Category Archives: Weekly features

December 12, 2011, 7:45 am

Columnar transpositions: Looking at the initial cycle

Welcome to Math Monday! Each Monday here at Casting Out Nines, we feature a mathematics-themed article. Today’s is a new installment in an ongoing virtual seminar on columnar transposition ciphers.

Let’s return to our ongoing look at the columnar transposition cipher. In the last article, we introduced the notion of cycles. A cycle can be thought of as a cluster of points which are moved around in a circular nature by a permutation. All permutations — including the permutation implemented by a columnar transposition cipher — break down into a product of disjoint cycles, and we can determine the order of a permutation (the smallest nonnegative power of the permutation that returns it to the identity) by finding the least common multiple of the lengths of the cycles in its disjoint cycle decomposition.

Since one of the main questions we are asking about CTC’s is about their order,…

November 28, 2011, 7:45 am

Cycles, and the cycle decomposition of a permutation

By Robert Talbert

Last week’s installment on columnar transposition ciphers described a formula for the underlying permutation for a CTC. If we assume that the number of columns being used divides the length of the message, we get a nice, self-contained way of determining where the characters in the message go when enciphered. Now that we have the permutation fully specified, we’ll use it to learn a little about how the CTC permutation works — in particular, we’re going to learn about cycles in permutations and try to understand the cycle structure of a CTC.

First, what’s a cycle? Let’s go back to a simpler permutation to get the basic concept. Consider the bijective function $p$ that maps the set $\{0,1,2,3,4, 5\} $ onto itself by the rule
$$p(0) = 4 \quad p(1) = 5 \quad p(2) = 0 \quad p(3) = 3 \quad p(4) = 2 \quad p(5) = 1$$
If you look carefully at the numbers here, you’ll see that some of…

November 21, 2011, 7:45 am

A formula under the hood of a columnar transposition cipher

By Robert Talbert

It’s been a couple of Math Mondays since we last looked at columnar transposition ciphers, so let’s jump back in. In the last post, we learned that CTC’s are really just permutations on the set of character positions in a message. That is, a CTC is a bijective function $ \{0, 1, 2, \dots, L-1\} \rightarrow \{0, 1, 2, \dots, L-1\}$ where $L$ is the length of the message. One of the big questions we left hanging was whether there was a systematic way of specifying that function — for example, with a formula. The answer is YES, and in this post we’re going to develop that formula.

Before we start, let me just mention again that all of the following ideas are from my paper “The cycle structure and order of the rail fence cipher”, which was published in the journal Cryptologia. However, the formula you’re about to see here is a newer (and I think improved) version of the one in the…

November 7, 2011, 7:45 am

Math Monday: Columnar transposition ciphers and permutations, oh my

By Robert Talbert

I hope you enjoyed Ed’s guest posts on NP-complete problems on TV the last couple of Mondays. It’s always great to hear from others on math that they are thinking about. This week it’s me again, and we’re going to get back to the notion of columnar transposition ciphers. In the first post about CTCs, we discussed what they are and in particular the rail fence cipher which is a CTC with two columns. This post is going to get into the math behind CTCs, and in doing so we’ll be able to work with CTCs on several different levels.

A CTC is just one of many transposition ciphers, which is one of the basic cryptographic primitives. Transposition ciphers work by shuffling the characters in the message according to some predefined rule. The way these ciphers work is easy to understand if we put a little structure on the situation.

First, label all the positions in the message from $0$ to …

October 24, 2011, 7:30 am

Math Monday: TV Lawyers Solve NP-Complete Problem (part 2)

By Robert Talbert

This is the second installment of a two-part article from guest blogger Ed Aboufadel. Thanks again, Ed, for contributing.

In Part I, we learned of an instance of the NP-complete problem subset-sum [1] that was solved by three lawyers on an episode of the USA Network show Suits [2]. The problem was to go through a set of deposits made to five banks in Liechtenstein and find a subset of deposits, where the total of the deposits was $152,375,242.18. Described as “simple mathematics” by one of the lawyers, the team solved the problem in a relatively short length of time. They couldn’t use a quick approximation algorithm for subset-sum, since they needed the sum to be exactly equal to their target amount. So, were they just lucky, smarter than the rest of us, or did they do something practically impossible?

