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Classification of bent functions by their Cayley graphs
=======================================================

The ``bent_function_cayley_graph_classification`` module defines:

 * the ``BentFunctionCayleyGraphClassification`` class;
   which represents the classification of the Cayley graphs
   within the extended translation class of a bent function; and
 * the ``BentFunctionCayleyGraphClassPart`` class,
   which represents part of a Cayley graph classification.

AUTHORS:

- Paul Leopardi (2016-08-02): initial version

EXAMPLES:

::

    The classification of the bent function defined by the polynomial x2 + x1*x2.

    sage: from boolean_cayley_graphs.bent_function import BentFunction
    sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
    sage: R2.<x1,x2> = BooleanPolynomialRing(2)
    sage: p = x2+x1*x2
    sage: f = BentFunction(p)
    sage: c = BentFunctionCGC.from_function(f)
    sage: dict(sorted(c.__dict__.items()))
    {'algebraic_normal_form': x0*x1 + x1,
     'bent_cayley_graph_index_matrix': [0 0 1 0]
     [1 0 0 0]
     [0 0 0 1]
     [0 1 0 0],
     'cayley_graph_class_list': ['CK', 'C~'],
     'dual_cayley_graph_index_matrix': [0 0 1 0]
     [1 0 0 0]
     [0 0 0 1]
     [0 1 0 0],
     'weight_class_matrix': [0 0 1 0]
     [1 0 0 0]
     [0 0 0 1]
     [0 1 0 0]}

REFERENCES:

The extended translation equivalence class and the extended Cayley equivalence class
of a bent function are defined by Leopardi [Leo2017]_.
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    Partial classification of the Cayley graphs within the
    extended translation equivalence class of a bent function.

    EXAMPLES:

    ::

        sage: from boolean_cayley_graphs.bent_function import BentFunction
        sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCP
        sage: R2.<x1,x2> = BooleanPolynomialRing(2)
        sage: p = x1+x2+x1*x2
        sage: f = BentFunction(p)
        sage: c1 = BentFunctionCGCP.from_function(f, c_stop=1)
        sage: print(c1)
        BentFunctionCayleyGraphClassPart.from_function(BentFunction(x0*x1 + x0 + x1, c_start=0, c_stop=1))
        sage: latex(c1)
        \text{\texttt{BentFunctionCayleyGraphClassPart.from{\char`\_}function(BentFunction(x0*x1{ }+{ }x0{ }+{ }x1,{ }c{\char`\_}start=0,{ }c{\char`\_}stop=1))}}

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        Constructor from an object or from class attributes.

        INPUT:

        - ``algebraic_normal_form`` -- a polynomial of the type
          returned by ``BooleanFunction.algebraic_normal_form()``,
          representing the ``BentFunction`` whose classification this is.
        - ``cayley_graph_class_list`` -- a list of ``graph6_string`` strings
          corresponding to the complete set of non-isomorphic Cayley graphs of
          the bent functions within the extended translation equivalence class
          of the ``BentFunction`` represented by ``algebraic_normal_form``,
          and their duals, if ``dual_cayley_graph_index_matrix`` is not ``None``,
        - ``bent_cayley_graph_index_matrix`` -- a ``Matrix` of integers,
          which are indices into ``cayley_graph_class_list`` representing the
          correspondence between bent functions and their Cayley graphs.
        - ``dual_cayley_graph_index_matrix`` -- a ``Matrix` of integers,
          which are indices into ``cayley_graph_class_list`` representing the
          correspondence between dual bent functions and their Cayley graphs.
        - ``weight_class_matrix`` -- a ``Matrix` of integers with value 0 or 1
          corresponding to the weight class of each bent function.
        - ``c_start`` -- an integer representing the Boolean vector
          corresponding to the first row of each matrix.

        OUTPUT:

        None.

        EFFECT:

        The current object ``self`` is initialized as follows.

        Each of
        - ``algebraic_normal_form``
        - ``cayley_graph_class_list``
        - ``bent_cayley_graph_index_matrix``
        - ``dual_cayley_graph_index_matrix``
        - ``weight_class_matrix``
        - ``c_start``
        is set to the corresponding input parameter.

        EXAMPLES:

        The partial classification of the bent function defined by the polynomial
        :math:`x_1 + x_2 + x_1 x_2` is copied from `c1` to `c2`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCP
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c1 = BentFunctionCGCP.from_function(f, c_stop=1)
            sage: c2 = BentFunctionCGCP(c1)
            sage: print(c1 == c2)
            True
        r
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        INPUT:

        - ``self`` -- the current object.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCP
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c1 = BentFunctionCGCP.from_function(f, c_stop=1)
            sage: print(c1)
            BentFunctionCayleyGraphClassPart.from_function(BentFunction(x0*x1 + x0 + x1, c_start=0, c_stop=1))
        �.from_function(BentFunction(z
, c_start=z	, c_stop=�)))r%r$�nrows�type�__name__�reprr )r(�c_stopr,r,r-�_repr_�s"
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        Constructor from the ``BentFunction`` ``bentf``.

