"That's How The Light Gets In" Tyler Jarvis

This article especially resonated with me because I am a bit of a perfectionist.  Sometimes our fear of failure really does get in the way of our progress.  Like I said in class yesterday, I used to be a lot less persistent with my study in math.  I would get frustrated if I didn't know how to do it immediately and didn't want to go through the failures necessary to figure it out on my own.  But as I realize the value of doing it on my own, and how I learn even more from my mistakes during the experience, even though it takes quite a bit longer I am much more motivated because I know that I will conceptualize it at a much deeper level.  I also feel the satisfaction of knowing that I had the ability to do it on my own, and I am even more motivated to work on the next problem.
I loved the steps that he listed and how he related it to math:
1) Admit and Accept Imperfection: Both in school and in life in general this holds true.  We live in a fallen state and will be until we are perfected through the Atonement.  So we might as well accept it so we can move on.  I love Julia Robinson's quote:
Monday—tried to prove theorem
Tuesday—tried to prove theorem
Wednesday—tried to prove theorem
Thursday—tried to prove theorem
Friday—theorem false3
It demonstrates how much patience we truly must have, in mathematics and in our everyday life.
2) Work hard to get your best approximation.
3) Get up and act on your best approximation.
4) Do it again.

I hope to implement all of this, both into my study habits and in my approach to life.  Great talk.  Thanks for sharing!

Study and the Prayer of Faith - James W. Cannon

I really loved how he talks about the experience that he had with prayer and his research.  I think it correlates well with the quote he includes at the end; that all truth is one and the same.  God is concerned with our pursuit of truth - both scientifically and morally.  Just as Cannon did not want to ask for help with scientific truth, sometimes we also don't ask for help in understanding gospel truths.  He did this because he wanted to satisfy his pride with the knowledge that he "did it on his own" (which still isn't true because everything we do is only possible through the grace of Jesus Christ).  Similarly, we feel that we can rely on ourselves day to day, when only we deceive ourselves into thinking that we are only a little bit bad (like how he thought he only had a little bit farther on that problem).  In reality, we have so much farther to go; infinitely farther.  When we realize this, we also understand how much more we must rely on Jesus Christ and His atonement in order for it to happen.  As we do this, we work with Him and accomplish a lot more than we could have if left to our own devices.

Great article on our search for truth - both temporal and spiritual.  Not all things may make logical or scientific sense, but we can know with a surety through the feelings of the Holy Ghost.  The feelings that we feel and the desires that we have for good, while reading the Book of Mormon, cannot be denied.

16.5, due December 9

1) it seems like we talked about most of this in class last time.  I feel like I understand all of the algorithms but I should probably just practice doing them.
2) I think it is cool that a lot of these methods are analogs to other processes that we have studied.

16.4 Due December 7

1) If 2P for all points is equal to infinity, then the factoring method that we learned a little bit ago would never work for computers right?  Because everything is in binary?  Unless we always just have different elliptic curves that are only encoded in binary mod 2?  What does it mean that elliptic curves are used over fields GF(2^n) with n large?  Say at least 150?
2)  I am just amazed that somehow this has practical use for computers.  What exactly else do they use elliptic curves for?

16.3 due December 4

1) I know that we started talking about this in class last time, but I am having a hard time understanding how elliptic curves can help us to factor a number. How do we know which elliptic curve we choose? What exactly is a singular curve?
2) I am shocked that there is yet again another factoring method.  There are probably many more that we haven't studied that only work in very specific circumstances.

The Imitation Game

Concerning the actual design of the enigma machine, I am still not completely sure how it works.  From what I understand, there are 26 rotors in the machine and at least one of the rotors shifts every time one letter is encrypted.  Then in order to decrypt, you would need to know the initial settings of the encryption machine, and it would simply go backwards?  Or I guess you would need to know the terminal settings so that it could go backwards?  I am not entirely sure?  And how does the key work?

