16.5, due on December 5

1. (Difficult): I don't remember why x congruent to y2*y1^-a decrypts for the ElGamal Cryptosystem and I forgot how to encrypt a message to a point using Elliptic Curves. The former I can review myself I think but a review of the latter in class would be helpful.

Also, I didn't understand the end of the ElGamal Digital Signatures when it says n=p-1 and when it talks about arbitrary assigned integers to points.

2. (Reflection): The  Elliptic Curve Diffie-Hellman Key Exchange I got! And elliptic addition/multiplication being analogous to regular multiplication/exponentiation makes some sense. That's good.

This is our last blog. Wow time flies!

16.4, due on December 3

1. (Difficult): What does an elliptic curve mod 2 look like? How does it relate to the graph of our other elliptic curves we've dealt with? (How did we know the intersection P=(0,1) on page 361?)

I didn't quite follow the new law of addition. How does it work?

Also, I think I understand the last example of addition in GF(4), but a picture would be really helpful so I know if I'm getting it our not.

2. (Reflection): It is crazy how math builds on each other. Elliptic curves mod 2 seem like such a crazy idea, but it's just a bunch of simple math concepts combined.

16.3, due on November 30

1. (Difficult): How is elliptic curve factoring like the p-1 factoring algorithm?

And why do we have to invert 599 when computing 8*7!?

And 8!P is not equal to infinity mod 761 because 8! is not a multiple of 777, but why must only multiples of 777 give inifinity mod 761 again; how is this related to cycles mod any number? I need to understand this example (starting on page 357) better to understand the general case I think.

And I have lots of questions about singular curves, so I'll just be brief and say I didn't really understand singular curves.

2. (Reflection): The elliptic curve factorization method seems pretty strong.

Also, so addition and multiplication on using points on an elliptic curve is analogous to multiplication and exponentiation using modular arithmetic. I think that's what the book is saying.

The last statement I think is a cool discovery that the p-1 and division trial methods are both encompassed by the elliptic curve factorization method. (360)

16.2, due on November 28

1. (Difficult): Half the numbers mod p are squares? Really? How do we know that?

Also, how can n*A = infinity? In fact the whole section on attacks for discrete logarithms for elliptic curves is fuzzy, especially because I really never understood the index calculus attack.

And, why do we take m=floor(x/K) instead of solving the equation x=mK+j like we normally would?

2. (Reflection): Just so I can solidify, when encoding using an elliptic curve we can make the probability of failure to encode the message as small as we would like. That's nice.

Lastly, elliptic curves are still very strange to me.

16.1, due November 26

1. (Difficult): What is a component of a cubic polynomial? And what does the top and bottom of the y-axis mean? Also, an example of change of variables would be good.

Lastly, what's the point of the Law of Addition in regards to elliptic curves? And the "point" infinity from which we define negatives and subtraction is still confusing. I'M NOT GETTING THE GEOMETRICAL MEANING BEHIND THE LAW OF ADDITION, INFINITY, SUBTRACTION AND ANYTHING ASSOCIATED WITH ELLIPTIC CURVES.

2. (Reflection): The computational summary at the end is nice. I would still like to understand this subject thoroughly by knowing why the formulas work by understanding the real geometrical meaning behind elliptic curves.

Furthermore, how is this going to help with cryptography? Is it going to be for more secret splitting or something like that where 3 people have parts of the secret and it takes two to get the secret (two points determine a third point and the elliptic curve)?

2.12, due on November 20

1. (Difficult): Actually I think I understood it all. I would like to try some examples myself though. Seeing some in class would also help.

2.(Reflection): Cryptology is cool because of stories like Rejewski and and the other Polish and British breaking the German's codes during WWII. Wow, they were patient and diligent to find all 105456 initial settings of the Enigmas! Cool stuff.

Non-math explanation and 19.3, due on November 19

1. (Difficult): How does a quantum computer transfer qubits without measuring them? Is this what a quantum computer does? What's the trick to designing the quantum computer?

Also, the notation is difficult. I'm not use to it.

2. (Reflection): Do you think we'll ever invent a good quantum computer?

*** I read some of the assignment, skimmed other parts, and skipped other parts (I'll have to go back to). It was just too much reading for me right now. ***

19.1 and 19.2, due on November 18

1. (Difficult): The whole quantum mechanics idea makes sense, but I can't explicitly explain it. It just kind of intuitively makes sense. So reading a photon changes the photon? So why do observed horizontal photons stay horizontal and vertical photons change?

