◈ STDF Basic Sciences Research Grant #50955

NCCAI: Number Theory
and Chaos Algorithms
for Computational & Artificial Intelligence

Number theory and chaos theory power the next generation of secure, efficient, and intelligent computational algorithms. We built NCCAI to develop, demonstrate, and distribute these algorithms openly — from Nile University and Cairo University.

3Predecessor Projects

9Research Tasks

24Month Duration

2Universities

🌐

Algorithm Repository

Browse and filter our full catalog of chaos, number theory, PRNG, and ANN algorithms — classified by domain, type, and hardware availability.

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▶

Interactive Execution

Run algorithms live in your browser. Adjust parameters and view phase-space attractors, time-series plots, and randomness histograms in real time.

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📚

Publications

Search and export citations from our full bibliography. Filter by year, type, domain, or author and download BibTeX or IEEE format.

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🎓

Workshops & Seminars

Access slides, recorded sessions, and materials from NCCAI workshops on chaos, number theory, and AI at Nile and Cairo Universities.

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Research Domains

🌀 Chaos Theory

🔢 Number Theory

🤖 Artificial Intelligence

🎲 PRNGs & Cryptography

⚡ FPGA Hardware

📊 Algorithm Evaluation

Project Partners

STDF Egypt

Nile University

Cairo University

Project Overview

About NCCAI

Project Description

The NCCAI platform is a publicly accessible web application functioning as a knowledge-sharing and algorithm demonstration repository. It showcases the interaction between number theory, chaos theory, and computational/artificial intelligence algorithms.

Developed under STDF Basic Sciences Research Grant #50955, the platform serves as the technical implementation of Task 2 in the NCCAI project Gantt chart.

Funding & Timeline

Grant: STDF #50955 — Basic Sciences Research

Start Date: January 2025

Duration: 24 months

PI: Prof. Ahmed Gomaa Ahmed Radwan

Institutions: Nile University, Cairo University

NILE UNIVERSITY CAIRO UNIVERSITY STDF EGYPT

Predecessor Projects

STDF #4276 — Nonlinear Dynamics & Chaos Cryptosystem (2013–2015)
Cairo Univ. #16-120 — Software & Hardware Speech Encryption (2017–2018)
STDF #45631 — Information Security Using Number Theory & Chaotic Systems (2022–2024)

Nine Research Tasks

Literature Review
Online Platform for Algorithms
Chaos Theory in CI Algorithm Design
Chaotic Time Series Modeling (ANN/PINN)
Number Theory in CI Algorithm Design
PRNGs & Cryptographic Algorithms
Algorithm Evaluation & Benchmarking
Hardware Realizations (FPGA)
Workshop Organization

Research Team

Meet the NCCAI Team

NCCAI Research

Research Areas

Six research domains at the intersection of number theory, chaos theory, and artificial intelligence — select an area to explore.

🌀

Chaos Theory in Machine Learning

Lyapunov exponents, phase space reconstruction, strange attractors, and chaos-enhanced neural architectures.

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🧠

ANN for Time Series & Chaotic Dynamics

LSTM, Echo State Networks, reservoir computing, and statistical models for chaotic time series forecasting.

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⚛️

Physics-Informed Neural Networks

Embedding PDEs into neural training for stable, generalizable prediction of chaotic dynamical systems.

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🔢

Number Theory & Residue Number System

CRT-based modular arithmetic, RNS AI accelerators, and photonic implementations for neural networks.

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🔐

Homomorphic Encryption

FHE schemes BFV, BGV, CKKS, TFHE — privacy-preserving computation on encrypted data for ML.

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⚡

FPGA Hardware Acceleration for AI

Systolic arrays, dataflow pipelines, FINN/hls4ml toolflows — 2,156+ images/sec on Zynq UltraScale+.

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Chaos Theory in Machine Learning

Integrating chaos-based concepts into ML pipelines — from data preprocessing and Lyapunov feature extraction to chaos-enhanced neural architectures and metaheuristic optimization.

