Galois ring isomorphism problem

TL;DR

This paper introduces the Galois ring isomorphism problem, extending the finite field isomorphism problem to Galois rings, enabling cryptographic constructions over rings modulo prime powers.

Contribution

It generalizes the finite field isomorphism problem to Galois rings, facilitating new cryptographic primitives over rings modulo prime powers.

Findings

Defines the Galois ring isomorphism problem

Shows how to lift residue field isomorphisms to Galois rings

Enables cryptographic primitives over rings modulo prime powers

Abstract

Recently, Dor\"oz et al. (2017) proposed a new hard problem, called the finite field isomorphism problem, and constructed a fully homomorphic encryption scheme based on this problem. In this paper, we generalize the problem to the case of Galois rings, resulting in the Galois ring isomorphism problem. The generalization is achieved by lifting the isomorphism between the corresponding residue fields. As a result, this generalization allows us to construct cryptographic primitives over the ring of integers modulo a prime power, instead of a large prime number.