Locally Reconstructable Non-malleable Secret Sharing
Non-malleable secret sharing (NMSS) schemes, introduced by Goyal and Kumar (STOC 2018), ensure that a secret m can be distributed into shares m1,...,mn (for some n), such that any t (a parameter <= n) shares can be reconstructed to recover the secret m, any t-1 shares doesnt leak information about m and even if the shares that are used for reconstruction are tampered, it is guaranteed that the reconstruction of these tampered shares will either result in the original m or something independent of m. Since their introduction, non-malleable secret sharing schemes sparked a very impressive line of research. In this work, we present a new feature of local reconstructablility in NMSS, which allows reconstruction of any portion of a secret by reading just a few locations of the shares. This is a useful feature, especially when the secret is long or when the shares are stored in a distributed manner on a communication network. In this work, we give a compiler that takes in any non-malleable secret sharing scheme and compiles it into a locally reconstructable non-malleable secret sharing scheme. To secret share a message consisting of k blocks of length r each, our scheme would only require reading r + log k bits (in addition to a few more bits, whose quantity is independent of r and k) from each partys share (of a reconstruction set) to locally reconstruct a single block of the message.