Welcome to the resource topic for 2023/1450
Title:
Post-Quantum Fully Homomorphic Encryption with Group Ring Homomorphisms
Authors: Christopher Leonardi, Maya Gusak
Abstract:Gentry’s groundbreaking work showed that a fully homomorphic, provably secure scheme is possible via bootstrapping a somewhat homomorphic scheme. However, a major drawback of bootstrapping is its high computational cost. One alternative is to use a different metric for noise so that homomorphic operations do not accumulate noise, eliminating the need for boostrapping altogether. Leonardi and Ruiz-Lopez present a group-theoretic framework for such a noise non-accumulating'' multiplicative homomorphic scheme, but Agathocleous et al. expose weaknesses in this framework when working over finite abelian groups. Tangentially, Li and Wang present a noise non-accumulating’’ fully homomorphic scheme by performing Ostrovsky and Skeith’s transform on a multiplicative homomorphic scheme of non-abelian group rings. Unfortunately, the security of Li and Wang’s scheme relies on the Factoring Large Numbers assumption, which is false given an adversary with a quantum computer. In this work, we seek to modify Li and Wang’s scheme to be post-quantum secure by fitting it into the Leonardi and Ruiz-Lopez framework for non-abelian rings. We discuss improved security assumptions for Li and Wang encryption and assess the shortcomings of working in a non-abelian setting. Finally, we show that a large class of semisimple rings is incompatible with the Leonardi and Ruiz-Lopez framework.
ePrint: https://eprint.iacr.org/2023/1450
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