I actually would like to talk to you about some of these questions, and others that I have including:

-How does the key work?  Our example in class was PGH PGH PGH.  Why was it sent three times?
-Why did eliminating the diagonal help them to solve the message quicker?  What exactly does that mean?
-I understand that if they could predict certain words that would be used in the cipher texts (like weather and Hitler) but how did knowing the ways that people would choose random words help them?  Like that woman who predicted that Cilly was the name of his girlfriend?  Why were they instructed to use 5 random letters at the beginning of every message?  It would make it harder to crack I presume?

I guess that movie was really interesting and moving, but as far as Cryptography goes, it just left me with more questions.  I guess I will see you again in your office!!

Oh and by the way, I was happy to at least know what a Beale cipher was, all because of that tedious, tedious (but fun) project!

16.1, Due November 30

1) Honestly, I am throughly confused.  I am not sure that I really understand how the elliptic curves are even used to factor integers.  I really don't even understand the difference between elliptic integrals and elliptic curves.
2)I think once I understand how this helps us to factor integers, that will be the most interesting part to me, because as of now it is just confusing.  I think it is pretty cool that from an two points on the curve, we can produce a third point on the curve.

18.1-18.2, Due November 24

1)I am not sure I totally understand parity.  What is a two dimensional parity code?  Hopefully we will actually go through all of these codes.  How is this correcting errors?  And what does it mean to go through a noisy channel?
2) It is pretty cool that they have actually found a way to correct errors.

November 20th

1)I don't think I fully understand how finding the period of a sequence will help us to somehow factor n?
2) I really loved the real-life examples that he provided.

19.1,19.2 Due November 181

1)What is a complex conjugate?  I am not sure I fully understand the notation and why a^2+b^2 would equal 1 in chapter 19.1.  Then in chapter two, if alice and Bob agreed on errors in the data previously, why would Bob have a 25% chance that he will measure the wrong value?  Is there a way to be more correct.
2)I really loved reading about the light photons.  Maybe I should be a physics major?

14.1-14.2, Due November 16

1)I understand the general idea but I am not sure I fully understand the algorithm in the Feige-Fiat-Shamir Identification scheme
2) It is interesting now to see the opposite of public key systems with zero-knowledge techniques.

Due November 13

• Which topics and ideas do you think are the most important out of those we have studied?
• What kinds of questions do you expect to see on the exam?
• What do you need to work on understanding better before the exam?

-RSA and the different attacks on RSA, as well as hash and signature functions

-I expect to see a lot of conceptual questions considering the fact that we need a computer to solve a lot of the problems.  I am just not sure how much I will need to memorize before going in.

-1-6 on the study guide, strengths and weaknesses, and memorizing some of the methods.

Dr. Bakker's Talk, November 12

"The Eight" Planet Solar Systems

The 8-planet solar system is part of what inspired people to study mathematics.

Tycho was the last naked-eye astronomer.  Kepler used his this data and studied it for 20 years to get his first, second, and third laws.

Lately gravitational singularities have been studied by mathematicians.

Two types of gravitational singularities: collision and non-collision.

Non-collision singularities have never been observed in the solar system.  Have collision singularities ever been observed in the solar system?

Comet Shoemaker in 1994 is the first ever observed collisions.

It came scarily close to us in October and there are more and more coming close to the earth.

12.1,12.2, Due November 11

1) I guess I don't fully understand how if each person has a different part of the secret, that if only a few members of the group collaborate, they can somehow guess or deduce the other part of the secret?  Also how can we have more than one solution?  The book talks about how if the determinant is not zero, then we have a unique solution, and that happens when we have distinct xk's.  But why would they not be distinct?  Sorry these are probably silly questions.

2) This secret sharing is an idea that I had never really considered before and I think it is really interesting.  Especially when they talked about how they used this same idea to control nuclear weapons in Russia because two out of the three people were needed to actually launch the missiles.  It would be scary to think that one person could have the control to release something like that.

