pqc/external/flint-2.4.3/fq_nmod/doc/fq_nmod.txt

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/*=============================================================================
This file is part of FLINT.
FLINT is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
FLINT is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FLINT; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2013 Mike Hansen
******************************************************************************/
*******************************************************************************
Context Management
*******************************************************************************
void fq_nmod_ctx_init(fq_nmod_ctx_t ctx, const fmpz_t p,
slong d, const char *var)
Initialises the context for prime~$p$ and extension degree~$d$,
with name \code{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 \code{var} is a null-terminated string
of length at least one.
int _fq_nmod_ctx_init_conway(fq_nmod_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 \code{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 \code{var} is a null-terminated string
of length at least one.
void fq_nmod_ctx_init_conway(fq_nmod_ctx_t ctx, const fmpz_t p,
slong d, const char *var)
Initialises the context for prime~$p$ and extension degree~$d$,
with name \code{var} for the generator using a Conway polynomial
for the modulus.
Assumes that $p$ is a prime.
Assumes that the string \code{var} is a null-terminated string
of length at least one.
void fq_nmod_ctx_init_modulus(fq_nmod_ctx_t ctx,
nmod_poly_t modulus,
const char *var)
Initialises the context for given \code{modulus} with name
\code{var} for the generator.
Assumes that \code{modulus} is and irreducible polynomial over
$\mathbf{F}_{p}$.
Assumes that the string \code{var} is a null-terminated string
of length at least one.
void fq_nmod_ctx_clear(fq_nmod_ctx_t ctx)
Clears all memory that has been allocated as part of the context.
long fq_nmod_ctx_degree(const fq_nmod_ctx_t ctx)
Returns the degree of the field extension
$[\mathbf{F}_{q} : \mathbf{F}_{p}]$, which
is equal to $\log_{p} q$.
fmpz * fq_nmod_ctx_prime(const fq_nmod_ctx_t ctx)
Returns a pointer to the prime $p$ in the context.
void fq_nmod_ctx_order(fmpz_t f, const fq_nmod_ctx_t ctx)
Sets $f$ to be the size of the finite field.
int fq_nmod_ctx_fprint(FILE * file, const fq_nmod_ctx_t ctx)
Prints the context information to {\tt{file}}. Returns 1 for a
success and a negative number for an error.
void fq_nmod_ctx_print(const fq_nmod_ctx_t ctx)
Prints the context information to {\tt{stdout}}.
void fq_nmod_ctx_randtest(fq_nmod_ctx_t ctx)
Initializes \code{ctx} to a random finite field. Assumes that
\code{fq_nmod_ctx_init} as not been called on \code{ctx} already.
void fq_nmod_ctx_randtest_reducible(fq_nmod_ctx_t ctx)
Initializes \code{ctx} to a random extension of a word-sized prime
field. The modulus may or may not be irreducible. Assumes that
\code{fq_nmod_ctx_init} as not been called on \code{ctx} already.
*******************************************************************************
Memory management
*******************************************************************************
void fq_nmod_init(fq_nmod_t rop, const fq_nmod_ctx_t ctx)
Initialises the element \code{rop}, setting its value to~$0$.
void fq_nmod_init2(fq_nmod_t rop, const fq_nmod_ctx_t ctx)
Initialises \code{poly} with at least enough space for it to be an element
of \code{ctx} and sets it to~$0$.
void fq_nmod_clear(fq_nmod_t rop, const fq_nmod_ctx_t ctx)
Clears the element \code{rop}.
void _fq_nmod_sparse_reduce(mp_ptr R, slong lenR, const fq_nmod_ctx_t ctx)
Reduces \code{(R, lenR)} modulo the polynomial $f$ given by the
modulus of \code{ctx}.
void _fq_nmod_dense_reduce(mp_ptr R, slong lenR, const fq_nmod_ctx_t ctx)
Reduces \code{(R, lenR)} modulo the polynomial $f$ given by the
modulus of \code{ctx} using Newton division.
void _fq_nmod_reduce(mp_ptr r, slong lenR, const fq_nmod_ctx_t ctx)
Reduces \code{(R, lenR)} modulo the polynomial $f$ given by the
modulus of \code{ctx}. Does either sparse or dense reduction
based on \code{ctx->sparse_modulus}.
void fq_nmod_reduce(fq_nmod_t rop, const fq_nmod_ctx_t ctx)
Reduces the polynomial \code{rop} as an element of
$\mathbf{F}_p[X] / (f(X))$.
