Test Preparation, Due on October 1

• Out of the topics we have studied thus far I think the different cryptosystems are most important. We talked a lot about them and spent considerable time going over how they work and how they are decrypted. I think they are important to know and will offer a nice background for the rest of the course.
• Some questions I expect to see on the exam are short descriptions of concepts. Many of the cryptosystems we covered are hard to test on paper with a simple calculator, but they can be described through words. I am expecting some short answer questions about the DES, AES, Viginere Cipher, Hill Cipher, and the like. I also suspect there to be some questions regarding modular arithmetic and finding inverses.
• One thing I need to work on to better be prepared for the exam is exactly what I wrote about above, understanding the big and little ideas involved with the different cryptosystems. I need to review the modes of operation as well. I feel fairly confident with the math, but my overall knowledge of the systems is lacking.

5.1-5.4, Due on September 29

One aspect that I am curious about is the phrase "highly unlinear." The S-boxes of the Rijndael System are repeatedly described this way and I've heard that phrase used in class as well. Does this phrase have a unique meaning or does it just mean what it sounds like? And why do they throw the highly in there? It seems that being linearity is a black and white topic, without any variation or degrees. I find the idea of multiple rounds to complicate a cryptosystem to be a very good idea. It complicates the message greatly while only complicating the algorithm slightly.
Section 5.2.5 on The Key Schedule was the hardest portion of the reading for me to follow. I can catch the bigger picture about what is happening with the different layers but the key schedule was very confusing. I understand that W(i) is a column vector but I don't know what happens when it is transformed with T(W(i-1)) and I don't understand what the r(i) represents. What exactly is the round key?

Due on September 27

1. The homework assignments have been taking me about 6 hours to do. I generally work with Kadi and Sharon for a few hours on Monday afternoon, and then I work on the remainder of the problems on Tuesday. Any that are still giving me trouble I ask you about enduring office hours on Wednesday. I have felt that the lecture and reading assignments have helped prepare me for them.
2. The homework has contributed the most to my learning in this class so far. It's hard to pick one thing, though, because they all work together for my learning. I get the least out of the reading, but perhaps that's because I do it first so the concepts are new. Lecture is the second most helpful, and trying out the concepts in the homework is the most helpful.
3. I suppose if there was one. thing I would suggest to make this class better it would be more descriptive notes on the board during lecture. I am often a few steps behind you when I'm copying down the notes and when I catch up I can't remember what you said and the notes do not give me any clues either. I just copy it down and worry that I will ask a question you just barely answered while I was copying down the previous section. It's not a big deal at all, but one way that would improve my learning.

3.11, Due on September 24

I really like Abstract Algebra (that's easy to say now) so naturally I enjoyed this section. I enjoy doing multiplication in modular arithmetic because I think it simplifies the numbers greatly. At first I was wondering how the modular arithmetic related to the cryptosystems but then it all made sense when I saw the correlation with the bytes key.  This was a good review of abstract algebra but I think it will ultimately make more sense when I put it to practice.
The hardest part of this section was the last part about the LFSR sequences. My linear algebra was sharp at one time but now it is in need of a review. I also found that last part to be somewhat detached from the rest of the chapter. I didn't see how it related. I also didn't follow all the example. I am curious to know what the connection between 2 to the 8th and the 8-bit bytes are. I know they are related but I'm not sure how.

4.5-4.8, Due on September 22

There were interesting and confusing parts of these sections. The first thing that struck me was that much of the DES breaking (and code breaking in general) is dependent upon memory. "This is possible with a lot of memory" seems to be a common phrase in this book. In this decade we have nearly limitless amounts of memory available (that's my own thought; I could be wrong) which will require the algorithms needed for crytosystems to be much more difficult. I was also interested in the password security section. I assumed passwords were more effective if they were >8 characters long and included symbols because they were harder to guess. Section 4.8 made me think that those types of passwords are harder to guess, but also they are easier to encrypt and harder for Eve to decrypt.
The modes of operation were very difficult to follow. I understand that they are variations of the DES, and Monday's lecture helped me understand the big picture of what was going on but I definitely got lost in the specifics of the different modes. I don't know why they are different or what it would look like to execute them all. I think the keys are still just strings of 1s and 0s.

4.1-4.2 and 4.4, Due on September 20

I enjoyed reading about the competition in the 1970s that eventually resulted in the design of the DES system. It was fun to read about whether IBM put a trapdoor into the system, or if the NSA put their own trapdoor into the system. Who can you trust when you have someone else come up with a crypto system for you?
Speaking of the DES, if I were to summarize all of my confusion into one word, it would be: Diagrams. I did not follow what was going on in the diagrams or the functions of the DES. I think I could follow most of the particulars if I knew what the big ideas were. I understand that some of the plaintext letters go to more than one ciphertext letters (although not all of them do) but I didn't get the big idea. One of my stumbling blocks is the concept of a 'bit.' Bit seems to have a very loose definition. The book says that a bit is a 1 or a 0. When we talk about a 64 bit key, do we mean the key is a 64 digit number composed entirely of 0s and 1s? It seems that sometimes bits refer to letters, but perhaps what they mean is that a letter is transformed into a number, and that number has only 0s and 1s.

