Python -- special method arithmetic using existing class methods

Question

It looks like you are confusing two concepts: strings that represent finite field polynomials and objects of the class gf2pim, which also represents finite field polynomials. You should instantiate a gf2pim object for each polynomial you wish to manipulate, and the operations on gf2pim objects should return other gf2pim objects. Currently you are trying to use the + operator on 2 strings, which is why your __add__ method is not being called. There are also several other problems with your definition of the gf2pim class. I've refactored your code a bit below, although it is still not perfect. I also left out the division methods, but I think what I did should get you on the right track. See section 3.4.8 of this link for more special method names for operator overloading.

import re
class gf2pim(object):#Your classes should generally inherit from object


    def __init__(self, binary):
        '''__init__ is a standard special method used to initialize objects.  Here __init__
        will initialize a gf2pim object based on a binary representation.'''
        self.bin = binary

    @classmethod
    def from_string(cls, string):
        return cls(cls._string_to_binary(string))

    def to_string(self):
        raw = "{0:b}".format(self.bin); raw = str(raw[::-1]); g = [] #reverse binary string for enumeration
        g = [i for i,c in enumerate(raw) if c == '1']
        processed = "x**"+" + x**".join(map(str, g[::-1]))
        if len(g) == 0: return 0 #return 0 if list empty
        return processed  #returns result in gf(2) polynomial form

    @classmethod
    def id(cls, lst):
        """returns modulus 2 (1,0,0,1,1,....) for input lists"""
        return [int(lst[i])%2 for i in range(len(lst))]

    @classmethod
    def _list_to_int(self, lst):
        """converts list to integer for later use"""
        result = self.id(lst)
        return int(''.join(map(str,result)))

    @classmethod
    def _string_to_list(cls, string):
        """performs regex on raw string and converts to list"""
        c = [int(i.group(0)) for i in re.finditer(r'\d+', string)]
        return [1 if x in c else 0  for x in xrange(max(c), -1, -1)]

    @classmethod
    def _string_to_binary(cls, string):
        """converts to base 2; bina,binb are binary values like 110100101100....."""
        x = cls._string_to_list(string)
        a = cls._list_to_int(x)
        bina = int(str(a),2)
        return bina  #

    def __add__(self,other):
        """
        __add__ is another special method, and is used to override the + operator.  This will only 
        work for instances of gf2pim and its subclasses.

        a,b are GF(2) polynomials like x**7 + x**3 + x**0 ...; returns binary string"""
        return gf2pim(self.bin^other.bin)

    def __sub__(self,other):
        """
        __sub__ is the special method for overriding the - operator

        same as addition in GF(2)"""
        return self.add(other)

    def __str__(self):
        return self.to_string()

if __name__ == '__main__':
    a = gf2pim.from_string("x**14 + x**1 + x**0")
    b = gf2pim.from_string("x**6 + x**2 + x**1")
    smadd1 = a+b
    print "smadd1   ",smadd1