Essential Maths for Zero Knowledge Proofs

Welcome to Essential Maths for Zero Knowledge Proofs. In this course, you'll explore the mathematical foundations that underpin ZKPs—from modular arithmetic to elliptic curves—through clear explanations and practical examples. Each module is designed to build your intuition and confidence, whether you're deepening your cryptography knowledge or applying these concepts in real-world protocols. Let's get started!

Essential Zero Knowledge Maths: Introduction

Introduction to the course and the basic concepts of zero knowledge proofs

Key Concepts include:

  • Security on ZK Protocols
  • Common Vulnerabilities
  • Auditing ZK Systems

Link for the accompanying slides here.

Essential Zero Knowledge Maths: Numbers and Terminology

This lesson covers core mathematical structures like modular arithmetic, groups, fields, and generators—key building blocks behind zero-knowledge proofs.

Key Concepts include:

  • Modular arithmetic
  • Groups and fields
  • Cyclic groups and generators
  • Finite fields
  • Multiplicative inverses and field constraints in ZK

Link for the accompanying slides here.

Essential Maths for Zero Knowledge Proofs: Complexity Theory

This lesson explores how we classify problems based on how hard they are to solve or verify, using ideas from complexity theory.

Key Concepts include:

  • Problem classification based on time/memory
  • P vs NP and their relevance to ZK
  • Verifiability vs solvability
  • Big O notation and efficiency
  • Application to SNARKs and STARKs

Link for the accompanying slides here.

Essential Maths for Zero Knowledge Proofs: Elliptic Curves

This lesson introduces elliptic curves—sets of points defined by a specific type of equation—and explains how their structure forms a group. These curves are foundational in cryptography, powering many systems including those used in zero-knowledge proofs.

Including

  • Elliptic curve equation: \( y^2 = x^3 + ax + b \)
  • Points on a curve as a mathematical group
  • Curve operations like point addition
  • Closure and identity (point at infinity)
  • Role of elliptic curves in cryptography and ZK

Link for the accompanying slides here.

Essential Maths for Zero Knowledge Proofs: Polynomials

This lesson explores how polynomials are used to represent information in zero-knowledge proofs. From roots and degrees to the Schwartz-Zippel lemma and interpolation, you’ll see how polynomials make it possible to verify complex claims efficiently—without revealing the underlying data.

This module will cover

  • Structure and degree of a polynomial
  • Polynomial roots and factorizations
  • Schwartz-Zippel lemma for testing equivalence
  • Interpolation and evaluation forms
  • Polynomials as proof objects in ZK systems
  • Role of randomness and commitments in verification

Link for the accompanying slides here.

Essential Maths for Zero Knowledge Proofs: Polynomial Commitment Schemes

This lesson introduces polynomial commitment schemes, which allow provers to commit to large, secret polynomials in a way that’s both succinct and verifiable.

We will investigate

  • Why we don’t send full polynomials in ZK
  • Commitments: succinct, one-way representations
  • Binding and Hiding
  • Random-point evaluation and generalization
  • Commitment types: KZG and FRI

Link for the accompanying slides here.

Essential Maths for Zero Knowledge Proofs: Further Techniques

This lesson introduces two advanced tools used in zero-knowledge proofs: error correction via Reed-Solomon codes and the inner product argument.

Key Concepts include:

  • Reed-Solomon codes for redundancy and recovery
  • Encoding messages as polynomial coefficients
  • Interpolation from noisy or partial evaluations
  • Inner product argument: secure transformation of claims
  • Use of vectors and dot products in proof constraints
  • Applications in Bulletproofs and other proving systems

Link for the accompanying slides here.

Essential Maths for Zero Knowledge Proofs: Outro

In this closing session, we highlight further resources to help you deepen your understanding of zero-knowledge proofs and related topics.

From recursive proofs and zkML to bootcamps and community spaces, we point you toward the next steps in your exploration.

  • GitHub repo with ZKMaths materials
  • Zero-knowledge machine learning (zkML) resources
  • Broader paths in cryptography and Web3 education

Link for the accompanying slides here.