Archive for October 13th, 2008

91115/11116 !

October 13, 2008

One of my favorite encryption techniques isn’t really an encryption at all: it fails in the basic sense that if you know the method of encryption, you can easily decrypt the message. But it does reduce every message to a fraction, and I like that. I can envision entire conversations consisting of phrases like “73/9”. Here’s how it works. You start by replacing each letter with a number: say, A=1, B=2,…,Z=26, and perhaps (space)=27, plus whatever punctuation that you want. Then you write your message as a continued fraction (that is, a nested fraction where each numerator is 1). For example, to write “HELLO” you’d let H=8, E=5, L=12, L=12, and O=15 and write:

8+\frac{1}{5+\frac{1}{12+\frac{1}{12+\frac{1}{15}}}}.

Then you’d simplify it before sending it. For example, the fraction above simplifies to:

8+\frac{1}{5+\frac{1}{12+\frac{1}{\frac{181}{15}}}} and then 8+\frac{1}{5+\frac{1}{12+\frac{15}{181}}}

all the way to \frac{91115}{11116}.

To decrypt, use division to write the fraction as a continued fraction (keeping in mind that words that end in “A” will turn out to be ambiguous unless you develop a way around it). Want to have some fun? Try decrypting \frac{2358}{169} (or the longer \frac{536341314626}{38440072317}).

I don’t know if the guy in the painting by Lesser Ury is really a spy, but I kind of think he looks like one.

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