Polynomial Commitments

Definition ∞ Polynomial commitments are cryptographic techniques that allow a party to commit to a polynomial function in a way that enables efficient verification of properties about that polynomial. They are used to prove that a computation was performed correctly without revealing the entire computation or its inputs. This technology is foundational for advanced cryptographic primitives like zero-knowledge proofs. It offers a way to verify computations in a trustless manner.
Context ∞ Polynomial commitments are a critical component in the advancement of zero-knowledge proof systems, particularly for scalability solutions in blockchain. Current discussions focus on their application in zk-rollups and other privacy-preserving technologies. Debates often address the efficiency, security, and proof generation time of different commitment schemes, such as KZG commitments. Future developments are expected to refine these schemes to achieve greater computational efficiency and broader applicability in decentralized systems.

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→Light Client Security

→Post-Quantum Cryptography

→Transparent Setup

FRI Proximity Tests Enable Transparent Logarithmic Data Availability Sampling

FRI-based Data Availability Sampling provides a transparent, post-quantum path to scalable light client verification with logarithmic communication overhead.

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→Decentralized

→Sublinear Space Complexity

→Proof System Design

Sublinear Zero-Knowledge Proofs Democratize Verifiable Computation and Privacy

Sublinear memory scaling for ZKPs breaks the computation size bottleneck, enabling universal verifiable privacy on resource-constrained devices.

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→State Burden Reduction

→Cryptographic Witness Size

→State Commitment Scheme

Verkle Trees Enable Practical Stateless Clients via Polynomial Commitments

Benchmarking Verkle Trees against SNARK-based Merkle structures proves vector commitments are the practical path to reducing state burden and fortifying decentralization.

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→Erasure Coding

→Polynomial Commitments

→Succinct Verification

Polynomial Commitments Secure Erasure Codes for Scalable Data Availability Sampling

Cryptographically-secured erasure codes enable light clients to verify data availability by sampling, resolving the scalability bottleneck for modular architectures.

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→Polynomial Commitments

→Succinct Arguments

→Arithmetic Circuit

Transparent Succinct Proofs Eliminate Trusted Setup and Large Proof Size

A novel Vector Hash Commitment achieves constant-size, transparent proofs, resolving the critical trade-off between ZK-SNARK succinctness and ZK-STARK setup-free security.

→Cryptographic Primitive

→Polynomial Evaluation

→Polynomial Commitments

Vector Commitments Enable Statelessness with Compact Verkle Trees

Vector commitments replace hash-based state structures, fundamentally enabling stateless clients by generating constant-sized cryptographic proofs.

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→Generalized Commitment Scheme

→Cryptographic Primitive

→Verifiable Computation

Formalizing Data Availability Sampling as a New Cryptographic Commitment Primitive

Researchers formalize Data Availability Sampling as a cryptographic primitive, introducing a new commitment scheme that rigorously secures light client verification.

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→Reed-Solomon Code

→Erasure Codes

→Computational Efficiency

FRIDA: FRI-based Data Availability Sampling without Trusted Setup

Leverages a novel property of the FRI proof system to construct a trustless, efficient data availability sampling scheme for modular blockchains.

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→Universal Setup

→Trusted Setup

→Constant Verifier Time

Poly-Universal Proofs Achieve Universal Setup and Updatable Security

This new polynomial commitment scheme decouples proof generation from circuit structure, enabling a single, secure, and continuously updatable universal setup.

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→Data Integrity

→Verifiable Computation

→Vector Commitments

Vector Commitments Enable Modular Blockchain Scalability and Asynchronous Security

A new Probabilistically Verifiable Vector Commitment scheme secures Data Availability Sampling, decoupling execution from data and enabling massive asynchronous scalability.