2.9-2.11, Due on September 17

One fascinating thing about this section for me was the idea that true randomness essentially only exists in nature. My father works for an electronics company that makes semiconductors so my curiosity was especially piqued when I read that the 'thermal noise from a semiconductor resistor is known to be a good source of randomness' (pg. 41). Computers help with more aspects of cryptography then I ever thought, but they haven't been able to generate true randomness yet.
The hardest part of these sections was the Linear Feedback Shift Register Sequences. I think that the idea is to create a ciphertext out of a one-way function in mod 2, but actually coming up with such a sequence and then decrypting it went over my head. I think that all the x variables are known pieces of the binary ciphertext, but I'm not sure. The matrix in the proposition on page 48 also confused me. This cipher reminds me of the Hill cipher because of the usage of linear algebra, although it appears to be much more complex.

2.5-2.8 and 3.8, Due on September 15

The most interesting thing for me in these sections was learning about the variety of crypto-systems. I was very interested to learn about the varieties of systems that have been invented, and then cracked. It seems that just as much ingenuity, creativity, and intelligence is required to make up a code system as is required to crack one. And I don't just mean cracking it once; I mean figuring out a systematic method for always breaking an encrypted message given the encryption method. When I first read about Kerckhoff's principle it seemed fairly insignificant to me, but now it is increasingly making more sense.
One thing that confused me about the inverse matrices was how the fractions related to their inverses. For example, the example in the book finds that 5 is the inverse of -2 mod 11. Then they make the conclusion that we can replace -1/2 by 5 in mod 11. How did they jump from -2 to -1/2? Another thing that was difficult for me was the ADFGX cipher. I didn't really understand how to formulate the matrices.

2.3, Due on September 13

This method- the Vignere Cipher- was fascinating. It's such a great variation of the substitution cipher that is easy to understand but very difficult to crack. I think it's interesting that two plaintext letters can map to the same ciphertext letter but still there is a unique encryption. Although I didn't understand it entirely, I also found the connection to vectors to be intriguing. One reason this class was attractive to me was that it appeared to be applied mathematics, and applied abstract algebra in particular. The Vignere Cipher is evidence of that application indeed.
The hardest part was forming and multiplying the probability vectors. I don't really understand what the letter frequency numbers mean. For example, the chart on page 17 says "a = .082" but I have no idea what the .082 represents. I know its higher than b=.015 so 'e' appears more frequently than 'b' but until I get some context for the numbers the vectors won't make much sense. (Upon writing that I am thinking that if I added up all the decimals they would sum to 1, meaning that there is a 8.2% chance that any given letter is an 'e'.) The other related hard part for me was figuring out why to multiply A (sub j) and A (sub i). Just in general the probability vectors were the hardest part for me.

Guest Lecturer, Due on September 10

The most interesting thing for me to learn about at the guest lecture was the extent that cryptography was used in the Church. I had no idea the Church leaders used it so much, although it makes sense that they would have a lot of information that they didn't want to be compromised. I was also intrigued by the pseudo-names used in the Doctrine and Covenants. In Alma 37:23 we read "I will prepare unto my servant Gazelem, a stone..." I had wondered what that meant and now it makes sense. (The guest lecturer mentioned in her lecture that Gazelem was a code name for Joseph Smith.) I was also intrigued by the Deseret language. I had not heard of that before.
The lecture wasn't too difficult for me to follow (it wasn't very mathematically dense), although it did leave me with some things to ponder about. I had never thought of using code words to minimize telegram expenses by minimizing words, and I never knew Lewis and Clark used cryptography to send messages back to D.C. Cryptography has been around for a long time and despite being somewhat primitive it has been very effective for a wide variety of organizations. Its uses definitely go beyond military purposes, which I did not think about before this lecture.

3.2-3.3, Due on September 3

1. The hardest part of this section for me was the very first part about the extended Euclidean Algorithm. I remember doing it for smaller numbers with a few 'backward' steps, but I did not follow the longer examples. I was confused with the book's notation about what x (naught) and y (naught) were.  The examples today were helpful and I think I can do it with the second method proposed by the book on the bottom of page 69. Another hard part was the situation when x is raised to a higher power then 1 (pg. 75). I can solve those problems with guessing, but I don't think that's the best approach. 
2. The most interesting part of the reading for me was the addition and multiplication tables in modular arithmetic. I am a highly visual learner and I love tables that compile so much information into a readable format. I like noticing patterns in the tables and using them to better understand exactly what is going on. In Combinatorics I used tables often to remember recursive relationships. 

Sorry this entry was late; I misread the reading assignments. I will not resubmit the assignment due on September 10, but if you want to read it again it's still up on my blog. Thanks.

2.1-2.2 and 2.4, Due on September 10

The most difficult part of this reading for me was understanding the affine functions. I think my knowledge of multiplicative inverses in modular arithmetic is a little rusty and I couldn't recall why 1/9 is the multiplicative inverse of 3 in mod 26. I understood most of the other aspects of affine functions but that one little part about inverses and the need for gcd (alpha, 26) =1 eluded me.
The frequency analysis and its accompanying parts was the most interesting. Looking at the frequency table on page 25 showed me how close some letter frequencies are (for example, h was .061 and r was .060). That type of analysis seems to be only relevant for finding e- the dominant favorite letter- and maybe t and a. I enjoyed reading about the two-letter frequencies and the patterns in the English language that help with the analysis. It made me realize how difficult this could be if a message was intercepted and the language was unknown. The substitution cipher seems much easier to crack if the language is known.