Cryptanalysis of Rédei- and Dickson permutations on arbitrary finite rings

Abstract

In the RSA public-key cryptosystem the encryption function is a permutation on the residue class ringℤ/Nℤ induced by some polynomialX ^k ∈(ℤ/Nℤ) [X], whereN is the product of two large primes. Variants of the RSA-scheme can be obtained, if this permutation is replaced by different types of rational permutations onℤ/Nℤ. A more general approach is the use of arbitrary finite rings instead of residue class rings in cryptography. Since the message space is finite, in either case cryptanalysis can be effected by superenciphering. A serious weakness of those PKCs is the existence of a large number of fixedpoints. But even if there are only few fixedpoints in the message space, the elements of considerable small cyclelength are much inconvenient. Anyway an analysis of the minimal cyclelength, i.e. the minimum of cyclelengths of elements different from fixedpoints, is necessary.

In this paper such an analysis will be carried out in the case of Rédei- and Dickson permutations on arbitrary finite rings. The results obtained provide a good basis to construct secure PKCs with best protection against superenciphering. Some of the problems and results in the special cases of finite fields and residue class rings have been stated earlier in the literature (see references).