469 lines
15 KiB
Plaintext
469 lines
15 KiB
Plaintext
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/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2011, 2012 Sebastian Pancratz
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Copyright (C) 2012, 2013 Andres Goens
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Copyright (C) 2013 Mike Hansen
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******************************************************************************/
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*******************************************************************************
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Context Management
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*******************************************************************************
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void fq_ctx_init(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)
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Initialises the context for prime~$p$ and extension degree~$d$,
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with name \code{var} for the generator. By default, it will try
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use a Conway polynomial; if one is not available, a random
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irreducible polynomial will be used.
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Assumes that $p$ is a prime.
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Assumes that the string \code{var} is a null-terminated string
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of length at least one.
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int _fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)
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Attempts to initialise the context for prime~$p$ and extension
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degree~$d$, with name \code{var} for the generator using a Conway
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polynomial for the modulus.
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Returns $1$ if the Conway polynomial is in the database for the
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given size and the initialization is successful; otherwise,
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returns $0$.
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Assumes that $p$ is a prime.
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Assumes that the string \code{var} is a null-terminated string
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of length at least one.
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void fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)
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Initialises the context for prime~$p$ and extension degree~$d$,
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with name \code{var} for the generator using a Conway polynomial
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for the modulus.
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Assumes that $p$ is a prime.
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Assumes that the string \code{var} is a null-terminated string
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of length at least one.
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void fq_ctx_init_modulus(fq_ctx_t ctx,
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fmpz_mod_poly_t modulus,
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const char *var)
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Initialises the context for given \code{modulus} with name
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\code{var} for the generator.
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Assumes that \code{modulus} is and irreducible polynomial over
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$\mathbf{F}_{p}$.
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Assumes that the string \code{var} is a null-terminated string
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of length at least one.
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void fq_ctx_clear(fq_ctx_t ctx)
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Clears all memory that has been allocated as part of the context.
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long fq_ctx_degree(const fq_ctx_t ctx)
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Returns the degree of the field extension
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$[\mathbf{F}_{q} : \mathbf{F}_{p}]$, which
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is equal to $\log_{p} q$.
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fmpz * fq_ctx_prime(const fq_ctx_t ctx)
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Returns a pointer to the prime $p$ in the context.
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void fq_ctx_order(fmpz_t f, const fq_ctx_t ctx)
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Sets $f$ to be the size of the finite field.
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int fq_ctx_fprint(FILE * file, const fq_ctx_t ctx)
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Prints the context information to {\tt{file}}. Returns 1 for a
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success and a negative number for an error.
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void fq_ctx_print(const fq_ctx_t ctx)
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Prints the context information to {\tt{stdout}}.
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void fq_ctx_randtest(fq_ctx_t ctx)
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Initializes \code{ctx} to a random finite field. Assumes that
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\code{fq_ctx_init} as not been called on \code{ctx} already.
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void fq_ctx_randtest_reducible(fq_ctx_t ctx)
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Initializes \code{ctx} to a random extension of a prime field.
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The modulus may or may not be irreducible. Assumes that
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\code{fq_ctx_init} as not been called on \code{ctx} already.
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*******************************************************************************
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Memory management
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*******************************************************************************
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void fq_init(fq_t rop, const fq_ctx_t ctx)
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Initialises the element \code{rop}, setting its value to~$0$.
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void fq_init2(fq_t rop, const fq_ctx_t ctx)
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Initialises \code{poly} with at least enough space for it to be an element
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of \code{ctx} and sets it to~$0$.
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void fq_clear(fq_t rop, const fq_ctx_t ctx)
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Clears the element \code{rop}.
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void _fq_sparse_reduce(fmpz *R, slong lenR, const fq_ctx_t ctx)
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Reduces \code{(R, lenR)} modulo the polynomial $f$ given by the
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modulus of \code{ctx}.
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void _fq_dense_reduce(fmpz *R, slong lenR, const fq_ctx_t ctx)
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Reduces \code{(R, lenR)} modulo the polynomial $f$ given by the
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modulus of \code{ctx} using Newton division.
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void _fq_reduce(fmpz *r, slong lenR, const fq_ctx_t ctx)
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Reduces \code{(R, lenR)} modulo the polynomial $f$ given by the
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modulus of \code{ctx}. Does either sparse or dense reduction
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based on \code{ctx->sparse_modulus}.
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void fq_reduce(fq_t rop, const fq_ctx_t ctx)
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Reduces the polynomial \code{rop} as an element of
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$\mathbf{F}_p[X] / (f(X))$.
