CNF-FSS and its Applications

Function Secret Sharing (FSS), introduced by Boyle, Gilboa and Ishai [BGI15], extends the classical notion of secret-sharing a value to secret sharing a function. Namely, for a secret function f (from a class F), FSS provides a sharing of f whereby succinct shares (“keys”) are distributed to a set of parties, so that later the parties can non-interactively compute an additive sharing of f(x), for any input x in the domain of f . Previous work on FSS concentrated mostly on the two-party case, where highly efficient schemes are obtained for some simple, yet extremely useful, classes F (in particular, FSS for the class of point functions, a task referred to as DPF – Distributed Point Functions [GI14,BGI15]). In this paper, we concentrate on the multi-party case, with p ≥ 3 parties and t-security (1 ≤ t < p). First, we introduce the notion of CNF-DPF (or, more generally, CNF-FSS), where the scheme uses the CNF version of secret sharing (rather than additive sharing) to share each value f(x). We then demonstrate the utility of CNF-DPF by providing several applications. Our main result shows how CNF-DPF can be used to achieve substantial asymptotic improvement in communication complexity when using it as a building block for constructing standard (t, p)-DPF protocols that tolerate t > 1 (semi-honest) corruptions. For example, we build a 2-out-of-5 secure (standard) DPF scheme of communication complexity O(N), where N is the domain size of f (compared with the current best-known of O(N) for (2, 5)-DPF). More generally, with p > dt parties, we give a (t, p)-DPF whose complexity grows as O(N) (rather than O( √ N) that follows from the (p − 1, p)-DPF scheme of [BGI15]). We also present a 1-out-of-3 secure CNF-DPF scheme, in which each party holds two of the three keys, with poly-logarithmic communication complexity. These results have immediate implications to scenarios where (multi-server) DPF was shown to be applicable. For example, we show how to use such a scheme to obtain asymptotic improvement (O(log N) versus O( √ N)) in communication complexity over the 3-party protocol of [BKKO20].