Private verification of Sudoku solution
Sudoku solution verification was introduced in the context of atomic Bitcoin transction:
Bitcoincore article -- Financial Crypto 2016 talk -- QED-it meetup
Informally, a chicken-and-egg problem was resolved this way: if solution is sent first then why pay for it, and if payment is made first then nothing prevents from sending just random bytes. It was shown how to create a hashlock transaction such that payment can be claimed with a hash preimage, and that preimages is verified to be a valid encryption key for a valid solution to the given Sudoku puzzle.
This verification idea could be best demonstrated with a few photos that are "worth a thousand words":
Actual solution verification was implemented as producing and verifying a zk-SNARK proof.
An alternative verification is introduced with this project, starting from the idea of polynomial set representation. We expect a circuit of reduced complexity, in terms of multiplication gates count.
Polynomial set representation was introduced in the context of communication-efficient "set reconciliation" problem solved by Minsky, Trachtenberg, and Zippel, 2003:
It was extended into bivariate polynomial representation of graphs, leading to a Schnorr-like proof systems for graph isomorhism and hamiltonicity with "large" non-binary challenges.
In this project, we experiment with a SNARK circuit for set characteristic polynomials as shown on the photos referenced earlier. In particular, we are interested in exact circuit complexity. For other practical purposes, this is a work-in-progress: we need to reorganise by introducing a gadget implementing methods to construct the circuit and assign the witness.
We plan to experiment with SNARK proofs for statements about graphs.