Logika a kryptografie

Wagner, Vojtěch

Title: Logic and cryptography Author: Bc.Vojtěch Wagner Department: Department of Algebra Supervisor: prof. RNDr. Jan Krajíček, DrSc. Abstract: This work is devoted to a study of a formal method of formalization of cryptographic constructions. It is based on defining a multi-sorted formal logic theory T composed of strings, integers and objects of sort k - k-ary functions. We allow some operations on them, formulate axioms, terms and formulas. We also have a special type of integers called the counting integers. It denotes the number of x from a given interval satisfying formula ϕ(x). It allows us to talk about probabilities and use terms of probability theory. The work first describes this theory and then it brings a formalization of the Goldreich-Levin theorem. The goal of this work is to adapt all needed cryptographic terms into the language of T and then prove the theorem using objects, rules and axioms of T. Presented definitions and principles are ilustrated on examples. The purpose of this work is to show that such theory is sufficiently strong to prove such cryptographic constructions and verify its correctness and security. Keywords: cryptography, protocol verifying, Soundness theorem, formal logic theory, the Goldreich-Levin theorem 1

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