.. _fq:

**fq.h** -- finite fields
===============================================================================

We represent an element of the finite field `\mathbf{F}_{p^n} \cong
\mathbf{F}_p[X]/(f(X))`, where `f(X) \in \mathbf{F}_p[X]` is a monic,
irreducible polynomial of degree `n`, as a polynomial in
`\mathbf{F}_p[X]` of degree less than `n`. The underlying data
structure is an :type:`fmpz_poly_t`.

The default choice for `f(X)` is the Conway polynomial for the pair `(p,n)`,
enabled by Frank Lübeck's data base of Conway polynomials using the
:func:`_nmod_poly_conway` function. If a Conway polynomial is not available,
then a random irreducible polynomial will be chosen for `f(X)`. Additionally,
the user is able to supply their own `f(X)`.

Types, macros and constants
-------------------------------------------------------------------------------

.. type:: fq_ctx_struct

.. type:: fq_ctx_t

.. type:: fq_struct

.. type:: fq_t

Context Management
--------------------------------------------------------------------------------


.. function:: void fq_ctx_init(fq_ctx_t ctx, const fmpz_t p, slong d, const char * var)

    Initialises the context for prime `p` and extension degree `d`,
    with name ``var`` for the generator.  By default, it will try
    use a Conway polynomial; if one is not available, a random
    irreducible polynomial will be used.

    Assumes that `p` is a prime.

    Assumes that the string ``var`` is a null-terminated string
    of length at least one.

.. function:: int _fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char * var)

    Attempts to initialise the context for prime `p` and extension
    degree `d`, with name ``var`` for the generator using a Conway
    polynomial for the modulus.

    Returns `1` if the Conway polynomial is in the database for the
    given size and the initialization is successful; otherwise,
    returns `0`.

    Assumes that `p` is a prime.

    Assumes that the string ``var`` is a null-terminated string
    of length at least one.

.. function:: void fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char * var)

    Initialises the context for prime `p` and extension degree `d`,
    with name ``var`` for the generator using a Conway polynomial
    for the modulus.

    Assumes that `p` is a prime.

    Assumes that the string ``var`` is a null-terminated string
    of length at least one.

.. function:: void fq_ctx_init_modulus(fq_ctx_t ctx, const fmpz_mod_poly_t modulus, const fmpz_mod_ctx_t ctxp, const char * var)

    Initialises the context for given ``modulus`` with name
    ``var`` for the generator.

    Assumes that ``modulus`` is an irreducible polynomial over the finite field `\mathbf{F}_{p}` in ``ctxp``.

    Assumes that the string ``var`` is a null-terminated string
    of length at least one.

.. function:: void fq_ctx_init_randtest(fq_ctx_t ctx, flint_rand_t state, int type)

    Initialises ``ctx`` to a random finite field, where the prime and degree is
    set according to ``type``. To see what prime and degrees may be output, see
    ``type`` in :func:`_nmod_poly_conway_rand`.

.. function:: void fq_ctx_init_randtest_reducible(fq_ctx_t ctx, flint_rand_t state, int type)

    Initializes ``ctx`` to a random extension of a prime field, where
    the prime and degree is set according to ``type``. If ``type`` is `0` the
    prime and degree may be large, else if ``type`` is `1` the degree is small
    but the prime may be large, else if ``type`` is `2` the prime is small but
    the degree may be large, else if ``type`` is `3` both prime and degree are
    small.

    The modulus may or may not be irreducible.

.. function:: void fq_ctx_clear(fq_ctx_t ctx)

    Clears all memory that has been allocated as part of the context.

.. function:: const fmpz_mod_poly_struct* fq_ctx_modulus(const fq_ctx_t ctx)

    Returns a pointer to the modulus in the context.

.. function:: slong fq_ctx_degree(const fq_ctx_t ctx)

    Returns the degree of the field extension
    `[\mathbf{F}_{q} : \mathbf{F}_{p}]`, which
    is equal to `\log_{p} q`.