9.1-9.4, due November 9

1) if you sign the message after it has gone through the hash, do we ever get the original message back? Or would we even want to? Because hash functions are not invertible.
2) it's so cool to understand now how the electronic signatures do work that I use in my everyday life.  I am wondering now why we always have to verify we are "not a computer" by typing in weird letters and numbers shown in an image above our text input box.  What does that have to do with our cryptography??

8.4-8.5,8.7 Due November 6

1)I am not sure I fully understand why we are concerned with collisions and the probability of one happening.  How does it apply in real world situations?  What is wrong with two numbers having the same hash value?  I thought the whole point of hash functions is that they are not invertible and it is likely that two values will have the same output.

2) I honestly just thought that the whole probability of two people having the same birthday was pretty interesting and at first unexpected.   But as you observe it further, then you see why it is so plausible.

8.1-8.2, Due November 4

1) I think I understand the main idea of hash functions but I would like to work through examples together in class.  Also, I didn't fully understand the proposition on pg. 221.
2) I think it is interesting that we are looking at a crypto system that is not invertible.  Does this mean that decryption is totally different?

7.3-7.5, Due November 2

1) I definitely had troubles understanding the lea.  I am also wondering why we are focusing specifically on log 4.  And then why is it comparing the values 6 and 16 to L2(9), which are the same mod 4.

2)  I thought exchange between Alice and Bob is interesting, where Alice can make predictions, send it to Bob.  Although it is in his possession, she is still the one to remove the lock.

6.5-6.7 and section 7.1 due October 28

1) I am just not 100% sure what kind of crypto system uses discrete logarithms.  Is stand RSA or something similar?  It's probably a silly question, but perhaps it will make more sense in the lecture.
2) It is interesting to me that we can only compute discrete logs for primes that are about as large as the primes that we can factor.

6.4.1 and 6.4.2, Due October 26

1) This might be a silly question, but I was wondering if it is hard to factor something, is it also hard to find the square root of that really big number?  Supposing that we really did want to check every prime up to its' square root?
I don't understand how they got the matrix they did on pg. 184.
2) I think its cool that the linear dependency will help us understand whether or not we have factorization.  But I would like to understand better how to look for linear dependencies in different moduli.

6.3, Due October 21

1)  I can't really follow the Miller-Rabin Primality Test.  What does it mean for something to be a pseudoprime for the base a?  Why would we even want to know if something is prime?  Why is that an advantage to us?  Does composite mean factorable?  Why is the success of these two methods not guaranteed?
2)  I think it is fascinating that we have these different tests to test if something is prime or not.  The fact that we can do it with such giant numbers is amazing, and that our computers can do it so quickly.  I am still trying to figure out why this would be of so much use to us.

3.9, Due Friday October 16th

1) Actually, the reading was pretty clear and straight forward.  I understand the actual operations but I am not exactly sure that I understand the reasoning and justification behind it.
2) I am just excited to see how we are going to use this in the context of cryptography.

6.2, Due October 14th

1)  How could someone come to know the first m/2 or the last m/4 digits of p if it is not part of the public key?  Same for d?  I don't fully understand the short plaintext attack on RSA.  And I still don't really understand why the fraction approximation of decimals works.  I understand how to do it, but I don't get why.  Do we need to understand that?
2) I am impressed with all of these attacks on RSA but I am wondering if they are ever REALLY used in real world.  Are these the only known attacks?  Are people still studying RSA and trying to find different attacks?  Or is it futile until we have quantum computers?

3.12, Due October 12

1)  This is a really cool idea but I am wondering how we will be able to find the closest fraction that looks like the decimal if the irrational number.  Is there something we can do on sage or on the computer to find fractions that seem to look a lot like those decimals?
2)  I will be interested to see how we use this in cryptography.  What are the applications?

