By Sihem Mesnager
This e-book supplies a close survey of the most effects on bent capabilities over finite fields, provides a scientific assessment in their generalizations, adaptations and functions, considers open difficulties in category and systematization of bent capabilities, and discusses proofs of numerous effects. This ebook uniquely presents an important finished assurance 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 necessary, in particular these drawn to mathematical foundations of cryptography. it may be used as a supplementary textual content for college classes on discrete arithmetic, Boolean services, or cryptography, and is acceptable for either easy periods for under-graduate scholars and complicated classes for experts in cryptography and mathematics.
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Additional resources for Bent Functions: Fundamentals and Results
X/ where the values of the trace are understood as the integers 0 and 1. 8. Let f W F2m ! F2 be a Boolean function. x/ C . 1/i X x2F2m And we finally record the following corollary. 9. Let 1 Ä r Ä 2n 1 be an integer and f W F2m ! F2 be a Boolean function. r; 2m 1/ D 1. 2 Fourier Transform and Walsh Hadamard Transform Let f be any complex valued function on Fn2 . f / is by definition the complex-valued mapping Fn2 ! y/. 1/x y ; x 2 Fn2 y2Fn2 where “ ” is a scalar product on Fn2 . y/. f // D 2n f .
F / D Hf where H is the so-called Hadamard matrix whose coefficient at row x 2 Fn2 and column y 2 Fn2 is Hx;y D . 1/x y . The Hadamard matrix is invertible and its inverse is given by H 1 D 21n H. 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.
Let n > 1 be a positive integer. Let f 1 ; ; ng P be a basis of F2n . x1 ; ; xn / 7! niD1 xi i . In this way, each variable taking value from F2n can be written as x D x1 1 C xn n . x1 ; ; xn /. mod 2n 1/. j/ which divides n. j/. j/ forms a linearly independent set over F2 , the set fx 7! j/g is also a linearly independent set over F2 . Moreover, the functions x 7! x/ are Boolean functions over F2n . mod 2n 1/). x/, 8x 2 F2n . j2 /; n of 2 modulo 2 1 which partition the set of integers modulo 2n 1.