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Algorizk

Research

Mathematical research designed to become practical technology.

Algorizk Labs investigates new mathematical structures, proof protocols, formal specifications, and hardware architectures for transparent, post-quantum, and high-performance cryptographic systems.

Public technical publication

Read the Algorizk Research Notes.

The public notes develop the mathematical foundations behind our work on kernel polynomials, folding, evaluation codes, and affine butterfly methods. Established, conditional, computational, and open results are distinguished explicitly.

Read the Research Notes
IACR ePrint 2026/1469 ↗FPGA Sumcheck case study →Public GitHub ↗

Publication boundary

The notes publish selected mathematics and research directions. Confidential client work, source code, implementation-specific intellectual property, and security-sensitive engineering details remain outside the public edition.

Active research programs

Research across representations, protocols, and implementation.

Each program addresses a different layer of the proof-system stack while remaining connected to a common objective: making advanced cryptographic computation more efficient and practical.

01Polynomial systems

Kernel Polynomials

A research program investigating structured representations of multilinear polynomials and their use in efficient, transparent proof systems.

Research directions

  • Structured bases for multilinear polynomials
  • Fast evaluation and transformation algorithms
  • Polynomial commitment constructions
  • Transparent and post-quantum proof systems

Research objective: The objective is to reduce the cost of representing, evaluating, and proving statements about large multilinear polynomials.

02Protocol research

Sumcheck-FRI Connections

Research into mathematical structures that connect multilinear protocols with FRI-style folding and proximity testing.

Research directions

  • FRI-compatible folding structures
  • Proximity gaps and soundness analysis
  • Multilinear-to-univariate representations
  • Post-quantum polynomial commitments

Research objective: This direction explores whether the strengths of Sumcheck and FRI can be combined in simpler, transparent, and hardware-efficient proof systems.

03Hardware acceleration

FPGA-Accelerated Sumcheck

Hardware-oriented research focused on reducing prover bottlenecks in Sumcheck through finite-field and FPGA optimization.

Research directions

  • Hardware-friendly protocol architecture
  • Finite-field arithmetic pipelines
  • Memory and communication optimization
  • FPGA performance and resource analysis

Research objective: The goal is to translate protocol-level structure into measurable reductions in proving time and hardware cost.

Read the FPGA Sumcheck case study
04Formal methods

Formal Verification of Proof Systems

Machine-checked formalization of mathematical structures and correctness-critical components underlying cryptographic protocols.

Research directions

  • Lean 4 formalization of Sumcheck
  • Multilinear-polynomial identities
  • Folding algorithms and correctness
  • Finite-field reasoning

Research objective: The objective is to develop reusable formal foundations that make cryptographic constructions easier to inspect, verify, and implement with confidence.

Explore formal verification

Research method

A recursive path from idea to validated result.

Research progresses through short cycles of mathematical formulation, construction, experimental verification, and refinement.

  1. 01

    Formulate

    Identify the mathematical structure, computational bottleneck, and precise research question.

  2. 02

    Construct

    Develop candidate definitions, algorithms, protocol components, and proof strategies.

  3. 03

    Test

    Use toy examples, symbolic checks, experiments, and prototypes to reject weak ideas quickly.

  4. 04

    Translate

    Convert validated results into technical specifications, implementation plans, or new research milestones.

Industrial relevance

Research connected to real computational constraints.

Theoretical results are evaluated against the requirements of real proof systems: prover cost, memory, communication, transparency, post-quantum security, and hardware efficiency.

  • Zero-knowledge proof systems
  • Polynomial commitment schemes
  • Verifiable computation
  • Post-quantum cryptography
  • High-performance provers
  • FPGA and hardware acceleration
  • Formal verification with Lean 4

Start a conversation

Have a difficult research problem?

Tell us what you are trying to prove, optimize, or implement. We can begin with a focused technical assessment of the problem, risks, and possible research directions.

Screened inquiries

Submit a concise, non-confidential description through the dedicated project inquiry form.

Direct contact information and a secure communication channel can be provided after the initial inquiry has been reviewed.

Submit a project inquiry