By Sihem Mesnager
This e-book supplies an in depth survey of the most effects on bent features over finite fields, offers a scientific review in their generalizations, adaptations and purposes, considers open difficulties in class and systematization of bent features, and discusses proofs of numerous effects. This e-book uniquely offers an important complete insurance of bent functions.It serves as an invaluable reference for researchers in discrete arithmetic, coding and cryptography. scholars and professors in arithmetic and computing device technological know-how also will locate the content material worthy, particularly these attracted 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 suitable for either easy periods for under-graduate scholars and complicated classes for experts in cryptography and mathematics.
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Additional info for Bent Functions: Fundamentals and Results
F /. Let f be a Boolean function defined on Fn2 . Then the Walsh Hadamard transform of f is the discrete Fourier transform of f , whose value at ! 2 Fn2 is defined as follows: X 8! / D . x/C! x/ C ! x/) where “ ” is the scalar product in Fn2 n Pn x2F2 defined as x y D iD1 xi yi . When dealing with Boolean functions, we rather resort to the Walsh transform that has nicer properties than the Fourier transform in most cases and is an easier tool to handle Boolean functions especially when we are interested in their cryptographic criteria.
1 Algebraic Normal Form The algebraic normal Form (in brief the ANF) is the classical representation of Boolean functions. It is the most used in cryptography and coding. The Algebraic Normal Form of an Boolean function f on Fn2 is the n-variable polynomial representation over F2 , of the form ! N/ denotes the power set of N D f1; ; ng. Every coordinate xi appears in this polynomial with exponents at most 1, because every bit in F2 equals its own square. x21 ˚ x1 ; ; x2n ˚ xn / and it is unique.
F2 ; x 7! x C x2 . x/ D 2jD0 1 aj xj I aj ; x 2 F2n . Let ˛0 ; ; ˛n 1 be a basis of the F2 -vectorspace F2n . 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.
Bent Functions: Fundamentals and Results by Sihem Mesnager