        INPUT:

        - ``bentf`` -- an object of class ``BentFunction``.
        - ``list_dual_graphs`` -- boolean (default: ``True``). 
          A flag indicating whether to list dual graphs.
        - ``c_start`` -- integer (default: 0).
          The smallest value of `c` to use for extended translates.
        - ``c_stop`` -- integer (default: ``None``).
          One more than largest value of `c` to use for extended
          translates. ``None`` means use all remaining values.
        - ``limited_memory`` -- boolean (default: ``False``).
          A flag indicating whether the classification might be
          too large to fit into memory.
        - ``algorithm`` -- string (default: ``default_algorithm``). 
          The algorithm used for canonical labelling.

        OUTPUT:

        An object of class BentFunctionCayleyGraphClassPart,
        initialized as follows.

        - ``algebraic_normal_form`` is set to ``bentf.algebraic_normal_form()``,
        - ``cayley_graph_class_list`` is set to a list of ``graph6_string`` stings
          corresponding to the complete set of non-isomorphic Cayley graphs
          of the bent functions within the extended translation equivalence
          class of ``bentf`` (and their duals, if ``list_dual_graphs`` is ``True``),
        - ``bent_cayley_graph_index_matrix`` is set to a matrix of indices
          into ``cayley_graph_class_list`` corresponding to these bent functions,
        - ``dual_cayley_graph_index_matrix`` is set to ``None``
          if ``list_dual_graphs`` is ``False``, otherwise it is set to
          a matrix of indices into ``cayley_graph_class_list`` corresponding
          to the duals of these bent functions, and
        - ``weight_class_matrix`` is set to the 0-1 matrix of weight classes
          corresponding to ``bent_cayley_graph_index_matrix``,
        - ``c_start`` is set to smallest value of `c` used for extended translates.

        Each entry ``bent_cayley_graph_index_matrix[c-c_start,b]`` corresponds to
        the Cayley graph of the bent function
        :math:`x \mapsto \mathtt{bentf}(x+b) + \langle c, x \rangle + \mathtt{bentf}(b)`.
        This enumerates all of the extended translates of ``bentf`` having ``c``
        from ``c_start`` to but not including ``c_stop``.

        EXAMPLES:

        A partial classification of the bent function defined by the polynomial
        :math:`x_1 + x_2 + x_1 x_2`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCPart
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c1 = BentFunctionCGCPart.from_function(f,c_start=2,c_stop=4)
            sage: dict(sorted(c1.__dict__.items()))
            {'algebraic_normal_form': x0*x1 + x0 + x1,
            'bent_cayley_graph_index_matrix': [0 1 0 0]
            [0 0 0 1],
            'c_start': 2,
            'cayley_graph_class_list': ['CK', 'C~'],
            'dual_cayley_graph_index_matrix': [0 1 0 0]
            [0 0 0 1],
            'weight_class_matrix': [0 1 0 0]
            [0 0 0 1]}

        A partial classification of the bent function defined by the polynomial
        :math:`x_1 + x_2 + x_1 x_2`, but with list_dual_graphs=False.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c2 = BentFunctionCGCPart.from_function(f,list_dual_graphs=False,c_start=0,c_stop=2)
            sage: dict(sorted(c2.__dict__.items()))
            {'algebraic_normal_form': x0*x1 + x0 + x1,
            'bent_cayley_graph_index_matrix': [0 1 1 1]
            [1 1 0 1],
            'c_start': 0,
            'cayley_graph_class_list': ['C~', 'CK'],
            'dual_cayley_graph_index_matrix': None,
            'weight_class_matrix': [1 0 0 0]
            [0 0 1 0]}
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        Test for equality between partial classifications.

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        INPUT:

        - ``other`` - BentFunctionCayleyGraphClassPart: another partial classification.

        OUTPUT:

        A Boolean value indicating whether ``self`` strictly equals ``other``.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCP
            sage: R2.<x0,x1> = BooleanPolynomialRing(2)
            sage: p = x0*x1
            sage: f1 = BentFunction(p)
            sage: c1 = BentFunctionCGCP.from_function(f1, c_stop=1)
            sage: f2 = BentFunction([0,0,0,1])
            sage: c2 = BentFunctionCGCP.from_function(f2, c_stop=1)
            sage: print(c2.algebraic_normal_form)
            x0*x1
            sage: print(c1 == c2)
            True
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    Classification of the Cayley graphs within the
    extended translation equivalence class of a bent function.

    EXAMPLES:

    ::

        sage: from boolean_cayley_graphs.bent_function import BentFunction
        sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
        sage: R2.<x1,x2> = BooleanPolynomialRing(2)
        sage: p = x1+x2+x1*x2
        sage: f = BentFunction(p)
        sage: c1 = BentFunctionCGC.from_function(f)
        sage: print(c1)
        BentFunctionCayleyGraphClassification.from_function(BentFunction(x0*x1 + x0 + x1))
        sage: latex(c1)
        \text{\texttt{BentFunctionCayleyGraphClassification.from{\char`\_}function(BentFunction(x0*x1{ }+{ }x0{ }+{ }x1))}}
    z_bent_function.csvz_cg_class_list.csvz
_matrices.csvc
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        Constructor from an object or from class attributes.

