Using a symmetric encryption algorithm E on a message M, users usually send E(M)

Question

Using a symmetric encryption algorithm E on a message M, users usually send E(M) to the recipient and then the recipient will compute E-1(E(M))=M to retrieve the message. For what symmetric encryption algorithms would it be equally secure to instead send E-1 (M) and have the recipient compute E(E-1(M))=M?

Suppose we have two symmetric keys K and L to encrypt M. We could compute K(L(M))=M?. Is there any secure encryption algorithms such that from K and L, we can compute a new key KL where KL(M)=K(L(M))?

I know XOR has both of the above properties if used on just one block. But my question is: does there exist any secure symmetric algorithms that have both properties and can be used for multiple blocks?

Explanation / Answer

As block ciphers are invertible, and since XOR is too - the main operation on the key stream for many stream ciphers - the resulting encryption/decryption modes of operation are often invertible. For stream ciphers that create a key stream that is XOR'ed with the plaintext, it is even true that E=E-1.

Some block cipher modes of operation however require padding. If padding is included in E then then you cannot perform E-1(M), because the padding would likely fail or result in a wrongly sized plaintext.

+ ECB mode is invertible, but requires padding
+ CBC mode is invertible, but requires padding
+ CFB, OFB and CTR modes generate a key stream are therefore invertible, they do not require padding

Any authenticated cipher is of course not invertible as E would include creation of the authentication tag, (and E-1 includes the verification of an authentication tag that isn't there).

Usually a (block) cipher has the same security for E as for E-1. It would however be advisable to use the cipher as intended, or verify (somehow) that the security conditions hold.

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