By Sihem Mesnager
This ebook supplies a close survey of the most effects on bent capabilities over finite fields, offers a scientific evaluate in their generalizations, diversifications and purposes, considers open difficulties in class and systematization of bent services, and discusses proofs of a number of effects. This booklet uniquely presents an important entire assurance of bent functions.It serves as an invaluable reference for researchers in discrete arithmetic, coding and cryptography. scholars and professors in arithmetic and laptop technological know-how also will locate the content material precious, specifically these drawn to mathematical foundations of cryptography. it may be used as a supplementary textual content for college classes on discrete arithmetic, Boolean capabilities, or cryptography, and is acceptable for either simple periods for under-graduate scholars and complex classes for experts in cryptography and mathematics.
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Additional resources for Bent Functions: Fundamentals and Results
The mapping u 7! ur is Proof. a; b/ WD u2U where 1 a permutation of U (we denote by x 7! r; 2m C1/ D 1. bu r 3 / . a; b/ D u2U the following decomposition of U: UDV[ V[ 2 V where V WD fu3 j u 2 Ug and is a generator of the cyclic group U. a/: Indeed, 2 2 is an element of V m m because 3 divides (2m C 1) (since m is odd) and the mapping v 7! 2 2 v 2 is a permutation on V. a 2 . a/: 2 2m C1 Now, note that since is a generator of U, Tr21 . r 3 / D Tr21 . r 3 / D 1. a/ if b D 1 if b 6D 1. a/ by means of P two ways the sum b2F?
7 (). Let m be an odd integer. 2m . a; a/: u2U Proof. c/ D 1 can be uniquely m represented as u Cu2 with u 2 U. 1=c/D1 In the last equality, we use the fact that the map c 7! 1=c is a permutation on F2m . 1=c/D1 from which we deduce the result. 5 Some Basic Notions in Number Theory In this section we present some basic background in number theory. In treating a number of arithmetical questions, the notion of congruence is extremely useful. Z=mZ/. Z=pZ/ is a cyclic group when p is a prime is equivalent to the existence of primitive roots.
Since in this book we are predominantly dealing with bent Boolean functions, emphasis here is laid upon binary polynomials. © Springer International Publishing Switzerland 2016 S. 1 Permutations Polynomials Permutation polynomials over finite fields have been an interesting subject of study for many years, and have applications in coding theory, cryptography, combinatorial design theory, and other areas of mathematics and engineering. Information about properties, constructions, and applications of permutation polynomials can be found in  and .