Those are very valid questions, and as a (former) professor, I know that most smart (and many 'not smart') students do ask those questions. Lecturers should do

Those are very valid questions, and as a (former) professor, I know that most smart (and many 'not smart') students do ask those questions. Lecturers should do their best to set up the motivation for each topic. This can fail because - There is not enough time. Unfortunately, formal courses have a fixed time, and the lecturer has to cover a certain amount of content in that time [1]. - Sometimes, its not actually possible to answer the 'why' questions with the amount of background that students have (or the lecturer has). The above is also true for books/written down notes as well. For instance, in your example of elliptic curves, the answer to your questions about "why the form y^2 = x^3 + ax + b" requires students to understand algebraic geometry first, where such curves are defined in general (over arbitrary fields etc), rather than over the real field as you have. If you read such a book, you will see why in the general setting, its completely natural to define such curves. I don't know exactly why elliptic curves are good for cryptography, but I would hazard that the reason is perhaps only apparent when you study them in the abstract - hence why you don't see this explanation in your resources. The writer might not even know themselves. [1] Yes yes, this is broken as well. In my ideal university there would be no semesters. Courses will be as short or as long as necessary to naturally end at an appropriate location.