Some Arithmetic Problems In Finite Fields

This thesis investigates some arithmetic problems with cryptographical background in finite fields and and consists of four chapters.In Chapter One, we improve and generalize some estimates on the number of rational points in algebraic varieties over finite fields, including Chevalley-Warning's theorem and Ax-Katz's theorem, and Wan's theorem. For example, by introducing the concept of "Φ—transformation", we extend a reduction formula for simple diagonal equations to more general cases; by applying Adolphson-Sperber's Newton polyhedron technique and exponential sums, we generalize Wan's theorem along two distinct directions.In Chapter Two, we provide the explicit formulae of the number of zeros of some classes of special hypersurfaces over finite fields. By introducing the concept of "GCD-connected set", we obtain a series of new results on the diagonal equations. We discover that S-equations are important in studying the number of zeros of equations just like the irreducible polynomials in studying the decomposition of polynomials. We also find a short proof to some theorems of Wang and Sun on the ladder equations.In Chapter Three, we discuss the bases over finite fields. By introducing the concept of "k—th multiplicative table", a new characterization of the dual bases over finite fields is given. Together with two enumeration theorems, we obtain a brief proof to several known results.In Chapter Four, we present how to factorize integers and design RSA-type public key cryptosystems by applying the groups formed by the rational points of conic and elliptic curves over finite fields, from which the general laws are concluded. In particular, we point out that RSA-type public key cryptosystems may resist the classical Wiener's attack by choosing suitable parameters.As for the cryptographical background of these problems, refer to the subsequent "Preface" .