Category Archives: Scholarship

November 28, 2011, 7:45 am

Cycles, and the cycle decomposition of a permutation

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…

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November 21, 2011, 7:45 am

A formula under the hood of a columnar transposition cipher

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…

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November 7, 2011, 7:45 am

Math Monday: Columnar transposition ciphers and permutations, oh my

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 …

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October 10, 2011, 10:53 am

What is a columnar transposition cipher?

We all have secrets to keep. Those secrets could be personal dirt we want to keep from others, or they could be something as mundane as our credit card numbers or medical histories. But all of us have information that we want to keep to ourselves or at least to a small circle of people whom we select. This is why the field of cryptology — the science of making and breaking coded messages, or more generally the notion of communicating in a secure way — is a viable and extremely active field of study these days.

I’ve been interested in cryptology ever since a student came to me in 1999 and asked me to direct an independent study on the subject for her. I’ve since taught topics courses in cryptology to math majors and to liberal arts students, and it always…

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March 12, 2010, 7:38 am

ICTCM underway

It’s a beautiful day here on the shores of Lake Michigan as the ICTCM gets underway. It’s a busy day and — to my never-ending annoyance — there is no wireless internet in the hotel. So I won’t be blogging/tweeting as much as I’d like. But here’s my schedule for the day.

  • 8:30 – Keynote address.
  • 9:30 – Exhibits and final preparations for my 11:30 talk.
  • 10:30 – “Developing Online Video Lectures for Online and Hybrid Algebra Courses”, talk by Scott Franklin of Natural Blogarithms.
  • 11:10 – “Conjecturing with GeoGebra Animations”, talk by Garry Johns and Tom Zerger.
  • 11:30 – My talk on using spreadsheets, Winplot, and Wolfram|Alpha|Alpha in a liberal arts calculus class, with my colleague Justin Gash.
  • 12:30 – My “solo” talk on teaching MATLAB to a general audience.
  • 12:50 – “Programming for Understanding: A Case Study in Linear Algebra”, talk by Daniel Jordan.
  • 1:30 – “Over a Decade of…

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September 29, 2008, 6:20 am

Monday GTD moment: Scholarship and GTD

This is the third installment of Monday GTD Moment, where I take a post to blog about Getting Things Done and how it applies in an academic setting. If you’re unfamiliar with GTD, here’s a good overview, and make sure to read David Allen’s book that started it all.

Last week I wrote about grading and GTD. I noted that grading is kind of a poor fit in traditional GTD. A prof can grade anywhere, so the idea of contexts fits awkwardly; and grading “tasks” are usually projects, although we think of them as tasks and although the next actions contained in those projects are usually nothing more than smaller projects. GTD wasn’t really made for the academic profession, and so the staple activities of academics don’t often fit well.

Another area similar to grading in its relatively poor fit within the canonical GTD philosophy is research, or more generally scholarship. By “scholarship” …

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August 8, 2008, 2:11 pm

What is a basic syllabus in educational technology?

So I’m plotting out my tactical plans for research and scholarship over the next year right now — my imagination being stoked by the completion of my Statement of Scholarship — and I’d like to go deeper into educational technology on a number of levels. I’d like not only to stay abreast of the rapidly-changing face of the technology being used in schools, but also the social implications of that technology, the legal issues behind it, and the technical nuts/bolts/bits of how this stuff works in the first place (including the computer network/programming side of things).

I’m just a user and a self-appointed pundit of ed tech, so I have no idea exactly where to start if I want really to go deeper on this subject. I do know that I’m going to swallow hard and read Digital Natives, Digital Immigrants by Prensky carefully (as opposed to skimmig it as I have done in the past) even though I…

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July 4, 2008, 6:07 pm

Letting teaching and research feed each other

Good article here at the Chronicle on balancing teaching with research, from a neuroscience professor who makes it work for him.

The reality of modern academe is that, no matter what your institutional affiliation, the time you can devote to research is being squeezed by multiple competing demands. No simple solution to that problem exists for any of us. But I have found that rethinking the nature of our professional commitments, such that teaching activities bleed into research ones (and vice versa), can be an effective way to reduce the time crunch. Academics describe their workload of scholarship, teaching, and service as if those were entirely separate entities. In reality, the line between teaching and research is usually much fuzzier.

Read the whole thing, in which Prof. Gendle writes at length about the potentially prosperous symbiosis between teaching and research. He points out …

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May 27, 2008, 11:49 am

Fun with finite fields

For those of you interested, I have a review of Finite Fields and Applications by Gary Mullen and Carl Mummert now posted at MAA Reviews. You can get to it here, although you have to be an MAA member to view it, or else pay $25/year for a nonmember subscription.

If you aren’t an MAA member and don’t want to pay, the bottom line of the review is: It’s a pretty good book. Very good for mathematicians, grad students, and advanced undergrads. Normal undergrads will need patience and perhaps a lot of help with the initial chapter, which is a lot of serious algebra which unfortunately doesn’t appear to make that much of an appearance in later chapters when the applications show up. And what’s with the three-paragraph treatment of AES? On the other hand, lots of neat stuff about Latin squares, including a cryptosystem based on mutually orthogonal Latin squares which I’d never seen before. …

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November 30, 2007, 5:37 am

A place for rejected math articles

If you’ve been submitting mathematics articles to refereed journals only to have them sent back to you every time, there’s hope. You can try submitting them to the new journal Rejecta Mathematica, which will consist only of papers which have been rejected from peer-reviewed journals. From their web site:

At Rejecta Mathematica, we believe that many previously rejected papers can nonetheless have a very real value to the academic community. This value may take many forms:

  • “mapping the blind alleys of science”: papers containing negative results can warn others against futile directions; 
  • “reinventing the wheel”: papers accidentally rederiving a known result may contain new insight or ideas; 
  • “squaring the circle”: papers discovered to contain a serious technical flaw may nevertheless contain information or ideas of interest;
  • “applications of cold fusion”: papers based on a…

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