Permutation Polynomials Modulo n, n 6= 2w and Latin Squares

Bhatta, Vadiraja G R and Shankar, B R (2009) Permutation Polynomials Modulo n, n 6= 2w and Latin Squares. International Journal of Mathematical Combinatorics, 2. pp. 58-65. ISSN 1937-1055

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Abstract

Our work is motivated by a recent paper of Rivest [6], concerning permutation polynomials over the rings Zn with n = 2w. Permutation polynomials over finite fields and the rings Zn have lots of applications, including cryptography. For the special case n = 2w, a characterization has been obtained in [6] where it is shown that such polynomials can form a Latin square (0 ≤ x, y ≤ n − 1) if and only if the four univariate Polynomials P(x, 0), P(x, 1), P(0, y) and P(1, y) are permutation polynomials. Further, it is shown that pairs of such polynomials will never form Latin squares. In this paper, we consider bivariate polynomials P(x, y) over the rings Zn when n 6= 2w. Based on preliminary numerical computations, we give complete results for linear and quadratic polynomials. Rivest’s result holds in the linear case while there are plenty of counterexamples in the quadratic case.

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