What is the Definition of Polynomial Commitment?

Polynomial commitment is a cryptographic primitive that allows a prover to commit to a polynomial in a concise manner. This commitment permits the prover to later open the polynomial at specific points, proving its value without revealing the entire polynomial. It forms a fundamental building block for advanced zero-knowledge proof systems, enabling efficient verification of complex computations. Such schemes ensure data integrity and privacy in various cryptographic protocols.

What is the Context of Polynomial Commitment?

Polynomial commitment schemes are central to ongoing research and development in scalable blockchain solutions, particularly for rollups and other layer-2 protocols that rely on zero-knowledge proofs. Discussions frequently involve comparing different commitment schemes, such as KZG or FRI, based on their efficiency, security assumptions, and proof size. The continuous refinement of these cryptographic tools is vital for enhancing the performance and privacy features of decentralized applications.

Polynomial Commitment ∞ News ∞ Feed 1

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∞Verifiable Computation

∞Post-Quantum Cryptography

∞Succinct Arguments

Phecda: Quantum-Resistant Transparent zkSNARKs for Verifiable Computation

This research introduces Phecda, a groundbreaking framework that constructs quantum-resistant transparent zkSNARKs through novel polynomial commitments and VOLE-in-the-Head arguments, enabling efficient, publicly verifiable computation against quantum threats.

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∞Distributed SNARKs

∞Folding Schemes

∞Circuit Scalability

Distributed SNARKs Achieve Scalable Proof Generation with Novel Folding Schemes

A new distributed SNARK system leverages folding schemes to drastically accelerate proof generation for large circuits, enhancing blockchain scalability.

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∞Proof Systems

∞Polynomial Commitment

∞Computational Integrity

Sublinear Memory Zero-Knowledge Proofs Democratize Verifiable Computation

A novel zero-knowledge proof system achieves sublinear memory scaling, fundamentally enabling privacy-preserving verifiable computation on ubiquitous resource-constrained devices.

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∞Proof Generation Speed

∞Arithmetic Circuit

∞Cryptographic Scalability

Optimal Prover Complexity Unlocks Linear-Time Zero-Knowledge Proof Generation

This breakthrough achieves optimal O(N) prover time for SNARKs, fundamentally solving the quasi-linear bottleneck and enabling practical, scalable verifiable computation.

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∞Data Integrity

∞Data Authentication

∞Dynamic Data Structures

Decoupled Vector Commitments Enable Dynamic Stateless Client Verification

Decoupled Vector Commitments bifurcate state and update history, achieving logarithmic proof size and constant-time verification for dynamic data.

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∞Decentralized Privacy

∞Cryptographic Assumption

∞Succinct Arguments

Lattice Zero-Knowledge Proofs Secure Scalable Blockchains Post-Quantum

Lattice cryptography enables a quantum-secure ZK proof system, future-proofing on-chain privacy and scalability against cryptographic collapse.

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∞Polynomial Commitment

∞Succinct Argument

∞Decentralized Scaling

Generic Folding Scheme Enables Efficient Non-Uniform Verifiable Computation

Protostar introduces a generic folding scheme for special-sound protocols, drastically reducing recursive overhead for complex, non-uniform verifiable computation.

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∞Homomorphic Property

∞Decentralized State

∞Verifiable Computation

Inner-Product Argument Vector Commitments Enable Constant-Time Proof Aggregation

This new Inner-Product Argument Vector Commitment achieves constant-time state verification, fundamentally unlocking truly scalable stateless clients.

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∞Prover Complexity

∞Cryptographic Primitive

∞ZK-SNARKs

Sublinear Memory ZK Proofs Democratize Verifiable Computation

A new space-efficient tree algorithm reduces ZK proof memory complexity from linear to square-root, enabling verifiable computation on all devices.

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∞Quantum Resistance

∞Zero-Knowledge Proofs

∞Succinct Arguments

Efficient Lattice Commitments Secure Post-Quantum Verifiable Computation

Greyhound introduces the first concretely efficient lattice-based polynomial commitment scheme, providing quantum-resistant security for all verifiable computation.

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∞Bitwise Operations

∞Verifiable Computation

∞Range Checks

New Lookup Argument Achieves Optimal Commitment Size for Universal ZK Circuits

Lasso introduces a sparse multilinear polynomial commitment scheme to make non-arithmetic ZK operations linear, unlocking the lookup singularity.

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∞Algebraic Geometry

∞Module Homomorphic

∞Zero-Knowledge Proofs

Lattice-Based Polynomial Commitments Achieve Post-Quantum Succinctness and Sublinear Verification

Greyhound is the first concretely efficient lattice-based polynomial commitment scheme, enabling post-quantum secure zero-knowledge proofs with sublinear verifier time.

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∞Model Performance

∞Succinct Arguments

∞Privacy Preservation

Zero-Knowledge Proof of Training Secures Private Decentralized Federated Learning Consensus

ZKPoT uses zk-SNARKs to verify model performance without revealing local data, achieving robust, scalable, and privacy-preserving decentralized consensus.

∞Algebraic Commitment

∞Recursive Proof System

∞Trustless Scaling

Hyper-Efficient Prover Unlocks Universal Transparent Zero-Knowledge Scaling

This new HyperPlonk scheme achieves linear prover time for universal transparent SNARKs, fundamentally accelerating verifiable computation for all decentralized applications.

