Can you crack the codes to find the names of three German mathematicians?

Here are those codes again:

1)  C.a1r3l1  Fdroime drroitcahr eGnaeugs sc iwlrcoytce  at hsea  D2i seqsuui swtoiNo n.eesa cAirtietm

2)  BeUrosfnnufe  )hen nio  owfpt oaf m'tltdhd rroucln2o rsiniuanm rtt foa s3hyaki aeie csa qe1n(enoea. n0rrdgia e0 icyoteRs0tte Aidvi

3)  bfxxhndsfdohergdylqudl

And here are the decrypted messages:

1)  Carl Friedrich Gauss wrote the Disquistiones Arithmeticae. Now use 2 as a cyclic generator mod 131.

2)  Bernhard Riemann's work on geometry laid the foundation for relativity. Use continued fraction of pi (sequence A001203) as a shift

## Solution

The key to cracking the first two codes is to realise that the characters have been rearranged. That is, the individual characters (letters and punctuation) in the message are unaltered, but their positions within the message had been modified. Hence the clue: permutation.

The first message was rearranged by interspersing characters from the beginning of the message, with characters from the end of the message. That is, in the scrambled message, the 1st character was unchanged ("C"), the 2nd character was the last character ".", the 3rd was the 2nd in the original, the 4th was the penultimate in the original, and so on.

The first message gives a clue for decrypting the second message: "use 2 as a cyclic generator mod 131."  What does this mean? A good place to start is the Wikipedia page on Cyclic Groups. In essence, by taking the remainder of powers of 2 modulo 131, we may generate all the numbers in the range 1 to 129. Here's a partial list:

[1, 2, 4, 8, 16, 32, 64, 128, 125, 119, 107, 83, ..., 15, 30, 60, 120, 109, 87, 43, 86, 41, 82, 33, 66]

This list tells us how to swap the characters around to decrypt the second message. The first and second characters remain unaltered, then we move the third character to position 4, the fourth to position 8, the fifth to position 16, and so on. (The second message is, by design, just the right length for this to work perfectly)

The second message tells us how to decrypt the third. We should use the sequence A001203, the simple continued fraction expansion of pi, as a Caesar shift (see answer #6 for details).

If you have made it this far, you may be interested in finding out more about the work of these three influential German mathematicians: Gauss, Riemann and Noether.

Python codes for decrypting these messages are attached below (with many thanks to Frazer Jarvis).

ċ
Sam Dolan,
31 May 2013, 01:20
ċ
Sam Dolan,
31 May 2013, 01:20
ċ
Sam Dolan,
31 May 2013, 01:21