Achieving security under concurrent composition is notoriously hard. Indeed, in the plain model, far reaching impossibility results for concurrently secure computation are known. On the other hand, some positive results have also been obtained according to various weaker notions of security (such as by using a super-polynomial time simulator). This suggest that somehow, ``not all is lost in the concurrent setting.\"
In this work, we ask what and exactly how much private information can the adversary learn by launching a concurrent attack? ``Can he learn all the private inputs in all the sessions? Or, can we preserve the security of some (or even most) of the sessions fully while compromising (a small fraction of) other sessions? Or is it the case that the security of all (or most) sessions is (at least partially) compromised? If so, can we restrict him to learn an arbitrarily small fraction of input in each session?\" We believe the above questions to be fundamental to the understanding of concurrent composition. Indeed, despite a large body of work on the study of concurrent composition, in our opinion, the understanding of what exactly is it that goes wrong in the concurrent setting and to what extent is currently quite unsatisfactory.
Towards that end, we adopt the knowledge-complexity based approach of Goldreich and Petrank [STOC\'91] to quantify information leakage in concurrently secure computation. We consider a model where the ideal world adversary (a.k.a simulator) is allowed to query the trusted party for some ``leakage\'\' on the honest party inputs. We obtain both positive and negative results, depending upon the nature of the leakage queries available to the simulator. Informally speaking, our results imply the following: in the concurrent setting, ``significant loss\" of security (translating to high leakage in the ideal world) in some of the sessions is unavoidable if one wishes to obtain a general result. However on the brighter side, one can make the fraction of such sessions to be an arbitrarily small polynomial (while fully preserving the security in all other sessions). Our results also have an implication on secure computation in the bounded concurrent setting [Barak-FOCS\'01]: we show there exist protocols which are secure as per the standard ideal/real world notion in the bounded concurrent setting. However if the actual number of sessions happen to exceed the bound, there is a graceful degradation of security as the number of sessions increase. (In contrast, prior results do not provide any security once the bound is exceeded.)
In order to obtain our positive result, we model concurrent extraction as the classical set-covering problem and develop, as our main technical contribution, a new sparse rewinding strategy. Specifically, unlike previous rewinding strategies which are very ``dense\'\', we rewind ``small intervals\'\' of the execution transcript and still guarantee extraction. This yields other applications as well, including improved constructions of precise concurrent zero-knowledge [Pandey et al.-Eurocrypt\'08] and concurrently secure computation in the multiple ideal query model [Goyal et al.-Crypto\'10].
In order to obtain our negative results, interestingly, we employ techniques from the regime of leakage-resilient cryptography [Dziembowski-Pietrzak-FOCS\'08].