# Card catalog (cryptology)

The **card catalog**, or "catalog of characteristics," in cryptography, was a system designed by Polish Cipher Bureau mathematician-cryptologist Marian Rejewski, and first completed about 1935 or 1936, to facilitate decrypting German Enigma ciphers.^{[1]}

## History[edit]

The Polish Cipher Bureau used the theory of permutations to start breaking the Enigma cipher in late 1932; the Bureau recognized that the Enigma machine's doubled-key (see Grill (cryptology)) permutations formed cycles, and those cycles could be used to break the cipher. With German cipher keys provided by a French spy, the Bureau was able to reverse engineer the Enigma and start reading German messages. At the time, the Germans were using only 6 *steckers*, and the Polish grill method was feasible. On 1 August 1936, the Germans started using 8 *steckers*, and that change made the grill method less feasible; the Bureau needed an improved method to break the German cipher.

Although the *steckers* would change which letters were in a doubled-key's cycle, the *steckers* would not change the number of cycles or the length of those cycles. Steckers could be ignored. Ignoring the mid-key turnovers, the Enigma machine had only 26^{3} distinct settings of the three rotors, and the three rotors could only be arranged in the machine 3!=6 ways. That meant there were only 105456 likely doubled-key permutations; the Bureau set about determining and cataloging the characteristic of each of those likely permutations. Each letter of the key could be one of partition number 13 = 101 possible values, and the 3 letters of the key meant there were 1030301 possible keys. On average, a key would find one setting of the rotors, but it might find several possible settings.

The Polish cryptanalyst could then collect enough traffic to determine all the cycles in a daily key; that usually took about 60 messages. The result might be:

He would use the lengths of the cycles (13^{2};10^{2}-3^{2};10^{2}-2^{2}-1^{2}) to look up the wheel order (II I III) and starting rotor positions in the card catalog, he would then use an Enigma to compute the un-*steckered* cycles:

By comparing the *steckered* cycles from the German traffic and the un-*steckered* cycles, the cryptanalyst can determine the *steckers*. In the example, the *CF* permutation has `(e)(z)`

and `(e)(w)`

. That requires that `e`

is un*stecker*ed and a W-Z *stecker*. The cycles can then be aligned on `e`

and W-Z to determine other *steckered* and un-*steckered* letters.

`(pjxroquctwzsy)(kvgledmanhfib)/(kxtcoigweh)...`

`(sjxroqtcuzwpy)(kngledamvhifb)/(kxucofgzeh)...`

`(!_____!_!**!_)(_!__=_!!!_!!_)/(__!__!_*=_)...`

Where `=`

is a known un-*steckered* letter, `*`

is a known *steckered* letter, and `!`

is a newly discovered *stecker*.
Repetition produces the *steckers* A-M, F-I, N-V, P-S, T-U, W-Z.

Preparation of the card catalog, using the cyclometer that Rejewski had invented about 1934 or 1935, was a laborious task that took over a year's time, but once the catalog was complete, obtaining Enigma daily keys was a matter of some fifteen minutes.^{[2]}^{[3]}

When the Germans changed the Enigma machine's "reflector," or "reversing drum," on 1 November 1937, the Cipher Bureau was forced to start over again with a new card catalog: "a task," writes Rejewski, "which consumed, on account of our greater experience, probably somewhat less than a year's time."^{[2]}

On 15 September 1938 the Germans completely changed the procedure for enciphering message keys, rendering the card-catalog method useless; this spurred the invention of Rejewski's cryptologic bomb and Henryk Zygalski's "perforated sheets."^{[4]}

## Notes[edit]

**^**Marian Rejewski, "The Mathematical Solution of the Enigma Cipher," pp. 284–87.- ^
^{a}^{b}Marian Rejewski, "Summary of Our Methods for Reconstructing ENIGMA and Reconstructing Daily Keys...", p. 242. **^**Marian Rejewski, "How the Polish Mathematicians Broke Enigma," p. 264.**^**Marian Rejewski, "Summary of Our Methods for Reconstructing ENIGMA and Reconstructing Daily Keys...", pp. 242–43.

## References[edit]

Methods and technology |
---|

Locations |

Personnel |

German Section cryptologists Wiktor Michałowski |

The Enigma cipher machine |
---|

- Władysław Kozaczuk,
*Enigma: How the German Machine Cipher Was Broken, and How It Was Read by the Allies in World War Two*, edited and translated by Christopher Kasparek, Frederick, MD, University Publications of America, 1984, ISBN 0-89093-547-5. - Marian Rejewski, "Summary of Our Methods for Reconstructing ENIGMA and Reconstructing Daily Keys, and of German Efforts to Frustrate Those Methods," Appendix C to Władysław Kozaczuk,
*Enigma*, 1984, pp. 241–45. - Marian Rejewski, "How the Polish Mathematicians Broke Enigma," Appendix D to Władysław Kozaczuk,
*Enigma*, 1984, pp. 246–71. - Marian Rejewski, "The Mathematical Solution of the Enigma Cipher," Appendix E to Władysław Kozaczuk,
*Enigma*, 1984, pp. 272–91.

This cryptography-related article is a stub. You can help Wikipedia by expanding it. |