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AbstractThe core of this thesis is the study of some mathematical objects or problems of interest incryptology. As much as possible, the author tried to emphasize the computational aspects of theseproblems. The topics covered here are indeed not only favorable to experimental investigations,but also to the quasi direct translation of the mathematical concepts involved into concretealgorithms and implementations.The first part is devoted to the study of a combinatorial conjecture whose validity entails theexistence of infinite classes of Boolean functions with good cryptographic properties. Althoughthe conjecture seems quite innocuous, its validity remains an open question. Nonetheless, theauthor sincerely hopes that the theoretical and experimental results presented here will give thereader a good insight into the conjecture.In the second part, some connections between (hyper-)bent functions — a subclass of Booleanfunctions —, exponential sums and point counting on (hyper)elliptic curves are presented. Bentfunctions and hyper-bent functions are known to be difficult to classify and to build explicitly.However, exploring the links between these different worlds makes possible to give beautifulanswers to theoretical questions and to design efficient algorithms addressing practical problems.The third and last part investigates the theory of (hyper)elliptic curves in a different direction.Several constructions in cryptography indeed rely on the use of highly specific classes of suchcurves which can not be constructed by classical means. Nevertheless, the so-called “complexmultiplication” method solves some of these problems. Class polynomials are fundamental objectsfor that method, but their construction is usually considered only for maximal orders. The modestcontribution of the author is to clarify how a specific flavor of their construction — the complexanalytic method — extends to non-maximal orders.