Can someone help me understand this?

Question

In the book Serious Cryptography

It says for a cipher's permutation to be secure

"Different keys should result in different permutations. Otherwise it becomes easier to decrypt without the key: if different keys result in identical permutations, that means there are fewer distinct keys than distinct permutations, and therefore fewer possibilities to try when decrypting without the key. In the Vigenere cipher, each letter from the key determines a substitution; there are 26 distinct letters, and as many distinct permutations."

Cam someone explain this please?

Answer 1

Since we're talking about classical cipher, I'll assume an alphabet will be the dataset we're working with; thusly, a letter from the alphabet will be a datum from the dataset.

A permutation defines the process that maps letters to possibly different letters. Two friends can agree on a single such permutation to exchange secret information.

If we want the encryption algorithm to be useful, different groups of people should be able to easily obtain different permutation by choosing a key.

However, if a different key can select the same permutation, then that key breaks the secrecy of the first key. Even worse, if a group of keys yields same (or even similar) permutation, then decrypting other messages encrypted using other keys in the group of keys becomes very easy. And this is very bad for an encryption algorithm.

Update for OP's comment

The above text discusses a "hypothetical" cipher (not Vigenere or any specific cipher) with weak keys, and explain some reason why it can be bad (it is bad regardless of what cipher we're talking about).