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Or the elliptic curve and the generator point are known to everyone?

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Yes.

The curve and generator point are part of the domain parameters and therefore made explicit if you use a named curve.

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  • $\begingroup$ Is the generator is known to everyone in the network or only between the sender and the receiver? $\endgroup$ – Venkatesh Feb 8 '17 at 10:02
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    $\begingroup$ Everybody knows the named curves, and the name of the curve and the domain parameters are considered public. Only the secret (usually $s$) of the private key is required to be secret. The point of the public key, well, I guess you can figure that one out :) - it's not necessarily known to everyone, but it's not secret either. $\endgroup$ – Maarten Bodewes Feb 8 '17 at 10:08
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Yes, base point is public parameter in elliptic curve cryptography. Suppose that, your base point is G and your private key is k. In this case, your public key would be kxG. Calculating kxG (by multiplying k and G) is fast whereas extracting k from public kxG is slow. In other words, you can calculate kxG in seconds wheras extracting k from kxG last almost infinite with brute force attack. There are approaches to calculate kxG fast such as double and add method (aka binary method).

That's the power of the discipline.

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