Given $E:y^{2} = x^{3}+9x$ over $\mathbb{Z}_{71}$, and $A = (0,0), \: B = (1,9)$, I'm asked to find $C=A\oplus B$. I just don't know how the direct sum of two points on an elliptic curve is defined, or if this symbol has a different meaning in the context of EC. Can somebody help me?

  • 2
    $\begingroup$ This is explained in detail at the wikipedia page, see section "Group Law". There are also given very helpful pictures for it, e.g., for the associativity of this new addition. Furthermore compare with your question here. $\endgroup$ – Dietrich Burde Nov 27 '15 at 12:44
  • $\begingroup$ I recommend Silverman and Tate's book Rational Points on Elliptic Curves, which gives a nice foundation for such calculations and important additional topics (such as the relationship to primality testing and factorization). $\endgroup$ – hardmath Nov 27 '15 at 13:48
  • $\begingroup$ If you haven't already done so, I recommend looking through previous Questions on this topic at Math.SE. Note that the "addition" of two points with integer coordinates will often produce fractional coordinates in $\mathbb{Q}$, which is obviated by working over the finite field $\mathbb{F}_{71}$. $\endgroup$ – hardmath Nov 27 '15 at 22:20

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