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*******************************************************************************
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Basic arithmetic
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*******************************************************************************
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void fq_add(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
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Sets \code{rop} to the sum of \code{op1} and \code{op2}.
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void fq_sub(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
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Sets \code{rop} to the difference of \code{op1} and \code{op2}.
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void fq_sub_one(fq_t rop, const fq_t op1, const fq_ctx_t ctx)
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Sets \code{rop} to the difference of \code{op1} and $1$.
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void fq_neg(fq_t rop, const fq_t op, const fq_ctx_t ctx)
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Sets \code{rop} to the negative of \code{op}.
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void fq_mul(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
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Sets \code{rop} to the product of \code{op1} and \code{op2},
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reducing the output in the given context.
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void fq_mul_fmpz(fq_t rop, const fq_t op, const fmpz_t x, const fq_ctx_t ctx)
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Sets \code{rop} to the product of \code{op} and $x$,
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reducing the output in the given context.
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void fq_mul_si(fq_t rop, const fq_t op, slong x, const fq_ctx_t ctx)
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Sets \code{rop} to the product of \code{op} and $x$,
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reducing the output in the given context.
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void fq_mul_ui(fq_t rop, const fq_t op, ulong x, const fq_ctx_t ctx)
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Sets \code{rop} to the product of \code{op} and $x$,
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reducing the output in the given context.
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void fq_sqr(fq_t rop, const fq_t op, const fq_ctx_t ctx)
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Sets \code{rop} to the square of \code{op},
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reducing the output in the given context.
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void _fq_inv(fmpz *rop, const fmpz *op, slong len, const fq_ctx_t ctx)
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Sets \code{(rop, d)} to the inverse of the non-zero element
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\code{(op, len)}.
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void fq_inv(fq_t rop, const fq_t op, const fq_ctx_t ctx)
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Sets \code{rop} to the inverse of the non-zero element \code{op}.
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void _fq_pow(fmpz *rop, const fmpz *op, slong len, const fmpz_t e,
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const fq_ctx_t ctx)
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Sets \code{(rop, 2*d-1)} to \code{(op,len)} raised to the power~$e$,
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reduced modulo $f(X)$, the modulus of \code{ctx}.
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Assumes that $e \geq 0$ and that \code{len} is positive and at most~$d$.
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Although we require that \code{rop} provides space for
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$2d - 1$ coefficients, the output will be reduced modulo
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$f(X)$, which is a polynomial of degree~$d$.
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Does not support aliasing.
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void fq_pow(fq_t rop, const fq_t op, const fmpz_t e, const fq_ctx_t ctx)
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Sets \code{rop} the \code{op} raised to the power~$e$.
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Currently assumes that $e \geq 0$.
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Note that for any input \code{op}, \code{rop} is set to~$1$
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whenever $e = 0$.
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void fq_pow_ui(fq_t rop, const fq_t op, const ulong e, const fq_ctx_t ctx)
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Sets \code{rop} the \code{op} raised to the power~$e$.
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Currently assumes that $e \geq 0$.
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Note that for any input \code{op}, \code{rop} is set to~$1$
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whenever $e = 0$.
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*******************************************************************************
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Roots
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*******************************************************************************
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void fq_pth_root(fq_t rop, const fq_t op1, const fq_ctx_t ctx)
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Sets \code{rop} to a $p^{th}$ root root of \code{op1}. Currently,
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this computes the root by raising \code{op1} to $p^{d-1}$ where
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$d$ is the degree of the extension.
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*******************************************************************************
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Output
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*******************************************************************************
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int fq_fprint_pretty(FILE *file, const fq_t op, const fq_ctx_t ctx)
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Prints a pretty representation of \code{op} to \code{file}.
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In the current implementation, always returns~$1$. The return code is
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part of the function's signature to allow for a later implementation to
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return the number of characters printed or a non-positive error code.
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int fq_print_pretty(const fq_t op, const fq_ctx_t ctx)
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Prints a pretty representation of \code{op} to \code{stdout}.
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In the current implementation, always returns~$1$. The return code is
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part of the function's signature to allow for a later implementation to
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return the number of characters printed or a non-positive error code.
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void fq_fprint(FILE * file, const fq_t op, const fq_ctx_t ctx)
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Prints a representation of \code{op} to \code{file}.
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For further details on the representation used, see
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\code{fmpz_mod_poly_fprint()}.
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void fq_print(const fq_t op, const fq_ctx_t ctx)
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Prints a representation of \code{op} to \code{stdout}.
