gf(2) Calculating the derivative
Since all even powers will create even coefficients (which in gf(2) are zero), they can be discounted.
Finding the derivative is simply masking the odd powers and shifting right one place.
f = b0000000000000000000000000000000000000000000000000001010000010101
f = 0x0000000000001415
f = x12 + x10 + x4 + x2 + 1
odd = Mask for odd bits
odd = b1010101010101010101010101010101010101010101010101010101010101010
odd = 0xaaaaaaaaaaaaaaaa
odd = x63 + x61 + x59 + x57 + x55 + x53 + x51 + x49 + x47 + x45 + x43 + x41 + x39 + x37 + x35 + x33 + x31 + x29 + x27 + x25 + x23 + x21 + x19 + x17 + x15 + x13 + x11 + x9 + x7 + x5 + x3 + x
a = odd & f
a = b0000000000000000000000000000000000000000000000000000000000000000
a = 0x0000000000000000
a = 0
f' = a >> 1
f' = b0000000000000000000000000000000000000000000000000000000000000000
f' = 0x0000000000000000
f' = 0