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I have a naive question, what is the nomenclature of exponentiation mean in ECC?

I was reading about exponential ElGamal, what does it mean if a generator point $G^x$ ? What does $G * \ldots * G$ actually mean?

I know what point double and point addition are, they are inline with scalar operations with the base point. How can two curve points be multiplied together?

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  • $\begingroup$ They don't. The authors simply have decided to take the points of an elliptic curve to be a multiplicative instead of an additive group. This is a different way of writing what is more commonly known as scalar multiplication. $\endgroup$ – SEJPM Mar 23 '18 at 12:29
  • $\begingroup$ hmm, this is a bit confusing, so in ECDH what does Y =xG represent? $\endgroup$ – user462003 Mar 23 '18 at 12:42
  • $\begingroup$ It probably means the same thing. Which source are you reading? What does its notation/definitions section say? $\endgroup$ – fkraiem Mar 23 '18 at 12:58
  • $\begingroup$ I was reading the basic scheme here: knot.kaist.ac.kr/seminar/archive/46/46.pdf $\endgroup$ – user462003 Mar 23 '18 at 14:09
  • $\begingroup$ The "exponentation" is just a repeated addition of the base point $g$, normally written $[n]g$ for $n$ in $\mathbb{Z}$, negative $n$ use the inverse of the corresponding positive $n$.. $\endgroup$ – Henno Brandsma Mar 23 '18 at 15:01
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In a group, where there is by definition only one operation, exponentiation means repeated application of the group operation, whatever that is. That is, if the group operation is noted $\circ$, $g$ is a group element, and $x$ is a positive integer, $g^x$ is short for $\underbrace{g\circ g\circ\dots\circ g}_\text{$x$ terms}$.

In elliptic curve groups, the operation is the point addition with which you are hopefully familiar.

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