Cubic Monomial Bent Functions: A Subclass of $\mathcal{M}$
SIAM Journal on Discrete Mathematics
EUROCRYPT'91 Proceedings of the 10th annual international conference on Theory and application of cryptographic techniques
New binomial bent functions over the finite fields of odd characteristic
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials
IEEE Transactions on Information Theory
A new class of monomial bent functions
Finite Fields and Their Applications
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Finite Fields and Their Applications
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In this paper we investigate the possibility of constructing bent functions over fields with odd characteristic. While in the binary case, and for n=2k, the bent property of monomials of the form Tr"1^n(ax^r^(^2^^^k^-^1^)) and binomials Tr"1^n(x^2^^^k^-^1+ax^r^(^2^^^k^-^1^)) were investigated in several papers, generalized bent functions f:GF(p^n)-GF(p) of the form Tr"1^n(@?"i"="1^ta"ix^r^"^i^(^p^^^k^-^1^)), p being an odd prime and n=2k, were not analyzed previously. In particular, the construction of vectorial (generalized) bent functions has not been addressed. It is shown that the necessary and sufficient bent conditions for both the single output function of the form f(x)=Tr"1^n(@?"i"="1^ta"ix^r^"^i^(^p^^^k^-^1^)) and the associated mapping F(x)=Tr"k^2^k(@?"i"="1^ta"ix^r^"^i^(^p^^^k^-^1^)), where F:GF(p^2^k)-GF(p^k), are very similar and can be expressed in terms of the image of a set V used in the direct sum decomposition of GF(p^2^k). Furthermore, it is observed that vectorial bent functions are easily constructed using the Maiorana-McFarland method.