zbMATH — the first resource for mathematics

A note on polynomials and functions in algebraic cryptography. (English) Zbl 0539.94018
The authors discuss the use of various special functions as enciphering functions in single-key and public-key cryptosystems. The rational functions over a finite field \(F_ q\) introduced by L. Rédei [Acta Sci. Math., Szeged 11, 85-92 (1946)] are proposed as enciphering functions and some of their relevant properties are described. Another idea suggested by the authors is to regard the set \(F^ r_ q\) of r- tuples of elements of \(F_ q\) as the cipher alphabet and to use permutation polynomial vectors to produce enciphering functions from \(F^ r_ q\) into itself. Specifically, the Dickson polynomial vectors [see Chapter 7 in the book by R. Lidl and H. Niederreiter: Finite fields (1984)] are proposed for this purpose. The well-known RSA- crypto-system is also analyzed. The authors show the interesting fact that if p and q are distinct odd primes, then any permutation \(P_ k\) of the residue class ring \(Z_{pq}\) of the form \(P_ k(x)=x^ k\) has at least 9 fixed points. The number of permutations \(P_ k\) which have the minimal number 9 of fixed points is determined.
Reviewer: H.Niederreiter

94A60 Cryptography
11T06 Polynomials over finite fields