The most difficult part of sections 3.4-3.6 for me was definitely the understanding of Euler's theorem primarily because the book progressed from one step to the next of the proof without what I felt was clear enough explanation or all of the steps to make sure that the notation and understanding of the next steps were clear. The large "pi" symbol, for instance, has always puzzled me and I was hoping for an explanation as to its meaning to no avail. The overall theory seems to be fairly easily understood where fermat's theorem states that a number to the power of a prime minus one is equal to one, but the proofs for that and then Euler's theorem were not clearly understood and did not enlighten the foundations of either. That being said, this is a critical component of public-key cryptography and I'm hoping it will be more clearly understood soon.

The overall theme of these three sections seemed to be finding shortcuts for complicated modular operations. Section 3.4, for instance, resulted in the application of systems for modular arithmetic which seem fairly obvious at first glance but have itneresting applications. Section 3.5 was as simple as its brevity but profoundly useful in reducing the number of operations for calculations of large exponentiation with modular operations which is especially important when designing computer software to tackle these areas. Section 3.6 built upon these ideas to present the foundations for public-key cryptosytems which are huge components for modern cryptography.