In the book, Applied Cryptography by Bruce Scheier, in the foundations chapter he defines symmetric encryption algorithms as
algorithms where the encryption key can be calculated from the decryption key and vice versa.
Then he goes on to say (caps added by me):
In most symmetric algorithms, the encryption key and decryption key are the same.
As per the definition it is clear that the encryption and decryption keys need not to be the same in a symmetric encryption algorithm.
Can someone please give me an example of such algorithm where the keys are not the same? Does any extra security added if one key is derived from the other or in other words why do we choose to have same key for encryption- decryption (other than the simplicity of use).