Overview

Many real-world datasets originate from underlying dynamical systems, and learning models can exhibit chaotic behavior during training. This research area explores how chaos-based concepts can be incorporated at every stage of the ML pipeline — from raw data through to model optimization.

Chaos Theory Fundamentals

Core concepts include sensitive dependence on initial conditions, topological mixing, and dense periodic orbits. Analysis tools — Lyapunov exponents, Poincaré sections, correlation dimension — quantify chaotic behavior. Discrete maps (Logistic, Henon), continuous attractors (Lorenz, Rossler, Chen), and hyperchaotic systems are compared across key dynamical properties.

Data Representation & Preprocessing

Feature extraction derives dynamic features using the Lyapunov Exponent and Phase Space Reconstruction via Takens' embedding theorem. Data transformation uses chaotic systems: discrete maps via topological transitivity, and continuous attractors whose outputs are combined with input data to expose underlying chaotic characteristics to the learning model.

Chaos in Neural Architectures & Optimization

Chaotic counterparts replace conventional architectural modules, or introduce noise injection to improve generalization. During training, Lyapunov exponents guide learning rate scheduling and regularization. Chaos-based metaheuristics (chaotic PSO, chaotic GA) improve hyperparameter search, while chaotic system outputs can initialize or guide weight updates.

ANN-Based Architectures for
Time Series & Chaotic Dynamics

From classical ARIMA models to LSTM, Echo State Networks, and reservoir computing — a systematic survey of neural forecasting for chaotic dynamical systems.

Overview

ANN models have grown in importance for chaotic time series forecasting due to their ability to model complex nonlinear dynamics and sensitivity to initial conditions. This area systematically examines how forecasting techniques have evolved from traditional statistical models to modern neural architectures, supported by real-world wind-speed SCADA experiments.

Classical Statistical Models

Linear models (ARIMA, AR, MA, ARMA, SARIMA) and state-space filtering techniques are reviewed alongside nonlinear extensions (SETAR, GARCH, Volterra series). Their benefits and drawbacks when applied to chaotic temporal data are analyzed through a comparative experiment on real-world wind-speed data quantifying how each model handles temporal complexity and nonlinearity.

Neural Architectures for Time Series

Fundamental architectures — MLP, CNN temporal models, LSTM, Bi-LSTM, GRU — are covered alongside hybrid approaches. Reservoir computing paradigms including Echo State Networks (ESN), Liquid State Machines (LSM), and structured ESN variants preserve expressive temporal representations while dramatically reducing training complexity compared to fully trained RNNs.

Modeling Chaotic Dynamics

Benchmark results on Lorenz, Rossler, Mackey-Glass, and other canonical chaotic systems demonstrate that reservoir-based approaches consistently excel at modeling chaotic dynamics due to their inherent temporal memory and nonlinear projection capabilities — outperforming classical and even some deep learning baselines on long-horizon chaotic forecasting tasks.

Physics-Informed
Neural Networks

Embedding physical laws — PDEs, boundary conditions, initial conditions — directly into neural training for stable, generalizable prediction of chaotic dynamical systems.

Overview

Chaotic dynamical systems exhibit extreme sensitivity to initial conditions, making long-term prediction a fundamental challenge. Physics-Informed Neural Networks (PINNs) offer a promising framework by integrating data-driven learning with physical constraints — the loss function embeds Partial Differential Equations alongside data fidelity, enforcing physical laws during training.

PINN Foundations

The total loss is L = L_data + λ·L_physics, where L_physics measures the PDE residual at collocation points. PINNs automatically embed PDEs, initial conditions, and boundary conditions — the network learns system dynamics while adhering to established physical laws. The Burger's equation serves as the canonical benchmark for training performance evaluation.

