Pick, Prove, Profit: The NIVC Singularity.

Incrementally Verifiable Computation (IVC) was first introduced by Valiant in his seminal 2008 work. This groundbreaking concept enables recursive proof systems to tackle long proving tasks by breaking them down into incremental steps. While transformative, the original implementations relied on a universal circuit, meaning the same circuit was repeatedly proven at each step. For zkVMs (zero-knowledge Virtual Machines), this approach introduced significant inefficiencies. Most of the circuit remains unused in any given iteration — for example, in a zkVM with 150 opcodes, only one opcode might be invoked at a time. This creates unnecessary overhead, slowing down performance and bloating memory requirements.

Then came Non-Uniform Incrementally Verifiable Computation (NIVC), introduced in 2022 by Kothapalli and Setty in SuperNova. NIVC elegantly addresses this inefficiency by allowing the proving cost to scale with the size of the circuit executed in each step, rather than the size of the full universal circuit. This made NIVC a perfect fit for zkVMs, significantly reducing per-step costs and aligning computational resources with actual usage. Early implementors, including our team at NovaNet, quickly recognized the transformative potential of NIVC for zkVM design.

However, even with NIVC’s optimizations, step-by-step proving with only one opcode at a time could still be slow for complex applications. This time, Kothapalli and Setty pushed the boundaries by developing Nebula, an advanced NIVC scheme tailored to enhance performance specifically for zkVMs. Nebula allows zkVMs to flexibly determine how many opcodes are proven in each step, dramatically improving proving speed and enabling finer control over memory usage. This flexibility extends zkVM compatibility to a broader range of devices, from high-performance servers to low-powered mobile hardware.

NovaNet's zkEngine, a cutting-edge implementation of Nebula, is in the works and open-sourced, further advancing the zkVM ecosystem. But the potential of NIVC extends far beyond zkVMs. In this article, I’ll explore how NIVC can transform not just zkVMs but also broader proving networks. By treating specialized circuits as individual “opcodes” within a larger NIVC framework, we unlock a wealth of possibilities. Some examples of specializes circuits include:

• ZKPs for AI: Circuits specifically designed for efficient zero-knowledge proofs of artificial intelligence or floating point computation.
• FHE and Lattices: Specialized schemes for Fully Homomorphic Encryption (FHE) and advanced cryptographic operations that are arguably quantum resistant.
• User-Focused Apps: Tiny circuits tailored for specific lightweight applications, ensuring privacy and scalability.
• ZKPs for blockchain scaling: Some schemes are really fast for specific problems.
• zkVM with unique strengths: THERE ARE schemes that are really fast for specific problems.

NIVC’s architecture promotes aggregation, enabling these specialized circuits to operate harmoniously within a larger network rather than each needing to find its own way on-chain. It supports standardized methods for handling foreign field arithmetic and can leverage hardware-accelerated provers to boost efficiency where needed. Moreover, the reward mechanism in an NIVC-powered prover network is built entirely on mathematical rigor and succinct verification, in stark contrast to schemes relying on re-execution or proof-of-work mechanisms.

In this article, I’ll delve into these topics and envision how NIVC can serve as a foundation for decentralized, scalable, and efficient proving network. The age of “pick, prove, profit” is here—let’s explore how NovaNet is using NIVC to revolutionize verifiable computation.

Why people spend months getting their verifier to fit into Groth16?

Building a zero-knowledge prover (ZKP) network isn’t just about designing efficient provers — it’s about making the entire proof pipeline work seamlessly, especially when integrating with constrained Layer 1 (L1) blockchains like Ethereum. One of the major bottlenecks in this process is composing multiple proving schemes into a final proof that fits within the constraints of a widely adopted verifier like Groth16.

L1 blockchains like Ethereum are highly constrained in terms of gas costs and computational capacity. Verifiers must be as minimal as possible to keep costs low. Most specialized ZKP scheme start off with expensive verifiers. Later, developers often spend weeks tweaking elements in their proofs to make them "fit" into the Groth16 verifier’s strict resource budget. This should be the work of the prover network rather than individual proving scheme architects. With NIVC this is exactly the case. The NIVC (Nebula) verifier is the only part that needs to be composed for on-chain verification.

