The Cryptographic Mathematics of ⒺⓃⒾⒼⓂⒶ This publication presents a historical perspective for informational and educational purposes, is the result of independent research, and does not necessarily reflect a position of NSA/CSS or any other US government entity. This publication is distributed free by the National Security Agency. If you would like additional copies, please email your request to
[email protected] or write to: Center for Cryptologic History National Security Agency 9800 Savage Road, Suite 6886 Fort George G. Meade, MD 20755 Color photographs, David S. Reynolds, NSA/CSS photographer Cover: General Heinz Guderian in armored command vehicle with an Engima machine in use, May/June 1940. German Federal Archive The Cryptographic Mathematics of Enigma Dr. A. Ray Miller Center for Cryptologic History National Security Agency Revised edition 2019 Dedicated to the Memory of the Allied Polish Cryptanalysts Marian Rejewski Jerzy Rozycki Henryk Zygalski he Enigma cipher machine had the confidence of German forces who depended upon its security. This misplaced con- Tfidence was due in part to the large key space the machine provided. This brochure derives for the first time the exact number of theoretical cryptographic key settings and machine configura- tions for the Enigma cipher machine. It also calculates the number of practical key settings Allied cryptanalysts were faced with daily throughout World War II. Finally, it shows the relative contribution each component of the Enigma added to the overall strength of the machine. ULTRA [decrypted Enigma messages] was the great- est secret of World War II after the atom bomb. With the exception of knowledge about that weapon and the prob- able exception of the time and place of major operations, such as the Normandy invasion, no information was held more tightly….