o
    Rh)                     @   s  d Z ddlZddlmZ ddlmZ ddlmZ ddlm	Z	 ddl
mZ ddlmZ dd	lmZmZ dd
lmZ ddlmZ ddlmZ edkrKdgZedkriddlZejd^ZZZeeeefdk rhdZndZdd Zdd Z dd Z!eeddgdG dd de	eZ"e" Z#Z$dS )z.Implementation of :class:`FiniteField` class.     N)GROUND_TYPES)doctest_depends_on)
int_valued)Field)ModularIntegerFactory)SimpleDomain)gf_zassenhausgf_irred_p_rabin)CoercionFailed)public)SymPyIntegerflintFiniteField.)r      c                    sd   t j  tjtjzd W n
 ty   Y dS w  fdd}fdd}||fS )Nr   )NNc                    s.   z| W S  t y    |  Y S w N	TypeErrorx)indexmodnmod a/home/air/sanwanet/backup_V2/venv/lib/python3.10/site-packages/sympy/polys/domains/finitefield.pyctx/   s
   z&_modular_int_factory_nmod.<locals>.ctxc                    
   |  S r   r   cs)r   	nmod_polyr   r   poly_ctx5      
z+_modular_int_factory_nmod.<locals>.poly_ctx)operatorr   r   r   r   OverflowErrorr   r   r    r   )r   r   r   r   r   _modular_int_factory_nmod"   s   r%   c                    sD   t jt|  t| tj fdd}fdd}||fS )Nc                    s*   z | W S  t y    |  Y S w r   r   r   )fctxr   r   r   r   A   s
   
z*_modular_int_factory_fmpz_mod.<locals>.ctxc                    r   r   r   r   )	fctx_polyfmpz_mod_polyr   r   r    H   r!   z/_modular_int_factory_fmpz_mod.<locals>.poly_ctx)r"   r   r   fmpz_mod_ctxfmpz_mod_poly_ctxr(   r$   r   )r&   r'   r(   r   r   _modular_int_factory_fmpz_mod;   s   

r+   c                 C   s   z| | } W n ty   td|  w d\}}}td ur4|  r4d}t| \}}|d u r4t| \}}|d u rAt| |||}d }|||fS )Nz"modulus must be an integer, got %s)NNFT)convertr
   
ValueErrorr   is_primer%   r+   r   )r   dom	symmetricselfr   r    is_flintr   r   r   _modular_int_factoryN   s   

r3   pythongmpy)modulesc                   @   s(  e Zd ZdZdZdZd ZZdZdZ	dZ
dZdZd<ddZedd	 Zed
d Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zd d! Zd"d# Zd=d$d%Zd=d&d'Zd=d(d)Zd=d*d+Z d=d,d-Z!d=d.d/Z"d=d0d1Z#d=d2d3Z$d=d4d5Z%d6d7 Z&d8d9 Z'd:d; Z(dS )>r   a	  Finite field of prime order :ref:`GF(p)`

    A :ref:`GF(p)` domain represents a `finite field`_ `\mathbb{F}_p` of prime
    order as :py:class:`~.Domain` in the domain system (see
    :ref:`polys-domainsintro`).

    A :py:class:`~.Poly` created from an expression with integer
    coefficients will have the domain :ref:`ZZ`. However, if the ``modulus=p``
    option is given then the domain will be a finite field instead.

    >>> from sympy import Poly, Symbol
    >>> x = Symbol('x')
    >>> p = Poly(x**2 + 1)
    >>> p
    Poly(x**2 + 1, x, domain='ZZ')
    >>> p.domain
    ZZ
    >>> p2 = Poly(x**2 + 1, modulus=2)
    >>> p2
    Poly(x**2 + 1, x, modulus=2)
    >>> p2.domain
    GF(2)

    It is possible to factorise a polynomial over :ref:`GF(p)` using the
    modulus argument to :py:func:`~.factor` or by specifying the domain
    explicitly. The domain can also be given as a string.

    >>> from sympy import factor, GF
    >>> factor(x**2 + 1)
    x**2 + 1
    >>> factor(x**2 + 1, modulus=2)
    (x + 1)**2
    >>> factor(x**2 + 1, domain=GF(2))
    (x + 1)**2
    >>> factor(x**2 + 1, domain='GF(2)')
    (x + 1)**2

    It is also possible to use :ref:`GF(p)` with the :py:func:`~.cancel`
    and :py:func:`~.gcd` functions.