Also, when Eve eavesdrops Bob will be wrong 25% of the time EVEN IF HE USES THE CORRECT BASES, correct? So really he has a 3/8 chance of getting the right bit while without Eve's eavesdropping he has 1/2 chance of getting the right bit. Correct?

2. (Reflection): I never would have thought of using quantum mechanics in cryptography!

14.1 and 14.2, due on November 14

1. (Difficult): Why doesn't Peggy want Victor to know which way she goes through the door? Didn't follow that in the opening door example.

I would like an example for 14.2.

Why is the second method in 14.2 faster than the first method?

About Arthur, I'm not understanding the part about the probability of square roots mod p or q or n.

2. (Reflection): Do ATM's work this way?

So much cryptography just seems like puzzles. They are intriguing. Difficult sometimes, but intriguing.

12.1 and 12.2, due on November 12

1. (Difficult): I'm not following why in section 12.1 the r's (or s) need to be random and even if they must be random, why can't we choose r's random so that each number is equally likely? I think I need an example or discussion because a worded explanation is confusing to me.

For section 12.2 the Shamir threshold scheme, why do we know p(x) = s(x)? And why is the formula for M so complicated? Why isn't is more like the linear system approach? And why do we need t pairs? I'm not quite getting this but sort of.

2. (Reflection): I like the part at the very end of 12.2 about sharing a secret among two companies A and B such that they need each other to find the secret. I think it's clever and beautiful.

Midterms 2 Study Quesitons, due on November 9

1. The midterm exam will be Monday and Tuesday, November 12 and 13, in the testing center, and will cover chapters 6, 7, 8, 9, and sections 3.4-3.10 and 3.12. For Friday November 9, as you study for the exam, write responses to some or all of the following questions.
• Which topics and ideas do you think are the most important out of those we have studied?  Factoring modulo n and finding squares modulo n are important because both of these help find each other and we can then use factoring to try and break RSA. Also, knowing the workings and attacks for RSA and discrete logarithms, like the ElGamal system, will be really important. 
• What kinds of questions do you expect to see on the exam? I expect to see a few shorter questions about the Chinese remainder theorem, Euler's theorem, Jacobi symbol calculations, ... and such math preliminary to understanding our major coding systems towards the beginning of the test. Then I expect questions about making and breaking RSA codes and discrete logarithm codes and variations of such in the second half of the test. Somewhere there will probably be a digital signature question and a primality test question too.
• What do you need to work on understanding better before the exam? I need to understand the continued fraction low exponent attack on RSA better and all the primality tests the most.

8.3 and 9.5, due on November 7

1. (Difficult): Hexadecimal notation is confusing to me. And I would like to see an example of SHA-1, even the diagrams are confusing. Also, I would like to see examples of the operations on strings of 32 bits. (pg225)

Lastly, what does "with appendix" mean again with regards to signature schemes?

2. (Reflection): DSA, another nice signature procedure. I kind of like the signature methods. I think they're clever.

9.1-9.4, due on November 5

1. (Difficult): 9.1/9.2- So now we don't care about giving m away, we just care about the signature? Because m is essentially public once (m,y) is made public.

9.2- Why is finding r more difficult than a discrete log problem.? Why does finding d allow Eve to compute alpha to the k?

9.4- Why does the birthday attack halve the number of bits under attack? Wasn't 2^30 arbitrary? Couldn't he find more or less places to change than 30?

2. (Reflection): What real world examples are there of blind signatures?

Also, I like the hash function, signatures scheme combo. Great solution so our signatures isn't forever long.

Finally, I like that Alice foiled Fred anyway. Haha!

8.4-8.5 and 8.7, due on November 2

1. (Difficult): In 8.4 where does the ln2 come from in the explanation of equation (8.1)? Why did we use that magic number?

Also, an n-bit function has output n or input n? Output n, right?

Finally, 8.5 and 8.7 mostly make sense but examples would help a lot.


2. (Reflection): I like thinking about problems like the birthday and license plate problem! I, for whatever reason, find it fascinating  Now maybe that's why I'm a math major, but I think everyone likes knowing these random things for some reason. That's why people like statistics and trivia. However, these brainteaser problems are more fun because it's not just about collecting information, it's about breaking the puzzle and deriving the answer. That's much more satisfying!

