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Using my understanding of finite field elliptic curve arithmetics, I've come to the conclusion that it is possible to divide a point by a scalar the same was we can multiply a point by a scalar. My method for division of 2 is simply multiply by the inverse of 2, however, I then tried to multiply the resulting point by 2 and I did not get the original point back. What am I missing here?

gtr32x
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1 Answers1

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You can do this, as long as the elliptic curve group has odd order.

In general $E(\Bbb F_q)$ has order $m$ with $(\sqrt q-1)^2\le m\le(\sqrt q+1)^2$ (Hasse's theorem). This order can be calculated by Schoof's algorithm. If $m$ is odd then $Q=[\frac{m+1}2]P$ is a solution of $[2]Q=P$. If $m$ is even then the equation $[2]Q=P$ may or may not be soluble. But, when $m$ is even, this trick will not find a solution.

Angina Seng
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  • Thanks for the answer! The trick does indeed work, I had just missed the appropriate parameter to mod over. – gtr32x Sep 12 '17 at 03:30