Designs and their codes
Two new classes of bent functions
EUROCRYPT '93 Workshop on the theory and application of cryptographic techniques on Advances in cryptology
Linear cryptanalysis method for DES cipher
EUROCRYPT '93 Workshop on the theory and application of cryptographic techniques on Advances in cryptology
Journal of Combinatorial Theory Series A
Discrete Mathematics
Handbook of Coding Theory
The Theory of Cyclic Codes and a Generalization to Additive Codes
Designs, Codes and Cryptography
Homogeneous Bent Functions, Invariants, and Designs
Designs, Codes and Cryptography
Improved fast correlation attacks using parity-check equations of weight 4 and 5
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
IEEE Transactions on Information Theory
New cyclic difference sets with Singer parameters
Finite Fields and Their Applications
Bent and hyper-bent functions over a field of 2l elements
Problems of Information Transmission
On quadratic approximations in block ciphers
Problems of Information Transmission
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
Quantum algorithms for highly non-linear Boolean functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Hyper-bent Boolean functions with multiple trace terms
WAIFI'10 Proceedings of the Third international conference on Arithmetic of finite fields
Upper bound for algebraic immunity on a subclass of Maiorana McFarland class of bent functions
Information Processing Letters
A new class of bent and hyper-bent Boolean functions in polynomial forms
Designs, Codes and Cryptography
On bent and highly nonlinear balanced/resilient functions and their algebraic immunities
AAECC'06 Proceedings of the 16th international conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
A new class of monomial bent functions
Finite Fields and Their Applications
Dickson polynomials, hyperelliptic curves and hyper-bent functions
SETA'12 Proceedings of the 7th international conference on Sequences and Their Applications
Reduction from non-injective hidden shift problem to injective hidden shift problem
Quantum Information & Computation
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Bent functions are those Boolean functions whose Hamming distance to the Reed-Muller code of order 1 equal 2n-1) - 2n/2-1 (where the number n of variables is even). These combinatorial objects, with fascinating properties, are rare. Few constructions are known, and it is difficult to know whether the bent functions they produce are peculiar or not, since no way of generating at random bent functions on 8 variables or more is known.The class of bent functions contains a subclass of functions whose properties are still stronger and whose elements are still rarer. Youssef and Gong have proved the existence of such hyperbent functions, for every even n. We prove that the hyper-bent functions they exhibit are exactly those elements of the well-known PIap class, introduced by Dillon, up to the linear transformations x ↦ δx, δ ∈ F2n*. Hyper-bent functions seem still more difficult to generate at random than bent functions; however, by showing that they all can be obtained from some codewords of an extended cyclic code Hn with small dimension, we can enumerate them for up to 10 variables. We study the non-zeroes of Hn and we deduce that the algebraic degree of hyper-bent functions is n/2. We also prove that the functions of class PIap are some codewords of weight 2n-1 - 2n/2-1 of a subcode of Hn and we deduce that for some n, depending on the factorization of 2n - 1, the only hyperbent functions on n variables are the elements of the class PIap#, obtained from PIap by composing the functions by the transformations x ↦ δx, δ ≠ 0, and by adding constant functions. We prove that non-PIap# hyper-bent functions exist for n = 4, but it is not clear whether they exist for greater n. We also construct potentially new bent functions for n = 12.