How to Determine the Total Number of Affine Ciphers for an Alphabet of 24 Letters

In cryptography and cryptanalysis, an affine cipher is a type of cipher that involves two parameters, the multiplicative key lsquo;arsquo; and the additive key lsquo;brsquo;, which are used to transform plaintext letters into ciphertext. The affine cipher is defined by a specific transformation:

Ex ax b mod n

where lsquo;nrsquo; is the size of the alphabet, lsquo;xrsquo; is the numerical representation of the plaintext letter, and lsquo;arsquo; and lsquo;brsquo; are the keys used in the transformation.

Steps to Calculate the Total Number of Affine Ciphers for an Alphabet of 24 Letters

To determine the total number of affine ciphers for an alphabet of 24 letters, we follow these steps:

1. Choosing the Multiplicative Key lsquo;arsquo;

The value of lsquo;arsquo; must be coprime to lsquo;nrsquo; for the cipher to be invertible. This means lsquo;arsquo; and lsquo;nrsquo; should not share any common factors other than 1. The count of such integers lsquo;arsquo; in the range 1 to (n-1) can be found using Euler's Totient Function, denoted as phi;(n).

For an alphabet size of 24 letters, the prime factorization of 24 is (2^3 times 3^1).

Calculate φ(24):

φ(24)  24 × (1 - 1/2) × (1 - 1/3)  8

2. Choosing the Additive Key lsquo;brsquo;

The value of lsquo;brsquo; can be any integer from 0 to (n-1). Therefore, there are (n) possible choices for lsquo;brsquo;.

3. Total Number of Affine Ciphers

The total number of affine ciphers can be calculated by multiplying the number of choices for lsquo;arsquo; by the number of choices for lsquo;brsquo;:

Text{Total Ciphers} φ(24) × 24 8 × 24 192

Example and Cleartext

Consider an affine cipher with an alphabet size of 24. An affine cipher can be thought of as mapping the plaintext symbols to the range of numbers 0 to 23, applying a transformation of the form mx b (mod n), and then translating back into symbols.

Example: The Caesar cipher is a special case where m 1 and only a shift is applied, wrapping around at the end of the alphabet.

For (n 24), we need to pick an m such that (gcd(m, n) 1). Otherwise, the encryption is not reversible. The number of such em{mrsquo}s is given by Euler's totient function. Using a variety of methods, we can conclude that (varphi(24) 8) choices of m.

For each em{m}, we may choose any value of em{b} between 0 to 23 inclusive. This results in the total number of possible ciphers being (8 times 24 192).

Are All These Ciphers Distinct?

Yes, all these ciphers are distinct. If (mx b mx b (mod n)) for all (x), then (m - m equiv b - b). However, the (x 0) case gives us (b - b equiv 0) and the (x 1) case gives us (m - m equiv 0). Therefore, distinct ((m, b)) pairs give distinct affine ciphers.

Conclusion

Thus, the total number of affine ciphers for an alphabet of 24 letters is 192. This includes the identity cipher (m 1, b 0). Adjusting for the identity cipher, there are effectively 191 unique ciphers.