2.8-2.11

1. Why is the does the LSB of the output of the trapdoor function on predictable (consecutive, no less) input give a pseudorandom number sequence? Wouldn't it depend on the function or is this characteristic of trapdoor functions?

2. All of this should be prefaced, "if I'm not mistaken..." According to Shannon's theory of entropy and information, a sequence with the highest entropy (and so information content) is a totally random one since it cannot be predicted. So why can a sequence generated from a relatively small number of bits be cryptographically secure? Isn't it impossible for such a sequence to have more information than its inputs? (I'll do you a favor and stay away from the sophomoric pontification: "Is anything really random?")

3.8, 2.5-2.8

1. The most difficult part for me was finding the decryption matrix with a known plaintext attack in section 3.7ish.

2. I was struck by the mention of cryptography in former pop-culture. The chain of secrecy is only as strong as its weakest link. By that, I mean, suppose you have really sensitive information that you are encrypting with RSA-128. That's pretty secure but not impossible to break. It might take some supercomputing resources or some mathematician resources. It then may be the case that one of the ends of the communication link is more vulnerable. If someone has access to the information, maybe they can give that access away. Maybe the going rate for hired goons is lower than mathematicians. Speaking of, mathematicians are generally known (as compared to engineers, say) for their idealism as their world is pure in a sense, perfectly described by certain principles. Devoid of the human element is maybe what I'm getting at. But that's not satisfactory when real security is on the line. I mean, don't some proofs of security rely on the existence of a perfect random oracle, which doesn't exist? Anyway, we can't neglect the human element. That sounds silly.

2.3

1. The most difficult part for me is following using the dot product for probabilities. The reasons follow...

2. It's interesting to me when a mathematical operation is taken out of the context that probably motivated its creation and put in a different one. Of course, we're a long way from geometric interpretations in any advanced math course. If we're looking for intuition (which I am), we might consider the 26-dimensional vector space which W exists in. A higher dot product would mean the vectors are more "coincident", right? As one shifts the vector, one is actually shifting the components into adjacent dimensions. Of course, I am conditioned to think of vector spaces when we use dot products. I am probably shoehorning this concept into an entirely unrelated framework that would not only give no more insight into the problem, it would also destroy creativity.

2.1-2.2, 2.4

1. Running frequency analysis on English plaintext is good and all, but in the real world, not only is English with punctuation encrypted, but all types of binary data, including compressed data (which has more entropy, I think). If something is entropy encoded, is it no longer susceptible to frequency analysis attacks?

2. I like how he said that everyone "knows" that substitution ciphers can be broken by frequency analysis. It's kind of like the budding cryptographer's mantra: "run some frequency analysis on it". Do all languages exhibit this kind of statistical exploitability? Chinese has a few thousand characters and the semantics and meanings are largely contextual. I suppose digrams and trigrams take that into account to a degree, but I wonder what Chinese codes looked like.

Lecture

1. Larrabee's cipher was (self-)proclaimed to be the greatest cipher invented or discovered. With Hogg's improvements. That seems a little weird. Probably the most difficult thing for me here is not falling into the same trap they did. Yeah, I know it's a cop out.

2. I was reading the liberal internet the other day (actually a few years ago), and some people were discussing their disgust with President Bush's wiretapping policy and the administration's appeal to the idea "if you have nothing to hide, why does privacy matter?" I suppose one could make the argument that we (as a society) conflate privacy with secrecy or security or something, but I wonder what their response would be to early Church leaders. Certainly they had a need to keep things secret from their enemies, but I think they are often labelled as aggressors hiding secret plans and not as victims trying to hide their location (or something). On the other hand, they openly suggested bribing politicians to get legislation through, so we can be sure those were different times.

3.2-3.3

1. I took Number Theory Summer term, so most of this is really fresh in my mind. I can never remember the proof that if (a,n)=d and ax=b (mod n), there are exactly d solutions distinct mod n to the congruence. Leaving fractions in congruence relations is new.

2. I was brought up with the doctrine that math was a tool and the motivation for inventing/discovering new math was solving problems. Certainly Newton's development of calculus jived with this. Recently in my math courses, I have heard professors say that math is something where you invent rules and see what ramifications those rules have. I also have had two (old) math professors talk about how the vast majority of dissertations are on such esoteric topics that there is no foreseeable application to any results obtained in the research. These two ideas have impressed me with a bit of cynicism. Am I naive in thinking that these "invented" rules are not arbitrary--that there is some real-world motivation for them? And if we assume there is no application of a set of different topics, what does it matter which one is researched, since it's all mind-games anyway?... Now the moral: When I learned about congruence and divisibility, I wavered between it being exclusively either obvious or (more often) inconsequential. Same with primes. But modern cryptography rests on these concepts and (essentially) truths. Did the mathematicians who developed it see the potential in it? I suppose they didn't, but why did these ideas disseminate among the mathematicians and become fixed in the curriculum?

Introduction

I am a senior majoring in Mathematics and minoring with specialty in Computer Science. Post calculus, I have taken linear algebra, ordinary diff. eq., partial diff. eq., abstract algebra, math 190, combinatorics, number, theory, theory of analysis 1, calc. of several variables, and discrete math (CS 236). I am taking this class because it is a part of my coherent set which also includes combinatorics and number theory. The coherence is that they will all contribute to work in information security. I don't have much experience with any CAS, but I have used Maple before. Dr. Vianey Villamizar is one of the better math instructors I've had. He is really good at exposing the motivation for certain constructions (right word?) like the Fourier transform, so that the material seems more approachable and accessible. If I were unable to come to your office hours, MWF 2-3pm, TTh anytime would work.