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

Micheli, Giacomo. Densities over global fields, arithmetic of subfield preserving maps and applications to cryptography. 2015, University of Zurich, Faculty of Science.

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.

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.