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I am having hard time grasping the math behind the elliptic curve cryptography and have to solve the following problem. Some general explanation is welcome too. Thank you in advance, and the problem is the following:

Given is the elliptic curve: $E:y^2 = x^3 + ax + b$, which is defined over $\mathbb{Z}_p$
How do you show that:
a) any point on $E$ with y-coordinate equal to zero has order $2$.
b) when $a = 0$ and $b\neq0$ , any point on $E$ with x-coordinate equal to zero has order $3$.

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  • $\begingroup$ If you know the definition of the group law in the curve then it is straightforward that any point on it with $\;y=0\;$ has order two. For (b) I guess you must take into account that you're working on characteristic $\;p\;$ . $\endgroup$ – DonAntonio Jan 20 '17 at 22:55
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Hints:

a) A point $P$ has order 2 if $P+P=\mathcal{O}$, and therefore $P=-P$. What is $-P$? Then equate the $y$-coordinates.

b) A point $Q$ has order 3 if $Q+Q+Q=\mathcal{O}$, and therefore $Q+Q=-Q$. Again write $-Q$ in terms of $Q$, and use the addition law to compute $Q+Q$. Then equate the $x$-coordinates of $Q+Q$ and $-Q$.

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