Manipulating elements with hardware

Abstract

You have already had some exposure to hardware appropriate for handling finite field elements. This is particularly true in the case of the CRC generator/checker circuit. By making a few modifications to the basic CRC checker such as removing the input, then for x⁴ + x³ + 1= 0 we have the circuit of Fig. 13.1. If this circuit is preloaded with a non-zero value then successive clocks will repeatedly cycle it through the sequence of field elements. By preloading with 2, we generate the sequence

2, 4, 8, 9, 11, 15, 7, 14, 5, 10, 13, 3, 6, 12, 1, 2, 4, … which is consistent with increasing powers of α for this field.