In general, most mathematics textbooks I read are formatted in the introduction, lemma, theorem, proof, corollary, and examples format. For these kinds of textbooks, I would generally ponder and try to generate proofs by myself before actually scrutinizing the real proof. If it takes me too long, I would read the first line of the proof or skim through it for inspiration. Usually, such as real analysis, the text is organized in a friendly manner so that most proofs can be naturally deduced from previous theorems and techniques.
However, while reading Introduction to Cryptography and Coding Theory by Wade Trapper, I found that it is almost impossible to use my usual approach because of the following reason:
- The algorithms proposed in the book are so ingenious and creative. I don’t think it is reasonable to generate them by myself without significant work.
- The algorithms themselves are not hard to validate once I read the exposition. For instance, the quadratic sieve method, the index calculus method, and the Miller-Rabin Primality Test can all be proved their correctness using basic concepts in number theory. However, it is almost impossible for me to generate them on my own.
- The book is not organized in the typical format. There are long expositions and everything including the applications, the efficacy and weakness of the algorithms, and historical importance are all condensed into a section. Many times, I’m unsure where to stop and ponder before the text spoils everything.
Similar to Cryptography, I think this problem can also occur in a lot of places in which the text emphasizes procedures and algorithms instead of theorems and proofs. In this case, how should I read the textbook and master the concepts?