First part of question: Pairing friendly curves exist and secp256k1 is not one of those. Does that exclusively mean one cannot find a Weil pairing for secp256k1 curve, or does that only imply that we cannot build useful PBC(pairing based cryptography) applications since the k value(embedding degree) is very large compared to the pairing friendly curves.
I am a beginner in the field and trying to understand these concepts by trial and error, but could not find a definitive answer to this question anywhere else: is it possible to get a weil pairing for secp256k1, or is it just impossible to compute due to large k.
second part of the question: If weil paring exists the problem of finding a discrete logarithm of the curve could be translated into finding discrete logarithm of an integer equation. The difficulty of the problem would not change but I am interested in inter-portability of this problem: from discrete log of curve to discrete log of an integer equation.