Distributed Verification of Mixing - Local Forking Proofs Model

One of generic techniques to achieve anonymity is to process messages through a batch of cryptographic mixes. In order to guarantee proper execution verifiable mixes are constructed: each mix provides a proof of correctness together with its output. However, if a mix is working on a huge number of messages at a time, the proof itself is huge since it concerns processing all messages. So in practice only a few verifiers would download the proofs and in turn we would have to trust what they are saying.

We consider a different model in which there are many verifiers, but each of them is going to download only a limited number of bits in order to check the mixes. Distributed character of the process ensures effectiveness even if many verifiers are dishonest and do not report irregularities found.

We concern a fully distributed and intuitive verification scheme which we call

local forking proofs

. For each intermediate ciphertext a verifier may ask for a proof that its re-encrypted version is in the output of the mix concerned. The proof shows that the re-encrypted version is within some subset of

k

ciphertexts from the output of the mix, and it can be performed with strong zero-knowledge or algebraic methods. They should work efficiently concerning communication complexity, if

k

is a relatively small constant.

There are many issues concerning stochastic properties of local forking proofs. In this paper we examine just one: we estimate quite precisely how many mixes are required so that if a local proof is provided for each message, then a plaintext hidden in an input message can appear on any position of the final output set.