        INPUT:

        - ``sobj`` -- BentFunctionCayleyGraphClassification: object to copy.

        - ``algebraic_normal_form`` -- a polynomial of the type
          returned by ``BooleanFunction.algebraic_normal_form()``,
          representing the ``BentFunction`` whose classification this is.
        - ``cayley_graph_class_list`` -- a list of ``graph6_string`` strings
          corresponding to the complete set of non-isomorphic Cayley graphs of
          the bent functions within the extended translation equivalence class
          of the ``BentFunction`` represented by ``algebraic_normal_form``,
          and their duals, if ``dual_cayley_graph_index_matrix`` is not ``None``,
        - ``bent_cayley_graph_index_matrix`` -- a ``Matrix` of integers,
          which are indices into ``cayley_graph_class_list`` representing the
          correspondence between bent functions and their Cayley graphs.
        - ``dual_cayley_graph_index_matrix`` -- a ``Matrix` of integers,
          which are indices into ``cayley_graph_class_list`` representing the
          correspondence between dual bent functions and their Cayley graphs.
        - ``weight_class_matrix`` -- a ``Matrix` of integers with value 0 or 1
          corresponding to the weight class of each bent function.

        OUTPUT:

        None.

        EFFECT:

        The current object ``self`` is initialized as follows.

        Each of
        - ``algebraic_normal_form``
        - ``cayley_graph_class_list``
        - ``bent_cayley_graph_index_matrix``
        - ``dual_cayley_graph_index_matrix``
        - ``weight_class_matrix``
        is set to the corresponding input parameter.

        EXAMPLES:

        The classification of the bent function defined by the polynomial
        :math:`x_1 + x_2 + x_1 x_2` is copied from `c1` to `c2`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c1 = BentFunctionCGC.from_function(f)
            sage: c2 = BentFunctionCGC(c1)
            sage: print(c1 == c2)
            True
        r
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        Sage string representation.

        INPUT:

        - ``self`` -- the current object.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c1 = BentFunctionCGC.from_function(f)
            sage: print(c1)
            BentFunctionCayleyGraphClassification.from_function(BentFunction(x0*x1 + x0 + x1))
        r/r0)r2r3r4r )
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        Read a Cayley graph class list from a csv file.

        The csv file is assumed to be created by the method
        save_cg_class_list_as_csv().

        INPUT:

        - ``file_name`` -- the name of the csv file.

        OUTPUT:

        A list of Cayley graphs in graph6_string format.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf2 = BentFunction([1,0,1,1])
            sage: c2 = BentFunctionCGC.from_function(bf2)
            sage: csv_name = tmp_filename(ext=".csv")
            sage: c2.save_cg_class_list_as_csv(csv_name)
            sage: cgcl_saved = BentFunctionCGC.cg_class_list_from_csv(csv_name)
            sage: print(cgcl_saved == c2.cayley_graph_class_list)
            True
            sage: os.remove(csv_name)

        rYN)�open�csv�
DictReader�append)re�	file_name�cg_list�csv_file�reader�rowr,r,r-�cg_class_list_from_csves#





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        Read three matrices from a csv file.

        The csv file is assumed to be created by the method
        save_matrices_as_csv().

        INPUT:

        - ``dim`` -- integer: the dimension of the bent function.
        - ``file_name`` -- the name of the csv file.

        OUTPUT:

        A tuple of matrices,

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf2 = BentFunction([1,1,0,1])
            sage: dim = bf2.nvariables()
            sage: c2 = BentFunctionCGC.from_function(bf2)
            sage: csv_name = tmp_filename(ext=".csv")
            sage: c2.save_matrices_as_csv(csv_name)
            sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
            sage: print(c2.bent_cayley_graph_index_matrix == ci_matrix)
            True
            sage: print(c2.dual_cayley_graph_index_matrix == di_matrix)
            True
            sage: print(c2.weight_class_matrix == wc_matrix)
            True
            sage: os.remove(csv_name)

        TESTS:

            Test the case where list_dual_graphs==False and dual_cayley_graph_index_matrix is None.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf = BentFunction([1,1,0,1])
            sage: dim = bf.nvariables()
            sage: c = BentFunctionCGC.from_function(bf, list_dual_graphs=False)
            sage: csv_name = tmp_filename(ext=".csv")
            sage: c.save_matrices_as_csv(csv_name)
            sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
            sage: print(c.bent_cayley_graph_index_matrix == ci_matrix)
            True
            sage: print(c.dual_cayley_graph_index_matrix == di_matrix)
            True
            sage: print(c.weight_class_matrix == wc_matrix)
            True
            sage: os.remove(csv_name)
        r7�dual_cayley_graph_indexNrqro�bent_cayley_graph_indexr)r�r�r��
fieldnamesr
�int)
rerir�rjr�r�r��	ci_matrix�	wc_matrix�	di_matrixr�rqror,r,r-�matrices_from_csv�s*@








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���
z7BentFunctionCayleyGraphClassification.matrices_from_csvc	CsZt�
|
|j�
}|��}|�|
|j�
}|��}|�||
|j�\}}}||||||d
�S)a�
        Constructor from three csv files.

