General Impossibility of Group Homomorphic Encryption in the Quantum World

TL;DR

This paper proves that constructing quantum-resistant abelian group homomorphic encryption schemes is impossible under minimal security assumptions, highlighting fundamental limitations in cryptography against quantum adversaries.

Contribution

It establishes a general impossibility result for abelian group homomorphic encryption in the quantum setting, introducing a new sampling probability analysis and discussing implications for existing schemes.

Findings

Quantum adversaries break existing group homomorphic schemes

Sampling generating sets of finite groups has bounded probability

Impossibility applies under minimal security assumptions

Abstract

Group homomorphic encryption represents one of the most important building blocks in modern cryptography. It forms the basis of widely-used, more sophisticated primitives, such as CCA2-secure encryption or secure multiparty computation. Unfortunately, recent advances in quantum computation show that many of the existing schemes completely break down once quantum computers reach maturity (mainly due to Shor's algorithm). This leads to the challenge of constructing quantum-resistant group homomorphic cryptosystems. In this work, we prove the general impossibility of (abelian) group homomorphic encryption in the presence of quantum adversaries, when assuming the IND-CPA security notion as the minimal security requirement. To this end, we prove a new result on the probability of sampling generating sets of finite (sub-)groups if sampling is done with respect to an arbitrary, unknown distribution. Finally, we provide a sufficient condition on homomorphic encryption schemes for our quantum attack to work and discuss its satisfiability in non-group homomorphic cases. The impact of our results on recent fully homomorphic encryption schemes poses itself as an open question.