Question

Suppose that you have an alphabet of 26 letters:

(a) How many possible simple substitution ciphers (Caesar Ciphers) are there? (this part here is just for context, im more asking about part b)

(b) A letter in the alphabet is said to be fixed if the encryption of the letter is the letter itself. How many simple substitution ciphers are there that leave

– no letter fixed?

– exactly two letters fixed?

Answer 1

Answer (b) - no letters fixed

Let A be the alphabet, of size a. That is: |A|=a.

Let C ( N = n ) be the count of ciphers of the alphabet of where the fixed letters (N) are equal to n, and similar for the various order relationships, C ( N = n ) , C ( N < n ) , C ( N ≤ n ) , C ( N ≥ n ) , C ( N > n ) , C ( N ≠ n ).

(a) C ( N ≥ 0 ) = a!

The count of all ciphers of alphabet, is the count of permutations of the alphabet.

- exactly two letters fixed

Use: C ( N = 2 ) = C ( N ≥ 1 ) − C ( N ≥ 2 )

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