Function secret sharing using fourier basis

Function secret sharing (FSS) scheme, formally introduced by Boyle et al. at EUROCRYPT 2015, is a mechanism that calculates a function f(x) for x∈ { 0, 1 } ⁿ which is shared among p parties, by using distributed functions f_i: { 0, 1 } ⁿ→ G (1 ≤ i≤ p), where G is an Abelian group, while the function f: { 0, 1 } ⁿ→ G is kept secret to the parties. We observe that any function f can be described as a linear combination of the basis functions by regarding the function space as a vector space of dimension 2 ⁿ and give a new framework for FSS schemes based on this observation. Based on the new framework, we introduce a new FSS scheme using the Fourier basis. This method provides efficient computation for a different class of functions (e.g., hard-core predicates of one-way functions), which may be inefficient to compute if we use the standard basis such as point functions. Our FSS scheme based on the Fourier basis is quite simple due to the fact that the Fourier basis is closed under the multiplication, while the previous constructions have to incorporate some complex mechanisms to overcome the difficulty.