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Cryptosystems based on polynomials over finite fields

Two cryptosystems using polynomials on finite fields are introduced; a permutation cipher based on permutation polynomials and a flexible secret sharing scheme. Polynomials are called permutation polynomials if they induce bijective functions. Metric and algebraic properties of permutation polynomials over a finite fields are investigated, especially properties associated with orbits and permutation cycles. Flexible secret sharing schemes have features that they can change a secret, enroll/disenroll members, increase/decrease threshold, without changing shares. A simple scheme using polynomials over finite fields is presented.