Chaos in Dynamical Systems

Core chaotic properties are reviewed: nonlinear dynamics, sensitive dependence on initial conditions, and the Lyapunov exponent as a quantitative measure of chaos. Benchmark systems (Lorenz, Van der Pol, Duffing, Kuramoto-Sivashinsky) serve as testbeds. PI-LSTM and PI-ESN hybrid architectures are evaluated on their ability to respect physical constraints while learning chaotic trajectories.

Challenges & Future Directions

Key challenges include training instability, long-term prediction error accumulation, and computational cost for high-dimensional systems. Future directions include improved collocation point sampling, adaptive loss weighting, uncertainty quantification, and scalable hybrid physics–ML models for high-dimensional chaotic regimes.

Number Theory &
Residue Number System in AI

RNS enables carry-free, fully parallel arithmetic — transforming how neural network hardware accelerators are designed, from moduli-based MAC units to photonic end-to-end implementations.

Overview

The Residue Number System represents integers as tuples of residues modulo pairwise coprime moduli, enabling carry-free, fully parallel arithmetic operations. This is a fundamental advantage for hardware-accelerated neural network inference — MAC operations execute independently across channels with no inter-channel dependencies.

RNS Fundamentals

RNS arithmetic covers addition, subtraction, and multiplication — all carry-free and fully parallel. Moduli selection criteria balance dynamic range and hardware efficiency. Conversion between binary and residue domains uses the Chinese Remainder Theorem (CRT). The Redundant RNS (RRNS) adds redundant moduli to enable fault detection and correction without binary domain conversion.

Partially & End-to-End RNS Accelerators

Partially RNS-based architectures apply residue arithmetic to selected stages — typically MAC operations — while remaining operations stay in the binary domain. End-to-end architectures operate the complete inference pipeline in the residue domain, including sign detection, magnitude comparison, and overflow mitigation. Both digital and photonic implementations are surveyed.

Reliability & Fault Tolerance

RRNS syndrome-based error detection and correction schemes provide hardware-level reliability without the overhead of binary conversion. System-level reliability approaches are compared across fault coverage, performance penalty, and implementation cost — critical considerations for AI accelerators deployed in safety-critical environments.

Homomorphic Encryption
for Privacy-Preserving AI

Computation on encrypted data without decryption — enabling privacy-preserving cloud ML, federated learning, and secure image processing using BFV, BGV, CKKS, and TFHE schemes.

Overview

Homomorphic Encryption enables computation directly on encrypted data without decryption — making it foundational for privacy-preserving cloud computing and federated machine learning. Operations on ciphertexts produce encrypted results that, when decrypted, equal the result of the same operations on the original plaintexts: Dec(HE.op(Enc(m))) = op(m).

FHE Scheme Structures

Major schemes are analyzed across key generation, encryption, and decryption processes: BFV and BGV for exact integer arithmetic based on RLWE; CKKS for approximate real-valued arithmetic suited to ML inference; TFHE for fast gate-by-gate boolean circuits; and the Gentry/DGHV schemes that established FHE foundations. The Paillier Partially HE scheme is also discussed.

Scheme Comparison

A comparative analysis covers noise growth, multiplicative depth, bootstrapping overhead, ciphertext size, and practical throughput. CKKS is most suitable for ML inference due to its approximate arithmetic; TFHE excels for boolean circuit evaluation; BFV/BGV are preferred for exact integer operations such as those in number-theoretic algorithms.

Applications

Applications span image processing (filtering on encrypted images), image encryption (HE combined with classical encryption), data hiding (steganography in the encrypted domain), and machine learning — privacy-preserving inference and training on encrypted datasets without exposing raw data or model parameters to the computing server.

FPGA Hardware Acceleration
for Artificial Intelligence

Systolic arrays, dataflow pipelines, FINN/hls4ml/Vitis AI toolflows — and a proof-of-concept CNN on AMD Zynq UltraScale+ achieving over 2,156 images per second.

Overview

FPGAs achieve a critical balance between ASIC efficiency and CPU/GPU flexibility. They leverage repeated MAC operations in neural networks using on-chip BRAM and specialized DSP blocks for parallel arithmetic, enabling deterministic low-latency inference — ideal for edge and embedded AI deployment where power and latency constraints are paramount.

