gf2-irreducibles 0.1.1

Lookup table for low-weight binary irreducible polynomials (GF(2^n)).

gf2-irreducibles 🔐

A zero-dependency, high-performance lookup library for low-weight binary irreducible polynomials over $GF(2)$.

This library contains a pre-computed database of polynomials for degrees $n$ where $2 \le n \le 10,000$. It is designed for researchers and engineers working in cryptography (ECC, finite field arithmetic) who need instant access to standard irreducible polynomials without expensive runtime testing.

🚀 Installation

pip install gf2-irreducibles

📖 Usage

Basic Lookup

Retrieve the minimal weight polynomial for a given degree.

import gf2_irreducibles as gf2

# Get the polynomial for degree 8
poly = gf2.get_poly(8)

print(f"Polynomial: {poly}")
# Output: x^8 + x^4 + x^3 + x^1 + 1

print(f"Integer:    {poly.to_int()}")
# Output: 283  (useful for bitwise operations)

print(f"Hex:        {poly.to_hex()}")
# Output: 0x11b

Checking Availability

if gf2.has_degree(123):
    print("Polynomial found!")

LaTeX Output for Papers

Useful for copying formulas directly into academic papers or reports.

# Generate LaTeX string for documentation
print(poly.to_latex())
# Output: x^{8} + x^{4} + x^{3} + x + 1

📚 Data Source

The polynomial data is derived from the technical report:

"Table of Low-Weight Binary Irreducible Polynomials" > Gadiel Seroussi > Hewlett-Packard Systems Laboratory, Report HPL-98-135, August 1998.

This library implements the data tables presented in the report, providing:

• [cite_start]Trinomials ($x^n + x^k + 1$) where they exist[cite: 27].
• [cite_start]Pentanomials ($x^n + x^{k_1} + x^{k_2} + x^{k_3} + 1$) where trinomials do not exist[cite: 28].

[cite_start]Among those of minimum weight, the polynomial listed is the one where the intermediate degrees are lowest (lexicographically first)[cite: 11].

⚡ Performance

• O(1) Lookup: Uses a static hash map. No irreducibility tests are performed at runtime.
• Zero Dependencies: Pure Python, standard library only.
• Lightweight: Minimal memory footprint.

🤝 Contributing

Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

📄 License

MIT