@spec add(t(), t()) :: t()

Adds two Fq12 elements.

@spec eq?(t(), t()) :: boolean()

Checks if two Fq12 elements are equal.

@spec field_div(t(), t()) :: t()

Divides two Fq12 elements (a / b = a * b^(-1)).

@spec frobenius(t()) :: t()

Computes the Frobenius endomorphism φ: Fq12 → Fq12 where φ(x) = x^p. This raises each coefficient to the power p and applies the precomputed basis transformation.

@spec from_integers([integer()]) :: t()

Creates an Fq12 element from integers.

@spec inv(t()) :: t()

Computes the modular inverse of an Fq12 element. Uses the FqP inverse implementation.

@spec is_one?(t()) :: boolean()

Checks if an Fq12 element is one.

@spec is_zero?(t()) :: boolean()

Checks if an Fq12 element is zero.

@spec mul(t(), t()) :: t()

Multiplies two Fq12 elements.

@spec new([Tezex.Crypto.BLS.Fq.t()]) :: t()

Creates an Fq12 element from a list of 12 Fq coefficients.

@spec new([Tezex.Crypto.BLS.Fq.t()], [integer()]) :: t()

Creates an Fq12 element from a list of coefficients and modulus coefficients. This is used for testing compatibility.

@spec one() :: t()

Returns the one element in Fq12.

@spec one([integer()]) :: t()

Returns the one element with custom modulus (for testing).

@spec optimized_final_exponentiation(t()) :: t()

Optimized final exponentiation for BLS12-381.

Instead of computing f^((p^12-1)/r) directly (~4314 bits), this computes:

1. f^(p^2 + 1) using 2 Frobenius operations + 1 multiplication
2. (f^(p^2 + 1))^(p^6 - 1) using 6 Frobenius operations + 1 inversion
3. result^((p^4-p^2+1)/r) using optimized exponentiation (~1268 bits)

This gives the same result as the naive approach but is much faster. Total: 8 Frobenius + 1 mul + 1 inv + 1 pow(1268 bits) vs 1 pow(4314 bits)

@spec pow(t(), integer()) :: t()

Raises an Fq12 element to a power.

@spec scalar_mul(t(), Tezex.Crypto.BLS.Fq.t()) :: t()

Multiplies an Fq12 element by a scalar Fq element.

@spec sub(t(), t()) :: t()

Subtracts two Fq12 elements.