Two Problems about Monomial Bent Functions
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In 2008, Langevin and Leander determined the dual function of three classes of monomial bent functions with the help of Stickelberger's theorem: Dillon, Gold and Kasami. In their paper, they proposed one very strong condition such that their method works, and showed that both Gold exponent and Kasami exponent satisfy this condition. In 2018, Pott et al. investigated the issue of vectorial functions with maximal number of bent components. They found one class of binomial functions which attains the upper bound. They also proposed an open problem regarding monomial function with maximal number of bent components. In this paper, we obtain an interesting result about the condition of Langevin and Leander, and solve the open problem of Pott et al.. Specifically, we show that: 1) for a monomial bent function over 𝔽_2^2k, if the exponent satisfies the first part of the condition of Langevin and Leander, then it satisfies the entire condition; 2) x^2^k+1 is the only monomial function over 𝔽_2^2k which has maximal number of bent components. Besides, as a byproduct, we dig out a very nice property about Kloosterman sums.
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