By Sihem Mesnager

This ebook offers an in depth survey of the most effects on bent features over finite fields, offers a scientific evaluation in their generalizations, adaptations and functions, considers open difficulties in category and systematization of bent services, and discusses proofs of a number of effects. This publication uniquely offers an important accomplished 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 beneficial, particularly these drawn to mathematical foundations of cryptography. it may be used as a supplementary textual content for collage classes on discrete arithmetic, Boolean features, or cryptography, and is suitable for either uncomplicated sessions for under-graduate scholars and complicated classes for experts in cryptography and mathematics.

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**Example text**

5. a; a/ D 0. Therefore, Eq. a/ D 4. mC1/=2 with a D ˙1. mC2/=2 C 1. Now, if i D 1, Eq. 5. 3). a/ D 4. 5). 5. a; a/ D 4, Eq. 5. mod 6/. Then 2m C 1 is a multiple of 9. Therefore, the mapping u 7! 4). t u Finally, we also have to express another type of exponential sums over U in terms of Kloosterman sums and cubic sums. This result will be useful in Chap. 9. 7 ([17]). 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!

2F2n Note that not all values of the Walsh Hadamard transform can have the same sign, except when the function is affine. / which implies that all these values are null except one. The Walsh transform satisfies also the Poisson summation formula, valid for every F2 -linear space E: X X . u/ D jEj . 1/a b . x/Ca x : x2bCE? x/ D a x ˚ " takes null value at every vector u ¤ a and takes value 2n . 1/" at a. The rank of a quadratic Boolean function is connected with the distribution of its Walsh–Hadamard transform values.

1/" at a. The rank of a quadratic Boolean function is connected with the distribution of its Walsh–Hadamard transform values. The following is due to Helleseth and Kumar. 2 ([10]). Let f be a quadratic Boolean function on F2n with rank 2t, 0 Ä t Ä b n2 c. 2. Now, let f be a Boolean function on F2n . ; k/ D X . x / ; n k x2F2n for ! 2 Walsh spectrum of quadratic function with rank 2t 1. /, ! 3 Some Classical Binary Exponential Sums Exponential sums are important tools in number theory for solving problems involving integer and real numbers in general.