Any specialized proving scheme, whether it's a zkSNARK, a STARK, or another system, has its own format and constraints. Integrating them requires transforming verifiers into circuits that can be used in the NIVC prover network. For many such cases this can also be the specialization of the prover network. At the network level we deal with foreign field work and other complications of proof composition transformations. This means that more specialized provers can get into production faster.

What ZK schemes are actually privacy preserving?

Zero-knowledge proofs (ZKPs) are often touted as the cornerstone of privacy in cryptographic systems, but not all ZKP schemes are inherently privacy-preserving. Many foundational schemes focus on succinctness and verifiability, ensuring that a computation is correct without revealing sensitive data. However, achieving true privacy—that is, ensuring no unintended information is leaked—often requires additional tailoring or wrapping.

In practice, many ZKP schemes are initially designed for performance or scalability and only later adapted to incorporate privacy-preserving features. For example, schemes optimized for efficient proving or verification may not initially obscure metadata or auxiliary inputs that could inadvertently compromise privacy. Addressing these gaps often involves building additional cryptographic layers or redesigning the proof structure to ensure zero-knowledge properties.

The rise of zkVMs presents an opportunity to address privacy at a higher level. A NIVC based zkVM (like NovaNet's zkEngine), with its modular and flexible architecture, can integrate specialized circuits designed to enforce privacy. These circuits can compose non-privacy-preserving schemes, ensuring that sensitive data remains confidential throughout the computation. This approach allows for the use of high-performance ZKPs without sacrificing privacy, as the zkVM abstracts and manages the complexities of privacy enforcement.

The updated HyperNova paper added folding for privacy and this has already been implemented. Its zero-knowledge property ensures that sensitive inputs are protected by default, making Nova a strong foundation for zkVMs aiming to support privacy-preserving computations. By leveraging such schemes, zkVMs can integrate a diverse range of ZKPs, each tailored for specific tasks, while maintaining end-to-end privacy guarantees.

As ZKP technology evolves, the ability to modularly enforce privacy at the zkVM level will become increasingly important. It ensures that even non-privacy-preserving schemes can contribute to privacy-critical applications, provided they operate within a system designed to safeguard sensitive information. The future of privacy in ZKPs lies not in expecting every scheme to be inherently private but in creating systems that can universally allow for easy addition of privacy.

I implemented a new scheme, now what?

Every year, countless new proving schemes and zero-knowledge proof (ZKP) innovations are published and implemented. These breakthroughs represent hours of research, design, and optimization, yet many of them never see practical use. Despite their potential, these schemes often remain experimental, sidelined due to challenges in integration, lack of incentives, and the difficulty of finding a place within the fragmented ecosystem of existing proof pipelines and blockchains.

Deploying a new scheme is not straightforward. Integration requires it to align with existing standards and workflows, a task that is both technically complex and time-consuming. Even when implemented, there are few avenues for developers to monetize their innovations. Beyond initial recognition in the form of research papers or open-source contributions, there’s often no sustainable model for rewarding creators. This lack of incentives discourages further development, leaving many promising schemes unused and undervalued.

NIVC (Non-Uniform Incrementally Verifiable Computation) offers a revolutionary solution by creating a framework where innovative proving schemes can thrive. With NIVC, specialized circuits can seamlessly integrate into a larger network without requiring extensive modifications or complex adjustments to fit into traditional pipelines. Instead of struggling for adoption, developers can focus on their core strengths — designing efficient, specialized provers — while the network handles proof composition and compatibility with end-verification systems like Groth16 or PlonK. This decoupling of specialization from composition enables creators to contribute their innovations without the added burden of navigating integration challenges.

