**TL;DR** Building off of existing theory and practical, we propose a novel protocol for efficient and robust decentralized coded computing using binary field SNARK constructions and coding-theoretic techniques. It leverages a tower of binary field extensions to natively capture various data types and utilizes block-level polynomial commitments and PLONKish arithmetization for efficient verification of the computations.

**Background** Coded computing has emerged as a promising approach for injecting redundancy into decentralized computations for robustness against faults and stragglers. However, existing solutions using zero-knowledge proofs (ZKPs) and fully homomorphic encryption (FHE) face challenges in terms of efficiency, flexibility and scalability.

**Proposal** Our protocol synergistically combines state-of-the-art techniques from binary field SNARKs and coding theory:

- Use a tower of binary field extensions \mathbb{F}_2 \subseteq \mathbb{F}_{2^2} \subseteq \mathbb{F}_{2^4} \subseteq \mathbb{F}_{2^8} \subseteq \cdots \subseteq \mathbb{F}_{2^{128}} to efficiently work with various data types
- Apply a block-level polynomial commitment scheme to commit coded boolean data with optimal rate and polylogarithmic proof size
- Adapt PLONKish techniques like product and permutation arguments over the binary fields to support expressive computations
- Introduce a shifting virtual polynomial for efficient rotations of coded data chunks
- Reconcile the different components via an interactive proof system with tower field arithmetic

**Illustration** Consider a multilinear polynomial f \in \mathbb{F}_2[X_1, \ldots, X_d] of degree \leq d. We encode the coefficients block-wise into a vector \vec{f} \in \mathbb{F}_{2^{\lceil \log{d} \rceil}}^{2^d/d} as follows:

- Partition the coefficients into 2^d/d blocks \{\vec{c}_i \in \mathbb{F}_2^d\}_{i=1}^{2^d/d}
- For each block i, evaluate g_i(X) = \sum_{j=0}^{d-1} c_{i,j} X^j at a fixed element \alpha_i \in \mathbb{F}_{2^{\lceil \log{d} \rceil}}
- Define \vec{f} = (g_1(\alpha_1), g_2(\alpha_2), \ldots, g_{2^d/d}(\alpha_{2^d/d}))

To avoid embedding overhead when committing, we directly work with the block-encoded vector \vec{f} which has length O(2^d/d) over the extension field instead of the full coefficient vector over \mathbb{F}_2 of length 2^d.

**Advantages** Our protocol achieves several advantages over prior works:

- Efficient prover times of \widetilde{O}(mN^2) field operations and verifier times of \widetilde{O}(N^2/\rho) field operations for N constraints and rate-\rho encoding, where m is the number of variables per constraint
- Proof sizes of O(N^2 \log k / \rho) bits that compares favorably to FRI-based systems like STARKs and RedShift
- Flexibility to work with multiple binary extension fields and compute natively over \mathbb{F}_2, enabling more compact constraint systems in some cases compared to R1CS
- Short structured reference strings that can be generated transparently without extra setup assumptions

**Applications** This protocol can be applied to efficiently verify computations expressed as boolean or arithmetic circuits in a decentralized setting with redundancy against faults. Potential use cases include:

- Privacy-preserving outsourcing of computations to a network of untrusted workers
- Scalable and robust multi-party computation (MPC) with low online communication
- Transparent and succinct proof systems for general computations with purely algebraic security assumptions

**Conclusion** We presented a high-performance cryptographic protocol for decentralized coded computing by combining binary field SNARKs and coding theory in a novel way. Our construction overcomes challenges in prior works and achieves asymptotic and concrete efficiency for a wide class of computations. This work expands the capabilities of zero-knowledge proof systems and enables exciting applications in privacy-enhancing technologies.