Boolean functions, algebraic curves and complex multiplication

Abstract : The first part is devoted to the study of a combinatorial conjecture whose validity entails the existence of infinite classes of Boolean functions with good cryptographic properties. Although the conjecture seems quite innocuous, its validity remains an open question. Nonetheless, the author sincerely hopes that the theoretical and experimental results presented here will give the reader a good insight into the conjecture. In the second part, some connections between (hyper-)bent functions -- a subclass of Boolean functions --, exponential sums and point counting on (hyper)elliptic curves are presented. Bent functions 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 beautiful answers 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 such curves which can not be constructed by classical means. Nevertheless, the so-called "complex multiplication" method solves some of these problems. Class polynomials are fundamental objects for that method, but their construction is usually considered only for maximal orders. The modest contribution of the author is to clarify how a specific flavor of their construction -- the complex analytic method -- extends to non-maximal orders.
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Theses
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https://pastel.archives-ouvertes.fr/pastel-00758378
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Submitted on : Wednesday, November 28, 2012 - 4:06:12 PM
Last modification on : Wednesday, February 20, 2019 - 2:40:30 PM
Long-term archiving on : Saturday, December 17, 2016 - 4:17:06 PM

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Jean-Pierre Flori. Boolean functions, algebraic curves and complex multiplication. General Mathematics [math.GM]. Télécom ParisTech, 2012. English. ⟨NNT : 2012ENST0003⟩. ⟨pastel-00758378⟩

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