A recursive solution for Turing’s H-M factor

Abstract

ABSTRACTTuring devised an electromechanical device, called a Bombe, to decipher German Enigma messages. His original design was considerably improved by the addition of Welchman’s diagonal board, allowing the use of much weaker menus and even menus without closures. Turing then made some extensive calculations to predict the performance of the improved Bombe. He provided a table, called the H-M factor, which allowed the expected number of stops to be predicted for menus containing a given number of letters and closures, but he gave no indication of how he constructed the table. This article is of technical interest and sheds some light on the H-M factor, including its variability with the tree-arrangement of the menu. It derives the H-M factor using a recursive technique for both the standard Bombe and the Jumbo Bombe, a later version incorporating the machine gun mechanism. These results may be of interest to those that code or run Bombe emulators.