Is there a concept of embedding degree for non-pairing based elliptic curves?

Question

From this post, I learned the concept of embedding degree. Intuitively, if embedding degree of an elliptic curve $E(F_p)$ is $k$, it means there is a way to transform points in $E(F_p)$ to $F_{p^k}$. Is the concept of embedding degree only valid for pairing-based elliptic curves, or does the same hold even for non-pairing based elliptic curves?

Answer 1

As pointed out by @yyyyyyy, every curve does have an embedding degree, i.e., there is some $k$ for which $p^k - 1$ is a multiple of $r$, the order one of the subgroups of a curve defined over $\mathbb{F}_p$.

There is a relevant result from Koblitz and Balasubramanian that establishes that the probability that the embedding degree of a random $n$-bit curve of prime order is "small" is vanishingly low: $$ \mathbf{Pr}[l \mid p^k - 1 \text{ and } k \le (\log p)^2] \le c_3 \frac{(\log 2^n)^9(\log \log 2^n)^2}{2^n} \,. $$ As such, only "special" curves that are explicitly designed to have small embedding degree $k$, i.e., pairing-friendly curves, are effectively computable; but the pairing does exist for all of them.