Fourier-based function secret sharing with general access structure

Function secret sharing (FSS) scheme is a mechanism that calculates a function f(x) for 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. Ohsawa et al. in 2017 observed 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 2ⁿ and gave new FSS schemes based on the Fourier basis. All existing FSS schemes are of (p, p)-threshold type. That is, to compute f(x), we have to collect f_i(x) for all the distributed functions. In this paper, as in the secret sharing schemes, we consider FSS schemes with any general access structure. To do this, we observe that Fourier-based FSS schemes by Ohsawa et al. are compatible with linear secret sharing scheme. By incorporating the techniques of linear secret sharing with any general access structure into the Fourier-based FSS schemes, we propose Fourier-based FSS schemes with any general access structure.