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I have a question about multiplication of two points belong to elliptic curve. I know every think about adding and scalar multiplication but not about multiplication of two points. Is there any method for this?

The reason that I need this is that in paper "Attributed-based encryption for fine-grained access control of encrypted data", we need choose group G and then a point g as the generator of this group. Then in setup algorithm, we should calculate g to the power of t1, while t1 is a number. So it means that we should multiply g into itself for t1 times. If we choose an elliptic field, how could we do this multiplication?

Thanks

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I have a question about multiplication of two points belong to elliptic curve. I know every think about adding and scalar multiplication but not about multiplication of two points.

Actually, when the paper talks about 'multiplying' two points, it's talking about what we more conventionally call 'point addition'; and hence what they call "g to the power to t1", we conventionally call "multiply the point g by the integer t1, that is, g added to itself t1 times".

It's not clear why they decided to refer to the elliptic curve as a multiplicative group and not an additive one. Their paper is from 2006; I had thought that, by that time, the additive convention was fairly prevalent. One possibility is that they looked at the convention of $\mathbb{G}_2$. The group $\mathbb{G}_2$ has an operation which is multiplication over a finite field (at least with any pairing operation you'd actually use), and hence is always written multiplicatively. Perhaps they decided to write the operations in $\mathbb{G}_1$ (the elliptic curve group) to be consistent with it.

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  • $\begingroup$ @tesoke: if you think my answer is helpful, please accept it (and upvote it if you're allowed) $\endgroup$
    – poncho
    May 5 '18 at 0:07
  • $\begingroup$ I did not know anything about this voting and so on. Thanks again. $\endgroup$
    – tesoke
    May 5 '18 at 2:01
  • $\begingroup$ Actually it is rather common in papers on schemes or protocols that involve pairings to use multiplicative notation for the elliptic curve groups. It hepls to make the description much more compact. Also if you consider bilinear groups just as abstract objects independent of how they are implemented it does not matter much which notation you actually use. $\endgroup$
    – DrLecter
    May 5 '18 at 18:31
  • $\begingroup$ Thanks but your comment is vague for me. Did You mean that it does not matter that we propose multiplication or power in our protocols?!!! $\endgroup$
    – tesoke
    May 6 '18 at 0:06
  • $\begingroup$ @tesoke: no, I'm saying it doesn't matter whether we call it multiplication or power $\endgroup$
    – poncho
    May 6 '18 at 2:00

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