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I am writing an exercise in cryptography about anonymous communication. One of the systems I have looked into is the mix net protocol. I want to prove that this is secure. I have seen many different ways of doing this, i.e. using a zero knowledge proof, but I don´t completely understand this.

Is there anyone who has a good explanation of why the mix net is secure?

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Security in mixnets usually means two things. Anonymity and integrity.

For anonymity a single honest mix server suffices. Even if all other mix servers cooperate by releasing their permutations a message cannot be tracked, since the honest one does not publish his permutation. As a result it is lost when it reaches it.

For integrity zero knowledge proofs can be used.

Imagine you have a 2x2 reencryption mixnet with inputs: $C_1 = Enc(m_1,r_1)$ and $C_2 = Enc(m_2,r_2)$.

The mixnet reencrypts the inputs and computes ${C'}_1 = Reenc(C_1)=Enc(m_1,r_1+r_1')$ and ${C'}_2 = Reenc(C_2)=Enc(m_2,r_2+r_2')$.

Then it selects a random bit $b$ and outputs $({C'}_1, {C'}_2)$ if $b=0$ and $({C'}_2, {C'}_1)$ if $b=1$.

In order to prove integrity one must prove that the output is a reencryption of the input without revealing the value of $b$. More specifically the statement $(C'_1 = Reenc(C_1) \wedge C'_2 = Reenc(C_2)) \bigvee (C'_1 = Reenc(C_2) \wedge C'_2 = Reenc(C_1))$ must be proved in zero knowledge.

This can be done using composition of the Chaum - Pedersen Protocol for DLOG equality.

For more details you can check Rivest's course in http://courses.csail.mit.edu/6.897/spring04/materials.html

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