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Recent research in quantum cryptography has led to the development of schemesthat encrypt and authenticate quantum messages with computational security. Thesecurity definitions used so far in the literature are asymptotic, game-based,and not known to be composable. We show how to define finite, composable,computational security for secure quantum message transmission. The newdefinitions do not involve any games or oracles, they are directly operational:a scheme is secure if it transforms an insecure channel and a shared key intoan ideal secure channel from Alice to Bob, i.e., one which only allows Eve toblock messages and learn their size, but not change them or read them. Bymodifying the ideal channel to provide Eve with more or less capabilities, onegets an array of different security notions. By design these transformationsare composable, resulting in composable security.  Crucially, the new definitions are finite. Security does not rely on theasymptotic hardness of a computational problem. Instead, one proves a finitereduction: if an adversary can distinguish the constructed (real) channel fromthe ideal one (for some fixed security parameters), then she can solve a finiteinstance of some computational problem. Such a finite statement is needed tomake security claims about concrete implementations.  We then prove that (slightly modified versions of) protocols proposed in theliterature satisfy these composable definitions. And finally, we study therelations between some game-based definitions and our composable ones. Inparticular, we look at notions of quantum authenticated encryption and QCCA2,and show that they suffer from the same issues as their classical counterparts:they exclude certain protocols which are arguably secure.

Composable and Finite Computational Security of Quantum Message Transmission