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In his paper on pairing based cryptography, Menezes claims that for $$ E_1: y^2+y = x^3+x+1 $$ the number of $\mathbb{F}_{2^m}$-points is $2^m +1 - (1+i)^m - (1-i)^m$. Whereas this is clear, it is not clear to me how he reaches this conclusion:

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In addition, I also do not see why it is relevant (and how one sees/arrives at this relevancy) to express $2^m +1 - (1+i)^m - (1-i)^m$ depending on m mod 8.

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This follows from the identity \begin{eqnarray}(1+i)^m+(1-i)^m&=&(\sqrt 2(\exp(i\pi/4))^m+(\sqrt 2(\exp(-i\pi/4))^m\\ &=&2^{m/2}(\exp(im\pi/4)+\exp(-im\pi/4))\end{eqnarray} which by Euler's formula is $$2^{m/2}2\cos(m\pi/4).$$

Now note that $2^{m/2}2=2\sqrt q$ and that $$\cos(m\pi/4)=\cases{1&$m\equiv 0\pmod 8$ \\ 1/\sqrt2&$m\equiv \pm1\pmod 8$\\ 0&$m\equiv\pm2\pmod 8$\\ -1/\sqrt2&$m\equiv\pm3\pmod 8$\\ -1&$m\equiv 4\pmod 8$}.$$

The reason for this expression is that checking the congruence class of $m\pmod 8$ and then computing a power of 2 is a simpler computation than $(1+i)^m+(1-i)^m$. These expressions also lend themselves more readily to confirming the necessary conditions for a pairing-friendly curve of various embedding degrees.

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  • $\begingroup$ Say I would want to do something similar for $E_2:y^2 = x^3-x+1$ over $\mathbb{F}_{3^m}$. How would I proceed? $\endgroup$ Jul 21 at 9:36
  • $\begingroup$ Your comment would be best asked as a follow on question. $\endgroup$
    – Daniel S
    Jul 21 at 9:57

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