Consider the following variants of the CSIDH squaring problem.

P1. Given $sE, E$ where $s$ is a random ideal class and $E$ is a random curve (reachable from initial $E_0$), find $s^2E$

P2. Given $sE_0$ where $s$ is a random ideal class and $E_0:y^2=x^3+x$ is a fixed initial curve, find $s^2E_0$.

Of course, there's also the decisional variant:

DP1. Given $sE,tE,E$ where $E$ is a random curve as above and $s,t$ are either i.i.d. sample of ideal class or $s^2=t$, decide which.

DP2. Given $sE_0,tE_0$ where $E_0:y^2=x^3+x$ and $s,t$ are either i.i.d. sample of ideal class or $s^2=t$, decide which.

What do we know about the hardness of (D)P1 compared to (D)P2? Are they even comparable?

I would also like to stress that, for another equivalent problem called the inverse problem stated in P3, fixing the base curve to $E_0$ would weakens P3 to P4 as easy as finding a quadratic twist. However, computing quadratic twists would only reduce (D)P2 to itself. So it is not so obvious whether the hardness of (D)P2 is weaken.

P3. Given $sE,E$ find $s^{-1}E$.

P4. Given $sE_0$ find $s^{-1}E_0$.

  • $\begingroup$ P1 and P2 look incomplete: what's the goal? Find s? As for DP1 and DP2, they're trivial as stated: sE = tE iff s = t. $\endgroup$ – Luca De Feo Jan 9 at 11:45
  • $\begingroup$ @LucaDeFeo oops, I've fixed the description. P1 and P2 are to find $s^2 E$ and the constraint in DP1, DP2 are $s^2=t$ rather than $s=t$... my bad $\endgroup$ – Taylor Huang Jan 11 at 2:27

If you have a solver for P1, then it can also solve P2, so the problems are comparable: P2 is easier.

P1 is assumed hard, and is used as the basis of, e.g. https://eprint.iacr.org/2020/1012.

P1 is known to be generically equivalent to P3 (see https://hal.archives-ouvertes.fr/hal-02373179 and https://eprint.iacr.org/2020/1012), however P2 is probably not equivalent to P4: the reduction does not work for these more constrained problems, and P4 is known to be easy, whereas P2 is likely not.

  • $\begingroup$ hi, thanks for your reply. but I think that P1 and P2 are only comparable in their worst-case hardness? since here $s$ is drawn randomly form Cl(O), we are talking about their average hardness. is it still comparable? $\endgroup$ – Taylor Huang Jan 27 at 7:45
  • $\begingroup$ just for clarification, by ''worst-case'' I meant that an P1 solver are required to solve for s^2E for each E that is given to him, while in the ''average-case'' such solver are only required to solve for s^2 E for most E. The main difference is that, in the average-case hardness of P1, we require s^2 E to not only be hard to solve at some instances of E but also that it is hard to solve for most E. Therefore, a reduction algorithm from P2 to P1 couldn't query the P1 solver on a fixed curve E0. $\endgroup$ – Taylor Huang Jan 28 at 2:40
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    $\begingroup$ So what counts is where E is drawn from, not where s is drawn from. But, yes, you're right, I'm talking about worst case. $\endgroup$ – Luca De Feo Jan 28 at 9:03
  • $\begingroup$ oh sure, what makes the difference was E. I guess I had a typo. $\endgroup$ – Taylor Huang Feb 1 at 3:05

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