Another Look at Extraction and Randomization of Groth's zk-SNARK

Abstract

Due to the simplicity and performance of zk-SNARKs they are widely used in real-world cryptographic protocols, including blockchain and smart contract systems. Simulation Extractability (SE) is a necessary security property for a NIZK argument to achieve Universal Composability (UC), a common requirement for such protocols. Most of the works that investigate SE focus on its strong variant which implies proof non-malleability. In this work we investigate a relaxed weaker notion, that allows proof randomization, while guaranteeing statement non-malleability, which we argue to be a more natural security property. First, we show that it is already achievable by Grothl6, arguably the most efficient and widely deployed SNARK nowadays. Second, we show that because of this, Grothl6 can be efficiently transformed into a black-box weakly SE NIZK, which is sufficient for UC protocols.

To support the second claim, we present and compare two practical constructions, both of which strike different performance tradeoffs:

Int-Grothl6 makes use of a known transformation that encrypts the witness inside the SNARK circuit. We instantiate this transformation with an efficient SNARK-friendly encryption scheme.

Ext-Groth16 is based on the SAVER encryption scheme (Lee et al.) that plugs the encrypted witness directly into the verification equation, externally to the circuit. We prove that Ext-Groth16 is black-box weakly SE and, contrary to Int-Groth16, that its proofs are fully randomiz able.