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In the binary case of Elliptic-Curve-Diffie-Hellman (ECDH) exchange protocol the domain parameters are $T = (m,f(x),a,b,G,n,h)$. What is the meaning of parameter $f(x)$?

You may find the definition of the protocol in wikipedia I've gone trough other books and materials but found nothing.

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  • $\begingroup$ please if you think this question is not suitable for the site provide some feedback so that i can improve it or removing it $\endgroup$ – Rodrigo Nov 5 '16 at 22:23
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    $\begingroup$ As it's listed as "in the binary case" on Wikipedia I'm fairly confident that this is the field-defining polynomial. $\endgroup$ – SEJPM Nov 6 '16 at 0:22
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This Wikipedia article is simply not complete. They refer to the NIST Special Publication 800-56A (http://csrc.nist.gov/publications/nistpubs/800-56A/SP800-56A_Revision1_Mar08-2007.pdf), which is now replaced by a new version (http://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-56Ar2.pdf).

It seems the Wikipedia article refers to the domain parameters as indicated on page 41 for ECC CDH, of the form $\left(q,FR,a,b,G,n,h\right)$. The explanation of $FR$ is stated on page 10 of the same document:

"Field Representation indicator (an ECC domain parameter); an indication of the basis used for representing field elements. FR is NULL if the field has odd prime order or if a Gaussian normal basis is used. If a polynomial basis representation is used for a field of order $2^m$, then FR indicates the reduction polynomial (a trinomial or a pentanomial)."

As SEJPM commented, this is the polynomial which defines the field $GF(q)$. Note that up to isomorphism there is only one field $GF(q)$, but there are many ways to represent it (by choosing different $FR$). Hence it is important to specify which $FR$ was chosen for compatibility.

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