gint: Gaussian Integers/Rationals + Gaussian-Integer RSA

gint provides two related numeric types:

• Zi – Gaussian integers, \(a + bi\) with \(a, b \in \mathbb{Z}\).

• Qi – Gaussian rationals, \(a + bi\) with \(a, b \in \mathbb{Q}\), represented exactly via fractions.Fraction.

Also provides Gaussian-integer RSA cryptography (teaching implementation only; not for real secrets)

Note: A Qi whose components both reduce to whole numbers is automatically returned as a Zi instead – Qi(4, 6) is a Zi(4, 6).

Quickstart

>>> from gint import Zi, Qi
>>>
>>> z1, z2, z3 = Zi(2, -3), Zi(1, 4), Zi(-8, 1)
>>> z1         # ==> Zi(2, -3)
>>> print(z1)  # ==> (2-3j)
>>> z12 = z1 * z2
>>> z12        # ==> Zi(14, 5)
>>> z12 / z1   # ==> aZi(1, 4)
>>> z12 / z3   # ==> Qi('-107/65', '-54/65')
>>> (z12 / z3) * z3  # ==> Zi(14, 5)
>>> 1 / Zi(1, 1)     # ==> Qi('1/2', '-1/2')
>>> Zi(1, 1)**-1     # ==> Qi('1/2', '-1/2')
>>> Zi.gcd(z12, z1)  # ==> Zi(2, -3)
>>> Zi.lcm(z12, z1)  # ==> Zi(14, 5)
>>> Qi(1.25, 3.4)    # ==> Qi('5/4', '17/5')
>>> print(Zi(5, 0))  # ==> 5
>>>
>>> from gint.crypto import generate_keypair, encrypt_text, decrypt_text
>>>
>>> public_key, private_key = generate_keypair(bits=256)
>>> ciphertext = encrypt_text("Gaussian primes are cool.", public_key)
>>> decrypt_text(ciphertext, private_key)  # ==> 'Gaussian primes are cool.'

API Reference

• Zi – Gaussian Integers
  • Zi
• Qi – Gaussian Rationals
  • Qi
• Crypto – Gaussian-Integer RSA
  • GaussianRSAPublicKey
  • GaussianRSAPrivateKey
  • GaussianRSACiphertext
  • generate_keypair()
  • encrypt_block()
  • decrypt_block()
  • block_capacity()
  • encrypt_bytes()
  • decrypt_bytes()
  • encrypt_text()
  • decrypt_text()
  • Further Reading