4.2 Decryption and Frequencies: How Might Frequencies Help?

Let A stand for text is decrypted. Let B stand for frequencies are unscrambled. By definition, unencrypted text has unscrambled frequencies. Therefore, if A is true, then B must be true. This can be suggestively written in formal, logical notation as $A\Rightarrow{}B$.

What we are now wondering about is whether we can turn things around: Is $A\Leftarrow{}B$ true? That is, if we somehow make B become true, then does that necessarily mean we made A become true?

Unfortunately, a little thought shows us that the answer is no! For example, consider an encryption key that does nothing except rename 'a' to 't' and vice versa. The effect on unigram frequencies of applying this key to text is to swap the frequencies of 'a' and 't'. Since plaintext "a theta" and ciphertext "t aheat" have exactly the same unigram frequencies, both have unscrambled unigram frequencies. (But note that the bigram frequencies are different.)

However, we should not give up hope because maybe, if we somehow make B become true --especially if we look at bigram frequencies-- then although in theory we aren't guaranteed that we've made A become true, perhaps in practice we are very likely to have made A become true. This turns out to be the case, an empirical ``fact'' that you will verify in Project 7.

In order to apply this idea, given a table of frequencies, we have to be able to recognize that it is unscrambled. We consider how to do that in the next subsection.

Roadmap
Section 3.7	Encipher plaintext $\Rightarrow$ scramble frequencies.
Section 3.7	Decipher ciphertext $\Rightarrow$ unscramble frequencies.
Section 4.2	(Hope) Unscramble frequencies $\Rightarrow$ decipher ciphertext
$\rightarrow$     Section 4.3	Q: How do we recognize that a table of frequencies is unscrambled?
Section 5	Q: What are legal and effective ways to rearrange frequencies?