Also, the BSGS system might be easier and faster to compute (and take the same storage space as the birthday attack) but in my opinion the birthday attack is easier to understand.

8.1 and 8.2, due October 31

1. (Difficult)- Does property 2 of hash functions mean nobody including Bob or Alice (whoever's decrypting) can find the preimage, or does it mean nobody other than Bob and/or Alice (whoever's decrypting) can find the preimage?

Also, if we have m' and h(m') can we check to see if h(m') really came from m' or are hash functions somewhat like a Decision Diffie-Hellman problem?

Lastly, I don't get the data integrity part of hash functions completely. It requires that we send m and h(m) and isn't sending the message without it being encrypted insecure? Or are we only concerned with data integrity and not secrecy when applying hash function to data integrity?

2. (Reflection)- Why are these called hash functions?

7.3-7.5, due on October 29

1. (Difficult): The bit commitment concept makes sense, but the details of bit b=x1 is a little fuzzy. An example would be nice.

Why wouldn't a solution to the decision Diffie-Hellman problem give a a solution to the computational Diffie-Hellman problem? Aren't they equivalent problems?

I'm not following the proofs for the propositions relating the two Diffie-Hellman problems and the ElGamal public key cryptosystem. Another go through it asking questions as a class would be helpful.

2.(Reflection): I really like the football analogy. Makes sense.

Also, I still think discrete logs are harder than big primes and RSA.

7.2, due on October 26

1. (Difficult) I could never follow the overall picture; I just understood parts throughout this section. For instance, I could follow step by step for the most part right at the beginning of section 7.2, but I couldn't see how that related to p=2. Sometimes I even missed steps like the proof of the lemma on page 209 I got all but the second step.

2. (Reflect) The examples helped...but I need a second or maybe even third go through this material. So far, I like RSA better. It makes more sense. Logs are confusing.

6.5-6.7 and 7.1, due on October 24

1. (Difficult): So the 6.5 method is factoring squares and small primes out of n by guessing, is that correct? And what does the book mean when it keeps saying "relation" in 6.5?

I think 6.6 and 7.1 make sense, but I would like to see examples.

What does #4 mean in 6.7 when it says "it is easy to find the functions Ek and Dk"? That it's easy to determine what they are if you have no idea just from the key? Or that you know the idea of Ek and Dk and it's easy to substitute in k? How is that different than #2?

The non-repudiation and authentication technique for public keys discussed in 6.7 is pretty clever and cool I think. Seeing an RSA example isn't essential, but would be helpful.


2.(Reflection): In 6.7 #4 seems a lot like #2 I think, but otherwise it's really clear that RSA works because it meets properties #1-#4. I recognized that before, but it's nice to see it written out.

The non-repudiation and authentication technique for public keys discussed in 6.7 is pretty clever and cool I think. Seeing an RSA example isn't essential, but would be helpful.

It will be interesting to see what else we learn about discrete logarithms, how we use them, and how we break them.

6.4.1 and 6.4.2, due on October 22

1. (Difficult): I'm still not understanding why the Universal Exponent Factorization Method and the Exponent Factorization Method work.

Also, in the Miller-Rabin test do we find a nontrivial factor of n using gcd(b0-1,n) only or do we get a nontrivial factor of n for any gcd(bk-1,n)?

And, on page 185 it talks about finding squares of the form 2j(in)^2+j^2 but then the in its example the book used squares of the form j(in)^2+j^2. What happened to the 2 in 2j(in)^2?

Lastly, why does gcd(x-y,n) give a nontrivial factor (in the Basic Principle) again?

2.(Reflection): We could only factor up to 20 digits the year after my dad was born. Now we can do up to 200 digits. We really have come a long way recently in the world of computers. I love it!

How is the quadratic sieve improved from the method these sections teach?

6.4 beginning, due October 19

1. (Difficult): I understand what the p-q factoring algorithm is saying, but I still don't understand why it works. How do the bj's make it work? Could we do an example in class?

2. (Reflection): It's really simple and often doesn't work because it's too time consuming, but I still think that the Fermat factorization method is pretty. Cool. It makes sense and you can still factor a pretty big number pretty quickly if it's close enough to a square.

