Zero-Knowledge Proofs – Study & Implementations

This repository contains my ongoing work studying and implementing the core mathematical and cryptographic building blocks behind modern zero-knowledge proof systems (SNARKs, PLONK-style systems, and related protocols).

The focus is on first-principles implementations: understanding the algebra and protocols deeply rather than relying on black-box libraries.

The code here is exploratory, educational, and evolves as my understanding improves.

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Repository Structure

Each directory focuses on a specific primitive or protocol commonly used in zero-knowledge systems.

fft/

Implementations of the Fast Fourier Transform over suitable fields and domains.

Used for:

• Polynomial evaluation and interpolation
• Fast polynomial multiplication
• Foundations for quotient polynomials in zk systems

Includes:

• Recursive FFT
• Inverse FFT
• Polynomial multiplication via FFT

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univariate_poly/

Core univariate polynomial operations over finite fields.

Includes:

• Polynomial arithmetic
• Evaluation
• Interpolation
• Utilities reused across multiple protocols

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multivariate_poly/

Multivariate polynomial representations and operations.

Relevant for:

• Multilinear extensions
• Sum-check protocol
• GKR-style protocols

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lagrange interpolation over prime fields/

Standalone implementations of Lagrange interpolation over finite fields.

Used for:

• Polynomial reconstruction from evaluations
• Foundations for commitments and proof systems

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shamir_secret_sharing/

Implementation of Shamir’s Secret Sharing scheme.

Covers:

• Polynomial-based secret sharing
• Threshold reconstruction
• Finite-field arithmetic

Serves as an early bridge between polynomial algebra and cryptographic protocols.

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sum_check/

Implementation of the Sum-Check protocol.

Used in:

• GKR
• Multilinear polynomial verification
• Interactive proof systems

Includes:

• Iterative sum-check rounds
• Polynomial reductions
• Verifier checks

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gkr/

Implementation of the Goldwasser–Kalai–Rothblum (GKR) protocol.

Focuses on:

• Verifying computations over layered arithmetic circuits
• Combining sum-check with polynomial techniques
• Scalable verification of computation

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kzg/

Implementations related to the Kate–Zaverucha–Goldberg (KZG) polynomial commitment scheme.

Includes:

• Univariate KZG commitments
• Multilinear KZG constructions
• Witness computation and verification logic

This module connects polynomial algebra directly to cryptographic commitments and pairings.

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impl simple circuit/

Early experiments with representing and evaluating arithmetic circuits.

Used to:

• Ground abstract protocols in concrete computation
• Prepare for circuit-based proof systems and arithmetization

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Goals of This Repository

• Build zero-knowledge primitives from scratch
• Understand how algebra, polynomials, and cryptography connect
• Explore how FFTs, commitments, and interactive proofs fit together
• Serve as a long-term reference while studying zk systems

This is not a production library.
Correctness, clarity, and understanding take priority over optimization.

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Planned / Future Additions

Possible future work includes:

• FFT-based polynomial division
• Multi-point KZG openings
• PLONK-style quotient polynomial construction
• Lookup arguments
• Constraint systems and arithmetization
• Comparisons with STARK-style constructions

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Notes

• Most modules assume familiarity with finite fields and basic cryptography
• Implementations may change as understanding improves
• Some code intentionally favors clarity over performance