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In many single-key, symmetric or conventional cryptosystems the elements of a finite field can be regarded as the characters of a plaintext and ciphertext alphabet. Some properties of polynomials or polynomial functions on finite fields can be used for constructing cryptosystems. This note demonstrates by way of examples that great care has to be taken in choosing polynomials for enciphering and deciphering. Often complex looking polynomial functions induce very simple permutations of the elements of a finite field and therefore are not suitable for the construction of cryptosystems. Also an indication is given of some further areas of research in algebraic cryptography.

On Cryptosystems Based on Polynomials and Finite Fields