6.3, due on October 17

1. (Difficult) I'm not quite following the why behind the Miller-Rabin Primality test. Specifically, why does bi mod p and q need to reach 1 at the same time in order for n to be prime (pg. 180)? And I didn't get how the book calculated how often a composite number registers as a prime using the Miller-Rabin Primality test (pg. 178).

2. (Reflection) Once I see some examples and practice more, I think the Jacobi calculations makes testing for primes really nice. It's pretty simple. Too bad there isn't such a simple way to factor.

3.10, due on October 15

1. (Difficult): There's a couple things I'm not following. In the proof of the very first proposition, how do we know a primitive root exists? Is there always one for every modulus? Or just for every prime modulus?

Also, why does j(p-1)/2 congruent to 0 (mod p-1) imply j is even?

And I believe the(2nd) proposition with all the properties of the Lengendre symbol, but I don't quite follow the why of all of them. Particularly the second one.

So to recap, using the Jacobi symbol -1 means the top number is not a square mod the bottom and +1 means the top number is a square mod the bottom if the top is a square for mod each of the prime factors of the bottom number. (What if the power of these prime factors isn't one?) Otherwise, the top number still isn't a square mod the bottom number. Is that correct?

2. (Reflection): The Lengendre symbol only works for modulus p, an odd prime. Good to remember.

Practicing this will be helpful.

Dr. Saari, The Mathematics of Voting

1. (Difficult): The difficult part of this talk was that the speaker talked about different weights (points for 1st, 2nd,... in a vote) and how they didn't matter, he could still come up with a voting scenario to get whatever outcome he wanted. At first I thought I knew what he meant by weights but wasn't sure because it intuitively seemed like having different weights would change the possible outcomes but once he reemphasized that that was what he meant by weights for I trusted him and understood the rest of the lecture pretty well.

The rest of the lecture was pretty straight forward.

2. (Reflection): I have thought a bit about the plurality voting problem before (when two choices, both are more preferred than the third but the third wins because the other two split the other votes basically). But it was really interesting to see how pair off voting secures any desired outcome and really the only secure method of voting is Borda's method (using weights for 1st place, 2nd place, 3rd place) - yet we still elect our  U.S. President (and other reps, propositions, ...) using a plurality voting method!

Additionally, I found it interesting that he found that plurality voting doesn't give the truly desired result 70% of the time. Wow!

And we have plurality elections coming up in November....

3.9, due on October 12

1. (Difficult): I got all of it except one thing. It said a was congruent to b mod pq. Shouldn't that be a^2 is congruent to b^2 mod pq? That's what it looks like in the example on the bottom of page 87 using 15 and 29.

And why is the factoring of n not quick? It looks like they did it quick in the books example using mod 77. Or does it just mean it's hard to factor quick for large n? That makes more sense.

2. (Reflection): I like that this uses the Chinese remainder theorem because I understand that. Now how are we going to use quadratic modular equations in cryptography?

6.2, due on October 10

1. (Difficult) The whole thing was difficult but especially the Short Plaintext part. I kept forgetting and/or wasn't understanding what the x's and y's were.

2.(Reflection) So long story short, under very specific cases RSA can be broken, but it's still really secure.

3.12, due on October 8

1. (Difficult): Continued fractions make sense but need solidifying. Can we see an example or two in class?
Also, I need to see an example of the simplified method at the end to understand what it's saying.

And out of curiosity (but less important to than my first two questions) I see that the quotients of computing the gcd is the same as the non-fraction part of the denominator, but why? And why are p-2 and p-1 ...defined as the 0's and 1's that they are?

2. (Reflection): So the algorithm for continued fraction is the floor of the original number x plus one over the floor of the reciprocal of the remainder(decimal part; what's left after discarding the floor of x) of x plus one over the floor of the reciprocal of the remainder of x plus...got it.

It will be interesting to see how we use this in cryptography.

6.1, due on October 5

1. (Difficult) I understand the logic of each step taken that explains how the public key works, but then I don't understand the big picture. I guess I just need to go through this again to piece the steps together better.

Also, I'm not following the logic of Claim 1 and Claim 2. Could you explain them in class?

2.(Reflection) The whole idea of a public key is cool. I'm not sure I could have thought of that. Also, this is when I wish I understood computer programming better because I think it would be really cool to work on this kind of stuff for the government or a company. Knowing the math is great but pounding out the details and getting a finished product that really does the stuff we're talking about would be neat. It would be cool to say, "Hey I coded that."