        INPUT:

        - ``file_name_prefix`` -- string: the common prefix to use for file names.

        OUTPUT:

        None.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf2 = BentFunction([1,1,0,1])
            sage: c2 = BentFunctionCGC.from_function(bf2)
            sage: prefix = tmp_filename()
            sage: c2.save_as_csv(prefix)
            sage: c2_saved = BentFunctionCGC.from_csv(prefix)
            sage: print(c2 == c2_saved)
            True
            sage: bent_function_csv_name = prefix + BentFunctionCGC.bent_function_csv_suffix
            sage: os.remove(bent_function_csv_name)
            sage: cg_class_list_csv_name = prefix + BentFunctionCGC.cg_class_list_csv_suffix
            sage: os.remove(cg_class_list_csv_name)
            sage: matrices_csv_name = prefix + BentFunctionCGC.matrices_csv_suffix
            sage: os.remove(matrices_csv_name)

        �r r!r"r#r$)	r�from_csv�bent_function_csv_suffixr r��cg_class_list_csv_suffixrPr��matrices_csv_suffix)	re�file_name_prefixrfr r!rir"r#r$r,r,r-r��s.%
�

�

��







�z.BentFunctionCayleyGraphClassification.from_csvTFcCstj
|
||d
�}||�
S)a<
        Constructor from the ``BentFunction`` ``bentf``.

        INPUT:

        - ``bentf`` -- an object of class ``BentFunction``.
        - ``list_dual_graphs`` -- boolean. a flag indicating
          whether to list dual graphs.
        - ``limited_memory`` -- boolean. A flag indicating
          whether the classification might be too large
          to fit into memory. Default is False.

        OUTPUT:

        An object of class BentFunctionCayleyGraphClassification,
        initialized as follows.

        - ``algebraic_normal_form`` is set to ``bentf.algebraic_normal_form()``,
        - ``cayley_graph_class_list`` is set to a list of ``graph6_string``
          strings corresponding to the complete set of non-isomorphic Cayley graphs
          of the bent functions within the extended translation equivalence
          class of ``bentf`` (and their duals, if ``list_dual_graphs`` is ``True``),
        - ``bent_cayley_graph_index_matrix`` is set to a matrix of indices
          into ``cayley_graph_class_list`` corresponding to these bent functions,
        - ``dual_cayley_graph_index_matrix`` is set to ``None``
          if ``list_dual_graphs`` is ``False``, otherwise it is set to
          a matrix of indices into ``cayley_graph_class_list`` corresponding
          to the duals of these bent functions, and
        - ``weight_class_matrix`` is set to the 0-1 matrix of weight classes
          corresponding to ``bent_cayley_graph_index_matrix``.

        Each entry ``bent_cayley_graph_index_matrix[c,b]`` corresponds to
        the Cayley graph of the bent function
        :math:`x \mapsto \mathtt{bentf}(x+b) + \langle c, x \rangle + \mathtt{bentf}(b)`.
        This enumerates all of the extended translates of ``bentf``.

        EXAMPLES:

        The classification of the bent function defined by the polynomial
        :math:`x_1 + x_2 + x_1 x_2`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c3 = BentFunctionCGC.from_function(f)
            sage: dict(sorted(c3.__dict__.items()))
            {'algebraic_normal_form': x0*x1 + x0 + x1,
            'bent_cayley_graph_index_matrix': [0 1 1 1]
            [1 1 0 1]
            [1 0 1 1]
            [1 1 1 0],
            'cayley_graph_class_list': ['C~', 'CK'],
            'dual_cayley_graph_index_matrix': [0 1 1 1]
            [1 1 0 1]
            [1 0 1 1]
            [1 1 1 0],
            'weight_class_matrix': [1 0 0 0]
            [0 0 1 0]
            [0 1 0 0]
            [0 0 0 1]}

        TESTS:

        The classification of the bent function defined by the polynomial
        :math:`x_1 + x_2 + x_1 x_2`, but with list_dual_graphs=False.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c4 = BentFunctionCGC.from_function(f,list_dual_graphs=False)
            sage: dict(sorted(c4.__dict__.items()))
            {'algebraic_normal_form': x0*x1 + x0 + x1,
            'bent_cayley_graph_index_matrix': [0 1 1 1]
            [1 1 0 1]
            [1 0 1 1]
            [1 1 1 0],
            'cayley_graph_class_list': ['C~', 'CK'],
            'dual_cayley_graph_index_matrix': None,
            'weight_class_matrix': [1 0 0 0]
            [0 0 1 0]
            [0 1 0 0]
            [0 0 0 1]}
        )rgrh)rrw)rerfrgrh�cpr,r,r-rw sa


�z3BentFunctionCayleyGraphClassification.from_functionNcCs�
tj
|
|d
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}|��
d}t||�
}|j}t|�
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}|j�	�|kr2t
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||�}
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||�}nd}t
||�}tt|�
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�
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}|||<qw|j��}t|�
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||f<||j||f|||f<|j||f|||f<q�q�qa|��}|��
|��
||||
||d�S)a�
        Constructor from saved class parts.

