Several Key Problems In Information Security And Big Data Storage

This thesis mainly considers two problems.The first one focuses on the study of permutation polynomials over finite fields,which have wide applications in cryptography,coding theory and combinatorial design theory.The second one mainly investigates the locally repairable code in data storage,which plays an important role in distributed storage systems.This thesis makes a further research on the problems from the perspective of combinatorics,and uses the related tools such as finite field theory,algebraic number theory and so on.In Chapter 1,we briefly introduce the backgrounds of the problems concerned with this thesis and summarize our contributions towards these topics.In Chapter 2,we investigate the permutation polynomials over finite fields.We prove two conjectures about the permutation trinomials with Niho exponents,which were proposed by Wu et al.,by using the method of treating squares and non-squares separately.Further,we construct two new classes of permutation trinomials by studying the number of solutions to special equations.And we generalize two examples proposed by Kyureghyan et al.to an infinite class.In Chapter 3,we mainly consider complete permutation polynomials and permutation poly-nomials with low differential uniformity.Four classes of monomial complete permutation poly-nomials and one class of trinomial complete permutation polynomials are presented,one of which confirms a conjecture proposed by Wu et al..Further,we make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al..In Chapter 4,we investigate the locally repairable code in distributed storage systems.We mainly focus on the upper bound for the dimension k and constructions of binary linear locally repairable codes.First,we derive an explicit upper bound for the dimension of such codes.Further,based on partial spreads and weakly independent sets,we get some new optimal binary locally repairable codes.In Chapter 5,we briefly introduce other problems considered in the PhD learning phase,such as traceability codes,regenerating codes,maximally recoverable codes.