Understanding Square and Cube Roots in secp224k1

So, there are four elliptic curves: secp160k1, secp192k1, secp224k1, and secp256k1. They were all generated together and share common setups. When trying to calculate square and cube roots, the first three work fine with their p-values. But secp224k1? Something's up there with its p-value.

Just to clarify, this formula only applies to the 'p' value, right? What's the point of that? I mean, square roots don't really apply to points on the curve, only to numbers. Plus p is outside the range of scalars (1 to n). Why's that necessary?

Right, the curve equation is y^2=x^3+b. So, to find x or y, you need that. The basic goal is to find a valid point on the curve. Pick an x, figure out y, or vice versa. Seems like it should be straightforward in all cases except secp224k1. What gives?

Can this really help in computing a private key from a public one? Like, could this break ECDSA?

I think you might be mixing up group scalars and field values. x can be zero and still give valid points on the curve if there's a square root to find for y. If y is zero, it's a similar deal, provided x satisfies the curve. But it gets complicated when the parameters don't allow it.

If that's true, then why would anyone create curves where these operations are fast, and then pick secp224k1 to make it trickier? Seems like a weird choice. If we can check if a square root exists easily, why not use the same p-value for all four curves?

There are complex methods for finding cube roots in finite fields. Some simple equations might work, but I doubt this info is published anywhere. The smallest pairs for square and cube roots are interesting, like (p+3)/8 for square roots.