Eisuke Koizumi

Toward Separating the Strong Adaptive Pseudo-freeness from the Strong RSA Assumption

The notion of pseudo-freeness of a group was introduced by Hohenberger, and formalized by Rivest in order to unify cryptographic assumptions. Catalano, Fiore and Warinschi proposed the adaptive pseudo-free group as a generalization of pseudo-free group. They showed that the RSA group (mathbb{Z}_N^times) is pseudo-free even if the adversary against pseudo-freeness is allowed to operate adaptively, provided that the adaptive behavior of the adversary is restricted by some specific parametric distribution. They also proposed the notion of strong adaptive pseudo-freeness in which the adaptive behavior of the adversary is not restricted. However, it remains open whether (mathbb{Z}_N^times) is also strongly-adaptive pseudo-free under the strong RSA (SRSA) assumption.

On the Complexity of Computing Discrete Logarithms over Algebraic Tori

This paper studies the complexity of computing discrete logarithms over algebraic tori. We show that the order certified version of the discrete logarithm over general finite fields (OCDL, in symbols) reduces to the discrete logarithm over algebraic tori (TDL, in symbols) with respect to the polynomial-time Turing reducibility. This reduction means that if the integer factorization can be computed in polynomial time, then TDL is equivalent to the discrete logarithm DL over general finite fields with respect to the Turing reducibility.

Making Cryptographic Primitives Harder

This paper studies a method for transforming ordinary cryptographic primitives to new harder primitives. Such a method is expected to lead to general schemes that make present cryptosystems secure against the attack of quantum computers. We propose a general technique to construct a new function from an ordinary primitive function f with a help of another hard function g so that the resulting function is to be new hard primitives. We call this technique a lifting of f by g. We show that the lifted function is harder than original functions under some simple conditions.