Staircase Codes for Secret Sharing With Optimal Communication and Read Overheads

Abstract

We study the communication efficient secret sharing (CESS) problem. A classical threshold secret sharing scheme encodes a secret into $n$ shares given to $n$ parties, such that any set of at least $t$ , $t , parties can reconstruct the secret, and any set of at most $z$ , $z , colluding parties cannot obtain any information about the secret. A CESS scheme satisfies the previous properties of threshold secret sharing. Moreover, it allows to reconstruct the secret from any set of $d\geq t$ , parties by reading and communicating the minimum amount of information. In this paper, we introduce three explicit constructions of CESS codes called Staircase codes . The first construction achieves optimal communication and read costs for a given $d$ . The second construction achieves optimal costs universally for all possible values of $d$ between $t$ and $n$ . The third construction, which is the most general, achieves optimal costs universally for all values of $d$ in any given set $\Delta \subseteq \{t, {\dots },n\}$ . The introduced Staircase codes can store a secret of maximal size, i.e., equal to $t-z$ shares, and they are all designed over a small finite field $GF(q)$ , for any prime power $q> n$ . However, Staircase codes may require dividing the secret and the shares into many symbols. We also describe how Staircase codes can be used to construct threshold changeable secret sharing with minimum storage cost, i.e., minimum share size.