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When we refer to symmetry in cryptographic algorithms such as symmetric or assymetric does the symmetry refers only to the equality or not of the keys, or does it also refer to the symmetry of encryption and decryption operations?

Also does symmetry when refering to keys means they are equal? Is there any symmetry between a private and public key? They might be different but they are linked in a way,in rsa for example.

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It only refers to the symmetry of the keys.

There are a number of block ciphers and modes (symmetric crypto) that have slightly different encrypt and decrypt routines. Yet they are still symmetric crypto. AES for example uses a number of building blocks for which there are inverse operations used to decrypt.

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To answer the second question – no, keys do not have to be equal.

One can consider an encryption scheme symmetrical, if it's "easy" to compute private (decryption) key from the public (encryption) one.

For most symmetrical algorithms used in practice the keys are equal though.

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