Assume we have a hardware device with build-in AES decryption and support for ECB and CBC mode (but no support for AES encryption). It only receives data and decrypts it. Minor modifications of the hardware are still possible, but you can't add AES encryption.
Now assume we want to add a MAC for ensuring integrity. CBC-MAC can't be used, as it requires the encryption mode of the block cipher.
Are there MACs you can construct with only the decryption mode of a block cipher?
I thought about using the CBC-MAC scheme and simply replace the AES encryption with the AES decryption, but I'm not sure this would result in a secure MAC.