Consider the following “back of the envelope” calculations. First,…

October 17, 2011, 7:30 am

Math Monday: TV Lawyers Solve NP-Complete Problem (part 1)

By Robert Talbert

For the next couple of weeks, Math Monday here at the blog will feature a guest blogger. Ed Aboufadel is Professor of Mathematics and chair of the Mathematics Department at Grand Valley State University, where I work. He’ll be writing a two-part series on a neat appearance of an NP-complete problem on network TV, adding yet another data point that mathematics is indeed everywhere. Thanks in advance, Ed!

On the new USA-network TV series Suits [1], Harvey Specter is a senior partner at the law firm of Pearson Hardman, and Mike Ross is his new associate. Mike never went to law school, but he combines a photographic, elephantine memory with near-genius intelligence to fake it well. Harvey is in on the deception, but none of the other partners know. During the eighth episode of the first season of Suits (broadcast August 11, 2011), Harvey and Mike, working with Louis Litt, a…

September 28, 2011, 4:00 am

Midweek recap, 09.28.2011

By Robert Talbert

Good stuff from the internet this past week:

John Golden (Mathhombre) shares an #edchat conversation about mathematical misconceptions of all sorts.
Maria Andersen (Teaching College Math) emphasizes a point you can’t make strongly enough: Failure is a NORMAL part of learning.
Ben Deaton (jbdeaton.com) has a great article of advice for surviving your first year of graduate school.
ProfHacker is starting a sweet-looking series of articles about using the command line interface. Talk about a lost art.
On the technical/math/crypto side, Bruce Schneier (Schneier on Security) describes three emerging cyber threats. None of them have to do with hackers squatting over a computer in their dorm rooms or parents’ basement.
Danny Caballero (Think Like a Physicist) follows up a post featured on last week’s midweek recap by announcing (somewhat reluctantly) that they’ve gone with…

September 21, 2011, 9:00 am

Midweek recap, 9.21.2011

By Robert Talbert

Interesting stuff from elsewhere on the web this week:

Danny Caballero, who does really interesting research on physics education at UC-Boulder, has just started up his own blog. Every post on it so far has been excellent, but his article “Which computational tool should we teach?” in particular is a great analysis of three major computational software tools from the standpoint of teaching physics students computational modeling.
John D. Cook examines the question, what point on the Earth’s surface is farthest from its center? Hint: The answer involves trigonometry and is not Mount Everest.
Dave Richeson looks at an interesting probability rule (of thumb).
My GVSU colleague David Coffey considers the role of remuneration in the motivation of teachers.
Profhacker introduces us to Codeacademy, a sort of Khan Academy for computer science. On a related note, my afternoon just …

August 26, 2011, 9:00 am

Friday Random 10, 8/26/11

By Robert Talbert

Don’t look now, but it’s the return of the Friday Random 10. Ten songs selected at random from my family’s, um, eclectic iTunes library. Notice how I say “my family’s” library, so as to deflect questions about why there are so many kids’ songs or Glee stuff coming up.

Candles (Glee cast version); Glee: The Music Presents the Warblers
Spiritual; Johnny Cash, Unchained
BWV Praeludium et Fuga in A; James Kibbie, Bach Organ Works: Preludes and Fugues
Sara; Fleetwood Mac, Greatest Hits
It Doesn’t Matter; Alison Krauss & Union Station, So Long So Wrong
Celebration; Kool & the Gang, Gold
Amazed; Lonestar, Lonely Grill
All the Way My Savior Leads Me; Rich Mullins, The World as Best I Remember It, vol. 2
Soul Refreshing; Robert Randolph & The Family Band, Unclassified
Jealous Hearted Man; Muddy Waters, Hard Again

Let’s focus this week on James Kibbie, a master organist a…

November 12, 2010, 4:04 pm

This week in screencasting: Optimization-palooza

By Robert Talbert

My calculus class hit optimization problems this week — or it might be better to say the class got hit by optimization problems. These are tough problems because of all their many moving parts, especially the fact that one of those parts is to build the model you plan to optimize. Most of my students have had calculus in high school, but too many calculus courses in high school as well as college focus almost primarily on algorithms for computation and spend little to no time with how to create a model in the first place. Classes that are so structured are doing massive harm to students in a number of ways, but that’s for another post or two.

Careful study of worked-out examples is an essential part of understanding optimization problems (though not the only part, and this alone isn’t sufficient). The textbook has a few of these. The professor can provide more, but class time really …