16.5 dec 9, 2015

It's neat that addition on an elliptic curve translates so easily to El Gamal cryptosystem. When I say easily, I mean that it's neat that the multiplication in El Gamal translates directly to addition on the elliptic curve.  I'm still confused on how we solve discrete logs and how we attack El Gamal elliptic Curve problems.  

16.4

First off, I gotta say-- Elliptic curves seemed so intimidating from the readings, but when we get to class and do the homework, it makes sense! Thanks for your help!

I'm still a little confused when subtracting points.  I know that you add the negative, but I'm not sure how to negate a point.

When adding points on the ec mod 2, do you just follow the same rules we've been using before?

16.3 Dec 4

It's neat to see some of the same tools we learned about originally coming back to help in a variety of situations like the gcd and the Chineese Remainder Theorem.

Can we pick any arbitrary equation and point mod n to factor n?  I was also confused in the second example how they started by trying to calculate 10!.  I don't know where the factorial came from as it wasn't in the first example at all. 

16.2 December 2

Combing discrete logs and elliptic curves sound like quite the problem! I don't get why we need to solve for certain exponents with elliptic curves.  Solving with elliptic curves in mod n seems about the same process at the regular elliptic curve problems  

16.1 Nov. 30th

I am confused about what an elliptic curve is.  I imaged it as the curve of an ellipse, but the graphs didn't make it look like that.  Also, I'm not sure how the addition rule of elliptical curves applies to cryptography.  

article and 19.3 Nov. 20

One thing that I am still confused about what a quantum computer does and how it is different from a normal computer.  Also, what do they mean about creating a superposition?  It's a neat that we can look at powers of a number and look and the modulus classes which creates a pattern. I'd never thought about looking for patterns that way before.  

19.1-19.2 Nov. 18

I've heard about quantum physics before and the whole world confused me. There is obvious structure and patterns in the world we live in so I don't get how we can make it so random and abstract as they do in the quantum world. How can we create laws to study quantum and then apply it to cryptography?

Test Studying Nov. 13

• Which topics and ideas do you think are the most important out of those we have studied?
• RSA
• Birthday Attack
• What kinds of questions do you expect to see on the exam?
• Decrypt this code
• how does RSA work?
• What do you need to work on understanding better before the exam?
• EVERYTHING!!! 
• 
  

12.1-12.2 due nov. 11

It's neat that you can split a secret message that three people need to use together to make it work! I understood the example for three people, but I got a little lost when moving from the specific case to the general case.  actually, I got really lost trying to move it into the general form.  I didn't understand where the matrices came from and where the s values nor what k was.  

9.1-9.4 Nov 9

The need for secure signatures makes sense.  It was funny to think of someone literally copying and pasting a signature onto a check.  With the digital age, it makes sense for an increase of security for signatures.

One thing I was confused about in the RSA method was why did Alice send the message as part of her signature? I thought that was the point of encryption to keep the message secret.  Does a hash function not adequately protect signatures or is that only a problem with the primes aren't big enough?  

8.4-8.5,8.7 November 6

I heard the birthday probability in a 7th grade classroom I observed earlier this year.  When I first heard that if there were 23 people in a room there is a 50% chance of having matching birthdays, I didn't believe it.  Then I saw the math and it makes sense!

What I got confused about was how the birthday method ties into hash functions and discrete logs.  I didn't understand the connection. 

8.1-8.2 Due Nov. 4

If a hash function can really make some of these encryption algorithms simpler, I'm really excited to learn about it!  

One thing that I'm confused about, is that I thought all functions have an inverse, so why is the hash function uninvertible?  

How do you determine if a hash function is strongly collision free or just weekly collision-free?

Also, I'm still struggling how to go from m=x0+x1q to a discreet log problem. 

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. 

Class Reflection Due September 28

1. I probably spend about an hour and a half on the homework assignments.  I feel like the homework and the reading assignments do prepare me for the homework.  It can still be challenging even with doing the reading and attending class, but usually I can figure it out with some effort.
2. Reading the section ahead of time has really helped my understanding in this class.  When I really read the section, even if I have questions, after attending the subsequent lecture, things make a lot more sense and start to click.
3. I need to stay on top of my homework and not get behind.  I need to make sure I plan adequate time to do my homework and the spend time really studying the reading before attending class.  

4.1-4.4 due September 21st

The simplified des type algorithm is really interesting! It takes the bit system we've been working with, but in a different way. It looks simple enough, but must be secure to have worked for so long. 

I had difficultly understanding the s boxes and how to come up with them. I was confused on how they are used. The process for encrypting and decrypting the sea permutations in section 4.4 look tedious. I don't understand how all he boxes and crossing. Arrows work. 

