Initial version of Galois field arithmetic in python. This uses log tables.

galois.py

@@ -0,0 +1,156 @@
+#!/usr/bin/python
+#
+# Native python routines for Galois field calculations.
+#
+# These calculations can be done much faster in C/C++ (and with vectorization), but
+# this code helps illustrate how Galois field math works.  Also, it's helpful if
+# you want to do small amounts of calculation without working with native C/C++
+# code.
+#
+
+import random, copy
+
+class GaloisException(Exception):
+    def __init__ (self, value):
+        self.value = value
+    def __str__ (self):
+        return repr(self.value)
+
+class GaloisNumber:
+    '''
+    Class to represent a number in a Galois (finite) field.
+
+    The class supports "normal" syntax for the addition, subtraction, multiplication,
+    division, additive inverse (-), and multiplicative inverse (~) operations.
+    '''
+    def __init__ (self, field, value=0):
+        self.field = field
+        self.assign (value)
+
+    def assign (self, v):
+        '''
+        Assign a new integer value to this Galois field number.  The number must be valid in
+        the field with which the GaloisNumber instance was defined.
+        '''
+        if v > self.field.size:
+            raise GaloisException ("Value {0} is outside field".format (value))
+        self.value = v
+
+    def __add__ (self, other):
+        if self.field != other.field:
+            raise GaloisException ("Field elements from different fields")
+        return GaloisNumber (self.field, self.value ^ other.value)
+
+    def __invert__ (self):
+        return self.field.invert (self)
+
+    def __neg__ (self):
+        # We want a shallow copy so that the field reference remains the same
+        return copy.copy (self)
+
+    def __mul__ (self, other):
+        if self.field != other.field:
+            raise GaloisException ("Field elements from different fields")
+        return self.field.multiply (self, other)
+
+    def __div__ (self, other):
+        if self.field != other.field:
+            raise GaloisException ("Field elements from different fields")
+        return self.field.divide (self, other)
+
+    def __sub__ (self, other):
+        return self + other
+
+    def __eq__ (self, other):
+        if self.field != other.field:
+            raise GaloisException ("Field elements from different fields")
+        return self.value == other.value
+
+    def __repr__ (self):
+        return self.field.fmt (self.value)
+
+class GaloisFieldLog:
+    '''
+    Pure python implementation of Galois (finite) field arithmetic routines.
+
+    There only needs to be one instantiation of the field for a given set of parameters,
+    but elements from different field instances with the same parameters may be mixed.
+    '''
+    field_widths = (4, 8, 12, 16)
+    poly_defaults = {4: 0x13, 8: 0x11d, 12:0x1053, 16: 0x1100b}
+    multiply_test_size = 10000
+    def __init__ (self, bits, primitive_polynomial = None, repr_prefix = 'G', alpha = 1):
+        '''
+        Create a Galois field using log/antilog tables for arithmetic.
+        '''
+        if bits not in self.field_widths:
+            raise GaloisException ("Field widths supported: {0}".format (self.field_widths))
+        self.bits = bits
+        self.size = (1 << bits)
+        self.prim = self.poly_defaults[bits] if not primitive_polynomial else primitive_polynomial
+        self.value_format = repr_prefix + '{:0>' + str(bits / 4) + 'x}'
+        self.alpha = alpha
+        # Set up the log and anti-log tables
+        self.log_tbl = [0] * self.size
+        self.antilog_tbl = [0] * (self.size - 1)
+        b = 1
+        for i in range (self.size - 1):
+            self.log_tbl[b] = i
+            self.antilog_tbl[i] = b
+            b <<= 1
+            if b >= self.size:
+                b ^= self.prim
+
+    def __eq__ (self, other):
+        return self.bits == other.bits and self.prim == other.prim and self.alpha == other.alpha
+
+    def fmt (self, v):
+        return self.value_format.format (v)
+
+    def multiply (self, v1, v2):
+        a = v1.value
+        b = v2.value
+        if a == 0 or b == 0:
+            return GaloisNumber (self, 0)
+        return GaloisNumber (self, self.antilog_tbl[(self.log_tbl[a] + self.log_tbl[b]) %
+                                                    (self.size - 1)])
+
+    def invert (self, v):
+        if v.value == 0:
+            return GaloisNumber(self, 0)
+        elif v.value == 1:
+            return GaloisNumber (self, 1)
+        else:
+            return GaloisNumber (self, self.antilog_tbl[self.size - 1 - self.log_tbl[v.value]])
+
+    def divide (self, v1, v2):
+        return self.multiply (v1, self.invert(v2))
+
+    def self_test (self):
+        mul_identity = GaloisNumber (self, 1)
+        v = GaloisNumber (self)
+        g_0 = GaloisNumber (self, 0)
+        g_1 = GaloisNumber (self, 1)
+        for i in range (self.size):
+            v.assign (i)
+            if i == 0: continue
+            assert v * ~v == mul_identity, "Multiplicative inverse failed at {}".format (i)
+            assert g_0 - v == -v, "Additive inverse failed at {}".format (i)
+            assert v * g_1 == v, "Multiplicative identity failed at {}".format (i)
+        vb = GaloisNumber (self)
+        for a in range (1, self.multiply_test_size):
+            v.assign (random.randint (1, self.size - 1))
+            vb.assign (random.randint (1, self.size - 1))
+            product = v * vb
+            assert product / v == vb, "Multiplication failed for {} * {}".format(v.value,vb.value)
+            assert product / vb == v, "Multiplication failed for {} * {}".format(v.value,vb.value)
+        return True
+
+if __name__ == '__main__':
+    for width in GaloisFieldLog.field_widths:
+        field = GaloisFieldLog (width)
+        if field.self_test ():
+            print "{0} bit field passed!".format (width)
+        g0 = GaloisNumber (field, 5)
+        g1 = GaloisNumber (field, 7)
+        print g0 + g1