A one-way function based on norm form equations

Abstract

Abstract. In this paper we present a new one-way function with collision resistance. The security of this function is based on the difficulty of solving a norm form equation. We prove that this function is collision resistant, so it can be used as a one-way hash function. We show that this construction probably provides a family of one-way functions.

The cross-correlation measure for families of binary sequences

“…Such a property is the existence of collisions in the given family. This notion appears, e.g., in [8], [35], [40], [41]; we will follow here Tóth's [40] presentation. Assume that N ∈ N, S is a given set (e.g., a set of certain polynomials or the set of all the binary sequences of a given length much less than N), to each s ∈ S we assign a unique binary sequence…”

“…There is another related notion appearing in the literature, namely, the notion of avalanche effect (see, e.g., [8], [15], [24], [40], [41]); here we will present Tóth's definition):…”

On a family of preimage-resistant functions

“…Bé r c z e s, Kö d mö n and P e t hő [7] constructed a family of preimage--resistant functions based on norm functions, well studied in the theory of diophantine equations. Bé r c z e s and Já rá s i [8] extended this result to a family based on index forms.…”

“…The aim of this paper is to continue the investigations of [7] and [8] on the preimage-resistance of functions defined over finite rings and improve their results in two directions. First, we are working on finite fields F q and not on finite rings Z m , where m is the product of two primes.…”

“…First, we are working on finite fields F q and not on finite rings Z m , where m is the product of two primes. For the security of the construction of [7], [8] m has to be hard to factorize, i.e., it must be at least 1024 bit long. In contrast the length of q can be considerably shorter, e.g., 256 or 512 bits.…”