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How did U.S. cryptanalysts break the Japanese Red and Purple cipher systems?

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A Vigenere cipher table displays the alphabet in both the top row and left column, with rows shifted to create a grid for encryption and decryption. Highlighted letters illustrate how to use the table. An explanation and example using the words cipher, key, and plaintext appears to the right. — cryptology In encrypting plaintext, the cipher letter is found at the intersection of the column headed by the plaintext letter and the row indexed by the key letter. To decrypt ciphertext, the plaintext letter is found at the head of the column determined by the intersection of the diagonal containing the cipher letter and the row containing the key letter.

U.S. cryptanalysts successfully broke the Japanese Red and Purple cipher systems through a combination of mechanical analysis and the exploitation of Japanese procedural weaknesses.

The Red Cipher

In 1930, the Japanese Foreign Office introduced its first rotor machine, designated "Red" by U.S. cryptanalysts. By 1935–36, a team from the U.S. Army Signal Intelligence Service (SIS), led by William F. Friedman, successfully cryptanalyzed the system. Their success drew heavily on previous experience analyzing Hebern rotor machines.

The Purple Cipher

In 1939, Japan introduced the "Purple" machine, which replaced rotors with telephone stepping switches. U.S. cryptanalysts broke this system due to two primary factors:
* Gradual Transition: The replacement of Red machines with Purple was slow, which provided an enormous number of "cribs"—known or suspected plaintext-ciphertext pairs—that aided the analysis.
* Systematic Key Generation: To avoid key distribution issues, the Japanese generated keys systematically, allowing U.S. cryptanalysts to eventually anticipate keys days in advance.

Friedman and his associates constructed functional analogs of the Purple machine, which were used throughout World War II to decrypt Japanese communications, a project collectively code-named Magic.