Densities over global fields, arithmetic of subfield preserving maps and applications to cryptography

Abstract

Abstract: The first part of the dissertation is devoted to the study of density questions in the case of global fields. For example, we compute the density of coprime m-tuples for the ring of integers of an algebraic number field. This is a generalization of a theorem by Cesaro over the rational integers: the natural density of the set of coprime pairs is 1/ζ(2), where ζ is the Riemann Zeta function. In Chapter 2 a function field version is provided. Chapter 3 deals with subfield pre- serving maps and analyses density questions in that context. In the second part we study some linear spanning sets for linear maps and then we specialize to the finite field case, giving also results on subfield preserving linear polynomials. The third part of the dissertation is devoted to a general construction for multiplicative Knapsack schemes. In particular, using previously developed tools, we show some applications of the construction, which consist of function field variants of the Naccache-Stern Knapsack Scheme, appearing in public key cryptography.