$(A,s) to find a short vector $$\mathbf{x }$$x such that $$\mathbf A \mathbf{x }\equiv \mathbf s \pmod {q}$$Ax≡s(modq). 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.

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