.. function:: const fmpz * fq_ctx_prime(const fq_ctx_t ctx)

    Returns a pointer to the prime `p` in the context.

.. function:: void fq_ctx_order(fmpz_t f, const fq_ctx_t ctx)

     Sets `f` to be the size of the finite field.

.. function:: int fq_ctx_fprint(FILE * file, const fq_ctx_t ctx)

    Prints the context information to ``file``. Returns 1 for a
    success and a negative number for an error.

.. function:: void fq_ctx_print(const fq_ctx_t ctx)

    Prints the context information to ``stdout``.


Memory management
--------------------------------------------------------------------------------


.. function:: void fq_init(fq_t rop, const fq_ctx_t ctx)

    Initialises the element ``rop``, setting its value to `0`.

.. function:: void fq_init2(fq_t rop, const fq_ctx_t ctx)

    Initialises ``poly`` with at least enough space for it to be an element
    of ``ctx`` and sets it to `0`.

.. function:: void fq_clear(fq_t rop, const fq_ctx_t ctx)

    Clears the element ``rop``.

.. function:: void _fq_sparse_reduce(fmpz * R, slong lenR, const fq_ctx_t ctx)

    Reduces ``(R, lenR)`` modulo the polynomial `f` given by the
    modulus of ``ctx``.

.. function:: void _fq_dense_reduce(fmpz * R, slong lenR, const fq_ctx_t ctx)

    Reduces ``(R, lenR)`` modulo the polynomial `f` given by the
    modulus of ``ctx`` using Newton division.

.. function:: void _fq_reduce(fmpz * r, slong lenR, const fq_ctx_t ctx)

    Reduces ``(R, lenR)`` modulo the polynomial `f` given by the
    modulus of ``ctx``.  Does either sparse or dense reduction
    based on ``ctx->sparse_modulus``.

.. function:: void fq_reduce(fq_t rop, const fq_ctx_t ctx)

    Reduces the polynomial ``rop`` as an element of
    `\mathbf{F}_p[X] / (f(X))`.


Basic arithmetic
--------------------------------------------------------------------------------


.. function:: void fq_add(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

    Sets ``rop`` to the sum of ``op1`` and ``op2``.

.. function:: void fq_sub(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

    Sets ``rop`` to the difference of ``op1`` and ``op2``.

.. function:: void fq_sub_one(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

    Sets ``rop`` to the difference of ``op1`` and `1`.

.. function:: void fq_neg(fq_t rop, const fq_t op, const fq_ctx_t ctx)

    Sets ``rop`` to the negative of ``op``.

.. function:: void fq_mul(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

    Sets ``rop`` to the product of ``op1`` and ``op2``,
    reducing the output in the given context.

.. function:: void fq_mul_fmpz(fq_t rop, const fq_t op, const fmpz_t x, const fq_ctx_t ctx)

    Sets ``rop`` to the product of ``op`` and `x`,
    reducing the output in the given context.

.. function:: void fq_mul_si(fq_t rop, const fq_t op, slong x, const fq_ctx_t ctx)

    Sets ``rop`` to the product of ``op`` and `x`,
    reducing the output in the given context.

.. function:: void fq_mul_ui(fq_t rop, const fq_t op, ulong x, const fq_ctx_t ctx)

    Sets ``rop`` to the product of ``op`` and `x`,
    reducing the output in the given context.

.. function:: void fq_sqr(fq_t rop, const fq_t op, const fq_ctx_t ctx)

    Sets ``rop`` to the square of ``op``,
    reducing the output in the given context.

.. function:: void fq_div(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

    Sets ``rop`` to the quotient of ``op1`` and ``op2``,
    reducing the output in the given context.

.. function:: void _fq_inv(fmpz * rop, const fmpz * op, slong len, const fq_ctx_t ctx)

    Sets ``(rop, d)`` to the inverse of the non-zero element
    ``(op, len)``.

.. function:: void fq_inv(fq_t rop, const fq_t op, const fq_ctx_t ctx)

    Sets ``rop`` to the inverse of the non-zero element ``op``.

