Function Secret Sharing for Mixed-Mode and Fixed-Point Secure Computation

Recently Boyle et al. (TCC 2019) proposed a new approach for secure computation in the {\em preprocessing model} building on {\em function secret sharing} (FSS). This approach can be used to realize any circuit containing gates that admit efficient FSS schemes. In this work, we make the following three technical contributions:

{\bf Improved Key Size.} The complexity of the preprocessing phase directly depends on the FSS key size. We improve the size of FSS keys for several existing FSS constructions through two important steps. First, we present a roughly $4\times$ reduction in FSS key size for the Distributed Comparison Function (DCF), i.e. ($f_\alpha(x) = \beta$ for all $x < \alpha$ and $0$, otherwise). Second, prior FSS schemes for many important function classes are obtained via reductions to multiple instances of DCF; for example, 2 instances for interval containment and $2m$ for splines with $m$ pieces. We significantly improve these reductions for public intervals and obtain {\em optimal} FSS schemes, i.e., through a {\em single instance of DCF}, thereby reducing the key sizes by up to $6-22\times$ for commonly used functions in mixed-mode secure computation such as ReLU and sigmoid.

{\bf FSS for New Function Families.} We present the first constructions of FSS schemes for arithmetic and logical right shift, as well as for bit-decomposition, where the output bits must be secret shared in a larger ring. These functions are crucial for many applications such as fixed-point arithmetic and machine learning.

{\bf FSS for Fixed-Point Arithmetic and Barrier.} One of the important functions in the realization of secure fixed-point arithmetic is that of multiply-then-truncate. While our work shows how to obtain a construction for this function in 2 rounds using sequential calls to FSS schemes for multiply and shift, we demonstrate a barrier towards improving this via FSS beyond what we achieve. Specifically, we show that a 1-round solution would require settling a major open problem in the area of FSS: namely, building an FSS for the class of bit-conjunction functions based on only symmetric-key cryptographic assumptions.