SLIDE: Shuffle Shamir Secret Shares Uniformly with Linear Online Communication and Guaranteed Output Delivery
We revisit shuffle protocols for Shamir secret sharing. Existing constructions either produce non-uniform shuffles or incur high communication and round complexity, sometimes exponential in the number of parties. We propose two new shuffle protocols that achieve uniform shuffling with communication complexity $O((k+l)n^2m\log m/\log k)$ for an $m$-by-$l$ matrix shared among $n$ parties, where $k\leq m$ is a tunable parameter. The first protocol is concretely efficient, while the second achieves the best-known $O(nml)$ online communication and $O(n)$ rounds. Experiments show significant improvements in online efficiency and total cost over prior work. Our key technical ingredient is a novel permutation sharing technique that represents permutations using smaller permutation matrices, making their application significantly more efficient. The first protocol applies independent secret permutations sequentially, while the second builds on shuffle correlation to achieve optimal online complexity. We further extend shuffle correlation to support guaranteed output delivery with linear online communication, yielding SLIDE, the first protocol to achieve both $O(nml)$ online communication and guaranteed output delivery. Our constructions rely only on basic Shamir secret sharing over any field of size greater than $n$. As shuffling is a fundamental primitive for MPC tasks such as sorting and oblivious data structures, our results enable more efficient and scalable secure computation in practice.