Multivariate basic function secret sharing from oblivious transfer

Function secret sharing (FSS) is a secret sharing technique for functions in a specific function class, mainly including distributed point function (DPF) and distributed comparison function (DCF). As an important basis for function secret sharing, DPF and DCF are the foundation for the extension of this technique to other more general and complex function classes. However, the function classes corresponding to the current DPF and DCF schemes are almost all unary function classes, and there is no efficient construction for multivariate function classes. The applications of FSS can be extended with the development of a multivariate scheme, e.g., a multi-keyword private information retrieval scheme can be constructed. To solve this problem, this paper presents a binary DCF scheme based on the “two-layer binary tree” structure. In a binary tree structure, each node computes the seed of its child nodes based on its own seed. The key technique is to realize the transition transfer of seeds by using oblivious transfer, to connect two unary structures. Theoretical analysis and experimental results show that our binary scheme changes from single-round communication in the original definition to multiround communication, and has great advantages in communication cost and computation efficiency. For the security parameter λ and input length n, the key size is reduced from to O(λn²) to O(λn) In addition, we explore the extensions and applications of the above method. In the batch computation, this paper uses oblivious transfer (OT) extension to realize the one-time transmission of multiple pairs of seeds and optimize its communication efficiency. By extending the structure from “two-layer” to “multi-layer”, a secret sharing scheme of multivariate mixed basic function is proposed based on the serial thought. Furthermore, by employing the parallel thought, a general 2-layer FSS structure from OT for multivariate mixed basic functions is explored to enhance the efficiency, where the first layer is composed of d parallel binary trees with d representing the input dimension, and the second layer is one binary tree of depth d. And the applications of our schemes in multi-keyword private information retrieval are presented.