Do all ciphers have equivalent decryption keys? Do all ciphers suffer from the problem of multiple equivalent decryption keys? Is the existence of equivalent keys an essential property for the security of a cipher?

If you could prove that a cipher had no equivalent keys, would that lead to a procedure for recovering the key under the known-plaintext attack model?
Do all ciphers suffer from the problem of multiple equivalent decryption keys?


No. The number of non-equivalent keys is bounded by the number of permutations. Since the number of permutations is very high there is a very big chance that ciphers do not have equivalent keys. This is especially true for ciphers with a high block size (AES with 128 bits). Even if an equivalent key would exist, it would probably be rather tricky to find a set of two equivalent keys. How tricky depends on the cipher of course.


  Is the existence of equivalent keys an essential property for the security of a cipher?


Therefore no.


  If you could prove that a cipher had no equivalent keys, would that lead to a procedure for recovering the key under the known-plaintext attack model?


No. Finding that a cipher does not have equivalent keys would enhance the security of the cipher, not decrease it.

More information about equivalent keys here.