2.9-2.11 due sept 18

Difficult: 

For the one time pad, I don't understand why the key is so expensive to produce. Can't a computer generate a list of random 0s and 1s without too much effort?  Also, I didn't understand how the BBS random generator operated (though it has a great name!). 

Reflection:

It was interesting to realize that companies often had to choose between security and speed.  I didn't realize that security influenced speed and vice versa when it came to encryption, but it makes sense. 

Section 3.8, 2.5-2.8 Due September 16

It was fun to read the Sherlock Homes story.  It's been said several times that a problem with substitution ciphers  English alphabet aren't very secure and it causes problem with authenticity.  In this story because Sherlock figured out the code and sent a message to the killer.  The killer came back to the scene of the crime when the police were there!  Be careful when sending secret messages!

Difficult:
I'm still struggling to understand what to do with a fraction in front of mod n.  I also don't remember how to calculate the determinate of a 3x3 matrix.  

2.1,2.2,2.4 Due September 11

Difficult: I didn't understand the math on our to decrypt affine ciphers.  I was especially confused about converting fractions into a specific number modn.  I didn't understand how they came up with and used the chart of the letters on the top and side with their frequency in the middle.  But it looks like it would be a convenient tool once understood.

Reflection:
It was neat to see how mods can help us decipher affine ciphers.  It was really neat to see the charts used to solve substitution ciphers more easily.  I will definitely use those next time I solve one of those! 

Codes and Ciphers in LDS History

I really enjoyed the lecture today!  It was so interesting to see how codes and ciphers were used in real life situations.  Sometimes it's hard for me to think of times when codes would be used, but I loved seeing how the church used them.  Something as simple as just encoding the key words or phrases really makes the letter difficult to understand.

On the whole, I understood what she was saying.  I was a little confused initially about the difference between a cipher and a code, but after she explained it, it made more sense.  I used to think they were interchangeable.   I didn't understand what the senator/Apostle was writing about that was so secretive that he needed a code.

It was very interesting! I'm glad we got to hear her speak today! 

3.2, 3.3 Due September 4

Difficult:
I had a hard time remembering mods and how they work.  I'm going to need to review them a little more before the rest of it makes sense like the division and multiplication rules.  Last time I really understood the Euclidean algorithm, but I don't understand the extended version.  I'm not quite sure where they are pulling their starting numbers from.  I'm not sure where to stop and how to start.


Reflection:  I learned about congrunces in my 290 class 5 years ago (freshman year was a long time ago!).  It's neat to take principles that we learned (but unfortunately I'm having a hard time remembering them!) and do more with them. 

1.1,1.2, 3.1 due September 2

Reflection: I had no idea that cryptography was used in so many different aspects of my life when identifying yourself online or when making purchases with your credit card or when using digital signatures.  The Euclidean algorithm is really neat!  i'm excited to use it!

Difficult:  I'm still not understanding how cryptography is used to make the above mentioned things work.  When I think of cryptography I think of the cypher we did in class-- different letters meaning different things.  I also don't understand where knowing what prime numbers are fits into our study of cryptography.  The information I read today feels pretty disjointed. 

Introduction, due on September 2

Wednesday September 2:  Answer the following Introduction questions in a blog entry.

• What is your year in school and major?
• I am a Senior studying Math Education
• Which post-calculus math courses have you taken?  (Use names or BYU course numbers.)
• I have taken Math 290, 313, 314, 334, 341, and 371.  This is my last math class at BYU
• Why are you taking this class?  (Be specific.)
• I am taking this class because it is required for my major.  I chose this class over number theory and graph theory because I heard such great reviews about it.   I also chose it because it sounds interesting! 
• Do you have experience with Maple, Mathematica, SAGE, or another computer algebra system?  Programming experience?  How comfortable are you with using one of these programs to complete homework assignments?
• I really don't have any experience using these programs.  I am a little uncomfortable with them, but I am willing to try. 
• Tell me about the math professor or teacher you have had who was the most and/or least effective.  What did s/he do that worked so well/poorly?
• Teachers that I have loved and learned a lot from in the class seemed to care about each of the students personally and want us to learn.  They are patient with us  They can explain things clearly.  It's frustrating to have professors who assume we already know what they are talking about.  My professors are all very knowledgeable, but some seem to have forgotten what it's like to learn it for the first time.  It's frustrating to have a professor tell the class to just stare at the problem long enough and then you'll figure it out.  
• Write something interesting or unique about yourself.
• I love to travel! I did two study abroads: one in at the Jerusalem Center, and the other one tour through Western Europe for my art history minor. 
• If you are unable to come to my scheduled office hours, what times would work for you?
• Sometimes I work during your office hours, but I could come in right after class on MWF.  I have class until 3:20 during the TA's office hours, so I should be able to go in for the last 40 minutes.