        INPUTS:

        - ``prefix_basename`` -- string. The prefix to use with mangled_name()
          to obtain the file names of the saved class parts.
        - ``dir`` -- string, optional. The directory where the parts
          are located. Default is None, meaning the current directory.
        - ``limited_memory`` -- boolean, default is False.
          A flag indicating whether the classification might be too large to
          fit into memory.

        OUTPUT:

        An object of class BentFunctionCayleyGraphClassification,
        constructed from the saved class parts.

        EXAMPLES:

        A classification of the bent function defined by the polynomial
        :math:`x_1 + x_2 + x_1 x_2`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassPart as BentFunctionCGCPart
            sage: import os.path
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: prefix = tmp_filename()
            sage: prefix_dirname = os.path.dirname(prefix)
            sage: prefix_basename = os.path.basename(prefix)
            sage: for row in range(4):
            ....:     c = BentFunctionCGCPart.from_function(f, c_start=row,c_stop=row+1)
            ....:     part_prefix = prefix_basename + "_" + str(row)
            ....:     c.save_mangled(
            ....:         part_prefix,
            ....:         dir=prefix_dirname)
            sage: cl1 = BentFunctionCGC.from_parts(
            ....:    prefix_basename,
            ....:    dir=prefix_dirname)
            sage: cl1.report(report_on_matrix_details=True)
            Algebraic normal form of Boolean function: x0*x1 + x0 + x1
            Function is bent.
            <BLANKLINE>
            Weight class matrix:
            [1 0 0 0]
            [0 0 1 0]
            [0 1 0 0]
            [0 0 0 1]
            <BLANKLINE>
            SDP design incidence structure t-design parameters: (True, (1, 4, 1, 1))
            <BLANKLINE>
            Classification of Cayley graphs and classification of Cayley graphs of duals are the same:
            <BLANKLINE>
            There are 2 extended Cayley classes in the extended translation class.
            <BLANKLINE>
            Matrix of indices of Cayley graphs:
            [0 1 1 1]
            [1 1 0 1]
            [1 0 1 1]
            [1 1 1 0]
            sage: for row in range(4):
            ....:     part_prefix = prefix_basename + "_" + str(row)
            ....:     BentFunctionCGCPart.remove_mangled(
            ....:         part_prefix,
            ....:         dir=prefix_dirname)
        )
�dirz_[0-9]*.sobjr
r7r8Nr�)r�mangled_name�glob�sortrr rrPr"�ncolsr]rrr
r#rTrW�dictr!rZr1r%r$rarbrc)reZprefix_basenamer�rhZmangled_part_prefixZfile_name_listZpart_nbr�partr rfrirjrkr"rgr#r$Zwhole_cg_indexZ
part_cg_indexrsrlZpart_crqror!r,r,r-�
from_parts�s~M

�






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�z0BentFunctionCayleyGraphClassification.from_partscCsH|
d
urdS|j|
jko#|j
|
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ko#|j|
jko#|j|
jko#|j|
jkS)a�
        Test for equality between classifications.

        WARNING:

        This test is for strict equality rather than mathematical equivalence.

        INPUT:

        - ``other`` - BentFunctionCayleyGraphClassification: another classification.

        OUTPUT:

        A Boolean value indicating whether ``self`` strictly equals ``other``.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R2.<x0,x1> = BooleanPolynomialRing(2)
            sage: p = x0*x1
            sage: f1 = BentFunction(p)
            sage: c1 = BentFunctionCGC.from_function(f1)
            sage: f2 = BentFunction([0,0,0,1])
            sage: c2 = BentFunctionCGC.from_function(f2)
            sage: print(c2.algebraic_normal_form)
            x0*x1
            sage: print(c1 == c2)
            True
        NFr�rxr,r,r-rzs!


�
�
�
�z,BentFunctionCayleyGraphClassification.__eq__c
sHt|j
�
}
|j}t|�
��f
d
d�t|
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D�
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�
f
dd�t|
�
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}|S)a�
        Obtain a representative bent function corresponding to each extended Cayley class.

        INPUT:

        - ``self`` -- the current object.

        OUTPUT:

        A list of tuples `(i_n,j_n)`, each of which is the first index into
        the matrix `self.bent_cayley_graph_index_matrix` that contains the entry `n`.
        The first index is determined by ``argwhere``.