3.5-3.7, due on October 3

1. (Difficult) I didn't follow the proof of Euler's Theorem. And I would like to go over the Three-Pass Protocol again to solidify it as well as the proof to the Proposition on page 84 to solidify it.

2. (Reflection) I'm glad I took number theory using another book because sense this book presents the same concepts in different ways, I can make more connections because I'm looking at the same math from different angles so to speak. I'm seeing different proofs for the same concepts. It's kinda cool.

3.4-3.5, due on October 1

1. I didn't understand the proof of the Chinese Theorem, but I did understand the lemma contained in it. Also, I'd like to see an example of applying the Chinese Theorem.

2. It's nice that I took number theory this past summer because then I already had an idea what the book was saying and I just had to remember it. If it were my first time thinking about the Chinese Theorem, it would have taken more work. In fact, number theory has helped a couple times this semester with cryptography.

5.1-5.4, due on September 26

1. (Difficult) It's hard to understand how you get the extra columns [W(i)'s] while constructing the key schedule without an example. If it's really important, an example in class would be nice.

Also, in the construction of the S-Box, why does x --> x^-1 make the S-Box achieve non-linearity?

Lastly, why do we regroup the lines at the end deriving our decryption process. It seems backwards. Or I'm tired.

2. So is this an unbreakable system? There's got to be one and this seems really really really strong. Again, it's pretty straightforward but so long it's hard to wrap my head around it.

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them? I've spent 5-5 1/2 hours on the last two assignments and I think 3-4 hours on the group coding assignment. And yes, lecture helps a lot! By giving concrete example to better understand the reading and its concepts. Example are great!
• What has contributed most to your learning in this class thus far? Lecture and reading have helped a lot with the theory and a bit with the nitty gritty of how they work but honestly I think the homework has solidified both the concepts and the procedures the most.
• What do you think would help you learn more effectively or make the class better for you? Ok, I need to start the homework sooner. That's on me, everything else is great. I will do that starting this week. 

Thanks!

3.11, due on September 21

1. Difficult: Why when p=2 is every nonzero polynomial in GF(2^n) a generating polynomial? And I didn't follow the linear algebra/matrix part (which is most of the section) in the section about LFSR.

2. I feel like I'm in abstract algebra again. That was a while ago...so far it's making sense.

4.5-4.8, due on September 19

1. (Difficult) What's difficult is wrapping my head around all of this. The concepts of how to do all these different codes makes sense but all the different parts to it make it so complex I don't think I could code a message with one of these codes if I had to. Do we have to?

2. (Reflection) There is no such thing as an unbreakable code is there?!

4.1, 4.2, and 4.4, due on September 17

1. I understand how to encrypt the almost DES code, but not decrypt. That's still hazy. Also, in section 4.4 I understand what it means that the DES system is not a group, but I don't completely follow the proof. I get lost at the lemma because I don't understand why r=0. And about the proof of why DES isn't closed under composition, I'm not following all the notation I don't think so I'm not getting it. But I believe it. What's D?

2. Again, this seems like a really fast hard to break system that's not too complicated, and yet there are so many steps! How do you wrap your head around it? With a computer I guess.

2.9-2.11, due on September 14

1.The last section is the confusing one. So the period of a sequence is the length of the segment of that sequence that is repeated? or that generates the rest of the sequence? Or both? And I almost got the matrix thing which helps us break the recurrence code but I'm shaky. So I don't know the vocab. well or how to break it well but I do know how to create a string of bits using a recurrence relation. It's just a formula based on previous values.

2. So...once we have all these numbers what do we do with them? I see that we're doing lots of math to generate these numbers but do they still just map to letters like we've been doing? How do we go from mod 2 to mod 26? I guess my reflection is still misconceptions but they are reflection/connection questions rather than these sections' specific content questions.

Another reflection, wow I'm glad we have computers to do all this! I don't want to. The theory is more interesting. :-)

3.8 and 2.5-2.8, due on September 12

1. This reading was very straight forward. No one concept was difficult. The tricky thing was the codes have so many steps to them, it's hard to wrap my head around it. I guess that is the point though, isn't it.

2. The funnest part of this reading was the Sherlock Holmes story. But that aside, I simply think the introduced codes are very creative.

