Cryptological Applications of Permutation Polynomials

Abstract

Abstract Permutation polynomials(PP) are of great interest in the study of finite fields and their applications in cryptography and coding. A polynomial P ( x ) = a 0 + a 1 x + a 2 x 2 +…+ a d x d with coefficients a 1 in a ring R is said to be a PP if P permutes the elements of R. The conditions for an arbitrary polynomial to be a PP over finite fields are however rather complicated. Lidl and Mullen [1] discuss a number of open problems regarding PP. Among these problems, is the problem of determining N d (q), the number of PP over F q of degree d where 1 ≤ d ≤ q − 2 and d Download full-size image q − 1. Recently Rivest[2] proposed a simple characterization of PP modulo n = 2 W . We deduce a result on the enumeration of Nd(q) based on this. In this paper we bring to focus certain properties of permutation polynomials (PP) that have been exploited for cryptological applications. We develop a methodology to completely cryptanalyse an encryption scheme proposed by Levine and Brawley [3] and similar symmetric key based schemes. We also consider the design of asymmetric cryptosystems based on permutation polynomials analogous to RSA. The function composition of permutation polynomials plays the role of modular exponentiation. We evaluate the complexity of this function composition based encryption. We show that PP based scheme is slower by a constant factor.