What is the link/relation, if any, between Zero Knowledge Proof (ZKP) and Homomorphic encryption?

  • 1
    $\begingroup$ ZKP can be used to mitigate or detect attacks against homomorphic-based multi-party protocols. $\endgroup$
    – cypherfox
    Commented Mar 23, 2018 at 1:40
  • $\begingroup$ RH has a ZKP that's not public. It involves a number with is coprime to 2 while also being coprime to all odd numbers. That sounds like it's not a number, but in this system it is. $\endgroup$ Commented Apr 2, 2018 at 3:14
  • $\begingroup$ @RyanMatthew What is RH? How/why does it work, can you include citations, etc. And maybe answer then? $\endgroup$
    – Ella Rose
    Commented Apr 2, 2018 at 3:20
  • $\begingroup$ And what does it mean that a zero knowledge proof is not public? $\endgroup$
    – Maeher
    Commented Apr 2, 2018 at 3:24

1 Answer 1


There are many.

  • Homomorphic encryption implies ZK proofs for NP. This is simply because homomorphic encryption implies one-way functions, which imply ZKP for NP.
  • Homomorphic encryption allows to compile any public-coin zero-knowledge proof into a designated-verifier non-interactive zero-knowledge proof; this was shown in the paper Non-interactive Zero-Knowledge from Homomorphic Encryption.
  • Zero-knowledge proofs are not known to imply anything regarding the existence of homomorphic encryption. In fact, only very strong forms of zero-knowledge proofs were very recently shown to imply public-key encryption.
  • Fully homomorphic encryption can be used to minimize the size of a non-interactive zero-knowledge proof, and reduce it to witness size + polynomial in the security parameter, see this paper.

This is just a sample of the many interplays between the two notions; if you had more specific relations in mind, please clarify.


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