        EXAMPLES:

        The result for the bent function defined by the polynomial :math:`x_1 + x_2 + x_1 x_2`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R2.<x1,x2> = BooleanPolynomialRing(2)
            sage: p = x1+x2+x1*x2
            sage: f = BentFunction(p)
            sage: c = BentFunctionCGC.from_function(f)
            sage: c.first_matrix_index_list()
            [(0, 0), (0, 1)]
        c
sg|]}
t�|
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�qSr,)
r�r@�index)
�ci_arrayr,r-rGhs
��zQBentFunctionCayleyGraphClassification.first_matrix_index_list.<locals>.<listcomp>c
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�qS)r
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�z=BentFunctionCayleyGraphClassification.first_matrix_index_listc
sN
	
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}|���|��}d|�
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        Print a report on the classification.

        The report includes attributes and various computed quantities.

        INPUT:

        - ``self`` -- the current object.
        - ``report_on_matrix_details`` -- Boolean (default: False).
           If True, print each matrix.
        - ``report_on_graph_details`` -- Boolean (default: False).
           If True, produce a detailed report for each Cayley graph.

        OUTPUT:

        (To standard output)
        A report on the following attributes of ``self``:

        - ``algebraic_normal_form``
        - ``cayley_graph_class_list``
        - If report_on_matrix_details is ``True``:
           - ``bent_cayley_graph_index_matrix``
           - ``dual_cayley_graph_index_matrix``
             (only if this is not ``None`` and is different from ``bent_cayley_graph_index_matrix``)
           - ``weight_class_matrix``
        - If report_on_graph_details is ``True``:
          details of each graph in ``cayley_graph_class_list``.

        EXAMPLES:

        Report on the classification of the bent function defined by
        the polynomial :math:`x_0 + x_0 x_1 + x_2 x_3`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R4.<x0,x1,x2,x3> = BooleanPolynomialRing(4)
            sage: p = x0+x0*x1+x2*x3
            sage: f = BentFunction(p)
            sage: c = BentFunctionCGC.from_function(f)
            sage: c.report(report_on_matrix_details=True, report_on_graph_details=True)
            Algebraic normal form of Boolean function: x0*x1 + x0 + x2*x3
            Function is bent.
            <BLANKLINE>
            Weight class matrix:
            [0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 1]
            [0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0]
            [1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1]
            [0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1]
            [0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0]
            [0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1]
            [1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0]
            [0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0]
            [0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0]
            [0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 1]
            [1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0]
            [0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0]
            [1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0]
            [1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1]
            [0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0]
            [1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0]
            <BLANKLINE>
            SDP design incidence structure t-design parameters: (True, (2, 16, 6, 2))
            <BLANKLINE>
            Classification of Cayley graphs and classification of Cayley graphs of duals are the same:
            <BLANKLINE>
            There are 2 extended Cayley classes in the extended translation class.
            <BLANKLINE>
            Matrix of indices of Cayley graphs:
            [0 1 0 0 0 1 0 0 0 1 0 0 1 0 1 1]
            [0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0]
            [1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1]
            [0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1]
            [0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0]
            [0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1]
            [1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0]
            [0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0]
            [0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0]
            [0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 1]
            [1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0]
            [0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0]
            [1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0]
            [1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1]
            [0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 0]
            [1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0]
            <BLANKLINE>
            For each extended Cayley class in the extended translation class:
            Clique polynomial, strongly regular parameters, rank, and order of a representative graph; and
            linear code and generator matrix for a representative bent function:
            <BLANKLINE>
            EC class 0 :
            Algebraic normal form of representative: x0*x1 + x0 + x2*x3
            Clique polynomial: 8*t^4 + 32*t^3 + 48*t^2 + 16*t + 1
            Strongly regular parameters: (16, 6, 2, 2)
            Rank: 6 Order: 1152
            <BLANKLINE>
            Linear code from representative:
            [6, 4] linear code over GF(2)
            Generator matrix:
            [1 0 0 0 0 1]
            [0 1 0 1 0 0]
            [0 0 1 1 0 0]
            [0 0 0 0 1 1]
            Linear code is projective.
            Weight distribution: {0: 1, 2: 6, 4: 9}
            <BLANKLINE>
            EC class 1 :
            Algebraic normal form of representative: x0*x1 + x0 + x1 + x2*x3
            Clique polynomial: 16*t^5 + 120*t^4 + 160*t^3 + 80*t^2 + 16*t + 1
            Strongly regular parameters: (16, 10, 6, 6)
            Rank: 6 Order: 1920
            <BLANKLINE>
            Linear code from representative:
            [10, 4] linear code over GF(2)
            Generator matrix:
            [1 0 1 0 1 0 0 1 0 0]
            [0 1 1 0 1 1 0 1 1 0]
            [0 0 0 1 1 1 0 0 0 1]
            [0 0 0 0 0 0 1 1 1 1]
            Linear code is projective.
            Weight distribution: {0: 1, 4: 5, 6: 10}

        REFERENCES:

        - [Leo2017]_.
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��zRBentFunctionCayleyGraphClassification.report.<locals>.graph_and_linear_code_reportz*Algebraic normal form of Boolean function:r7�Functionr�r�zbent.r�zWeight class matrix:z3SDP design incidence structure t-design parameters:r9r:T)
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�z,BentFunctionCayleyGraphClassification.report�(�cCsJ
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        Print a table of Cayley classes in LaTeX format.

