Two Constructions Of Minimal Linear Codes And Their Applications In Secret Sharing Schemes

Boolean functions have important applications in cryptography and coding theory.Some special linear codes can be constructed by using Boolean functions.How to construct linear codes with some special properties is an important question in coding theory.A minimal linear code refers to a linear code in which all codewords are minimal codewords.Boolean functions can be used to construct minimal linear codes over finite field GF(q).Minimal linear codes are widely used in secret sharing schemes and secure multi-party computation in cryptography.Most of the known minimal linear codes satisfy the Ashikhmin-Barg bound,i.e.wtmin/wtmin>(q-1)/q,in which wtmin and wtmax denote the minimum and maximum non-zero weight in the linear code,respectively.The problem of how to construct minimal linear codes breaking the Ashikhmin-Barg bound has only progressed in recent years.This thesis presents a class of linear codes based on Boolean functions over finite fields in odd characteristic and the following results are obtained:(1)The Walsh transformation of Boolean functions is used to analyze the underlying functions in this class of linear codes.The relationship between the weight distribution of this class of linear codes and the Walsh spectrum value of the underlying function is given.A method of using the underlying function to determine the minimality of this class of linear codes and a set of corresponding constraints are obtained by classifying and discussing the parameters of this class of linear codes.Based on this set of constraints,characteristic functions and polynomial functions are used to construct two classes of underlying functions that meet the conditions,and the corresponding minimal linear codes are also obtained.Furthermore,this thesis presents the complete weight distribution of the first class of minimal linear codes and the partial weight distribution of the second class of minimal linear codes.It is proved that these two classes of linear codes are minimal linear codes that can break the Ashikhmin-B arg bound.(2)Two classes of minimal linear codes constructed are used to realize two secret sharing schemes with different characteristics in this thesis.The minimal linear codes constructed in this paper have multiple weight distributions,so the access structure of the realized secret sharing schemes is more complex.Furthermore,we analyze the parameters and usage scenarios of the two secret sharing schemes separately due to the fact that the secret sharing schemes based on minimal linear codes have practical applications.The number of all minimal access sets is given,and some complete minimal access sets are listed.