I am trying to understand multiplication in finite fields. I have understood the usual multiplication method for multiplying two polynomials in a finite field. But I am unable to understand the algorithm using arrays, a portion of which from the training notes is attached. Also I have not understood the underlied step in the attached photo. Shifting of cells for multiplication by x is clear....but I have not understood the process if this is not the case. Any help towards understanding this concept would be appreciated enter image description here
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1It's going to save a lot of guesswork for Readers if you include some details of what you do understand. Asking Readers to parse "training notes" and infer what part of finite field multiplication is clear and what part is unclear is a burden. – hardmath Apr 05 '19 at 16:34
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In the attached photograph, I have understood the steps upto the text "what happens if this is not the case". I want to understand the algorithm for multiplication of polynomials in finite field using array method – SAK Apr 05 '19 at 16:37
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The picture shows an example of multiplication, but does not include the value of $f(x)$ as an irreducible polynomial over $\mathbb Z \bmod 3$. See this brief introduction to posting mathematical notation here. – hardmath Apr 05 '19 at 16:41
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$f(x)$ as an irreducible polynomial is given in the second last step as $x^3-x^2+1$ – SAK Apr 05 '19 at 16:43
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Request also explain the third last underlined step – SAK Apr 05 '19 at 16:44
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From cd=$2x^3+x^2+x+(f(x))$ how do we get cd=$2x^3+x^2+x-2f(x)+(f(x))$ – SAK Apr 05 '19 at 16:49
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Let us continue this discussion in chat. – hardmath Apr 05 '19 at 16:50
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The Question currently lacks context. Have a look at this previous Question, Left-multiplication of a finite field element - Matrix representation for a closely related discussion. Addition with elements represented by arrays/components is very straightforward. – hardmath Apr 05 '19 at 18:48