        For a given classification, print, in LaTeX format, the table
        of selected properties of the Cayley classes of that classification.

        INPUT:

        - ``self`` -- the current object.
        - ``width`` -- integer (default: 40): the table width.
        - ``rows_per_table`` -- integer (default: 6). 
          The number of rows to include before starting a new table.

        OUTPUT:

        (To standard output.) A table in LaTeX format.

        EXAMPLES:

        Print the table of Cayley classes for the classification of the bent
        function defined by the polynomial :math:`x_0 + x_0 x_1 + x_2 x_3`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R4.<x0,x1,x2,x3> = BooleanPolynomialRing(4)
            sage: p = x0+x0*x1+x2*x3
            sage: f = BentFunction(p)
            sage: c = BentFunctionCGC.from_function(f)
            sage: c.print_latex_table_of_cayley_classes()
            \small{}
            \begin{align*}
            \def\arraystretch{1.2}
            \begin{array}{|cccl|}
            \hline
            \text{Class} &
            \text{Parameters} &
            \text{2-rank} &
            \text{Clique polynomial}
            \\
            \hline
            0 &
            (16, 6, 2, 2) &
            6 &
            \begin{array}{l}
            8t^{4} + 32t^{3} + 48t^{2} + 16t + 1
            \end{array}
            \\
            1 &
            (16, 10, 6, 6) &
            6 &
            \begin{array}{l}
            16t^{5} + 120t^{4} + 160t^{3}
            \,+
            \\
            80t^{2} + 16t + 1
            \end{array}
            \\
            \hline
            \end{array}
            \end{align*}
        cSs\td
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zIBentFunctionCayleyGraphClassification.print_latex_table_of_cayley_classescCs�
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        Print a table comparing Cayley graphs with graphs from Tonchev's codes.

        Tonchev's codes are binary projective two-weight codes
        as published by Tonchev [Ton1996]_, [Ton2007]_.

        INPUT:

        - ``self`` -- the current object.
        - ``width`` -- integer (default: 40): the table width.
        - ``algorithm`` -- string (default: ``default_algorithm``).
          Algorithm used for canonical labelling.

        OUTPUT:

        (To standard output.) A table in LaTeX format.

        .. NOTE::

            The comparison displayed in this table really only makes sense for
            bent functions in 6 dimensions.

        EXAMPLES:

        The classification for the bent function defined by the polynomial
        :math:`x_0 x_1 + x_2 x_3 + x_4 x_5`.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: R6.<x0,x1,x2,x3,x4,x5> = BooleanPolynomialRing(6)
            sage: p = x0*x1 + x2*x3 + x4*x5
            sage: f = BentFunction(p)
            sage: c = BentFunctionCGC.from_function(f) # long time (60 seconds)
            sage: c.print_latex_table_of_tonchev_graphs() # long time (depends on line above)
            \begin{align*}
            \def\arraystretch{1.2}
            \begin{array}{|ccl|}
            \hline
            \text{Class} &
            \text{Parameters} &
            \text{Reference}
            \\
            \hline
            0 & [35,6,16] & \text{Table 1.156 1 (complement)}
            \\
            0 & [35,6,16] & \text{Table 1.156 2 (complement)}
            \\
            1 & [27,6,12] & \text{Table 1.155 1 }
            \\
            \hline
            \end{array}
            \end{align*}

        REFERENCES:

        - [Ton1996]_.

        - [Ton2007]_.
        r�r�z\begin{array}{|ccl|}r�r�r�z\text{Reference}r�c
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��	

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        Plot the matrix attributes to figure files.

        Use ``matrix_plot`` to plot the matrix attributes
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        and ``weight_class_matrix`` to corresponding figure files.

        INPUT:

        - ``self`` -- the current object.
        - ``figure_name`` -- string. 
          The prefix to use in the file names for the figures.
        - ``cmap`` -- string (default: ``'gist_stern'``). 
          The colormap to use with ``matrixplot``.