    >>> from sympy import cancel, gcd
    >>> cancel((x**2 + 1)/(x + 1))
    (x**2 + 1)/(x + 1)
    >>> cancel((x**2 + 1)/(x + 1), domain=GF(2))
    x + 1
    >>> gcd(x**2 + 1, x + 1)
    1
    >>> gcd(x**2 + 1, x + 1, domain=GF(2))
    x + 1

    When using the domain directly :ref:`GF(p)` can be used as a constructor
    to create instances which then support the operations ``+,-,*,**,/``

    >>> from sympy import GF
    >>> K = GF(5)
    >>> K
    GF(5)
    >>> x = K(3)
    >>> y = K(2)
    >>> x
    3 mod 5
    >>> y
    2 mod 5
    >>> x * y
    1 mod 5
    >>> x / y
    4 mod 5

    Notes
    =====

    It is also possible to create a :ref:`GF(p)` domain of **non-prime**
    order but the resulting ring is **not** a field: it is just the ring of
    the integers modulo ``n``.

    >>> K = GF(9)
    >>> z = K(3)
    >>> z
    3 mod 9
    >>> z**2
    0 mod 9

    It would be good to have a proper implementation of prime power fields
    (``GF(p**n)``) but these are not yet implemented in SymPY.

    .. _finite field: https://en.wikipedia.org/wiki/Finite_field
    FFTFNc                 C   s   ddl m} |}|dkrtd| t|||| \}}}|| _|| _|| _| d| _| d| _|| _	|| _
|| _t| j| _d S )Nr   )ZZz*modulus must be a positive integer, got %s   )sympy.polys.domainsr8   r-   r3   dtype	_poly_ctx	_is_flintzerooner/   r   symtype_tp)r1   r   r0   r8   r/   r   r    r2   r   r   r   __init__   s   zFiniteField.__init__c                 C      | j S r   )rB   r1   r   r   r   tp      zFiniteField.tpc                 C   s4   t | dd }|d u rddlm} || j | _}|S )N	_is_fieldr   )isprime)getattrsympy.ntheory.primetestrI   r   rH   )r1   is_fieldrI   r   r   r   is_Field   s
   zFiniteField.is_Fieldc                 C   s
   d| j  S )NzGF(%s)r   rE   r   r   r   __str__   r!   zFiniteField.__str__c                 C   s   t | jj| j| j| jfS r   )hash	__class____name__r;   r   r/   rE   r   r   r   __hash__   s   zFiniteField.__hash__c                 C   s"   t |to| j|jko| j|jkS )z0Returns ``True`` if two domains are equivalent. )
isinstancer   r   r/   )r1   otherr   r   r   __eq__   s
   


