Lower Bounds on FSS from Dynamic Data Structures

Abstract

In Function Secret Sharing (FSS), a dealer with a given function f: {0,1}ⁿ → 𝔾 from n bits to a commutative group 𝔾 such that f is in a function class ℱ shares succinct keys with two properties. Evaluating each key separately on a common input x results in additive shares of f(x) and any subset of the keys does not provide information on f. Two-party FSS schemes which are reducible to One-way Functions (OWF) have applications in cryptography, complexity, and in practical data security systems. 
We establish a two-way transformation between a two-party FSS scheme for a function class ℱ, which is black-box reducible to an OWF, or even black-box reducible to a family of Pseudo-Random Functions (PRF) and a dynamic data structure that supports range queries on ℱ. A data structure of this type enables dynamically adding functions to a multiset of functions F ⊆ ℱ, and answering range queries on the output of F, i.e., returning ∑_{f ∈ F} f(x) for a query x. The data structures are defined in one of several models which abstract RAM.
The correspondence together with known lower bounds on the update time and the query time in data structures leads to the first non-trivial lower bounds on FSS schemes which are black-box reducible to PRF. These lower bounds apply to FSS schemes with polynomial key size and include:  
- For ℱ^d_{box}, the class of all functions which assign a constant group element β ∈ 𝔾 to any input in a specified d-dimensional box and 0 to all other inputs: if the key sharing function, Gen, runs in time polynomial in n and the evaluation function is Eval then:  
- If d ≥ 2 and 𝔾 = ℤ₂ then Eval’s running time is Ω ((n^{3/2})/(log³ n)). 
- If d ≥ 2 and 𝔾 is cyclic such that log |𝔾| = (1 + ε) n then Eval’s running time is Ω ((n/(log n)) ²). 
- If d > 2 is a constant and further, Gen and Eval correspond to operations on data structures in the Oblivious Group Model (this includes all known FSS from OWF techniques), then the product of Eval’s time and the key size is Ω(n^{d-1}).  
- For ℱ_{mono}, the class of all monomials ax^b ∈ 𝔽_{2ⁿ}[X] such that b ≤ B, assuming n^{ω(1)} ≤ B ≤ 2^{n/4}: if Gen runs in polynomial time, then Eval’s running time is Ω ((n √{log B})/(log² n)).