For educational purposes I would need to work on an elliptic curve that has a small field, but holds the safety futures of a real curve. Is that possible to have such a curve!?
For example for the curve of secp256k1, is that possible to find small $P$, $N$ and $G$ so that the mini curve does still have safety features of the real curve? By safety features I mean those features that come from the difficulty of solving the discrete logarithm problem. Of course I know that a curve with a small field is vulnerable through brute-force attacks.
To do so, first I read about properties of the prime $P$. It seems that it is a SAFE prime which is equal to $3 \bmod 4$. If I search for a small prime with the same two properties am I getting the correct mini curve?
How are $G$ and $N$ decided then?
I need a curve that lets me do point manipulations (adding, doubling, halving,...) with plotable easy-to-work-with integers, not those 78-digits of secp256k1.