zFiniteField.__eq__c                 C   rD   )z*Return the characteristic of this domain. rN   rE   r   r   r   characteristic   rG   zFiniteField.characteristicc                 C   s   | S )z*Returns a field associated with ``self``. r   rE   r   r   r   	get_field     zFiniteField.get_fieldc                 C   s   t | |S )z!Convert ``a`` to a SymPy object. )r   to_intr1   ar   r   r   to_sympy  s   zFiniteField.to_sympyc                 C   sF   |j r| | jt|S t|r| | jt|S td| )z0Convert SymPy's Integer to SymPy's ``Integer``. zexpected an integer, got %s)
is_Integerr;   r/   intr   r
   r[   r   r   r   
from_sympy
  s
   zFiniteField.from_sympyc                 C   s*   t |}| jr|| jd kr|| j8 }|S )z,Convert ``val`` to a Python ``int`` object.    )r_   r@   r   )r1   r\   avalr   r   r   rZ     s   
zFiniteField.to_intc                 C   s   t |S )z#Returns True if ``a`` is positive. )boolr[   r   r   r   is_positive  s   zFiniteField.is_positivec                 C      dS )z'Returns True if ``a`` is non-negative. Tr   r[   r   r   r   is_nonnegative  rY   zFiniteField.is_nonnegativec                 C   re   )z#Returns True if ``a`` is negative. Fr   r[   r   r   r   is_negative"  rY   zFiniteField.is_negativec                 C   s   | S )z'Returns True if ``a`` is non-positive. r   r[   r   r   r   is_nonpositive&  rG   zFiniteField.is_nonpositivec                 C      |  | jt||jS z.Convert ``ModularInteger(int)`` to ``dtype``. )r;   r/   from_ZZr_   K1r\   K0r   r   r   from_FF*     zFiniteField.from_FFc                 C   ri   rj   )r;   r/   from_ZZ_pythonr_   rl   r   r   r   from_FF_python.  rp   zFiniteField.from_FF_pythonc                 C      |  | j||S z'Convert Python's ``int`` to ``dtype``. r;   r/   rq   rl   r   r   r   rk   2     zFiniteField.from_ZZc                 C   rs   rt   ru   rl   r   r   r   rq   6  rv   zFiniteField.from_ZZ_pythonc                 C      |j dkr| |jS dS z,Convert Python's ``Fraction`` to ``dtype``. r9   Ndenominatorrq   	numeratorrl   r   r   r   from_QQ:     
zFiniteField.from_QQc                 C   rw   rx   ry   rl   r   r   r   from_QQ_python?  r}   zFiniteField.from_QQ_pythonc                 C   s   |  | j|j|jS )z.Convert ``ModularInteger(mpz)`` to ``dtype``. )r;   r/   from_ZZ_gmpyvalrl   r   r   r   from_FF_gmpyD  s   zFiniteField.from_FF_gmpyc                 C   rs   )z%Convert GMPY's ``mpz`` to ``dtype``. )r;   r/   r   rl   r   r   r   r   H  rv   zFiniteField.from_ZZ_gmpyc                 C   rw   )z%Convert GMPY's ``mpq`` to ``dtype``. r9   N)rz   r   r{   rl   r   r   r   from_QQ_gmpyL  r}   zFiniteField.from_QQ_gmpyc                 C   s,   | |\}}|dkr| | j|S dS )z'Convert mpmath's ``mpf`` to ``dtype``. r9   N)to_rationalr;   r/   )rm   r\   rn   pqr   r   r   from_RealFieldQ  s   zFiniteField.from_RealFieldc                 C   s,   dd | j | j| fD }t|| j| j S )z7Returns True if ``a`` is a quadratic residue modulo p. c                 S      g | ]}t |qS r   r_   .0r   r   r   r   
<listcomp>[      z)FiniteField.is_square.<locals>.<listcomp>)r?   r>   r	   r   r/   )r1   r\   polyr   r   r   	is_squareX  s   zFiniteField.is_squarec                 C   sz   | j dks	|dkr|S dd | j| j| fD }t|| j | jD ]}t|dkr:|d | j d kr:| |d   S q dS )zSquare root modulo p of ``a`` if it is a quadratic residue.

        Explanation
        ===========
        Always returns the square root that is no larger than ``p // 2``.
        ra   r   c                 S   r   r   r   r   r   r   r   r   i  r   z&FiniteField.exsqrt.<locals>.<listcomp>r9   N)r   r?   r>   r   r/   lenr;   )r1   r\   r   factorr   r   r   exsqrt^  s   zFiniteField.exsqrt)Tr   ))rR   
__module____qualname____doc__repaliasis_FiniteFieldis_FFis_Numericalhas_assoc_Ringhas_assoc_Fieldr/   r   rC   propertyrF   rM   rO   rS   rV   rW   rX   r]   r`   rZ   rd   rf   rg   rh   ro   rr   rk   rq   r|   r~   r   r   r   r   r   r   r   r   r   r   r   l   sN    X


	








)%r   r"   sympy.external.gmpyr   sympy.utilities.decoratorr   sympy.core.numbersr   sympy.polys.domains.fieldr   "sympy.polys.domains.modularintegerr    sympy.polys.domains.simpledomainr   sympy.polys.galoistoolsr   r	   sympy.polys.polyerrorsr
   sympy.utilitiesr   sympy.polys.domains.groundtypesr   __doctest_skip__r   __version__split_major_minor_r_   r%   r+   r3   r   r7   GFr   r   r   r   <module>   s<      