Permutation Polynomials And Their Combinatorial Applications

Abstract

newline From the 19th century, the theory of permutation polynomial over finite fields, newlinethat are arose in the work of Hermite and Dickson, has drawn general at- newlinetention. The last three decades has seen rapid progress on the research on newlinepermutation polynomials due to their diverse applications in cryptography, newlinecoding theory, finite geometry, combinatorics and many more areas of math- newlineematics and engineering. For this reason, the study of permutation poly- newlinenomials is important. This thesis is mainly devoted to the study of some newlineproblems on permutation polynomials and their combinatorial applications. newlineThe arrangement of this thesis is as follows. newlineIn first chapter, we introduce basic definitions and some useful lemmas newlinethat are helpful to understand entire thesis. newlineAlthough various type of different classes of permutation polynomials newlinewith simple appearance or good properties have been found day by day, newlinebut the general problems of characterization of permutation polynomials of newlineprescribed forms are still challenging. We present some new classes of per- newlinemutation pentanomials by using some known results in the second chapter. newlineThird chapter is associated with an equivalent form of Kloosterman poly- newlinenomials conjecture that are posed by Hollmann and Xiang in 2004 in a deeper newlineconnection to permutation polynomials. The study of Kloosterman polyno- newlinemials are interesting because every Kloosterman polynomial give rise to a newlinenew Kloosterman sum identity. newlinePlanar functions are of central interest in cryptography, coding theory newlineand design theory. It is not hyperbole to say that being planar is a very newlinerestrictive property. It is not easy to find planar functions. There is even newlineno classification known of all monomials planar functions. Thus, finding new classes of planar functions are in particular interesting. In this way, we have newlinecontributed a new class of planar functions over finite fields in fourth chapter. newlineIn finite field theory, to apply permutation polynomials in other area of newlinemathematics and engineering is also