EUROCRYPT '01 Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques: Advances in Cryptology
Hyper-bent functions and cyclic codes
Journal of Combinatorial Theory Series A
The divisibility modulo 24 of Kloosterman sums on GF(2m), m odd
Journal of Combinatorial Theory Series A
A New Family of Hyper-Bent Boolean Functions in Polynomial Form
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
IEEE Transactions on Information Theory
Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials
IEEE Transactions on Information Theory
New cyclic difference sets with Singer parameters
Finite Fields and Their Applications
A New Family of Hyper-Bent Boolean Functions in Polynomial Form
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
Hyper-bent Boolean functions with multiple trace terms
WAIFI'10 Proceedings of the Third international conference on Arithmetic of finite fields
Binary kloosterman sums with value 4
IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
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Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. These combinatorial objects, with fascinating properties, are rare. The class of bent functions contains a subclass of functions the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer. (Hyper)-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. Few constructions of hyper-bent functions defined over the Galois field ${\mathbb F}_{2n}$ (n = 2m ) are proposed in the literature. The known ones are mostly monomial functions. This paper is devoted to the construction of hyper-bent functions. We exhibit an infinite class over ${\mathbb F}_{2n}$ (n = 2m , m odd) having the form $f(x) = Tr_1^{o(s_1)} (a x^{s_1}) + Tr_1^{o(s_2)} (b x^{s_2})$ where o (s i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 n *** 1 which contains s i and whose coefficients a and b are, respectively in ${\mathbb F}_{2^{o(s_1)}}$ and ${\mathbb F}_{2^{o(s_2)}}$. We prove that the exponents $s_1={3(2^m-1)}$ and $s_2={\frac {2^n-1}3}$, where $a\in {\mathbb F}_{2n}$ ($a\not=0$) and $b\in{\mathbb F}_4$ provide a construction of hyper-bent functions over ${\mathbb F}_{2n}$ with optimum algebraic degree. We give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums and the cubic sums involving only the coefficient a .