ZVER -FHE ZkVERifier

We create and demonstrate a proving system that can run inside FHEnix, an EVM blockchain that is capable of Fully Homomorphic Encryption (FHE). As the FHE acts on the two operations, addition ("+") and multiplication ("*") allowing calculations of both operations on encrypted data, the inverse operations are quite expensive in terms of computational resources (Gas). Exponentiations is quite unreasonable and calculation of a Bilinear Pairing Function is prohibitive. To make reasonable compromise, we arithmetize general computations into polynomial evaluation and execute them on-chain. This allows for small "proofs" to execute and "show" the results instead of proving them by checking commitments and calculations on a pre-calculated quasi-random point using Bilinear Pairing functions.

Looking at the obstacle in calculating the Bilinear Pairing Function, we shall leave the Succinctness of ZK proofs to future improvements, and try to settle for secrecy only.

Now, we can simply play out the calculation and verify its results.

Generally, we could enter the secret inputs into a FHEnix program and let it calculate the final conditions. If we do this, we are sacrificing 2 guarantees:

There is no guarantee that the program will not leak the private data. There is no guarantee that the program will ever finish (the Halting Problem). Instead, we will create a simple finite-state Pushdown Automoaton, in which each operation is either push, multiply or add. At the end, if the result is , the verification passes.