On non-abelian homomorphic public-key cryptosystems

TL;DR

This paper introduces the first homomorphic cryptosystems for non-abelian groups, enabling secure computations on encrypted data within these complex algebraic structures, expanding the scope beyond previously known abelian cases.

Contribution

It constructs homomorphic cryptosystems for non-abelian groups, specifically for any fixed solvable group H, a novel advancement in cryptography.

Findings

First homomorphic cryptosystems for non-abelian groups

Construction applicable to any fixed solvable group H

Extends homomorphic encryption beyond abelian groups

Abstract

An important problem of modern cryptography concerns secret public-key computations in algebraic structures. We construct homomorphic cryptosystems being (secret) epimorphisms f:G --> H, where G, H are (publically known) groups and H is finite. A letter of a message to be encrypted is an element h element of H, while its encryption g element of G is such that f(g)=h. A homomorphic cryptosystem allows one to perform computations (operating in a group G) with encrypted information (without knowing the original message over H). In this paper certain homomorphic cryptosystems are constructed for the first time for non-abelian groups H (earlier, homomorphic cryptosystems were known only in the Abelian case). In fact, we present such a system for any solvable (fixed) group H.