A new class of three-weight linear codes from weakly regular plateaued functions

Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes.Comment: The Extended Abstract of this work was submitted to WCC-2017 (the Tenth International Workshop on Coding and Cryptography

Linear codes with few weights from non-weakly regular plateaued functions

Linear codes with few weights have significant applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. There are a number of methods to construct linear codes, one of which is based on functions. Furthermore, two generic constructions of linear codes from functions called the first and the second generic constructions, have aroused the research interest of scholars. Recently, in \cite{Nian}, Li and Mesnager proposed two open problems: Based on the first and the second generic constructions, respectively, construct linear codes from non-weakly regular plateaued functions and determine their weight distributions. Motivated by these two open problems, in this paper, firstly, based on the first generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions and determine the weight distributions of the constructed codes. Next, based on the second generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions belonging to $\mathcal{NWRF}$ (defined in this paper) and determine the weight distributions of the constructed codes. We also give the punctured codes of these codes obtained based on the second generic construction and determine their weight distributions. Meanwhile, we obtain some optimal and almost optimal linear codes. Besides, by the Ashikhmin-Barg condition, we have that the constructed codes are minimal for almost all cases and obtain some secret sharing schemes with nice access structures based on their dual codes.Comment: 52 pages, 34 table

Security in Locally Repairable Storage

In this paper we extend the notion of {\em locally repairable} codes to {\em secret sharing} schemes. The main problem that we consider is to find optimal ways to distribute shares of a secret among a set of storage-nodes (participants) such that the content of each node (share) can be recovered by using contents of only few other nodes, and at the same time the secret can be reconstructed by only some allowable subsets of nodes. As a special case, an eavesdropper observing some set of specific nodes (such as less than certain number of nodes) does not get any information. In other words, we propose to study a locally repairable distributed storage system that is secure against a {\em passive eavesdropper} that can observe some subsets of nodes. We provide a number of results related to such systems including upper-bounds and achievability results on the number of bits that can be securely stored with these constraints.Comment: This paper has been accepted for publication in IEEE Transactions of Information Theor

Quantum secret sharing with qudit graph states

We present a unified formalism for threshold quantum secret sharing using graph states of systems with prime dimension. We construct protocols for three varieties of secret sharing: with classical and quantum secrets shared between parties over both classical and quantum channels.Comment: 13 pages, 12 figures. v2: Corrected to reflect imperfections of (n,n) QQ protocol. Also changed notation from $(n,m)$ to $(k,n)$, corrected typos, updated references, shortened introduction. v3: Updated acknowledgement