how can one find the inverse of an element over the Galois field, GF(2^8)?
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The 255 elements of G(256) different from the null element can be written as , where $\displaystyle \alpha$ is a 'primitive root' of the field. A basic property is that the are periodic of period 255 so that the inverse element of is :..
... as example we consider the GF(256) generated by the polynomial...
(1)
... one 'primitive root' of which is . We have...
(2)
Observing table (2) we can deduce that the inverse of $\displaystyle x$ is , the inverse of is and so one...
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$\displaystyle \chi$ $\displaystyle \sigma$