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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

##### MSC:
 94A60 Cryptography 11T06 Polynomials over finite fields