*******************************************************************************
Basic arithmetic
*******************************************************************************
void fq_nmod_add(fq_nmod_t rop, const fq_nmod_t op1, const fq_nmod_t op2,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to the sum of \code{op1} and \code{op2}.
void fq_nmod_sub(fq_nmod_t rop, const fq_nmod_t op1, const fq_nmod_t op2,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to the difference of \code{op1} and \code{op2}.
void fq_nmod_sub_one(fq_nmod_t rop, const fq_nmod_t op1, const fq_nmod_ctx_t ctx)
Sets \code{rop} to the difference of \code{op1} and $1$.
void fq_nmod_neg(fq_nmod_t rop, const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Sets \code{rop} to the negative of \code{op}.
void fq_nmod_mul(fq_nmod_t rop, const fq_nmod_t op1, const fq_nmod_t op2,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to the product of \code{op1} and \code{op2},
reducing the output in the given context.
void fq_nmod_mul_fmpz(fq_nmod_t rop, const fq_nmod_t op, const fmpz_t x,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to the product of \code{op} and $x$,
reducing the output in the given context.
void fq_nmod_mul_si(fq_nmod_t rop, const fq_nmod_t op, slong x,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to the product of \code{op} and $x$,
reducing the output in the given context.
void fq_nmod_mul_ui(fq_nmod_t rop, const fq_nmod_t op, ulong x,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to the product of \code{op} and $x$,
reducing the output in the given context.
void fq_nmod_sqr(fq_nmod_t rop, const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Sets \code{rop} to the square of \code{op},
reducing the output in the given context.
void _fq_nmod_inv(mp_ptr *rop, mp_srcptr *op, slong len, const fq_nmod_ctx_t ctx)
Sets \code{(rop, d)} to the inverse of the non-zero element
\code{(op, len)}.
void fq_nmod_inv(fq_nmod_t rop, const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Sets \code{rop} to the inverse of the non-zero element \code{op}.
void _fq_nmod_pow(mp_ptr *rop, mp_srcptr *op, slong len, const fmpz_t e,
const fq_nmod_ctx_t ctx)
Sets \code{(rop, 2*d-1)} to \code{(op,len)} raised to the power~$e$,
reduced modulo $f(X)$, the modulus of \code{ctx}.
Assumes that $e \geq 0$ and that \code{len} is positive and at most~$d$.
Although we require that \code{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.
void fq_nmod_pow(fq_nmod_t rop, const fq_nmod_t op, const fmpz_t e,
const fq_nmod_ctx_t ctx)
Sets \code{rop} the \code{op} raised to the power~$e$.
Currently assumes that $e \geq 0$.
Note that for any input \code{op}, \code{rop} is set to~$1$
whenever $e = 0$.
void fq_nmod_pow_ui(fq_nmod_t rop, const fq_nmod_t op, const ulong e,
const fq_nmod_ctx_t ctx)
Sets \code{rop} the \code{op} raised to the power~$e$.
Currently assumes that $e \geq 0$.
Note that for any input \code{op}, \code{rop} is set to~$1$
whenever $e = 0$.
*******************************************************************************
Roots
*******************************************************************************
void fq_nmod_pth_root(fq_nmod_t rop, const fq_nmod_t op1,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to a $p^{th}$ root root of \code{op1}. Currently,
this computes the root by raising \code{op1} to $p^{d-1}$ where
$d$ is the degree of the extension.
*******************************************************************************
Output
*******************************************************************************
int fq_nmod_fprint_pretty(FILE *file, const fq_nmod_t op,
const fq_nmod_ctx_t ctx)
Prints a pretty representation of \code{op} to \code{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.
int fq_nmod_print_pretty(const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Prints a pretty representation of \code{op} to \code{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.
void fq_nmod_fprint(FILE * file, const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Prints a representation of \code{op} to \code{file}.
For further details on the representation used, see
\code{nmod_poly_fprint()}.
void fq_nmod_print(const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Prints a representation of \code{op} to \code{stdout}.
For further details on the representation used, see
\code{nmod_poly_print()}.
char * fq_nmod_get_str(const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Returns the plain FLINT string representation of the element
\code{op}.
char * fq_nmod_get_str_pretty(const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Returns a pretty representation of the element~\code{op} using the
null-terminated string \code{x} as the variable name.
*******************************************************************************
Randomisation
*******************************************************************************
void fq_nmod_randtest(fq_nmod_t rop, flint_rand_t state,
const fq_nmod_ctx_t ctx)
Generates a random element of $\mathbb{F}_q$.
void fq_nmod_randtest_not_zero(fq_nmod_t rop, flint_rand_t state,
const fq_nmod_ctx_t ctx)
Generates a random non-zero element of $\mathbb{F}_q$.
void fq_nmod_randtest_dense(fq_nmod_t rop, flint_rand_t state,
const fq_nmod_ctx_t ctx)
Generates a random element of $\mathbb{F}_q$ which has an
underlying polynomial with dense coefficients.
