I need to find the inverse of generator of finite field $\mathbb{F}_{2^4}$ with irreducible polynomial , $f(x)=x^4+x+1$ i.e. if $g=0010$ is the generator of this field then how to find $g^{-1}$?
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What do you mean with $g=0010$? – Bernard May 20 '15 at 17:56
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What Robert Israel writes (+1). If you want to learn a bit more about doing arithmetic operations in this field take a look at this Q&A I prepared for referrals like this. Your $g$ is denoted $\gamma$ there, so $g^{-1}=g^{14}$ is $\gamma^{14}=\gamma^3+1=1001$. – Jyrki Lahtonen May 20 '15 at 18:05
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@Jyrki Lahtonen :thank you very much,it helped me to understand the logic. – AJAY KUMAR May 20 '15 at 18:24
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@Bernard:actually g=0010 was binary string and a generator of finite field. – AJAY KUMAR May 20 '15 at 18:26
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I think what you mean is: your field $GF_{2^4}$ is represented as $\mathbb F[\alpha]$ where $\mathbb F = \mathbb Z_2$ and $\alpha^4 + \alpha + 1 = 0$, and your generator is $\alpha$ (or it might be $\alpha^2$, depending on how you're representing things). Well, multiply that equation by $\alpha^{-1}$ and you see $\alpha^{3} + 1 + \alpha^{-1} = 0$, so $\alpha^{-1} = \alpha^3 + 1$. A similar idea will work if you're looking for $(\alpha^2)^{-1}$.
Robert Israel
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