        OUTPUT:

        (To figure files:
        ``figure_name`` + ``"_bent_cayley_graph_index_matrix.png"``,
        ``figure_name`` + ``"_dual_cayley_graph_index_matrix.png"``,
        ``figure_name`` + ``"_weight_class_matrix.png"``) Plots of the corresponding
        matrix attributes.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import glob
            sage: import os
            sage: bf = BentFunction([1,1,0,1])
            sage: dim = bf.nvariables()
            sage: c = BentFunctionCGC.from_function(bf)
            sage: figure_name = tmp_filename()
            sage: c.save_matrix_plots(figure_name)
            sage: figure_list = glob.glob(figure_name+"*.png")
            sage: for figure in figure_list:
            ....:     print(
            ....:         "_bent_cayley_graph_index_matrix.png" in figure or
            ....:         "_dual_cayley_graph_index_matrix.png" in figure or
            ....:         "_weight_class_matrix.png" in figure)
            ....:     print(os.path.isfile(figure))
            ....:     os.remove(figure)
            True
            True
            True
            True
            True
            True
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�cmap�
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�z7BentFunctionCayleyGraphClassification.save_matrix_plotscCsz|j}d
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        Save the Cayley graph class list to a csv file.

        INPUT:

        - ``file_name`` -- the name of the csv file.

        OUTPUT:

        None.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf2 = BentFunction([1,0,1,1])
            sage: c2 = BentFunctionCGC.from_function(bf2)
            sage: csv_name = tmp_filename(ext=".csv")
            sage: c2.save_cg_class_list_as_csv(csv_name)
            sage: cgcl_saved = BentFunctionCGC.cg_class_list_from_csv(csv_name)
            sage: print(cgcl_saved == c2.cayley_graph_class_list)
            True
            sage: os.remove(csv_name)
        �cayley_graph_indexrYr��
r�)r�rYN)r!r�r��
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��"�z?BentFunctionCayleyGraphClassification.save_cg_class_list_as_csvc	Cs�t|j
�
}|��}d
|}|j}|j}|j}gd�
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t|
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|
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�

q>q8Wd�
dS1smw



Y
dS)a�
        Save the matrices bent_cayley_graph_index_matrix,
        dual_cayley_graph_index_matrix and weight_class_matrix to a csv file.

        INPUT:

        - ``file_name`` -- the name of the csv file.

        OUTPUT:

        None.

        EXAMPLES::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf2 = BentFunction([0,1,1,1])
            sage: dim = bf2.nvariables()
            sage: c2 = BentFunctionCGC.from_function(bf2)
            sage: csv_name = tmp_filename(ext=".csv")
            sage: c2.save_matrices_as_csv(csv_name)
            sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
            sage: print(c2.bent_cayley_graph_index_matrix == ci_matrix)
            True
            sage: print(c2.dual_cayley_graph_index_matrix == di_matrix)
            True
            sage: print(c2.weight_class_matrix == wc_matrix)
            True
            sage: os.remove(csv_name)

        TESTS:

            Test the case where list_dual_graphs=False and dual_cayley_graph_index_matrix is None.

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf = BentFunction([1,0,1,1])
            sage: dim = bf.nvariables()
            sage: c = BentFunctionCGC.from_function(bf, list_dual_graphs=False)
            sage: csv_name = tmp_filename(ext=".csv")
            sage: c.save_matrices_as_csv(csv_name)
            sage: (ci_matrix,di_matrix,wc_matrix) = BentFunctionCGC.matrices_from_csv(dim, csv_name)
            sage: print(c.bent_cayley_graph_index_matrix == ci_matrix)
            True
            sage: print(c.dual_cayley_graph_index_matrix == di_matrix)
            True
            sage: print(c.weight_class_matrix == wc_matrix)
            True
            sage: os.remove(csv_name)

        r7)rorqr�rNr�r�r�)
rr rPr"r#r$r�r�r�r�r�rTr�)r(r�rfrirjr�r�r�r�Zmatrix_filer
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�
���"�z:BentFunctionCayleyGraphClassification.save_matrices_as_csvcCsFt|�
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|j�

|�|
|j�

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d
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        Save the classification as three csv files with a common prefix.

        INPUT:

        - ``self`` -- the current object.
        - ``file_name_prefix`` -- string: the common prefix to use for file names.

        OUTPUT:

        None.

        EXAMPLES:

        ::

            sage: from boolean_cayley_graphs.bent_function import BentFunction
            sage: from boolean_cayley_graphs.bent_function_cayley_graph_classification import BentFunctionCayleyGraphClassification as BentFunctionCGC
            sage: import os
            sage: bf2 = BentFunction([0,0,0,1])
            sage: c2 = BentFunctionCGC.from_function(bf2)
            sage: prefix = tmp_filename()
            sage: c2.save_as_csv(prefix)
            sage: c2_saved = BentFunctionCGC.from_csv(prefix)
            sage: print(c2 == c2_saved)
            True
            sage: bent_function_csv_name = prefix + BentFunctionCGC.bent_function_csv_suffix
            sage: os.remove(bent_function_csv_name)
            sage: cg_class_list_csv_name = prefix + BentFunctionCGC.cg_class_list_csv_suffix
            sage: os.remove(cg_class_list_csv_name)
            sage: matrices_csv_name = prefix + BentFunctionCGC.matrices_csv_suffix
            sage: os.remove(matrices_csv_name)
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