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Mix-nets can be used to shuffle vectors of shared secrets. This operation can be an important building block for solving combinatorial problems where constraints are secret to different participants. A main contribution of this paper is to show how participants in the mix-net can provide Zero-Knowledge proofs to convince each other that they do not tamper with the shuffled secrets, and that inverse permutations are correctly applied. The approach is related to the proof of knowing an isomorphism between large graphs. We also make a detailed review and comparison with rationales and analysis of Chaum's and Merritt's mix-nets. Mix-nets offer only computational security since participants get encrypted versions of all the shares. Information theoretically secure algorithms can be obtained using secure arithmetic circuit evaluation. The arithmetic circuit previously proposed for shuffling a vector of size k was particularly slow. Here we also propose a new arithmetic circuit for performing the operation in O(k^2) multiplications and requiring k-1 shared random numbers with different domains. Another contribution is to provide more efficient arithmetic circuits for combinatorial optimization problems, exploiting recent secure primitives. Examples are shown of how these techniques can be used in the Secure Multi-party Computation (SMC) language [Sil04b]. SMC's procedures for generating uniformly distributed random permutations are also detailed.

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