Secure Modulo Zero-Sum Randomness as Cryptographic Resource

We propose a new cryptographic resource, which we call modulo zero-sum randomness, for several cryptographic tasks. The modulo zero-sum randomness $X_1, \ldots, X_m$ is distributed randomness among $m$ parties, where $X_1, \ldots, X_m$ are independent of each other but $\sum X_i =0$ holds. By using modulo zero-sum randomness, we show that multi-party secure computation for some additively homomorphic functions is efficiently realized without the majority honest nor secure communication channels (but public channel). We also construct secret sharing protocols without secure communication channels. Moreover, we consider a new cryptographic task multi-party anonymous authentication, which is realized by modulo zero-sum randomness. Furthermore, we discuss how to generate modulo zero-sum randomness from some information theoretic assumption. Finally, we give a quantum verification protocol of testing the property of modulo zero-sum randomness

A Zero-One Law for Secure Multi-Party Computation with Ternary Outputs (full version)

There are protocols to privately evaluate any function in the honest-but-curious setting assuming that the honest nodes are in majority. For some specific functions, protocols are known which remain secure even without an honest majority. The seminal work by Chor and Kushilevitz [CK91] gave a complete characterization of Boolean functions, showing that each Boolean function either requires an honest majority, or is such that it can be privately evaluated regardless of the number of colluding nodes. The problem of discovering the threshold for secure evaluation of more general functions remains an open problem. Towards a resolution, we provide a complete characterization of the security threshold for functions with three different outputs. Surprisingly, the zero-one law for Boolean functions extends to Z_3, meaning that each function with range Z_3 either requires honest majority or tolerates up to $n$ colluding nodes