Khalimov G. Theoretical basis of universal hashing on algebraic curves

The aim is to develop a theory of universal hash functions for rational algebraic curves to construct a system demonstrable persistent authentication. The object of study is a message authentication processes in information systems and information technology systems under tight requirements for demonstrable resilience and minimize costs for authentication. Purpose of the study is the main tenets of the theory of universal hashing to construct provably secure message authentication and unconditional in information systems and information technology systems in terms of the offender, to the modification and tampering with messages. Theoretical positions universal hashing on rational functions of algebraic curves that allowed to resolve the basic contradiction of authentication between probability of a collision, expenses of key space and length hashing messages has developed. The method of universal hashing over a functional field of algebraic curves has offered, assessment of the collision's probability of universal hashing as well as asymptotic upper boarder of collisions has obtained. Functional fields on the maximum curves of the first, second and third genus are constructed. A theoretical study of the algebraic curves is conducted, methods of point's quantity calculation of the Fermat and Hurwitz curves has designed, construction methods of the uncommon curve and maximum curves of Hurwitz are developed, and best curves for universal hashing has defined. The method of hash functions calculation based on calculation via multi-parametrical Horner's scheme is developed, the algorithms of hashing on maximum curves within highest first, second and third genus on Fermat and Hurwitz curves, on a curve of Suzuki with a great number of points and the reduced calculation complexity has constructed. Methods of cascade universal hashing, practical recommendations of application of the universal hashing methods on algebraic curves, software of the analysis and construction universal hashing on rational functions of curves are developed.