Formally Verified Arguments of Knowledge

This library aims to provide a modular and composable framework for formally verifying succinct non-interactive arguments of knowledge (SNARKs). This is done as part of the verified-zkevm effort.

In the first stage of this library's development, we plan to formalize interactive (oracle) reductions (the information-theoretic core of almost all SNARKs today), and prove completeness and soundness for a select list of protocols (see the list of active formalizations below).

For each protocol, we aim to provide:

• An executable specification of the protocol, constructed modularly using our composition and lifting interfaces,
• Proofs of completeness & round-by-round knowledge soundness via generic theorems about composition and lifting.

In the future, we plan to verify functional equivalence of the executable spec (or modifications thereof) for certain protocols (e.g., sum-check, FRI, or WHIR), with the extracted code from Rust implementations of the same protocols.

Library Structure

The core of our library is a mechanized theory of Interactive Oracle Reductions (see OracleReduction):

1. An IOR (called OracleReduction in our formalization) is an interactive protocol between a prover and a verifier to reduce a relation on some public statement & private witness to another relation .
2. The verifier may not see the messages sent by the prover in the clear, but can make oracle queries to these messages using a specified oracle interface;

• For example, one can view the message as a vector, and query entries of that vector. Alternatively, one can view the message as a polynomial, and query for its evaluation at some given points.

3. We can (sequentially) compose multiple IORs with compatible relations together (e.g. and ), preserving the security properties in most cases. We can also lift an IOR from a simple context (i.e., the input & output pairs of statement & witness) into a larger, more complex context, creating a virtual oracle reduction.

• These two operations, sequential composition and lifting, allow us to build existing protocols (e.g. Plonk) from common sub-routines (e.g. zero check, permutation check, quotienting), and derive security of the constructed protocol from that of the components.

4. Once we have specified the desired IOR, the next step is to turn it into a non-interactive arguments of knowledge. This is often achieved in two steps:

• We first apply the (interactive) BCS transform, which materializes the message oracles in an IOR using appropriate (functional) commitment schemes. Roughly speaking, a functional commitment scheme consists of a commitment algorithm, along with an associated opening argument for the correctness of the specified oracle interface.
• The interactive BCS transform then replaces every oracle message from the prover with a commitment to that message, and runs an opening argument for each oracle query the verifier makes to the prover's messages. These opening arguments may be batched.
• We then apply the Fiat-Shamir transform, which collapses interaction via letting the prover derive the verifier's random challenges through querying a hash function (modeled as a random oracle). We will formalize the duplex-sponge version of Fiat-Shamir, which utilizes cryptographic sponges for efficiency in practice.

5. Our formalization follows the emerging view of IORs as the central information-theoretic object underlying modern SNARKs. We also follow the latest understanding & abstraction for the interactive BCS and Fiat-Shamir transformation. See BACKGROUND for an (in-progress) summary of the relevant history.

Using the theory of interactive oracle reductions, we then formalize various proof systems in ProofSystem.

Active Formalizations (last updated: 7 August 2025)

The library is currently in development. Alongside general development of the library's underlying theory, the following cryptographic components are actively being worked on:

• The Sum-Check Protocol
• Spartan
• Merkle Trees
• FRI and coding theory pre-requisites
• STIR and WHIR
• Binius

VCV-io, ArkLib's main dependency alongside mathlib is also being developed in parallel. We are also starting work on the Bluebell probabilistic program logic in (our fork of) iris-lean.

Roadmap & Contributing

We welcome outside contributions to the library! Please see CONTRIBUTING and, the list of issues for immediate tasks, and the ROADMAP for a list of desired contributions.

If you're interested in working on any of the items mentioned in the list of issues or the roadmap, please see verified-zkevm.org, contact the authors, or open a new issue.

Release Schedule

New releases are planned around the middle of each month, in line with the Lean and mathlib release cycles.