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I have read a lot about the discrete logarithm problem of ecc, but I still do not understand the problem as follows:

We have domain parameters: (p, E, P, n, h), where n is the order group of P.

The process of creating privatekey - publickey is as follows:

  1. Select a random number k under [1, n - 1]
  2. Calculate Q = kP
  3. Return Q is publickey, k is privatekey.

As far as I know, domain paramters and Q will be public. But, suppose I was the eavesdropper, I know P produces 1000 points (as the parameter n), because Q = kP so Q will be a point in point set generated by P, so I can generate order group P and see the position of Q for privatekey k (since k is in [1, n-1]).

For example: P generate 0P, P, 2P, 3P, ... 1000P. I choose k under [1,999] is 120 -> Q = 120P -> position of Q in the order group P is 120 -> k = 120.

So if I know P and Q, then I know privatekey then ??? So where is it safe?

I suppose so because NIST gives you some recommended domain parameters and I think the order group has to be pre-calculated to save on computational cost.

Thank you for everyone's help!

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    $\begingroup$ Consider that in a practical parametrization, $n$ and $k$ have at least 48 decimal digits, rather than 3 in your example. $\endgroup$ – fgrieu Jul 3 '17 at 11:49
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The poblem of computing the private key $d$ from the public key $Q$ is indeed referred to as $ECDLP$. It is crucial that you choose the domain parameters so that the $ECDLP$ is intractable. Concerning your choice of $d$ ... it is too small and can thus be computed efficiently. Imagine $d$ not having 7 Bits like in your example but 192-, 384- or even 512 Bits. You wouldn't get very far using your method (or any other currently known and implementable algorithm).

In order for the $ECDLP$ to be intractable, $d$ must be adequately chosen so that you cannot just brute-force all values for $d$ until you find a $d$ so that $dP = Q$. This is why curves that are considered as secure today use large values for $n$ and $p$ and create a random value $d$ in the range $[1\dots n-1]$.

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    $\begingroup$ Correction: crypto libraries do not set the msbit of $d$ in order to make brute-force search (or even a square-root time search) harder; it's easy to show it doesn't. Now, there are libraries that do set the msbit, but those libraries do it for subtler reasons (dealing with potential side channel attacks) $\endgroup$ – poncho Jul 3 '17 at 11:22
  • $\begingroup$ Oh good to know! I edited the msb part! $\endgroup$ – TrinityTonic Jul 3 '17 at 16:59

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