Hardness of SIS and LWE with Small Parameters

David Balbás

Madalina Bolboceanu, Zvika Brakerski, Renen Perlman +1 more

Adeline Roux-Langlois

Lattice-based cryptography is a branch of cryptography exploiting the presumed hardness of some well-known problems on lattices. Its main advantages are its simplicity, efficiency, and apparent security against quantum computers. The principle of the security proofs in lattice-based cryptography is to show that attacking a given scheme is at least as hard as solving a particular problem, as the Learning with Errors problem (LWE) or the Small Integer Solution problem (SIS). Then, by showing that those two problems are at least as hard to solve than a hard problem on lattices, presumed polynomial time intractable, we conclude that the constructed scheme is secure.In this thesis, we improve the foundation of the security proofs and build new cryptographic schemes. We study the hardness of the SIS and LWE problems, and of some of their variants on integer rings of cyclotomic fields and on modules on those rings. We show that there is a classical hardness proof for the LWE problem (Regev's prior reduction was quantum), and that the module variants of SIS and LWE are also hard to solve. We also give two new lattice-based group signature schemes, with security based on SIS and LWE. One is the first lattice-based group signature with logarithmic signature size in the number of users. And the other construction allows another functionality, verifier-local revocation. Finally, we improve the size of some parameters in the work on cryptographic multilinear maps of Garg, Gentry and Halevi in 2013.

Adeline Roux-Langlois, Damien Stehlé

Abstract. Most lattice-based cryptographic schemes are built upon the assumed hardness of the Short Integer Solution (SIS) and Learning With Errors (LWE) problems. Their efficiencies can be drastically improved by switching the hardness assumptions to the more compact Ring-SIS and Ring-LWE problems. However, this change of hardness assumptions comes along with a possible security weakening: SIS and LWE are known to be at least as hard as standard (worst-case) problems on euclidean lattices, whereas Ring-SIS and Ring-LWE are only known to be as hard as their restrictions to special classes of ideal lattices, corresponding to ideals of some polynomial rings. In this work, we define the Module-SIS and Module-LWE problems, which bridge SIS with Ring-SIS, and LWE with Ring-LWE, respectively. We prove that these average-case problems are at least as hard as standard lattice problems restricted to module lattices (which themselves generalize arbitrary and ideal lattices). As these new problems enlarge the toolbox of the lattice-based cryptographer, they could prove useful for designing new schemes. Importantly, the worst-case to average-case reductions for the module problems are (qualitatively) sharp, in the sense that there exist converse reductions. This property is not known to hold in the context of Ring-SIS/Ring-LWE: Ideal lattice problems could reveal easy without impacting the hardness of Ring-SIS/Ring-LWE. 1

Divesh Aggarwal, Jin Ming Leong, Alexandra Veliche

Yuehua Chen, Qipeng Liu, Mark Zhandry

We show polynomial-time quantum algorithms for the following problems: (*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a constant. (*) Learning with errors (LWE) problem given LWE-like quantum states with polynomially large moduli and certain error distributions, including bounded uniform distributions and Laplace distributions. (*) Extrapolated dihedral coset problem (EDCP) with certain parameters. The SIS, LWE, and EDCP problems in their standard forms are as hard as solving lattice problems in the worst case. However, the variants that we can solve are not in the parameter regimes known to be as hard as solving worst-case lattice problems. Still, no classical or quantum polynomial-time algorithms were known for the variants of SIS and LWE we consider. For EDCP, our quantum algorithm slightly extends the result of Ivanyos et al. (2018). Our algorithms for variants of SIS and EDCP use the existing quantum reductions from those problems to LWE, or more precisely, to the problem of solving LWE given LWE-like quantum states. Our main contribution is solving LWE given LWE-like quantum states with interesting parameters using a filtering technique.