How to find $E_{p^2}$ of an elliptic curve $E_p$ defined over finite field $F_p$ where $p$ is a prime number?

  • $\begingroup$ Perhaps googling "Hensel's Lemma" can help. $\endgroup$ – DonAntonio Mar 2 '13 at 11:46
  • $\begingroup$ If Ep:Y^2 = X^3 + 2X + 3 where p=7, now how should I find E(p^2)? $\endgroup$ – ANKIT Mar 2 '13 at 12:34

Well, so we have $\,y^2=x^3+2x+3\,$ , so what I had in mind is to check particular cases, say:

$$y=1\Longrightarrow f(x):=x^3+2x+2=0\Longrightarrow x= 3\,\,\,\text{is a root}\,\,\pmod 7$$

Now, as you can see here , since $\,f'(3)\neq 0\,$ (i.e., a simple root), we can define

$$t=-\frac{f(3)}{7}f'(3)^{-1}=-5\cdot 29=-5=2\pmod 7\Longrightarrow s=3+2\cdot 7=17$$

is a root of $\,f(x)\pmod{7^2}\,$ , which means the pair $\,(17,1)\,$ satisfies the ellitpci curve in $\,\Bbb F_{7^2}\,$ ...

Note that the other root modulo $\,7\,$ of this case, which is $\,2\,$ , is a double one, so Hensel's lemma cannot help to lift it modulo $\,49\,$ as $\,f'(2)=0\pmod 7\,$ ...

  • $\begingroup$ Nice deduction Don +1 $\endgroup$ – mrs Mar 2 '13 at 18:18
  • $\begingroup$ @DonAntanio shouldn't it be (17,1) that satisfies the curve? $\endgroup$ – ANKIT Mar 4 '13 at 8:13
  • $\begingroup$ Of course, @ANKIT: I mixedthose two. Thanks. $\endgroup$ – DonAntonio Mar 4 '13 at 10:35
  • $\begingroup$ @DonAntonio thanks very much. $\endgroup$ – ANKIT Mar 4 '13 at 18:04

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