Incorrect is_primitive_poly attribute of "large" finite fields

[mhostetter]

As discovered in #185, the is_primitive_poly attribute of GF(2^256) isn't correct. It says False, however the specified polynomial is primitive (according to a Google search... can't currently test with galois due to #187).

In [1]: import galois                                                                                                                        

In [2]: GF = galois.GF(2**256, irreducible_poly="x^256 + x^241 + x^178 + x^121 + 1", primitive_element="x", verify=False)                    

In [3]: print(GF.properties)                                                                                                                 
GF(2^256):
  characteristic: 2
  degree: 256
  order: 115792089237316195423570985008687907853269984665640564039457584007913129639936
  irreducible_poly: x^256 + x^241 + x^178 + x^121 + 1
  is_primitive_poly: False
  primitive_element: x

In [4]: poly = GF.irreducible_poly; poly                                                                                                     
Out[4]: Poly(x^256 + x^241 + x^178 + x^121 + 1, GF(2))

In [5]: alpha = GF.primitive_element; alpha                                                                                                  
Out[5]: GF(2, order=2^256)

# The polynomial is primitive because poly(alpha) = 0
In [6]: poly(alpha, field=GF)                                                                                                                
Out[6]: GF(0, order=2^256)

The issue is here. Upon manually debugging those lines, the polynomial evaluation was 1 and not 0, which lead the equality to be False. Likely there is an issue/bug in _function_python().

galois/galois/_fields/_gf2m.py

Lines 28 to 35 in c294e5c

	# Determine if the irreducible polynomial is primitive
	if cls._is_primitive_poly is None:
	# TODO: Clean this up
	coeffs = cls.irreducible_poly.coeffs.view(np.ndarray)
	x = np.array(cls.primitive_element, dtype=cls.dtypes[-1], ndmin=1)
	add = cls._func_python("add")
	multiply = cls._func_python("multiply")
	cls._is_primitive_poly = cls._function_python("poly_evaluate")(coeffs, x, add, multiply, cls.characteristic, cls.degree, cls._irreducible_poly_int)[0] == 0

[mhostetter]

After #189, the above runs as:

In [1]: import galois                                                                                                                                    

In [2]: GF = galois.GF(2**256, irreducible_poly="x^256 + x^241 + x^178 + x^121 + 1", primitive_element="x", verify=False)                                

In [3]: print(GF.properties)                                                                                                                             
GF(2^256):
  characteristic: 2
  degree: 256
  order: 115792089237316195423570985008687907853269984665640564039457584007913129639936
  irreducible_poly: x^256 + x^241 + x^178 + x^121 + 1
  is_primitive_poly: True
  primitive_element: x