6.1, October 9

1) I don't exactly understand how Bob makes n and e public.  What does it mean to have a public key crypto system?  In what way would he make it public?  And would it make it easier to decrypt?  Less unsafe?  And although we have been practicing working with really really big exponents, how in the world are we going to deal with ones this big..?
2) I think it is cool that we can use this basic number theory for encryption.  I was trying to imagine how we would use Euler's equation for a cipher, but I like to see the application.  I also think it is funny their is a method called PGP (Pretty Good Privacy).  Why does it not matter if e-mail is that secure?  Why do we encrypt e-mails?  For security?  Or to make sure what is sent stays true to what was intended to be sent?

3.6-3.7, Due October 7

1) I honestly don't really understand the proof for format's theorem but I think it is really cool how it works and how it is so helpful for us to evaluate somewhat complicated expressions in mod(n).  I am also not completely understanding the general idea of Euler's theorem.  What is that notation on pg. (81) half-way down?

2) I love learning all of these new ways to re-write expressions and magically simplify.  I also think it is cool that such ancient discoveries have proved to be so useful in cryptography and with computers.  No longer is it just cool for math-sakes.  It actually has application.

3.4-3.5, Due October 5

1)  I guess I still don't fully understand what it means to just write one congruence into a system of congruences mod factors of n.  If this system of congruences is not like an addition of the two congruences, how do they relate to one another.  It is a pretty cool idea.
2)  I love to see all the different things that we can do modulo.  I also think it's cool to see the cross between linear algebra and this class.  I think overtime we see the material cross over between subjects, it just makes it that much more meaningful and applicable.

Due October 2, 2015

As I was looking over the list of things that we have learned over the semester, I honestly feel that most of what we have learned is important.  But I think for me, a lot of the number theory that we have learned is important (modulo, divisibility, GCD, Finite Fields, etc.)  I think it is important to really solidify our understanding for these different concepts so that we can actually apply them to the ciphers.  I think that it is important to understand generally how DES and AES work because they are actually used today, or in more recent years.  However, these are some of the hardest topics for me to understand.

I expect to see questions that are conceptual, asking about the actual cipher, strengths, weaknesses, and how to encrypt and decrypt.  I think this because a lot of the ciphers that we have learned about require computers to decipher.  I also expect to see questions that require us to practice various algorithms and procedures, like the Euclidean algorithm,  finding solutions to linear congruences, etc.

I REALLY need help understanding ECB,CBC, and CTR.  I missed a few on the homework and I have a very superficial understanding.  I also don't really remember the reasoning behind decryption of LSFR.  Meet in the middle attacks!  Those are the biggest things.  However, I do need to continue reviewing everything.

5.1-5.4, Due September 30

1) Ok, I know you explained this in class but for some reason I don't understand XOR.  I mean, I guess I understand the operation, but I am not 100% sure why it works.  Also, in the S-Box, is the matrix by which you multiply just randomly chosen?
2) I am so impressed that someone even came up with this cipher.  I am curious about how secure it is?  Are we still using AES today?

Questions Due Monday September 28th, 2015

I spend 1-2 hours on each reading assignment.  The lecture definitely prepares me for the homework. I am frustrated because I feel like I have a very superficial understanding of the reading, but I probably need to take more time to really internalize and digest what I am reading instead of rushing through it.  That will probably be one of my goals for the rest of the semester.

I think the exploration of the homework is where I really learn the very most because that is where I need to make sure I understand the concepts enough to actually problem solve.  This class is my favorite this semester, and when I finally understand it and successfully decrypt or solve it is that much more fulfilling.

I think the biggest thing I need to work on is giving more time to the readings, but it is hard because I feel like I have such limited time with all of my homework for my class and work.  But I will still try to manage my time more so that I can take more time to read and understand.

4.1, 4.2, and 4.4 Due Monday September 21

1)  The notation is pretty confusing to me.  I think that I am following the diagram, but I am wondering if the first half of the message, or in their example case, the first 6 bits are also dependent on the second half, or the the last 6 bits.  Because it looks like it is adding terms from both the L and the R (after it has been through the function) in order to get the next R bit.  I follow the diagram but I don't completely understand the notation.  I also don't completely understand the meaning of a permutation.

