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146 CHAPTER 8. POLYALPHABETIC CIPHERS
fails with such few letters. 9
How can we make such long keywords One way would be to use phrases,
or paragraphs from agreed upon books. This works well, but becomes harder
to remember. Apparently, when people write to the National Security Agency
(NSA), the cryptographic arm of the US Government, saying “I’ve discovered
a great new and unbreakable code for you to use”, this is most frequently the
method they describe. 10
Another method we will call the Double Vigenère. We carefully pick two
keywords and encipher the message twice, using each keyword once.
Examples:
(1) Encipher aaaaaaaaaaaaaa. First use the keyword IT and then re-encipher
with the keyword WAS.
plaintext a a a a a a a a a a a a a a a a a a a a
key 1 I T I T I T I T I T I T I T I T I T I T
partial ciphertext I T I T I T I T I T I T I T I T I T I T
key 2 W A S W A S W A S W A S W A S W A S W A
ciphertext E T A P I L E T A P I L E T A P I L E T
IT and WAS act like a 6-letter keyword!
(2) Encipher banana banana banana banana using LAKE and then OCEANS.
How long is the combination keyword 11
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
What is the pattern How long a “keyword” does the use of two keywords
produce The combination keyword will start to repeat whenever each individual
keyword starts to repeat. The last letter enciphered before this repetition
starts must be enciphered with the last letter of each keyword. So each keyword
has repeated some number of times up to this point. The number of letters
enciphered so far must then be a multiple of each keylength, and so the smallest
multiple of each is when this first occurs.
The smallest multiple of each has a formal name, the least common multiple
or lcm. These Double Vigenère ciphers act like a single Vigenère cipher
whose keyword is as long as the least common multiple of the original keywords.
9 Another way to strengthen the polyalphabetic ciphers would be to avoid Caesar ciphers
as the components but instead use better mixed alphabets. We will look at this in Exercise
8.10.
10 This method was first discovered in the 1500’s by Giacolomo Cardano. He, however,
started the key over each time for each word in the message. Cardano is best remembered in
mathematics for discovering – or stealing, depending on whose version of history you believe
– the cubic formula (like the quadratic formula but for equations of degree three).
11 They act like a 12 letter keyword. ACBEL SZGCA KWACB ELSZG CAKW.