Staircase codes for secret sharing with optimal communication and read overheads

We study the communication efficient secret sharing (CESS) problem. 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, colluding parties cannot obtain any information about the secret. A CESS scheme satisfies the previous properties of threshold secret sharing. In addition, it has minimum communication and read costs when the user contacts d, d ≥ t, shares. The intuition behind the possible savings on these costs 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 two explicit constructions of CESS codes called Staircase Codes. The first construction achieves optimal communication and read costs for a fixed d, d ≥ t. The second construction achieves optimal costs universally for all possible values of d, t ≤ d ≤ n. Both constructions can be designed over a field GF(q), for any prime power q > n.