Cryptology Key Generators with Long Periods

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Periods

We generate long periods by superposing some different short periods. These could by realized by gear wheels with appropriate numbers of teeth.

Assume q wheels that have n_i different positions resp. Each wheel moves by one position for each encryption step. If we describe the state of the complete system by the q-tuple of positions, then the period of the state changes of this system is the least common multiple of the single periods. For an optimal effect one usually makes the single periods pairwise coprime.

The first reported occurrence of this idea is ascribed to VERNAM's colleague MOREHOUSE. VERNAM originally used as key a punch tape glued together to an endless loop. MOREHOUSE suggested the combination of several such tapes.

The combination of several gear wheels is ascribed to Parker HITT (1877–1971, US Army cryptologist in World War I), who apparently also was the first to formulate the insight:

»No message is safe in cipher ... unless the key phrase is comparable in length with the message itself.«

Examples

Historical examples used the periods

• 17, 19, 21, 23 by Arvid DAMM, resulting period 156 009,
• 11, 15, 17, 19 for the Enigma A, resulting period 53 295,
• 17, 19, 21, 23, 25 for HAGELIN's C-35, resulting period 3 900 225,
• 17, 19, 21, 23, 25, 26 for HAGELIN's C-38, resulting period 101 405 850,
• 29, 31, 37, 41, 43, 47 for newer machines by HAGELIN, resulting period 2 756 205 443 ≈ 3x10⁹,
• 47, 53, 59, 61, 64, 65, 67, 69, 71, 73 for the »Geheimschreiber« by Siemens & Halske, resulting period 893 622 318 929 520 960 ≈ 9x10¹⁷,
• 23, 26, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61 for the »Schlüsselzusatz« by Lorenz, resulting period 16 033 955 073 056 318 658 ≈ 1.6x10¹⁹.