Staircase Codes for Secret Sharing with Optimal Communication and Read Overheads

We study the communication efficient secret sharing (CESS) problem introduced by Huang, Langberg, Kliewer and Bruck. A classical threshold secret sharing scheme randomly encodes a secret into $n$ shares given to $n$ parties, such that any set of at least $t$, $t<n$, parties can reconstruct the secret, and any set of at most $z$, $z<t$, parties cannot obtain any information about the secret. Recently, Huang et al. characterized the achievable minimum communication overhead (CO) necessary for a legitimate user to decode the secret when contacting $d\ge t$ parties and presented explicit code constructions achieving minimum CO for $d=n$. The intuition behind the possible savings on CO is that the user is only interested in decoding the secret and does not have to decode the random keys involved in the encoding process. In this paper, we introduce a new class of linear CESS codes called Staircase Codes over any field $GF(q)$, for any prime power $q>n$. We describe two explicit constructions of Staircase codes that achieve minimum communication and read overheads respectively for a fixed $d$, and universally for all possible values of $d,t\le d\le n$.