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Abstract
You will recall that in our example of GF(16) we introduced ‘logarithms’ for non-zero field elements that could be used like conventional logarithms to convert multiplication into addition. This is certainly of practical significance, since addition modulo 15 is easily implemented on a chip, while polynomial multiplication followed by division with remainder (the method used to define multiplication) is both more complicated and slower. On the other hand, the existence of such logarithms is closely linked to a remarkable property possessed by finite fields that is of great theoretical importance. If logarithms exist, then every non-zero element of the field is an integer power of the element with a logarithm 1. In the case of GF(16) every non-zero element is a power of the element we have denoted by 2.
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