Converts the canonical representation [0, p-1] to the symmetric representation [-(p-1)/2, (p-1)/2] for display purposes.

Reduces the Gaussian integer (a + bi) modulo p to the [0, p-1] range.

Performs Gaussian multiplication (self * other) and reduces modulo p. (a1 + b1i) * (a2 + b2i) = (a1*a2 - b1*b2) + (a1*b2 + a2*b1)i

Performs Gaussian modular exponentiation: (a + bi)^e mod p. Uses the efficient Square-and-Multiply algorithm.

Decrypts the Gaussian ElGamal ciphertext (nu1, nu2) using the private key 'a'. Message: mu = nu2 * tau^-1 mod p, where tau = nu1^a mod p

Encrypts a Gaussian integer message (mu) using the Gaussian ElGamal scheme. Ciphertext: (nu1, nu2)

Performs standard (non-modular) Gaussian division: self / other. Returns: - A GaussianInteger if the division results in a

Computes (a^d) % n using modular exponentiation.

Multiplies two ciphertexts component-wise (nu1, nu2) * (nu1', nu2') mod p.

Performs the Miller-Rabin primality test.

Checks for equality.

Calculates the Gaussian modular inverse z^-1 mod p using the Norm method: z^-1 = conjugate(z) * Norm(z)^(-1) mod p.

Returns the conjugate (a - bi). The optional p is for consistency with other modular methods, but this is a standard arithmetic operation.

Returns the norm (a^2 + b^2), which is an integer.

Returns the norm (a^2 + b^2) mod p, which is an integer.