Public key Exceeding Mod P? A Clarification Request On The Discrete Logarithm Problem
It is important to note that the Elliptic Curve Discrete Logarithm Problem (ECDLP) is slightly different from the normal Discrete Logarithm Problem (DLP).
In DLP, the ^ operator is actually exponentiation and it is operating on integers so mod p is just the normal modulus. However for ECDLP, ^ is actually elliptic curve point multiplication. This is different from exponentiation and is a much more involved process. You also cannot mod p with a point because they are not integers. While there is a modulo, it is part of the point multiplication process itself.
The compressedness of the parameters you use does not really matter. In the end, the compression is just the encoding of a point. You need to parse this encoding to determine the X and Y coordinates of the point; both uncompressed and compressed allow you to determine them. The uncompressed just gives you the X and Y coordinates directly while the compressed one gives you the X and whether Y is even or odd so that you can compute the correct Y coordinate. When you do the point multiplication, you are not handling the encoded point as an integer and just multiplying it with x. You are doing elliptic curve point multiplication which is way more complicated.