Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials

Abstract

This paper is devoted to the classification of hyperbent functions, i.e., bent functions which are bent up to a primitive root change. We first exhibit an infinite class of monomial functions which are not hyperbent. This result means that Kloosterman sums at point 1 on F 2 m cannot be zero, unless m = 4. For the functions with multiple trace terms, we express their spectrum by means of Dickson polynomials. We then introduce a new tool to describe these hyperbent functions, whose efficiency is proving by the characterization of a class of binomial bent functions.

A New Modified MARS Cryptosystem Based on Niho Exponent with an Enhanced S-Box Generation

“…Let f : F 2 n → F 2 be a Boolean function such that f (x) = Tr n 1 (ax s ) for a given positive integer s and for some a ∈ F 2 n [17]. Functions of this form are called monomial functions.…”

Permutation polynomials EA-equivalent to the inverse function over GF (2 n )

“…Tr(x −1 +λx) = 0 [10,8]. Characterizing the set of those λ ∈ F 2 n with K(λ) = 0 is raised as an open problem in [8].…”

“…Characterizing the set of those λ ∈ F 2 n with K(λ) = 0 is raised as an open problem in [8]. The complete kloosterman sums are also powerful for us to characterize the permutation polynomials of the type x −1 + L(x) over F 2 n , especially we need the following results.…”

“…Based on these reasons, it is appropriate to choose the permutation polynomials affine or extend affine (EA) equivalent to the inverse function on F 2 2n as the S-boxes of block ciphers. For example, the S-box of AES is affine equivalent to the inverse function over F 2 8 .…”

Generic Construction of Boolean Functions with A Few Walsh Transform Values of Any Possible Algebraic Degree

“…One distinguishes two kinds of constructions of bent functions: primary constructions, which do not need to use previously constructed functions for designing new ones, and secondary constructions (of new functions from known ones). The non-exhaustive lists of references dealing with primary constructions of bent Boolean functions are [4,9,10,5,20,21,23,27,31].…”