Broadcast Secret-Sharing, Bounds and Applications

Abstract

Consider a sender S and a group of n recipients. S holds a secret message m of length l bits and the goal is to allow S to create a secret sharing of m with privacy threshold t among the recipients, by broadcasting a single message c to the recipients. Our goal is to do this with information theoretic security in a model with a simple form of correlated randomness. Namely, for each subset A of recipients of size q, S may share a random key with all recipients in A. (The keys shared with different subsets A must be independent.) We call this Broadcast Secret-Sharing (BSS) with parameters l, n, t and q. Our main question is: how large must c be, as a function of the parameters? We show that n−t q l is a lower bound, and we show an upper bound of ( n(t+1) q+t − t)l, matching the lower bound whenever t = 0, or when q = 1 or n − t. When q = n − t, the size of c is exactly l which is clearly minimal. The protocol demonstrating the upper bound in this case requires S to share a key with every subset of size n − t. We show that this overhead cannot be avoided when c has minimal size. We also show that if access is additionally given to an idealized PRG, the lower bound on ciphertext size becomes n−t q λ + l − negl(λ) (where λ is the length of the input to the PRG). The upper bound becomes ( n(t+1) q+t − t)λ + l. BSS can be applied directly to secret-key threshold encryption. We can also consider a setting where the correlated randomness is generated using computationally secure and non-interactive key exchange, where we assume that each recipient has an (independently generated) public key for this purpose. In this model, any protocol for non-interactive secret sharing becomes an ad hoc threshold encryption (ATE) scheme, which is a threshold encryption scheme with no trusted setup beyond a PKI. Our upper bounds imply new ATE schemes, and our lower bound becomes a lower bound on the ciphertext size in any ATE scheme that uses a key exchange functionality and no other cryptographic primitives. 2012 ACM Subject Classification Security and privacy → Information-theoretic techniques