Algebraic Methods for Constructing One-way Trapdoor Functions

In this dissertation, we consider an extension of the discrete logarithm problem to the case of a semigroup acting on a finite set: the Semigroup Action Problem (SAP). New protocols and one-way trapdoor functions based on the difficulty of such problems are proposed. Several instances are studied both from a conceptual and cryptographic point of view. We discuss the application of existing generic algorithms to the resolution of an arbitrary SAP. The Pohlig-Hellman reduction leads to the notion of c-simplicity in semirings. Generic square-root attacks lead to semigroups with a negligible portion of invertible elements. After having described the situation when linear algebra over fields can be used, an application of the theory of finite csimple semirings produces an example of SAP where no such known reduction applies. An extension of the Elliptic Curve Discrete Logarithm Problem (ECDLP) is defined using the Frobenius homomorphism of elliptic curves over finite fields. Actions induced by the Chebyshev polynomials are studied in different algebraic structures such as Fq, Z/nZ and Matn(Fq). Those are shown to be equivalent to known hard problems such as FACTORING and DLP in finite fields. Finally, non-associative operations lead to the study of the SAP in Paige loops, i.e., finite simple non-associative Moufang loops. To my parents, and to Sandrine.