This paper is devoted to hyperbent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin–Gong family, which was also used by Wang and coworkers to obtain another similar extension, fits in a much more general setting. To this end, we first explain how the original restriction for Charpin–Gong criterion can be weakened before generalizing the Mesnager approach to arbitrary Dillon-like exponents. Afterward, we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of Mesnager and Wang and coworkers, but also to characterize the hyperbentness of several new infinite classes of Boolean functions. We go into full details only for a few of them, but provide an algorithm (and the corresponding software) to apply this approach to an infinity of other new families. Finally, we compare the asymptotic and practical performances of different characterizations, including these in terms of hyperelliptic curves, and actually build hyperbent functions in cases which could not be attained through naive computations of exponential sums.

This paper is devoted to hyper-bent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin-Gong family, which was also used by Wang et al. to obtain another similar extension, fits in a much more general setting. To this end, we first explain how the original restriction for Charpin-Gong criterion can be weakened before generalizing the Mesnager approach to arbitrary Dillon-like exponents. Afterward, we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of Mesnager, and Wang et al., but also to characterize the hyper-bentness of new infinite classes of Boolean functions.

Recently, Dziembowski et al. introduced the notion of non-malleable codes (NMC), inspired from the notion of non-malleability in cryptography and the work of Gennaro et al. in 2004 on tamper proof security. Informally, when using NMC, if an attacker modifies a codeword, decoding this modified codeword will return either the original message or a completely unrelated value. The definition of NMC is related to a family of modifications authorized to the attacker. In their paper, Dziembowski et al. propose a construction valid for the family of all bit-wise independent functions. In this article, we study the link between the second version of the Wire-Tap (WT) Channel, introduced by Ozarow and Wyner in 1984, and NMC. Using coset-coding, we describe a new construction for NMC w.r.t. a subset of the family of bit-wise independent functions. Our scheme is easier to build and more efficient than the one proposed by Dziembowski et al.