Category Archives: Abstract algebra

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.

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…

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…

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 …

September 19, 2011, 8:00 am

Math Monday: What is casting out nines?

http://www.flickr.com/photos/artnoose/

Last week in this post, I asked for requests for math topics you’d like to read about. One person wrote in and asked:

Why don’t you enlighten us about the name “Casting Out Nines?” I learned a system in grade school with the same name –it was a way of checking multiplication and long division answers. Long before calculators.