.. function:: void fq_gcdinv(fq_t f, fq_t inv, const fq_t op, const fq_ctx_t ctx)

     Sets ``inv`` to be the inverse of ``op`` modulo the modulus
     of ``ctx``.  If ``op`` is not invertible, then ``f`` is
     set to a factor of the modulus; otherwise, it is set to one.

.. function:: void _fq_pow(fmpz * rop, const fmpz * op, slong len, const fmpz_t e, const fq_ctx_t ctx)

    Sets ``(rop, 2*d-1)`` to ``(op,len)`` raised to the power `e`,
    reduced modulo `f(X)`, the modulus of ``ctx``.

    Assumes that `e \geq 0` and that ``len`` is positive and at most `d`.

    Although we require that ``rop`` provides space for
    `2d - 1` coefficients, the output will be reduced modulo
    `f(X)`, which is a polynomial of degree `d`.

    Does not support aliasing.

.. function:: void fq_pow(fq_t rop, const fq_t op, const fmpz_t e, const fq_ctx_t ctx)

    Sets ``rop`` the ``op`` raised to the power `e`.

    Currently assumes that `e \geq 0`.

    Note that for any input ``op``, ``rop`` is set to `1`
    whenever `e = 0`.

.. function:: void fq_pow_ui(fq_t rop, const fq_t op, const ulong e, const fq_ctx_t ctx)

    Sets ``rop`` the ``op`` raised to the power `e`.

    Currently assumes that `e \geq 0`.

    Note that for any input ``op``, ``rop`` is set to `1`
    whenever `e = 0`.



Roots
--------------------------------------------------------------------------------


.. function:: int fq_sqrt(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

    Sets ``rop`` to the square root of ``op1`` if it is a square, and return
    `1`, otherwise return `0`.

.. function:: void fq_pth_root(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

    Sets ``rop`` to a `p^{th}` root root of ``op1``.  Currently,
    this computes the root by raising ``op1`` to `p^{d-1}` where
    `d` is the degree of the extension.

.. function:: int fq_is_square(const fq_t op, const fq_ctx_t ctx)

    Return ``1`` if ``op`` is a square.

Output
--------------------------------------------------------------------------------


.. function:: int fq_fprint_pretty(FILE * file, const fq_t op, const fq_ctx_t ctx)

    Prints a pretty representation of ``op`` to ``file``.

    In the current implementation, always returns `1`.  The return code is
    part of the function's signature to allow for a later implementation to
    return the number of characters printed or a non-positive error code.

.. function:: int fq_print_pretty(const fq_t op, const fq_ctx_t ctx)

    Prints a pretty representation of ``op`` to ``stdout``.

    In the current implementation, always returns `1`.  The return code is
    part of the function's signature to allow for a later implementation to
    return the number of characters printed or a non-positive error code.

.. function:: int fq_fprint(FILE * file, const fq_t op, const fq_ctx_t ctx)

    Prints a representation of ``op`` to ``file``.

    For further details on the representation used, see
    :func:`fmpz_mod_poly_fprint`.

.. function:: void fq_print(const fq_t op, const fq_ctx_t ctx)

    Prints a representation of ``op`` to ``stdout``.

    For further details on the representation used, see
    :func:`fmpz_mod_poly_print`.

.. function:: char * fq_get_str(const fq_t op, const fq_ctx_t ctx)

    Returns the plain FLINT string representation of the element
    ``op``.

.. function:: char * fq_get_str_pretty(const fq_t op, const fq_ctx_t ctx)

    Returns a pretty representation of the element ``op`` using the
    null-terminated string ``x`` as the variable name.


Randomisation
--------------------------------------------------------------------------------


.. function:: void fq_randtest(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

    Generates a random element of `\mathbf{F}_q`.

.. function:: void fq_randtest_not_zero(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

    Generates a random non-zero element of `\mathbf{F}_q`.

.. function:: void fq_randtest_dense(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

    Generates a random element of `\mathbf{F}_q` which has an
    underlying polynomial with dense coefficients.

