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Find the values of \begin{align} (a) \ x^{12}*x^{10} \ as \ a \ sum \ of \ power \ of \ x \\ (b) (x^{2}+1)*(x^{3}+1) \ as \ single \ power \ of \ x \end{align} in $ GF(16) $ from the following table: enter image description here

$$ $$ I have tried as- (a) $ x^{12}*x^{10}=(x^{3}+x^{2}+x+1)*(x^{2}+x+1) $ , but now i can not multiply. Any help is appreciated

MAS
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    Expand as usual $(x^{3}+x^{2}+x+1)*(x^{2}+x+1) = x^5+ x^4+x^3+x^4+x^3+x^2+\ldots$ and simplify using $2 = 0$ and your table. Since $16 = 2^4$ the maximum needed power is $x^3$ – reuns May 04 '17 at 03:46
  • Also the second question is a way to solve the 1st one easily – reuns May 04 '17 at 03:50
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    See here for examples of using discrete logarithm tables of finite fields. Your table appears there as well. The point is that because $x^{15}\equiv1$ you get $$x^{12}\cdot x^{10}=x^{22}=x^{15}\cdot x^7\equiv x^7\equiv ?$$ Similarly, reading the right half of the table gives $$(x^2+1)(x^3+1)\equiv x^8\cdot x^{14}=x^{22}=\cdots.$$ – Jyrki Lahtonen May 04 '17 at 04:32

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