  We investigate quantum authentication schemes constructed from quantum
error-correcting codes. We show that if the code has a property called purity
testing, then the resulting authentication scheme guarantees the integrity of
ciphertexts, not just plaintexts. On top of that, if the code is strong purity
testing, the authentication scheme also allows the encryption key to be
recycled, partially even if the authentication rejects. Such a strong notion of
authentication is useful in a setting where multiple ciphertexts can be present
simultaneously, such as in interactive or delegated quantum computation. With
these settings in mind, we give an explicit code (based on the trap code) that
is strong purity testing but, contrary to other known strong-purity-testing
codes, allows for natural computation on ciphertexts.