.. function:: void fq_rand(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

    Generates a high quality random element of `\mathbf{F}_q`.

.. function:: void fq_rand_not_zero(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

    Generates a high quality non-zero random element of `\mathbf{F}_q`.


Assignments and conversions
--------------------------------------------------------------------------------


.. function:: void fq_set(fq_t rop, const fq_t op, const fq_ctx_t ctx)

    Sets ``rop`` to ``op``.

.. function:: void fq_set_si(fq_t rop, const slong x, const fq_ctx_t ctx)

    Sets ``rop`` to ``x``, considered as an element of
    `\mathbf{F}_p`.

.. function:: void fq_set_ui(fq_t rop, const ulong x, const fq_ctx_t ctx)

    Sets ``rop`` to ``x``, considered as an element of
    `\mathbf{F}_p`.

.. function:: void fq_set_fmpz(fq_t rop, const fmpz_t x, const fq_ctx_t ctx)

    Sets ``rop`` to ``x``, considered as an element of
    `\mathbf{F}_p`.

.. function:: void fq_swap(fq_t op1, fq_t op2, const fq_ctx_t ctx)

    Swaps the two elements ``op1`` and ``op2``.

.. function:: void fq_zero(fq_t rop, const fq_ctx_t ctx)

    Sets ``rop`` to zero.

.. function:: void fq_one(fq_t rop, const fq_ctx_t ctx)

    Sets ``rop`` to one, reduced in the given context.

.. function:: void fq_gen(fq_t rop, const fq_ctx_t ctx)

    Sets ``rop`` to a generator for the finite field.
    There is no guarantee this is a multiplicative generator of
    the finite field.

.. function:: int fq_get_fmpz(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

    If ``op`` has a lift to the integers, return `1` and set ``rop`` to the lift in `[0,p)`.
    Otherwise, return `0` and leave `rop` undefined.

.. function:: void fq_get_fmpz_poly(fmpz_poly_t a, const fq_t b, const fq_ctx_t ctx)

.. function:: void fq_get_fmpz_mod_poly(fmpz_mod_poly_t a, const fq_t b, const fq_ctx_t ctx)

    Set ``a`` to a representative of ``b`` in ``ctx``.
    The representatives are taken in `(\mathbb{Z}/p\mathbb{Z})[x]/h(x)` where `h(x)` is the defining polynomial in ``ctx``.

.. function:: void fq_set_fmpz_poly(fq_t a, const fmpz_poly_t b, const fq_ctx_t ctx)

.. function:: void fq_set_fmpz_mod_poly(fq_t a, const fmpz_mod_poly_t b, const fq_ctx_t ctx)

    Set ``a`` to the element in ``ctx`` with representative ``b``.
    The representatives are taken in `(\mathbb{Z}/p\mathbb{Z})[x]/h(x)` where `h(x)` is the defining polynomial in ``ctx``.

.. function:: void fq_get_fmpz_mod_mat(fmpz_mod_mat_t col, const fq_t a, const fq_ctx_t ctx)

    Convert ``a`` to a column vector of length ``degree(ctx)``.

.. function:: void fq_set_fmpz_mod_mat(fq_t a, const fmpz_mod_mat_t col, const fq_ctx_t ctx)

    Convert a column vector ``col`` of length ``degree(ctx)`` to an element of ``ctx``.


Comparison
--------------------------------------------------------------------------------


.. function:: int fq_is_zero(const fq_t op, const fq_ctx_t ctx)

    Returns whether ``op`` is equal to zero.

.. function:: int fq_is_one(const fq_t op, const fq_ctx_t ctx)

    Returns whether ``op`` is equal to one.

.. function:: int fq_equal(const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

    Returns whether ``op1`` and ``op2`` are equal.

.. function:: int fq_is_invertible(const fq_t op, const fq_ctx_t ctx)

    Returns whether ``op`` is an invertible element.

.. function:: int fq_is_invertible_f(fq_t f, const fq_t op, const fq_ctx_t ctx)

    Returns whether ``op`` is an invertible element.  If it is not,
    then ``f`` is set of a factor of the modulus.