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For further details on the representation used, see
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\code{fmpz_mod_poly_print()}.
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char * fq_get_str(const fq_t op, const fq_ctx_t ctx)
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Returns the plain FLINT string representation of the element
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\code{op}.
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char * fq_get_str_pretty(const fq_t op, const fq_ctx_t ctx)
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Returns a pretty representation of the element~\code{op} using the
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null-terminated string \code{x} as the variable name.
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*******************************************************************************
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Randomisation
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*******************************************************************************
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void fq_randtest(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)
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Generates a random element of $\mathbb{F}_q$.
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void fq_randtest_not_zero(fq_t rop, flint_rand_t state,
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const fq_ctx_t ctx)
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Generates a random non-zero element of $\mathbb{F}_q$.
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void fq_randtest_dense(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)
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Generates a random element of $\mathbb{F}_q$ which has an
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underlying polynomial with dense coefficients.
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*******************************************************************************
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Assignments and conversions
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*******************************************************************************
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void fq_set(fq_t rop, const fq_t op, const fq_ctx_t ctx)
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Sets \code{rop} to \code{op}.
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void fq_set_ui(fq_t rop, const ulong x, const fq_ctx_t ctx)
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Sets \code{rop} to \code{x}, considered as an element of
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$\mathbb{F}_p$.
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void fq_set_fmpz(fq_t rop, const fmpz_t x, const fq_ctx_t ctx)
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Sets \code{rop} to \code{x}, considered as an element of
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$\mathbb{F}_p$.
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void fq_swap(fq_t op1, fq_t op2, const fq_ctx_t ctx)
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Swaps the two elements \code{op1} and \code{op2}.
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void fq_zero(fq_t rop, const fq_ctx_t ctx)
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Sets \code{rop} to zero.
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void fq_one(fq_t rop, const fq_ctx_t ctx)
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Sets \code{rop} to one, reduced in the given context.
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void fq_gen(fq_t rop, const fq_ctx_t ctx)
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Sets \code{rop} to a multiplicative generator for the finite field.
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|
*******************************************************************************
|
||
|
|
|
||
|
|
Comparison
|
||
|
|
|
||
|
|
*******************************************************************************
|
||
|
|
|
||
|
|
int fq_is_zero(const fq_t op, const fq_ctx_t ctx)
|
||
|
|
|
||
|
|
Returns whether \code{op} is equal to zero.
|
||
|
|
|
||
|
|
int fq_is_one(const fq_t op, const fq_ctx_t ctx)
|
||
|
|
|
||
|
|
Returns whether \code{op} is equal to one.
|
||
|
|
|
||
|
|
int fq_equal(const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
|
||
|
|
|
||
|
|
Returns whether \code{op1} and \code{op2} are equal.
|
||
|
|
|
||
|
|
int fq_is_invertible(const fq_t op, const fq_ctx_t ctx)
|
||
|
|
|
||
|
|
Returns whether \code{op} is an invertible element.
|
||
|
|
|
||
|
|
int fq_is_invertible_f(fq_t f, const fq_t op, const fq_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_trace(fmpz_t rop, const fmpz *op, slong len, const fq_ctx_t ctx)
|
||
|
|
|
||
|
|
Sets \code{rop} to the trace of the non-zero element \code{(op, len)}
|
||
|
|
in $\mathbf{F}_{q}$.
|
||
|
|
|
||
|
|
void fq_trace(fq_t rop, const fq_t op, const fq_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_norm(fmpz_t rop, const fmpz *op, slong len, const fq_ctx_t ctx)
|
||
|
|
|
||
|
|
Sets \code{rop} to the norm of the non-zero element \code{(op, len)}
|
||
|
|
in $\mathbf{F}_{q}$.
|
||
|
|
|
||
|
|
void fq_norm(fq_t rop, const fq_t op, const fq_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_frobenius(fmpz *rop, const fmpz *op, slong len, slong e,
|
||
|
|
const fq_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_frobenius(fq_t rop, const fq_t op, slong e, const fq_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_bit_pack(fmpz_t f, const fq_t op, mp_bitcnt_t bit_size,
|
||
|
|
const fq_ctx_t ctx)
|
||
|
|
|
||
|
|
Packs \code{op} into bitfields of size \code{bit_size}, writing the
|
||
|
|
result to \code{f}.
|
||
|
|
|
||
|
|
void fq_bit_unpack(fq_t rop, const fmpz_t f, mp_bitcnt_t bit_size,
|
||
|
|
const fq_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}.
|