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## Harvard’s Math 152

I’ve enjoyed reading the Math 152 Weblog associated with the Discrete Mathematics course of the same name at Harvard.  Many of the posts should be of interest to Metroplex Math Circle attendees, but two recent posts struck me as being very similar to topics from our Fall 2008 lectures.

Math 152 helps you get jobs…

In this post a student talks about an interview he had with a quantitative trading firm which asked him to do a discrete path problem which was very similar to those Richard Rusczyk shared with us in his Math and Finance lecture.

Reading Project:  Groups, Factoring and Cryptography

This post built upon ideas that were introduced to Math Circle participants by both Dr. Bennette Harris and Alicia Prieto Lagarica in their lectures on cryptography.  It was particularly interesting that this Harvard student was able to apply his understanding of discrete math to the practical applications of RSA encryption just as Dr. Harris taught.

## Dr. Bennette Harris Recap The second lecture of the season provided lessons in both applied mathematics and number theory. Dr. Bennette Harris began with an easy to follow introduction to binary and computer operations including XOR. He then defined the terms “encoding” and “encryption” and gave several examples of early encryption methods.

Dr. Harris described the difference between “strong” and “weak” encryption methods. This led to a deep dive into the popular RSA encryption method which depends on extremely large prime numbers. Dr. Harris used this as an opportunity to give the students a sense for mind-boggling large numbers and some of the very creative and efficient ways of determining whether or not a given number is a prime. In this part of the discussion, Dr. Harris taught the students various techniques in modular arithmetic. Modular arithmetic is a tool that is not just useful in encryption but is critical for success in math contests and other pursuits.

Dr. Harris used several techniques to engage students including a series of 10 problems that he worked into the lecture for the students to solve. He also demonstrated the UBASIC program which very quickly found tremendously large prime numbers.

Dr. Harris has provided his slides from the lecture and answers to the problems. Members of the Metroplex Math Circle e-mail group can download these files from the group site. To join the e-mail group simply click below. Click to join MetroplexMathCircle

## Dr. Bennette Harris Today! Please join us today for a lecture by Dr. Bennette Harris on “Computer Data Encryption – Decrypted.” We will meet at 2:00 in our usual location: room 2.410 of the Engineering and Computer Sciences South (ECSS) building on the UT Dallas campus. The building is at the corner of Drive A and Rutford.

As we learned from Richard Rusczyk last week, computer science relies far more on the discreet math taught at Math Circles and used in contests rather than the standard curriculum. If a student wants the option of pursuing a career in technology, learning the topics introduced by Dr. Harris can be a great head start.

## Dr. Bennette Harris – Problem Set 2

Here is the continuation of Dr. Bennett Harris’ problems to warm us up before his lecture on September 27th. Congratulations to Dominic for being the first to answer the previous set of problems in the comments. Another happy discovery is the fact that WordPress, which hosts this site, supports the use of $\LaTeX$ code!

### Problems

A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.

6. Crack this secret message: “uifsfaxjmmacfabmnptuaopaqsppgtaupebz”

7. Code the alphabet a=0, b=1, …, z=25, space=26. We will encrypt each letter of the alphabet by mapping it to the letter 3 farther along in the alphabet. Thus, a maps to d, b maps to e, and so forth. The message “a bat is fat” would be encrypted as “dcedwclvciodw” in this scheme. Find a mathematical function f(x) such that y=f(x) gives the correct encryption for every value x = 0,1,2,…,26. Test your function by encrypting “the toy boat floats”.

8. The purpose of these two questions is to provide points of comparison for the questions that follow:
How many particles are there in the known universe?
How many microseconds have elapsed since the beginning of the universe?

9. How long (approximately will do) would it take to calculate $10000^{1234567890}$ at 1,000,000 calculations per second? How many digits does the answer have? If p is 200 digits long, and if q is 200 digits long, about how many digits are there in pq?

10. The number $2^{521}$ – 1 is known to be prime. Estimate how long it would take a computer to demonstrate this by repeated division, at 1,000,000 divisions per second.

## Dr. Bennette Harris – Problem Set 1

This Saturday’s (9/27) Math Circle lecture by Dr. Bennett Harris on “Computer Data Encryption – Decrypted” promises to be a great combination of number theory and applied math. To warm up the students, Dr. Harris forwarded some problems that I will post in two parts. Feel free to offer solutions in the comments or to just work them on your own. Full solutions will be made available at the next Math Circle.

### Problems

A solution for each of the following should either give the correct answer, or a technique for determining the answer in reasonable time with the assistance of a calculator.

2. What is the largest number you must test to demonstrate that 83 is prime?

3. What is the smallest 5-digit prime? How could you find this number?

4. Is 1234567890 prime? What about 123456789?

5. Assume an alphabet with 26 letters (plus a blank space). A substitution cipher is a one-to-one mapping of this alphabet onto itself. How many such substitution ciphers are there?