7.2 Computing Discrete Logs

• The hardest part to understand was how the Pohlig-Hellman Algorithm works. It didn't make to much sense to me how we broke up x(p-1/q) into two parts.
• I thought it was interesting to see how we can compute some logarithms using the different algorithms.

6.5-6.7 & 7.1 The RSA Challenge, An Application to Treaty Verification, The Public Key Concept, and Discrete Logarithms

• These sections were pretty interesting. I enjoyed hearing about the challenge to break the RSA code even though the winnings weren't that large. I also thought it was interesting learning about digital signatures, although the book briefly touched on it.
• The hardest thing to learn was about how the digital signature works. Other than that the discrete logarithms weren't too hard to understand but I can see that it'll get much more complicated later.

6.4.1 The Quadratic Sieve

• It's pretty neat that we can find the factors by combining rows in a system of equations mod 2. It was cool to see the small amount of linear algebra because I really enjoy linear algebra.
• The hard part to understand was how to pick the equations in the system of equations. It wasn't completely clear to me to know which equations would be useful (really how to pick them).

6.4 Factoring

• It was interesting to see different ways to factor n. It was interesting that if p-1 or q-1 is a product of all small primes then n is realitively easy to factor.
• I didn't understand very well how the p-1 factoring algorithm works. I was confused on what B! is.

6.3 Primality Testing

• It was interesting to see how to test if a number is composite. I found it interesting that with the Miller-Rabin and the Solovay-Strassen Primality Tests, we can only say that a number is either composite or most likely prime. We can at most say that a number is probably prime.
• The hardest part to understand was why b_k fluctuates between being congruent to 1 mod the factors of n. In the example b1 was congruent to 1 mod 3 and mod 11 by not congruent to 1 mod 17 until it hit b3.

3.10 Legendre and Jacobi Symbols

• This section was kind of confusing when it talked about the different rules for the Legendre symbol. I was confused on how if (a/n)=+1 a could be a square mod n. It took me a second to realize that if n is composite then a doesn't have to be a square of the primes and therefore (a/n)=+1 doesn't necessarily mean that a is a square mod n. Only if n is a prime.
• I thought it was interesting however to find easier ways to find if a is a square mod n.

3.9 Square Rotts Mod n

• In this reading I was confused on how they obtained x is congruent to plus of minus 29 in the Example. But after a while I figured out that -29 is congruent to -4 mod 11. I was trying to figure out how 29 was congruent to +4. This confused me. I also don't see how this would be useful if finding the four roots mod n is computationally equivalent to factoring n.
• It was interesting finding the congruencies of squares.

3.12 Continued Fractions 6.2 Attacks on RSA

• Continued Fractions was interesting in the fact that we could approximate pi very well in a few amounts of iterations. It was also interesting to see that we could use continued fractions to factor n.
• The hard part of the reading was understanding the fine details of the attacks on RSA, especially the timing attacks.

6.1 The RSA Algorithm

• The hardest part of the reading was understanding how knowing n and phi of n gives us p and q (the prime factors of n).
• The most interesting part of the reading was seeing how abstract algebra is used to send a pretty secure message. It's neat to see how prime numbers are so useful.

3.6-3.7 Fermant and Euler & Primitive Roots

• I found the three-pass protocol interesting. I thought it's cool how you can send the key because it's tricky and reminded me of a game.
• I found the proofs for Fermant's and Euler's Theorems hard to understand. I understand the principles but how to prove them is difficult.

3.4-3.5 The Chinese Remainder Theorem & Modular Exponentiation

• I thought that the most interesting part was the technique to simplify the expression x^(a) (mod n). I had seen something similar but this is method is more practical when n is a large number.
• I thought the hardest part to understand was the Chinese Remainder Theorem. I was confused on how to find x when you have multiple congruencies.

Exam 1

• I think the most important topic that we've learned is probably DES since it's the only crypto system that is still used today.
• On the exam I expect to see some substitution ciphers, some shift ciphers, and some problems with the DES like we had on the homework. Hopefully, the equations will be on the test.
• I need to work on understanding finite fields and memorize the techniques for the specific crypto systems.