7.2 due Friday October 30

I was really confused in the first example. I followed them when they said 9^5 was congruent to 1 mod 11, but I got lost when they said that x had to be even so magically it was six. Why do we use that formula if it doesn't give us the answer? If it needs to be even, why do we choose six instead of 4??

The other methods were really wordy and I got lost on the explanation.   I am certain though that hey will be helpful! I saw from the homework that when p gets to be large, guessing a lot of numbers take a lot of time! Brute force is not the most practical way to solved r x. 

6.5-6.7 and section 7.1. Due October 28

Difficult:
How do we get x and y when using RSA for verification (6.6)? Are those arbitrary or are they calculated from the givens? In other encryption algorythms besides the ones discussed in the reading, is there always a trap door? I'm still confused on what a discreet log is and how it's related to mods.


Reflection:
It's neat to see an aspect of mathematics that is still improving and changing, such as methods to factor large primes.  I have often thought of math as something static and unchanging.  2+2 is always going to equal 4.  But things are continuing to evolve helping us solve more problems!

6.4.1-6.4.2 October 26

The example at hte beginning of 6.4.1 on the quadratic sieve didn't make sense.  I didn't understand where they were were getting their numbers.  It seemed like they just fell from the sky.  It started with 9398^2 is congruent to 5^5*19.  After that, I didn't see how they got 19095^2 is congruent to 2^2*5*11*13*19 all (mod 3837523). It was neat to see how the gcd helped us find another factor.

6.4 October 23

Does Fermet's factorization method work for any number?  Can I write any number as the difference of two primes squared? Is there a method of how to find those primes?  Also, I don't understand where the p-1 theorem comes from or why it works.  I was also confused on how to do the steps of the method. 

6.3 due October 21

I had forgotten that there were ways to prove primality without having to find a factor of the prime. I'm still struggling to understand modular exponentiation. I don't why or how Miller Rabin test works 

3.9 Due October 16

Reflection:
I remember solving a problem earlier in the book (or maybe it was a different class) how the only solutions for x^2 was congruent to plus or minus 1 mod something.  It was neat to learn more of the math behind it.

Difficult:
I was trying to follow the example of finding the square root of 5 mod 11.  Is (p+1)/4=3 always a true statement?  Is that why when we are trying to find a square root, like in our previous example we raise 5^3?  I didn't understand why that happens.  Also, does a square root always exist in either the positive or the negative? I also didn't understand the statement at the end about when x=+/-a, or +/-b to find the factors of n becuase p and q are both prime so how do a and b help us?  

6.2 October 14th

It was neat to see how continued fractions can help with factoring.  Although I don't know how to go from continued fractions to finding the factors.  It was really interesting to see how we can use some of the same types of attacks from earlier in the semester like a plain text attack.  It seems like every attack on RSA comes from someone who wasn't thinking ahead.  It surprises me that someone using this complicated of a system would choose an e that wasn't long enough or pick a message that was too short or pick q and p that are too close together. In general, I understood generally why each approach could work, but in practice, I don't understand how the math works and how to use those weaknesses to crack RSA. 

3.12 due October 12

I never knew that you could calculate fractional approximations for irrational numbers. I thought they just happened to find that 22/7 was a very accurate estimation. It's neat to see the process of how to calculate those approximations.

I didn't understand the relationship between the continued fractions and the greatest common factor. I also didn't catch how they found the fractions 1, 10/9, 2461/2215, 2471/2224 from 12345/11111. 

6.1 Due October 9

Reflection:  It was great to understand RSA a little more!  I hear it mentioned in class all the time so it was great to study it more.  It was neat to see the quadratic formula being used to find the roots.  It's neat that the formula learned in 9th grade can still prove useful to us.

Difficult:  I still don't quite understand how Eve knows half of the components to decrypt the message but she can't figure it out with the computers we have today.  I don't get how all the different parts work together, I think I got lost in all the variables.  

3.6-3.7 due October 7th

I'm not understanding what this symbol is meaning. 

I'm also struggling to understand what is and what isn't a primitive root. The examples it gave in the book for three squared makde sense, but I don't see why three cubed isn't a primitive root as well. 

I think it's so neat how we can use mods to figures out of a number is prime! Who knew this little quirky beatbox looking at groups of numbers could rub so useful! 

Test review October 2nd

--Which topics and ideas do you think are the most important out of
those we have studied?

-- it's been interesting to learn about the strengths and weaknesses of each cipher and when they are used in real life settings. It makes the math more applicable. 

--What kinds of questions do you expect to see on the exam?

-- I'm expecting to see different plain texts and cipher texts and then being asked to decrypt them. Also I'm expecting to answer questions about the different types of encryption methods 

--What do you need to work on understanding better before the exam?

-- I'm still struggling to understand all the stuff about polynomials. I don't know how to find the generator and then how to write everything in terms of a generator.