2.3, due on September 10

1. The difficult part about this section was understanding why the first method of finding the key works. I know how to follow the steps of computation and do it but I don't understand why when i=j the letters have been shifted the same amount...oh ok. I just looked at it again I think I got it. But this is still the hard part. The idea of why the first key finding method works is so big I'd like to be walked through it again to get my mind around it.

2. This is a really cool encryption process because letters don't keep mapping to the same things. That's what my group was trying to get around when we derived our code for the last hw assignment. We didn't do quite as good a job as a Vigenere cipher but we still made a good code. It's kind of fun breaking codes too. It would have taken me a long time to come up with a way to break the Vigenere cipher. It's pretty ingenious what other people have come up with. Cool.

And thanks with last lecture doing examples and letting us try an example. That clarifies a lot.

2.1-2.2 and 2.4, due on September 7

1. (Difficult): The most difficult part of this is decrypting the affine code. I understand encrypting it so I should  understand decrypting; maybe I just need another look at it and some more practice. How does the book know how many possible keys there are from the numbers? Not sure about that.

2. (Reflection): The most interesting part of these sections is the frequency table. I knew from common sense to look for special combinations when decrypting a substitution code but to have the probabilities and then consider the probabilities of commuted order of letters...wow lots of detail what an art! And a cool puzzle.

1. (Difficult) Most of the lecture was very easy to follow. The trickiest code was when she showed us the one dependent on a repeating word, like "milk" or "fifteen", mapped it's letters to a message and then mapped those letters again to other letters. I didn't completely follow all that but if I had more time to study it I'm sure I could get it. Clever idea.

2. (Reflection) I knew about the D&C name substitutions but didn't know about all of the encrypted letters sent from the Congress representatives to Utah. That's very clever some of the codes they sent. Especially when the decoded words were in a completely different letter. My favorite though was that they used the Hawaiian alphabet as a code and of course only Utah had someone who could read it because of missionary work. Yeah missionary work! I feel like BYU would have someone who could understand pretty much any foreign language thrown at us too. :-)

3.2 and 3.3, due on August 31

1. (Difficult) Working backwards through the Euclidean algorithm and using the extended Euclidean algorithm is easy enough in theory. It's just tricky in practice because it's so easy to make an algebraic mistake. Also, finding inverses in modular arithmetic makes sense in theory but I'd be uncomfortable right now if I saw it on a test having no practice. Inverse practice will be helpful.

2. (Reflective) So a recap just for me. If the gcd(divisor, n) = 1 then divide and use fractions like normal. Otherwise, fractions don't work and before dividing a congruence relation first divide by the gcd. So far these sections are making sense. Of course, I've worked in modular arithmetic before.

1.1-1.2 and 3.1, due on August 29

1. (Difficult) I think these sections would make a lot more sense if there were some examples. For instance, "Here's the plaintext: It is said that one day... and here it is ciphertext: Kn ki iuky npun ... and the encryption key is: k=i n=t, i=s, ..." because I thought I got it and now I'm not sure. Are the encrpyt and decrypt texts or keys?

Also, how is the algorithm of encryption different than the key? Because Kerckhoff's principle says Eve knows the algorithm of encryption but not the key. What's the difference? 

2. (Reflection) I did not know that cryptography was such an art. I should have, but never thought about it. There is a lot of theory, techniques, vocabulary, ... about it. Wow!

Introduction, due on August 29

Year and major: Senior and Math/MathEd

Upper math courses: All calc classes plus linear algebra, both theory of analysis classes, abstract algebra, ode's, adv. topics in applied math (513R), survey of geometry, history and philosophy of math, complex analysis, and number theory.

Why this class: It sounded interesting and I needed another upper level math class.

Most/least effective teacher: At BYU the most effective teacher I had orchestrated class discussion really, really well. Meaning, there was no such thing as a dumb question and it was ok to make mistakes and slow down in class. So everyone really delved into the subject and we did a lot of math- lots of examples- during class.  The least effective teacher was just the opposite. He never checked for understanding and talked over our heads.

Computer programs: I've used Java, Fortran, C, C++, and Matlab but not really Mathematica, Maple, or SAGE. So I'm not that comfortable but I think I can pick up quickly on SAGE.

Unique Me: I and all my siblings are girls.

Office Hours: I can't make yours but anytime 10 to 12:50 would work, M-F.