Suppose $E$ be an Elliptic Curve over a field $F_q$ and $q=p^n$ where $p=$ prime. We know that the Elliptic Curve group $E(F_q)$ under addition is an Abelian/Commutative Group of order, $\#E(F_q)=q+1-t$ where $|t|\leq2\sqrt{q}$, and the structure of this group is either cyclic or almost cyclic.

Since the point addition and doubling formulas need multiplicative inverse to be computed so how can it be proven that all the elements in $E(F_q)$ have multiplicative inverse?

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    $\begingroup$ It sounds like you are confusing two groups. The formulas for addition/doubling are about the group of the elliptic curve. The required inverses are in the multiplicative group of non-zero elements of $F_q$. Those inverses exist by virtue of $F_q$ being a field. $\endgroup$ – Jyrki Lahtonen Apr 6 '15 at 12:29
  • $\begingroup$ But if I am not wrong the point addition and doubling require the elements of the elliptic curve group and generate a point in that group. So it is all about the elements of the elliptic curve group. $\endgroup$ – user110219 Apr 6 '15 at 12:34
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    $\begingroup$ Yes, the points you are adding are on the elliptic curve. But the formulas use the coordinates of those points. And the coordinates are elements of the finite field. $\endgroup$ – Jyrki Lahtonen Apr 6 '15 at 12:42
  • $\begingroup$ Oh God, I mixed the point with coordinate. Thank you and now it is clear to me. $\endgroup$ – user110219 Apr 6 '15 at 13:02
  • $\begingroup$ No problem. Glad to hear that the problem was resolved. $\endgroup$ – Jyrki Lahtonen Apr 7 '15 at 10:59

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