Special functions
--------------------------------------------------------------------------------


.. function:: void _fq_trace(fmpz_t rop, const fmpz * op, slong len, const fq_ctx_t ctx)

    Sets ``rop`` to the trace of the non-zero element ``(op, len)``
    in `\mathbf{F}_{q}`.

.. function:: void fq_trace(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

    Sets ``rop`` to the trace of ``op``.

    For an element `a \in \mathbf{F}_q`, multiplication by `a` defines
    a `\mathbf{F}_p`-linear map on `\mathbf{F}_q`.  We define the
    trace of `a` as the trace of this map.  Equivalently, if `\Sigma`
    generates `\operatorname{Gal}(\mathbf{F}_q / \mathbf{F}_p)` then the trace of
    `a` is equal to `\sum_{i=0}^{d-1} \Sigma^i (a)`, where `d =
    \log_{p} q`.

.. function:: void _fq_norm(fmpz_t rop, const fmpz * op, slong len, const fq_ctx_t ctx)

    Sets ``rop`` to the norm of the non-zero element ``(op, len)``
    in `\mathbf{F}_{q}`.

.. function:: void fq_norm(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

    Computes the norm of ``op``.

    For an element `a \in \mathbf{F}_q`, multiplication by `a` defines
    a `\mathbf{F}_p`-linear map on `\mathbf{F}_q`.  We define the norm
    of `a` as the determinant of this map.  Equivalently, if `\Sigma` generates
    `\operatorname{Gal}(\mathbf{F}_q / \mathbf{F}_p)` then the trace of `a` is equal to
    `\prod_{i=0}^{d-1} \Sigma^i (a)`, where
    `d = \text{dim}_{\mathbf{F}_p}(\mathbf{F}_q)`.

    Algorithm selection is automatic depending on the input.

.. function:: void _fq_frobenius(fmpz * rop, const fmpz * op, slong len, slong e, const fq_ctx_t ctx)

    Sets ``(rop, 2d-1)`` to the image of ``(op, len)`` under the
    Frobenius operator raised to the e-th power, assuming that neither
    ``op`` nor ``e`` are zero.

.. function:: void fq_frobenius(fq_t rop, const fq_t op, slong e, const fq_ctx_t ctx)

    Evaluates the homomorphism `\Sigma^e` at ``op``.

    Recall that `\mathbf{F}_q / \mathbf{F}_p` is Galois with Galois group
    `\langle \sigma \rangle`, which is also isomorphic to
    `\mathbf{Z}/d\mathbf{Z}`, where
    `\sigma \in \operatorname{Gal}(\mathbf{F}_q/\mathbf{F}_p)` is the Frobenius element
    `\sigma \colon x \mapsto x^p`.

.. function:: int fq_multiplicative_order(fmpz * ord, const fq_t op, const fq_ctx_t ctx)

    Computes the order of ``op`` as an element of the
    multiplicative group of ``ctx``.

    Returns 0 if ``op`` is 0, otherwise it returns 1 if ``op``
    is a generator of the multiplicative group, and -1 if it is not.

    This function can also be used to check primitivity of a generator of
    a finite field whose defining polynomial is not primitive.

.. function:: int fq_is_primitive(const fq_t op, const fq_ctx_t ctx)

    Returns whether ``op`` is primitive, i.e., whether it is a
    generator of the multiplicative group of ``ctx``.


Bit packing
--------------------------------------------------------------------------------


.. function:: void fq_bit_pack(fmpz_t f, const fq_t op, flint_bitcnt_t bit_size, const fq_ctx_t ctx)

    Packs ``op`` into bitfields of size ``bit_size``, writing the
    result to ``f``.

.. function:: void fq_bit_unpack(fq_t rop, const fmpz_t f, flint_bitcnt_t bit_size, const fq_ctx_t ctx)

    Unpacks into ``rop`` the element with coefficients packed into
    fields of size ``bit_size`` as represented by the integer
    ``f``.