2) After looking at the DES algorithm and reading a little bit about it, I am impressed that people came up with such a secure and somewhat fast encryption method.  I also think it is pretty cool that although the DES algorithm is too long to discuss in the previous chapters, they could teach the same idea through a more simple example.  I am curious to learn more about differential analysis and how it works in order to break certain encryptions.

2.9-2.11, due Friday September 18

1)  To be completely and utterly honest, I just don't understand binary numbers and how they work.  I am still trying to understand 2.8 which explains them and how you can represent everything in 0's and 1's.  Why is it more advantageous for us to do this?  For one time pads, I understand how it is so efficient and secure, but isn't the key extremely long?  I mostly just don't understand the binary system which I think is causing the big gap in my understanding for the rest of the sections.
2)  I do think it is interesting that this can work and somehow we can encrypt and decrypt so speedily.  I  am happy because I understand modulo much better now than I did while taking Math 290 and now I can see why it is actually so useful and applicable to us in our world today.

3.8, 2.5-2.8 Due September 16th

1) I don't understand how the frequency analysis is usually successful for play fair ciphers because the blocks of two letters are two small.  Then for block ciphers, do you just pick a random matrix M to begin with?  I also can't remember how to take the determinant of a 3x3 matrix.  But I can look that up before class.  I am confused on the description of this whole cipher.  Binary numbers????
2) The play fair cipher was clever to me.  I like to see the creativity in all of the different cipher methods.  The story about Sherlock holmes was awesome and refreshing.

2.3, Due on September 14th

1. I understand the first method of decryption but I don't understand their explanation of why it works.  For finding the key length, why would the number of coincidences tell us that the number of displacements is also the number of digits in the key?  Similarly, once we find how many digits are in the key, why do we count by that? (i.e. 1, 6, 11, etc.)  I do not even understand the second method of decryption and how to calculate it, but I will read that section again before class.

2.  I just think it is so amazing to see the correlation between cryptography and linear algebra.  I never imagined that I would be using the dot product to decrypt a message.  I also think it is pretty amazing that the decryption solutions have been found for all of these encryptions.

2.1,2.2, and 2.4, Due on September 11

1.  I feel like I have a very superficial understanding of shift ciphers and how they relate to modular arithmetic.  I don't understand how they found the multiplicative inverse for 9(mod26) is 3.  Are there multiple multiplicative inverses?  I also don't completely understand how you can have a function or an encryption that is not one to one.  Why do we need to make sure that the gcd(z,26)=1?

2.  I think it is pretty amazing that you can make a cipher text so hard to break if the adversary only has the cipher text available.  But then again, if you just know a little bit of plaintext with the cipher text, depending on the cipher, description becomes so much easier so suddenly.  I still don't understand the Affine Ciphers completely, but I would like to know why they are so much more secure.  I would like to continue practicing decrypting substitution ciphers since it uses so much wit and cleverness.

Guest Speaker Lecture, Due on September 11

1.  To be honest, the hardest thing for me to understand was the fact that someone actually forgot the code names and what they stood for.  Personally, I thought that they would take better care of that information.  It is also ironic that someone stepped forward explaining that they did remember the missing names, but that information was filed away and they didn't find it until 1987.  Especially when this is referring to scripture, you would think they would take better care of that information.
2.  I loved hearing all of this history because until the lecture, I had no clue that cryptography would have a connection to the church but it totally makes sense now.  One of my favorite ciphers was the one where there is a grid with three letters in each square.  Each letter has either none, one, or two '.  I thought it was really clever and it looks a lot cooler and more foreign when you don't know the code. Overall, the lecture just made me a lot more interested in Cryptography and it's place not just in history, but in our society now.  It helped me to see more relevance of the subject and helped instill a desire within me to keep working hard, even though it is sometimes difficult for me and it is just the beginning of the semester!

3.2 and 3.3, Due on September 4

1.  I simply could not follow the extended Euclidean algorithm.  I am not sure if it is because I don't understand the operation, but even after going over their example several times, I am having a hard time conceptualizing it or even understanding the steps and logic behind each operation.  Why is it useful to us?