Neural Networks from a Hardware Perspective

Core computation patterns — convolution, pooling, batch normalization, fully connected layers — are analyzed for hardware mapping. Systolic array architectures and sliding-window topologies are the primary strategies for CNN layers. Precision considerations (INT8, fixed-point, floating-point) are critical for area efficiency and throughput — quantization to INT8 can deliver 4× improvements over FP32.

Design Approaches

Two primary strategies: fully custom hardware design generating systolic arrays and spatial dataflow pipelines per network; and hardware-software co-design partitioning compute-intensive kernels between programmable logic and ARM/RISC-V processors. Both require coordinated optimization: model compression (quantization, pruning), loop unrolling, tiling, and double buffering for memory hierarchy management.

Automated Frameworks & Proof of Concept

FINN, hls4ml, FpgaConvNet, and Vitis AI convert PyTorch/TensorFlow models into synthesizable hardware or prebuilt IP cores, dramatically reducing RTL engineering effort. A proof-of-concept single-layer CNN on AMD Zynq UltraScale+ (ZCU102) using FpgaConvNet achieved over 2,156 images/second with exceptionally low computational error, validating the effectiveness of automated FPGA toolflows.

Scientific Output

Research Publications

Journal papers, conference proceedings, and technical reports from the NCCAI research team.

NCCAI Interactive Lab

Experiments & Execution

Interactive simulations and testing ground for chaos-based logic, image encryption, and hardware-accelerated algorithms.

Community & Knowledge

Workshops & Seminars

Advancing the field of Artificial Intelligence through collaborative learning, hands-on hardware sessions, and expert-led seminars.

Research Community

Community Forum

Collaborative space for Chaos theory discussions, technical support, and research exchange.

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NCCAI: Number Theory and Chaos Algorithms for Computational & Artificial Intelligence

Algorithm Repository

Interactive Execution

Publications

Workshops & Seminars

About NCCAI

Project Description

Funding & Timeline

Predecessor Projects

Nine Research Tasks

Meet the NCCAI Team

Research Areas

Chaos Theory in Machine Learning

ANN for Time Series & Chaotic Dynamics

Physics-Informed Neural Networks

Number Theory & Residue Number System

Homomorphic Encryption

FPGA Hardware Acceleration for AI

Chaos Theory in Machine Learning

Overview

Chaos Theory Fundamentals

Data Representation & Preprocessing

Chaos in Neural Architectures & Optimization

Key Topics

Area Info

ANN-Based Architectures forTime Series & Chaotic Dynamics

Overview

Classical Statistical Models

Neural Architectures for Time Series

Modeling Chaotic Dynamics

Key Topics

Area Info

Physics-InformedNeural Networks

Overview

PINN Foundations

Chaos in Dynamical Systems

Challenges & Future Directions

Key Topics

Area Info

Number Theory &Residue Number System in AI

Overview

RNS Fundamentals

Partially & End-to-End RNS Accelerators

Reliability & Fault Tolerance

Key Topics

Area Info

Homomorphic Encryptionfor Privacy-Preserving AI

Overview

FHE Scheme Structures

Scheme Comparison

Applications

Key Topics

Area Info

FPGA Hardware Accelerationfor Artificial Intelligence

Overview

Neural Networks from a Hardware Perspective

Design Approaches

Automated Frameworks & Proof of Concept

Key Topics

Area Info

Research Publications

Experiments & Execution

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Workshops & Seminars

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NCCAI: Number Theory
and Chaos Algorithms
for Computational & Artificial Intelligence

ANN-Based Architectures for
Time Series & Chaotic Dynamics

Physics-Informed
Neural Networks

Number Theory &
Residue Number System in AI

Homomorphic Encryption
for Privacy-Preserving AI

FPGA Hardware Acceleration
for Artificial Intelligence