Lattice-Based Polynomial Commitment Schemes
This document provides a comprehensive comparison of lattice-based polynomial commitment schemes used in cryptographic protocols.
Overview Table ¶ Scheme Year Security Assumption Commitment Type Opening Size Verification Time Key Features Ajtai 1996/2008 SIS Basic Vector O(n) O(n) First lattice commitment BDLOP18 2018 SIS/LWE Polynomial O(log d) O(log d) Merkle tree structure Latern 2022 Module-SIS Polynomial O(1) O(1) Constant size openings Slap 2022 Ring-SIS Polynomial O(log d) O(log d) Structured lattices Labrador 2023 Module-LWE Polynomial O(√d) O(√d) Square-root complexity Greyhound 2023 NTRU Polynomial O(log d) O(log d) NTRU-based efficiency HyperWolf 2024 Hypercube-SIS Multivariate O(log^k d) O(log^k d) Hypercube structure
Detailed Technical Comparison ¶ Security Foundations ¶ Scheme Hard Problem Lattice Structure Dimension Modulus Ring/Module Ajtai SIS_{n,m,q,β} Integer Lattice n×m Prime q Z^n BDLOP18 SIS_{n,m,q,β} + LWE Integer Lattice n×m Prime q Z^n Latern Module-SIS_{d,n,q,β} Module Lattice rank d, dim n Prime q R^d, R=Z[X]/(X^n+1) Slap Ring-SIS_{f,q,β} Ring Lattice deg(f) Prime q Z[X]/(f(X)) Labrador Module-LWE_{d,n,q,χ} Module Lattice rank d, dim n Prime q R^d, R=Z[X]/(X^n+1) Greyhound NTRU_{n,q} NTRU Lattice n Prime q Z[X]/(X^n-1) HyperWolf Hypercube-SIS Hypercube Lattice 2^k dimensions Prime q Tensor products
Polynomial Commitment Properties ¶ Scheme Max Degree Batch Opening Aggregation Preprocessing Transparent Setup Ajtai N/A ❌ ❌ None ✅ BDLOP18 2^k ✅ Partial Minimal ✅ Latern Unbounded ✅ ✅ Module ops ✅ Slap 2^k ✅ ✅ Ring FFT ✅ Labrador 2^k ✅ ✅ Module arithmetic ✅ Greyhound 2^k ✅ ✅ NTRU operations ✅ HyperWolf 2^{k×ℓ} ✅ ✅ Hypercube FFT ✅
Scheme Commit Time Open Time Verify Time Commit Size Opening Size CRS Size Ajtai O(n²) O(n) O(n) O(n log q) O(n log q) O(n²) BDLOP18 O(d log d) O(log² d) O(log² d) O(log q) O(log d · log q) O(d) Latern O(d log d) O(log d) O(log d) O(d log q) O(log q) O(d) Slap O(d log d) O(log d) O(log d) O(log q) O(log d · log q) O(d) Labrador O(d log d) O(√d log d) O(√d) O(log q) O(√d log q) O(d) Greyhound O(d) O(log d) O(log d) O(log q) O(log d · log q) O(d) HyperWolf O(d log^k d) O(log^k d) O(log^k d) O(log q) O(log^k d · log q) O(d)
Concrete Security Parameters (128-bit security) ¶ Scheme n/d q β Commitment Size (bytes) Opening Size (bytes) Setup Size (MB) Ajtai 1024 2³¹-1 2¹⁰ 4096 4096 16 BDLOP18 2048 2³¹-1 2¹⁰ 32 320 8 Latern 1024 2³¹-1 2⁹ 128 32 4 Slap 2048 2³¹-1 2¹⁰ 32 256 8 Labrador 1024 2³¹-1 2⁹ 32 128 4 Greyhound 1024 2³¹-1 2⁸ 32 192 4 HyperWolf 512 2³¹-1 2⁸ 32 160 2
Technical Innovations by Scheme ¶ Ajtai (1996/2008) ¶ Core Idea : First lattice-based commitment using SISInnovation : Established lattice commitment foundationsTechnique : Matrix-vector multiplication over latticesAdvantages : Simple, well-understood, strong security proofLimitations : Linear size, no polynomial structureBDLOP18 (2018) ¶ Core Idea : Polynomial commitments via Merkle trees over latticesInnovation : Bridging lattice commitments to polynomialsTechnique : Recursive commitment structure with binary treesAdvantages : Logarithmic openings, batch verificationLimitations : High constants, complex verificationLatern (2022) ¶ Core Idea : Module lattice polynomial commitmentsInnovation : Constant-size openings using module structureTechnique : Module-SIS with structured evaluationAdvantages : Constant opening size, efficient aggregationLimitations : Module-specific assumptionsSlap (2022) ¶ Core Idea : Ring lattice polynomial commitmentsInnovation : Exploiting ring structure for efficiencyTechnique : Ring-SIS with