I had an easier time understanding the basic idea behind congruences, but I guess I don't yet see how it will be helpful to us in this course or in real life.  I am sure that will become more obvious as the class goes on.

I had a hard time understanding how they solved the last example on pg. 73.  How is 4/11111 the same as working with mod 12345?

I also don't completely understand how to work with fractions.  Is there a logical way to keep track of when I need certain integers to be relatively prime or when their gcd(a,n)=1.


2. When we studied congruences in Math290 I was curious as to what discipline or sector of math to which it is helpful.  I am excited to see how it is a part of cryptography and keeping messages safe.

I think my favorite part was in the second example on page 73 when they show how 7 is congruent to 18 which is congruent to 29 which is congruent to 40 and as we find all the different congruencies it makes it possible for us to divide.  It's an interesting way to think and sort of reminds me of how our number system is base 10, but how you can do math in different bases.

1.1-1.2 and 3.1, due on September 2

1)  In section 1.1.1, I didn't completely understand the description of airplanes determining if another airplane is a friend or a foe.  It says, "send a random message to the plane, which encrypts the message automatically and sends it back.  Only a friendly airplane is assumed to have the correct key.  Compare the message from the plane with the correctly encrypted message.  if they match the plane is friendly.  If not, it's the enemy."
This could be very elementary, so bear with me, but I guess I just don't understand the system.  Anytime you send a message to a plane, it will be sent to their plane and they will encrypt it using their encryption machine and send it back?  Or do they send their response in an encrypted message? I guess I just don't understand what exactly the cipher text would be in this case... the plain text that you just sent them now encrypted?  Or is their response that is encrypted?

I am also not 100% sure on the difference between a key and an algorithm.  What does it mean to keep the algorithm the same but to just change the key?  Does algorithm just mean the type of key?  Like the equation for it?

In sections 3.1.2

I am having a hard time wrapping my head around the Prime Number Theorem.  Why are we trying to prove that pi(x) is the number of primes less than x?  How does this prove that there are infinitely many prime numbers?

I have a surface understanding of the Lemma on pg. 65, and I definitely could not replicate the proof.

Lastly, I just can't understand the Theorem and Corollary on pg. 68, but I will continue trying to read and understand it before class tomorrow.  I don't understand it to the point that I can't even really identify the reason I don't understand it.



2) I loved chapter 1 because it really illustrated the application of this material in our world today.  I am very surprised because I didn't realize how cryptography is so widely used and severely needed in our world of technology, despite the fact that I use it every day.  I like that in 1.2 it broke the applications into four different, clear sections.  I have used and appreciate each part.  I am also interested to see the connection between cryptography and games.

In chapter 3, I felt like I was back in my Math 290 class.  The proofs are simple and beautiful for the most part and I find myself wanting to work out the proof myself before I read the solution.  I am very intrigued by the Euclidean algorithm and I am curious as to how it was discovered.  I am amazed at how quick it is and the fact that you don't need to factor.

Introduction, Due on September 2

Haley Rasmussen - Math 485 Mathematical Cryptography

1. I am a Math Education Major.  I am a Senior and will student-teach Fall 2016.
2. I have taken Math 313, 334, 290
3. I am taking this class because it seems interesting and fulfills a credit for my major.  I also have little to no experience with programming, so I would like to learn.
4. I don't have any experience and at the moment I am not very comfortable at all, but I am willing to put in the time to learn and work with you or the T.A.
5. The best and most effective math professor that I had was extremely interested in the student and ensuring that they understood.  They didn't just talk at us the whole time, but helped us to work through it ourselves so that we could learn more completely.

The worst math professor that I had was completely disconnected from the student, lectured at us the whole time, and that was it.

6.  I majored in Ballet at the University of Utah for two years but I decided to transfer to BYU so that I could become a Math Teacher.

7.  I can't make it to your office hours, but I could come MWF 2:00-3:00 pm or later in the afternoon at about 4:00 pm