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Let me ask one question. Maybe, you know an answer. Thanks in advance for any response.

Let's fix an elliptic curve $E$ over the field $\mathbb{Q}$ of rationals without complex multiplication, i.e., its endomorphism ring is just the ring $\mathbb{Z}$ of integers. I want to generate an elliptic curve with the same $j$-invariant over an arbitrary finite prime field $\mathbb{F}_p$ of some cryptographic size, typically ~$256$ bits. And as usual, the points group $E(\mathbb{F}_p)$ must contain a large prime subgroup of little cofactor. Of course, we cannot use the complex multiplication method to find such a curve. Is it possible to find it by iterating only $p$ (not $j$) for a long, but realistic time? Seemingly, in the classical standards on ECC such as NIST's one the two values $p$, $j$ were simultaneously iterated to find a desired curve more rapidly. I need to keep the $j$-invariant, because my curve $E/\mathbb{Q}$ has very interesting properties useful in ECC that are induced to $E/\mathbb{F}_p$ regardless of a concrete prime $p$.

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Yes, this is straightforward enough. Write down you elliptic curve with coefficients in $\mathbb Q$. For any $p$ you can then use the same equation with the coefficients interpreted as elements of $\mathbb F_p$. It is then highly feasible to count the number of points on this curve over the prime field using the Schoof-Elkies-Atkin algorithm and then testing the resulting point count for primality.

Of course if there is a small torsion subgroup of your rational curve, you should divide that subgroup order out of finite field curve group unless it trivialises in the finite field.

ETA: Note that we believe that SEA will run in time $O(\log^4 p)$ and would perhaps expect to run it maybe a couple of hundred times to find a curve with prime order with $p\approx 2^{256}$

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  • $\begingroup$ Daniel S, thanks for your answer. How long does this brute force last? It must be of polynomial time. $\endgroup$ Mar 22 at 12:28
  • $\begingroup$ @DimitriKoshelev See my edit above. The computation is more than practical as a one-off. $\endgroup$
    – Daniel S
    Mar 22 at 12:34
  • $\begingroup$ Maybe, there are some strict mathematical estimates for running time of the given generation process. I need to refer to them in my future article if I decide to write it. $\endgroup$ Mar 22 at 12:41
  • $\begingroup$ The heuristic estimate in SEA is that roughly half of the small primes $\ell$ are Elkies primes with respect to $p$. It might be that this can be established for almost all $p$. If you want to avoid this heutirtic, then the naive Schoof algorithm can be perfromed in time roughly $\log^5p$: en.wikipedia.org/wiki/Schoof%27s_algorithm#Complexity $\endgroup$
    – Daniel S
    Mar 22 at 13:07
  • $\begingroup$ Mathematically speaking, I need any answer to the following question: What is the average proportion of prime order reductions at primes $p$ of ~$256$ bits if $E/\mathbb{Q}$ is a torsion-free fixed elliptic curve? If this proportion is quite large, then we actually can find a desired curve $E/\mathbb{F}_p$ by using the SEA algorithm. $\endgroup$ Mar 22 at 13:18

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