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∞Verifiable Computation

∞Proof System

∞Trustless Verification

Logarithmic-Depth Commitments Enable Truly Stateless Blockchain Verification

A new Logarithmic-Depth Merkle-Trie Commitment scheme achieves constant-time verification, enabling light clients to securely validate state without storing it.

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∞Commitment Scheme

∞Sigma Protocol

∞Sublinear Verifier

Lattice-Based Polynomial Commitments Achieve Post-Quantum Succinctness and Efficiency

Greyhound is the first concretely efficient polynomial commitment scheme from standard lattice assumptions, securing ZK-proof systems against future quantum threats.

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∞Opening Consistency

∞Interactive Oracle Proofs

∞Cryptoeconomics

Opening-Consistent IOPs Enable Trustless Erasure Code Commitments

This research introduces Erasure Code Commitments, a new primitive constructed via a novel IOP compiler, solving data availability without a trusted setup or high overhead.

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∞Cryptographic Primitive

∞Sub-Circuit Parallelization

∞Plonk Protocol Optimization

Distributed Zero-Knowledge Proofs Scale Zkrollups with Constant Communication

A distributed Plonk protocol minimizes inter-prover communication to a constant factor, eliminating the zkRollup prover bottleneck and unlocking massive Layer 2 scalability.

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∞Cryptographic Primitive

∞Logarithmic Complexity

∞Trustless Verification

Vector Commitments Enable Sublinear State Verification for Stateless Clients

A new polynomial vector commitment scheme transforms light clients into secure, stateless verifiers, dramatically improving blockchain decentralization and user security.

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∞Approximate Range Proof

∞Zero-Knowledge Proofs

∞Cryptographic Primitives

Post-Quantum Zero-Knowledge Proofs Achieve Shorter, Faster Verification

Lantern introduces a direct polynomial product proof for vector norms, slashing post-quantum ZKP size for practical privacy applications.

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∞Complexity Preserving

∞Verifiable Computation

∞Proof Carrying Data

Universal Zero-Knowledge Proofs Eliminate Program-Specific Trusted Setup

A universal circuit construction for SNARKs decouples the setup from the program logic, establishing a single, secure, and permanent verifiable computation layer.

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∞Private Smart Contracts

∞Distributed Proving

∞Scalable Cryptography

Collaborative zk-SNARKs Enable Private, Decentralized, Scalable Proof Generation

Scalable collaborative zk-SNARKs use MPC to secret-share the witness, simultaneously achieving privacy and 24× faster proof outsourcing.

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∞Two Dimensional Sampling

∞Data Recovery Threshold

∞Verifiable Data

Data Availability Sampling Secures Modular Blockchain Scalability

Modular architecture decouples core functions, using Data Availability Sampling and erasure coding to enable trust-minimized, mass-scale rollups.

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∞Succinct Verification

∞Distributed Consensus

∞Data Availability Sampling

Erasure Code Commitments Secure Data Availability Sampling Consistency

This new cryptographic primitive guarantees a commitment binds to a valid erasure codeword, solving data inconsistency in modular blockchain scaling.

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∞Zero-Knowledge Proofs

∞Transparent Setup

∞Code-Based Cryptography

Lattice-Based Zero-Knowledge Proofs Secure Computation against Quantum Threat

The research introduces quantum-resistant zero-knowledge proof systems leveraging hard lattice problems, ensuring long-term privacy and verifiability for decentralized architectures.

Polynomial Commitment

Definition

Context

Phecda: Quantum-Resistant Transparent zkSNARKs for Verifiable Computation

Distributed SNARKs Achieve Scalable Proof Generation with Novel Folding Schemes

Sublinear Memory Zero-Knowledge Proofs Democratize Verifiable Computation

Optimal Prover Complexity Unlocks Linear-Time Zero-Knowledge Proof Generation

Decoupled Vector Commitments Enable Dynamic Stateless Client Verification

Lattice Zero-Knowledge Proofs Secure Scalable Blockchains Post-Quantum

Generic Folding Scheme Enables Efficient Non-Uniform Verifiable Computation

Inner-Product Argument Vector Commitments Enable Constant-Time Proof Aggregation

Sublinear Memory ZK Proofs Democratize Verifiable Computation

Efficient Lattice Commitments Secure Post-Quantum Verifiable Computation

New Lookup Argument Achieves Optimal Commitment Size for Universal ZK Circuits

Lattice-Based Polynomial Commitments Achieve Post-Quantum Succinctness and Sublinear Verification

Zero-Knowledge Proof of Training Secures Private Decentralized Federated Learning Consensus

Hyper-Efficient Prover Unlocks Universal Transparent Zero-Knowledge Scaling

Logarithmic-Depth Commitments Enable Truly Stateless Blockchain Verification

Lattice-Based Polynomial Commitments Achieve Post-Quantum Succinctness and Efficiency

Opening-Consistent IOPs Enable Trustless Erasure Code Commitments

Distributed Zero-Knowledge Proofs Scale Zkrollups with Constant Communication

Vector Commitments Enable Sublinear State Verification for Stateless Clients

Post-Quantum Zero-Knowledge Proofs Achieve Shorter, Faster Verification

Universal Zero-Knowledge Proofs Eliminate Program-Specific Trusted Setup

Collaborative zk-SNARKs Enable Private, Decentralized, Scalable Proof Generation

Data Availability Sampling Secures Modular Blockchain Scalability

Erasure Code Commitments Secure Data Availability Sampling Consistency

Lattice-Based Zero-Knowledge Proofs Secure Computation against Quantum Threat

Incrypthos

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