Circuit Design for AI Tasks
For AI-specific operations, such as verifying the output of a 3-layer Multi-Layer Perceptron (MLP) with 100 neurons per layer using 16-bit fixed-point arithmetic and ReLU activation functions, the circuit complexity is substantial. Current ZK technology makes proving complex neural networks extremely expensive. A realistic assessment indicates that such a network would require millions of constraints when accounting for:
Matrix multiplications dominate circuit complexity, requiring approximately 10^4 constraints for a 100×100 matrix operation with 16-bit precision.
ReLU activation functions contribute approximately 30 constraints per neuron when implemented with efficient range proofs.
Practical implementations use circuit optimization techniques like constraint merging and batch normalization approximation.
Matrix multiplications dominate circuit complexity, requiring approximately 10^4 constraints for a 100×100 matrix operation with 16-bit precision.
ReLU activation functions contribute approximately 30 constraints per neuron when implemented with efficient range proofs.
Practical implementations use circuit optimization techniques like constraint merging and batch normalization approximation.
For larger models, the constraint count grows quadratically with network width and linearly with depth, making ZKP generation computationally infeasible for models beyond ~10^5 parameters without applying model compression or sharding techniques.
We acknowledge that current ZK technology makes proving complex neural networks extremely expensive. While we present theoretical constraint counts, practical implementations face significant challenges, particularly for models beyond basic MLPs. Our research focuses on addressing these fundamental limitations through circuit optimizations and modular approaches.
To address these scalability limitations, we're developing a hierarchical verification approach for larger AI models leveraging Substrate's off-chain worker infrastructure. This approach would partition neural networks along natural boundaries (layers, activation functions) and establish verifiable interfaces between components. Each component would generate independent proofs through a technique we call "commitment chaining," where the output commitment of one component becomes the input commitment of the next. We hypothesize this approach could reduce proof generation complexity from O(n²) to approximately O(n log n), potentially making it feasible to verify significantly larger models through Substrate's parallel off-chain worker execution.
Our research roadmap includes empirical validation of this approach with models in the 10-15M parameter range while maintaining cryptographic security through Substrate's secure execution environment.
For matrix multiplications, we will explore the Strassen algorithm with recursive decomposition, potentially reducing the asymptotic complexity from O(n³) to O(n^2.807) for large matrices.
For ReLU activations, we're investigating range-constraint optimizations using binary decomposition approaches, which could reduce the per-neuron constraint count.
We plan to establish comprehensive benchmarks on reference hardware configurations to quantify exact proof generation times and optimization benefits.
For matrix multiplications, we will explore the Strassen algorithm with recursive decomposition, potentially reducing the asymptotic complexity from O(n³) to O(n^2.807) for large matrices.
For ReLU activations, we're investigating range-constraint optimizations using binary decomposition approaches, which could reduce the per-neuron constraint count.
We plan to establish comprehensive benchmarks on reference hardware configurations to quantify exact proof generation times and optimization benefits.
Proof Systems
zk-SNARKs
Generate compact proofs (288 bytes) with 128-bit security, verified in 2 milliseconds through either EVM pre-compiled contracts or native Substrate verification pallets.
zk-STARKs
Produce larger proofs (100KB) with 256-bit, post-quantum security, requiring no trusted setup, suitable for transparency-critical applications.
zk-SNARKs
Generate compact proofs (288 bytes) with 128-bit security, verified in 2 milliseconds through either EVM pre-compiled contracts or native Substrate verification pallets.
zk-STARKs
Produce larger proofs (100KB) with 256-bit, post-quantum security, requiring no trusted setup, suitable for transparency-critical applications.
The Proof Behind Champions
Those who compete at the edge of human precision now back the technology that defines digital truth.
