By Sihem Mesnager
This ebook offers a close survey of the most effects on bent capabilities over finite fields, offers a scientific review in their generalizations, diversifications and purposes, considers open difficulties in class and systematization of bent capabilities, and discusses proofs of a number of effects. This e-book uniquely offers an important accomplished insurance of bent functions.It serves as an invaluable reference for researchers in discrete arithmetic, coding and cryptography. scholars and professors in arithmetic and desktop technological know-how also will locate the content material worthy, in particular these drawn to mathematical foundations of cryptography. it may be used as a supplementary textual content for collage classes on discrete arithmetic, Boolean services, or cryptography, and is suitable for either simple periods for under-graduate scholars and complicated classes for experts in cryptography and mathematics.
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Extra resources for Bent Functions: Fundamentals and Results
4, thus yielding an alternative and more concrete proof of it. A well-known result by Chou, Gomez-Calderon and Mullen  describes the cardinality of the preimage of an arbitrary element. 26]). Let F2m be the finite field with 2m elements and 1 Ä r Ä 2n 1 be an integer. x/ D y. 28]. The proof heavily relies on the study of the map F2n ! F2m x 7! x C x 1 / D xr C x r : The following property is then a corollary to the above results. 6 (). Let 1 Ä r Ä 2n induces a permutation of 1 be an integer.
J/. 17/ D f17; 23; 25g. The coset leaders modulo 26 are: 0; 1; 2; 4; 5; 7; 8; 13; 14 and 17. function f from F3n ! j/ to F3 . If we do not identify Fnp with Fpn , the p-ary function has a representation as a unique multinomial in x1 ; ; xn , where the variables xi occur with exponent at most p 1. This is called the multivariate representation or algebraic normal form (ANF). The algebraic degree of a p-ary function is the global degree of its multivariate representation. x; y/ of elements in Fpm .
Every element x 2 F2n can Pn 1 be uniquely decomposed as: x D iD0 xi ˛i with xi 2 F2 . x0 ; iD0 xi ˛i g are independent. j/ is the Hamming weight of the binary expansion of j. 8 1 Generalities on Boolean Functions and p-Ary Functions Consequently, the algebraic degree of f given in polynomial form is equal to the maximum 2-weight of an exponent j for which aj ¤ 0 if D 0 and to n if D 1. t. aj 6D 0g. The computation of the algebraic degree is not always simple in general when the function is given in its univariate representation (in contrast with the ANF).