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This number comes from the fact there are 12 numbers that are coprime with 26 that are less than 26 (these are the possible values of a).
Each value of a can have 26 different addition shifts (the b value) ; therefore, there are 12*26 or 312 possible keys.
To give a small level of encryption, this table can be randomized and shared with the recipient.
A becomes N, B becomes O, and so on up to M, which becomes Z, then the sequence continues at the beginning of the alphabet: N becomes A, O becomes B, and so on to Z, which becomes M.
The encryption function for a single letter is where modulus m is the size of the alphabet and a and b are the key of the cipher.
The value a must be chosen such that a and m are coprime.
In classical cryptography, a permutation cipher is a transposition cipher in which the key is a permutation.
To apply a cipher, a random permutation of size E is generated (the larger the value of E the more secure the cipher).