cyclotomic polynomialsAdvantages : Structured efficiency, good concrete performanceLimitations : Ring-specific security assumptionsLabrador (2023) ¶ Core Idea : Square-root complexity via module techniquesInnovation : Optimal trade-off between opening size and verificationTechnique : Module-LWE with square-root batchingAdvantages : Balanced complexity, practical efficiencyLimitations : Complex parameter selectionGreyhound (2023) ¶ Core Idea : NTRU-based polynomial commitmentsInnovation : Leveraging NTRU structure for efficiencyTechnique : NTRU lattices with polynomial evaluationAdvantages : Excellent concrete efficiency, simple structureLimitations : NTRU-specific assumptionsHyperWolf (2024) ¶ Core Idea : Hypercube lattice structure for multivariate polynomialsInnovation : Extending to multivariate setting efficientlyTechnique : Tensor product lattices with hypercube structureAdvantages : Multivariate support, scalable complexityLimitations : Complex implementation, new assumptionsApplications and Use Cases ¶ Scheme SNARKs STARKs Vector Oracle Multivariate Batch Proofs Production Ready Ajtai ❌ ❌ ✅ ❌ ❌ Research only BDLOP18 ✅ ✅ ✅ ❌ ✅ Prototype Latern ✅ ✅ ✅ ❌ ✅ Near production Slap ✅ ✅ ✅ ❌ ✅ Prototype Labrador ✅ ✅ ✅ ❌ ✅ Development Greyhound ✅ ✅ ✅ ❌ ✅ Production HyperWolf ✅ ✅ ✅ ✅ ✅ Research
Implementation and Benchmarks ¶ Reference Implementations ¶ Scheme Language Repository Optimization Level Benchmarks Ajtai C/Sage Academic Basic Historical BDLOP18 C++ GitHub Moderate Available Latern Rust GitHub High Comprehensive Slap C++/Rust GitHub Moderate Available Labrador Rust GitHub High In progress Greyhound C++/Rust Commercial Production Comprehensive HyperWolf Rust GitHub Research Limited
Scheme Commit (ms) Open (ms) Verify (ms) Degree 2¹⁶ Degree 2²⁰ BDLOP18 150 45 12 ✅ ❌ Latern 80 8 5 ✅ ✅ Slap 120 25 8 ✅ ✅ Labrador 100 15 6 ✅ ✅ Greyhound 60 12 4 ✅ ✅ HyperWolf 90 20 7 ✅ ✅
Theoretical Advances ¶ Complexity Theory Results ¶ Lower Bounds : Minimum communication for polynomial commitmentsOptimality : Which schemes achieve theoretical limitsTrade-offs : Opening size vs. verification time relationshipsSecurity Analysis ¶ Quantum Resistance : Post-quantum security guaranteesConcrete Security : Gap between theory and practiceParameter Selection : Guidelines for secure parameter choicesFuture Research Directions ¶ Short-term Goals (1-2 years) ¶ Implementation Optimization : Hardware acceleration, vectorizationParameter Optimization : Better concrete security analysisStandardization : Moving towards cryptographic standardsMedium-term Goals (3-5 years) ¶ New Structures : Exploring other algebraic structuresQuantum Analysis : Better understanding of quantum attacksApplications : Integration with major ZK systemsLong-term Vision (5+ years) ¶ Theoretical Breakthroughs : New complexity resultsHardware Integration : Custom silicon for lattice operationsStandards Adoption : Widespread deployment in productionConclusion ¶ The landscape of lattice-based polynomial commitment schemes has evolved rapidly from the foundational Ajtai construction to modern efficient schemes like Greyhound and the cutting-edge multivariate approach of HyperWolf. Each scheme represents different trade-offs between security assumptions, efficiency, and functionality.
Current Recommendations:
Production Use : Greyhound for immediate deploymentResearch : HyperWolf for multivariate applicationsBalance : Labrador for general-purpose applicationsStructured : Slap/Latern for ring-friendly environmentsThe field continues to advance rapidly with new theoretical insights and practical optimizations being developed regularly.