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Okay, I have a ECDSA public key $P$. The curve is arbitrary and not necessarily secure.

The finite field of curve, $q$, is a $315$ bit prime. I know the following information regarding key:

  1. The private key, $x$ is of the form $2^t \mod{n},$ $t \in ]0, k[$, where $n$ is the order of subgroup (a 300 bit prime) and $k=2^{225}$.
  2. The CM field discriminant endomorphism of order $5$ is available to me.
  3. The point negation endomorphism, however, does not apply to range.

From 2 and 3 I can deduce that the number of $t$ value to be checked should be $k/5$ because the Discriminant is $\sqrt{-5}$.

I want to know which algorithm can perform best.

Disclaimer: The above problem is a homework question but I had not made any progress regarding this in 40 days, other than the above mentioned cm discriminant, that is why I am asking here to get some hints.

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