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Zmodn: Practical Modular Arithmetic Using NumPy

Zmodn provides a Python class for representing and performing operations in the ring of integers modulo n (ℤ/nℤ). It's built on NumPy for efficient array operations, making it suitable for applications in cryptography, computer algebra, and other fields requiring modular arithmetic.

Features

  • Perform modular arithmetic operations (addition, subtraction, multiplication, division, exponentiation)
  • Compute modular inverses
  • Handle matrix operations in modular arithmetic, including multiplication and inversion
  • Compare elements in modular arithmetic
  • Seamless integration with NumPy functions

Installation

Install Zmodn using pip:

pip install -e .

Quick Start

Here's a basic example of using the Zmodn class:

from zmodn import Zmodn

# Create Zmodn objects
a = Zmodn([1, 2, 7], 5)
b = Zmodn([3, 4, 2], 5)

print(a)  # Output: [1 2 2] (mod 5)
print(a + b)  # Output: [4 1 4] (mod 5)
print(a * b)  # Output: [3 3 4] (mod 5)
print(a.mod_inv())  # Output: [1 3 3] (mod 5)

Detailed Usage

Creating Zmodn Objects

# Single element
x = Zmodn(3, 7)  # 3 (mod 7)

# List of elements
y = Zmodn([1, 2, 3], 5)  # [1 2 3] (mod 5)

# Matrix
z = Zmodn([[1, 2], [3, 4]], 6)  # [[1 2] [3 4]] (mod 6)

Arithmetic Operations

a = Zmodn([1, 2], 7)
b = Zmodn([3, 4], 7)

print(a + b)  # Addition: [4 6] (mod 7)
print(a - b)  # Subtraction: [5 5] (mod 7)
print(a * b)  # Multiplication: [3 1] (mod 7)
print(a / b)  # Division: [5 4] (mod 7)
print(a ** 2)  # Exponentiation: [1 4] (mod 7)
print(-a)  # Negation: [6 5] (mod 7)

Matrix Operations

m1 = Zmodn([[1, 2], [3, 4]], 5)
m2 = Zmodn([[2, 3], [1, 4]], 5)

print(m1 @ m2)  # Matrix multiplication: [[4 1] [0 0]] (mod 5)
print(m1.inv())  # Matrix inversion: [[4 3] [2 4]] (mod 5)

Modular Inverse

x = Zmodn([2, 3], 5)
print(x.mod_inv())  # Modular inverse: [3 2] (mod 5)

Comparison Operations

a = Zmodn([1, 2], 7)
b = Zmodn([1, 3], 7)

print(a == b)  # False
print(a < b)   # True
print(a >= b)  # False

Notes and Limitations

  • The mod_inv() method works for individual elements and vectors, but not for matrices. Use inv() for matrix inversion.
  • All operations assume that Zmodn objects have the same modulus.
  • The class includes type checking and error handling for invalid inputs.

Documentation

For more detailed information, please refer to the full documentation.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

License

This project is licensed under the Creative Commons Zero v1.0 Universal license. See the LICENSE file for details.

Citation

If you use Zmodn in your research, please cite it as follows:

@software{Sanchez_Alejandro_2020,
    title = {{Zmodn}: Practical Modular Arithmetic Using NumPy},
    author = {Sánchez, Alejandro},
    month = {11},
    year = {2023},
    url = {https://github.com/asanchezyali/Zmodn},
}

Contact

For questions or issues, please open an issue on GitHub or contact the maintainer directly. You can also join our Discord community for discussions, support, and updates: Math & Code Discord Server Join us to connect with other users, get help, and stay updated on the latest developments!