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What's the security of Elliptic Curve Cryptography over finite fields of word-sized characteristic $p\approx2^{50\pm10}$? We are talking about $\Bbb F_q$ where $q=p^k$ for some suitable $k$.

Furthermore, I would like to know about the possibility of pairings over fields of such word-sized characteristics.

References appreciated.

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    $\begingroup$ You might want to take a look at this recent paper which proposes a curve with over the field $\mathbf F_q$ with $q=9767^{19}$. Section 2.2 talks a little about the extension degree. $\endgroup$ – corpsfini Feb 21 at 12:26
  • $\begingroup$ Thanks, this is awesome! However, would you happen to have any reference for characteristic closer to the values I mentioned? Furthermore, I would like some references on pairings over fields of word-sized characteristics too. $\endgroup$ – Rascalniikov Feb 21 at 18:35

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