Perhaps most significantly, NIVC introduces a model where developers are rewarded for their contributions. By joining the network, each proving scheme becomes a specialized "opcode," tasked with handling specific computations, such as zkAI, cryptographic operations, or lightweight circuits for mobile applications. These circuits earn value every time they are invoked in the network, providing ongoing incentives for creators. NIVC thus transforms the ZKP landscape, making it not just a hub for innovation but a sustainable ecosystem where new schemes don’t just ask, “Now what?” but instead, “What’s next?”

Mathematics over re-execution: a NIVC approach to consensus

Consensus mechanisms are the backbone of decentralized systems, ensuring agreement among participants about the state of the network. Traditional consensus approaches often rely on computationally expensive processes like re-execution, where nodes independently verify the correctness of a computation by repeating it. While effective, this method incurs significant overhead, both in terms of resources and time, and scales poorly as complexity grows. Moreover, the re-execution may not be feasible for resource constrained devices.

NIVC (Non-Uniform Incrementally Verifiable Computation) introduces a fundamentally better way to reach consensus. Instead of relying on re-execution, NIVC employs pure mathematics to achieve succinct, verifiable proofs of various computations. This shift eliminates the need for redundant calculations and reduces the computational load on the network. Provers in an NIVC-powered system produce compact proofs that cryptographically guarantee the correctness of their computations. Verifiers only need to check the proof, a process that is orders of magnitude more efficient than full re-execution.

This mathematical approach has profound implications for scalability and efficiency. By focusing on proofs rather than re-execution, NIVC enables networks to handle more complex computations while maintaining low latency and high throughput. Specialized provers within the network can focus solely on their tasks, knowing that their outputs will seamlessly integrate into a broader consensus mechanism driven by cryptographic validation. This architecture not only optimizes performance but also opens the door for more diverse applications, as the network can accommodate a wide range of computational tasks without burdening verifiers with excessive workloads.

Moreover, the purity of NIVC's mathematical consensus ensures greater security and transparency. The proofs are verifiable by anyone, fostering trust without requiring extensive computation. This also means that they can succinctly verify that they are being rewarded correctly for their contributions. This model contrasts sharply with re-execution-based systems or proof-of-work schemes, which can be resource-intensive and do not ensure succinct verification.

By replacing re-execution with mathematics, NIVC represents a paradigm shift in how consensus is achieved in prover networks. It offers a scalable, efficient, and secure alternative that aligns with the future of decentralized computation, paving the way for faster adoption and innovation.

The path to the NIVC singularity

In the zk ecosystem, the concept of the "lookup singularity" envisions a world where every computational problem can be efficiently expressed and resolved through succinct lookup arguments. It represents a convergence of speed, efficiency, and universality in proof systems. Similarly, the "NIVC Singularity" charts a new frontier where every computation, no matter how specialized or diverse, can seamlessly integrate into a unified prover network. This vision isn’t just theoretical — it’s a practical roadmap for the future of zero-knowledge computation.

NIVC transforms the way we think about proving systems by enabling a modular, composable network of specialized circuits. It fosters an environment where provers focus exclusively on their tasks, and the network handles aggregation, finalization, privacy, and integration. This architecture doesn’t just streamline processes; it creates a feedback loop of innovation. As more specialized circuits join the network, the efficiency, diversity, and utility of the overall system grow, accelerating the pace of adoption and making zero-knowledge technology accessible to new domains.

The NIVC Singularity represents the moment when proving networks reach critical mass, achieving the perfect blend of scalability, efficiency, and incentivization. At this point, computation becomes a collaborative effort, where provers and verifiers work in harmony, powered not by brute force or re-execution but by the elegance of pure mathematics. As proof systems evolve, this singularity will redefine the boundaries of what’s possible with verifiable computation.

In a world shaped by the NIVC Singularity, innovation is rewarded, complexity is conquered, and computation becomes truly universal. The future is no longer about isolated breakthroughs but about building a network where every contribution, no matter how niche, strengthens the whole. The age of “pick, prove, profit” is here, and it’s driven by the potential of NovaNet's NIVC based prover network.