On the one-way algebraic homomorphism

In this paper we discuss the relation between a one-way group homomorphism and a one-way ring homomorphism. Let U, V be finite abelian groups with #U = n. We show that if there exists a one-way group homomorphism f: U → V, then there exists a one-way ring homomorphism F: Z_n ⊕ U → Z_n ⊕ Im f. We also give examples of such ring homomorphisms which are one-way under a standard cryptographic assumption. This implies that there is an affirmative solution to an extended version of the open question raised by Feigenbaum and Merrit: Is there an encryption function f such that both f(x + y) and f(x · y) can be efficiently computed from f(x) and f(y) ? A multiple signature scheme is also given as an application of one-way ring homomorphisms.