Just started to research more into Monero recently. I was one of the people who used to think Monero was a "bad guy" thanks to the way tech journalists and mainstream media report and portrays it. After making a point to read about it and took the time to research, test and experience it. Monero seems to be the only crypto and chain that fully aligns with the ethos of Satoshi Nakamoto!
A fun project would be: factor Mersenne numbers! Naive trial division -> optimize your code by only trying valid candidates (you’ll learn about quadratic residues, Jacobi symbol etc.) -> You’ll soon realize that trial division only goes so far, motivated to learn new approaches. The P−1 method may impress you, it’s powerful. Enjoy coding that. -> But then, you’ll be frustrated by even stronger opponents here and there beyond M100, which the P−1 method can’t factor. Now you’ll be so ready, even determined, to learn ECM. At this point, you’ll find using elliptic curves is actually not so difficult, because it’s just like P−1. And yay, 20- or 30- digit numbers are no longer your enemies, they’re just small fish, elliptic curves rule!
Before you know it, you’ll have a clear, intuitive vision about “an elliptic curve over a finite field”. Try to understand why ECM works. Try to count the number of points. Everything you experience with ECM is related to ECC too.
I’m not that good; know enough to know that I don’t really know much. Just a hobbyist; posted ℍappy ℍamilton Day! here.