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I would like to ask how to modify Pohlig Hellman algorithm if I need to work with polynomials $GF(2^{60})$ I know how this algorithm works with numbers, but I am not able to imagine how to do some operations with polynomials. E.g. if I need to do $g^{\frac{p-1}{q^e}}$. How can I do this with polynomial?

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A field like $GF(2^n)$ is represented by the residue classes of polynomials modulo $f(x)$ where $f \in GF(2)[x]$ is an irreducible polynomial of degree $n$ with binary coefficients. There are $2^n$ such polynomials $$a(x)=a_0+a_1 x+ \cdots+ a_{n-1} x^{n-1}.$$

Addition is by mod 2 addition of coefficients. When you multiply two polynomials, you take the remainder of the product modulo $f(x)$.

You can use square and multiply (see here) to compute the power $$a(x)^k=a(x)^{k_0+2 k_1+ 2^2 k_2+ \cdots+ 2^t k_t},$$ taking care to reduce modulo $f(x)$ from time to time, so the degrees of the squares don't get too large.

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