*******************************************************************************
Assignments and conversions
*******************************************************************************
void fq_nmod_set(fq_nmod_t rop, const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Sets \code{rop} to \code{op}.
void fq_nmod_set_ui(fq_nmod_t rop, const ulong x, const fq_nmod_ctx_t ctx)
Sets \code{rop} to \code{x}, considered as an element of
$\mathbb{F}_p$.
void fq_nmod_set_fmpz(fq_nmod_t rop, const fmpz_t x, const fq_nmod_ctx_t ctx)
Sets \code{rop} to \code{x}, considered as an element of
$\mathbb{F}_p$.
void fq_nmod_swap(fq_nmod_t op1, fq_nmod_t op2, const fq_nmod_ctx_t ctx)
Swaps the two elements \code{op1} and \code{op2}.
void fq_nmod_zero(fq_nmod_t rop, const fq_nmod_ctx_t ctx)
Sets \code{rop} to zero.
void fq_nmod_one(fq_nmod_t rop, const fq_nmod_ctx_t ctx)
Sets \code{rop} to one, reduced in the given context.
void fq_nmod_gen(fq_nmod_t rop, const fq_nmod_ctx_t ctx)
Sets \code{rop} to a multiplicative generator for the finite field.
*******************************************************************************
Comparison
*******************************************************************************
int fq_nmod_is_zero(const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Returns whether \code{op} is equal to zero.
int fq_nmod_is_one(const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Returns whether \code{op} is equal to one.
int fq_nmod_equal(const fq_nmod_t op1, const fq_nmod_t op2,
const fq_nmod_ctx_t ctx)
Returns whether \code{op1} and \code{op2} are equal.
int fq_nmod_is_invertible(const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Returns whether \code{op} is an invertible element.
int fq_nmod_is_invertible_f(fq_nmod_t f, const fq_nmod_t op,
const fq_nmod_ctx_t ctx)
Returns whether \code{op} is an invertible element. If it is not,
then \code{f} is set of a factor of the modulus.
*******************************************************************************
Special functions
*******************************************************************************
void _fq_nmod_trace(fmpz_t rop, mp_srcptr *op, slong len,
const fq_nmod_ctx_t ctx)
Sets \code{rop} to the trace of the non-zero element \code{(op, len)}
in $\mathbf{F}_{q}$.
void fq_nmod_trace(fq_nmod_t rop, const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Sets \code{rop} to the trace of \code{op}.
For an element $a \in \mathbb{F}_q$, multiplication by $a$ defines
a $\mathbb{F}_p$-linear map on $\mathbb{F}_q$. We define the
trace of $a$ as the trace of this map. Equivalently, if $\Sigma$
generates $\Gal(\mathbb{F}_q / \mathbb{F}_p)$ then the trace of
$a$ is equal to $\sum_{i=0}^{d-1} \Sigma^i (a)$, where $d =
\log_{p} q$.
void _fq_nmod_norm(fmpz_t rop, mp_srcptr *op, slong len, const fq_nmod_ctx_t ctx)
Sets \code{rop} to the norm of the non-zero element \code{(op, len)}
in $\mathbf{F}_{q}$.
void fq_nmod_norm(fq_nmod_t rop, const fq_nmod_t op, const fq_nmod_ctx_t ctx)
Computes the norm of \code{op}.
For an element $a \in \mathbb{F}_q$, multiplication by $a$ defines
a $\mathbb{F}_p$-linear map on $\mathbb{F}_q$. We define the norm
of $a$ as the determinant of this map. Equivalently, if $\Sigma$ generates
$\Gal(\mathbb{F}_q / \mathbb{F}_p)$ then the trace of $a$ is equal to
$\prod_{i=0}^{d-1} \Sigma^i (a)$, where
$d = \text{dim}_{\mathbb{F}_p}(\mathbb{F}_q)$.
Algorithm selection is automatic depending on the input.
void _fq_nmod_frobenius(mp_ptr *rop, mp_srcptr *op, slong len, slong e,
const fq_nmod_ctx_t ctx)
Sets \code{(rop, 2d-1)} to the image of \code{(op, len)} under the
Frobenius operator raised to the e-th power, assuming that neither
\code{op} nor \code{e} are zero.
void fq_nmod_frobenius(fq_nmod_t rop, const fq_nmod_t op, slong e,
const fq_nmod_ctx_t ctx)
Evaluates the homomorphism $\Sigma^e$ at \code{op}.
Recall that $\mathbb{F}_q / \mathbb{F}_p$ is Galois with Galois group
$\langle \sigma \rangle$, which is also isomorphic to
$\mathbf{Z}/d\mathbf{Z}$, where
$\sigma \in \Gal(\mathbf{F}_q/\mathbf{F}_p)$ is the Frobenius element
$\sigma \colon x \mapsto x^p$.
*******************************************************************************
Bit packing
*******************************************************************************
void fq_nmod_bit_pack(fmpz_t f, const fq_nmod_t op, mp_bitcnt_t bit_size,
const fq_nmod_ctx_t ctx)
Packs \code{op} into bitfields of size \code{bit_size}, writing the
result to \code{f}.
void fq_nmod_bit_unpack(fq_nmod_t rop, const fmpz_t f, mp_bitcnt_t bit_size,
const fq_nmod_ctx_t ctx)
Unpacks into \code{rop} the element with coefficients packed into
fields of size \code{bit_size} as represented by the integer
\code{f}.