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The connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, Lisoněk exploited it to reformulate the Charpin–Gong characterization of a large class of hyper-bent functions in terms of numbers of points on hyperelliptic curves. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin–Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by Mesnager. Doing so, a polynomial time and space test for the hyper-bentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong.

In this paper we study the number of carries occurring while performing an addition modulo . For a fixed modular integer t, it is natural to expect the number of carries occurring when adding a random modular integer a to be roughly the Hamming weight of t. Here we are interested in the number of modular integers in producing strictly more than this number of carries when added to a fixed modular integer . In particular it is conjectured that less than half of them do so. An equivalent conjecture was proposed by Tu and Deng in a different context. Although quite innocent, this conjecture has resisted different attempts of proof and only a few cases have been proved so far. The most manageable cases involve modular integers t whose bits equal to are sparse. In this paper we continue to investigate the properties of , the fraction of modular integers a to enumerate, for t in this class of integers. Doing so we prove that has a polynomial expression and describe a closed form for this expression. This is of particular interest for computing the function giving and studying it analytically. Finally, we bring to light additional properties of in an asymptotic setting and give closed-form expressions for its asymptotic values. [ABSTRACT FROM AUTHOR]