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A Framework for Fully-Simulatable H-Out-Of-N Oblivious Transfer

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A Framework for Fully-Simulatable H-Out-Of-N Oblivious Transfer 1 A Framework For Fully-Simulatable h-Out-Of-n Oblivious Transfer Zeng Bing, Tang Xueming, and Chingfang Hsu ✦ Abstract—We present a framework for fully-simulatable h-out-of-n where i1 < i2 <...<ih 6 n. The receiver expects to oblivious transfer (OT n) with security against non-adaptive malicious h get the messages mi1 ,mi2 ,...,mih without leaking any adversaries. The framework costs six communication rounds and costs information about his private input, i.e., the h positive at most 40n public-key operations in computational overhead. Com- integers he holds. The sender expects all new knowledge pared with the known protocols for fully-simulatable oblivious transfer that works in the plain mode (where there is no trusted common learned by the receiver from their interaction is at most reference string available) and proven to be secure under standard h messages. Obviously, the OT most literature refer to is 2 n model (where there is no random oracle available), the instantiation OT1 and can be viewed as a special case of OTh . based on the decisional Diffie-Hellman assumption of the framework is Considering a variety of attack we have to confront the most efficient one, no matter seen from communication rounds or n in real environment, a protocol for OTh with security computational overhead. against malicious adversaries (a malicious adversary Our framework uses three abstract tools, i.e., perfectly binding com- mitment, perfectly hiding commitment and our new smooth projective may act in any arbitrary malicious way to learn as much hash. This allows a simple and intuitive understanding of its security. extra information as possible) is more desirable than We instantiate the new smooth projective hash under the lattice the one with security against semi-honest adversaries assumption, the decisional Diffie-Hellman assumption, the decisional N- (a semi-honest adversary, on one side, honestly does th residuosity assumption, the decisional quadratic residuosity assump- everything told by a prescribed protocol; on one side, tion. This indeed shows that the folklore that it is technically difficult to records the messages he sees to deduce extra information instantiate the projective hash framework under the lattice assumption is not true. What’s more, by using this lattice-based hash and lattice- which is not supposed to be known to he). Using Goldre- n ich’s compiler [24], [27], we can gain the former version based commitment scheme, we gain a concrete protocol for OTh which is secure against quantum algorithms. from the corresponding latter version. However, the resulting protocol is prohibitive expensive for practical Index Terms—oblivious transfer (OT) protocols. use, because it is embedded with so many invocations of zero-knowledge for NP. Thus, directly constructing the protocol based on specific intractability assumptions NTRODUCTION 1 I seems more feasible. 1.1 Oblivious transfer The first step in this direction is independently made BLIVIOUS transfer (OT), first introduced by [49] and by [44] and [1] which respectively presents a two- 2 arXiv:1005.0043v3 [cs.CR] 1 Apr 2011 O later defined in another way with equivalent effect round efficient protocol for OT1 based on the decisional [16] by [18], is a fundamental primitive in cryptography Diffie-Hellman (DDH) assumption. Starting from these and a concrete problem in the filed of secure multi-party works and using the tool smooth projective hashing, computation. Considerable cryptographic protocols can [32] abstracts and generalizes the ideas of [1], [44] to 2 be built from it. Most remarkable, [27], [31], [34], [56] a framework for OT1 . Besides DDH assumption, the proves that any secure multi-party computation can be framework can be instantiated under the decisional N-th based on a secure oblivious transfer protocol. In this residuosity (DNR) assumption and decisional quadratic paper, we concern a variant of OT, h-out-of-n oblivious residuosity (DQR) assumption [32]. n n Unfortunately, these protocols (or frameworks) are transfer (OTh ). OTh deals with the following scenario. only half-simulatable not fully-simulatable. By saying a A sender holds n private messages m1,m2,...,mn. A protocol is fully-simulatable, we means that the protocol receiver holds h private positive integers i1,i2,...,ih, can be strictly proven its security under the real/ideal Zeng Bing is with the College of Computer Science and Technology, Huazhong model simulation paradigm. The paradigm requires that University of Science and Technology, Wuhan City, Hubei 430074 China (e- for any adversary in the real world, there exists a corre- mail:[email protected]). Tang Xueming is with the College of Computer Science and Technology, sponding adversary simulating him in the ideal world. Huazhong University of Science and Technology, Wuhan City, Hubei 430074 Thus, the real adversary can not do more harm than China (e-mail:[email protected]). the corresponding ideal adversary does. Therefore the Chingfang Hsu is with the College of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan City, Hubei 430074 security level of the protocol is guaranteed not to be China (e-mail:[email protected]). lower than that of the ideal world. Undesirably, a half- 2 2 simulatable protocol for OT1 only provides a simulator corrupts the parties preset before the running of the in the case the receiver is corrupted such as [1], [44] or protocol. in the case the sender is corrupted such as [32]. Though constructing protocols for fully-simulatable 2 Considering security, requiring a protocol to be fully- OT1 with security against malicious adversaries has n simulatable is necessary. Specifically, a fully-simulatable been studied well, constructing protocols for such OTh protocol provides security against all kinds of attacks, hasn’t. We note that there are some works aiming to ex- n especially the future unknown attacks taken by any tend known cryptographic protocols to OTh . [42] shows n adversary whose computational resource is fixed when how to implementation OTh using log n invocation of 2 constructing the protocol (generally, it is assumed that OT1 under half-simulation. A similar implementation n the adversaries run arbitrary probabilistic polynomial- for adaptive OTh can be seen in [43]. What’s more, the time) [7], [24], while a not fully-simulatable protocol same authors of [42], [43] propose a way to transform a doesn’t. For example, the protocols proposed by [1], singe-server private-information retrieval scheme (PIR) [32], [44] suffer the selective-failure attacks, in which a into an oblivious transfer scheme under half-simulation malicious sender can induce transfer failures that are too [45]. With the help of a random oracle, [30] shows dependent on the messages that the receiver requests how to extend k oblivious transfers (for some security [45]. parameter k) into many more, without much additional Constructing fully-simulatable protocols for OT with effort. However, the Random Oracle Model is risky. security against malicious adversaries naturally becomes First, [10] shows that a scheme is secure in the Random the focus of the research community. [6] first presents Oracle Model does not necessarily imply that a particular such a fully-simulatable protocol. In detail, the OT is an implementation of it (in the real world) is secure, or even n adaptive h-out-n oblivious transfer (denoted by OTh 1 that this scheme does not have any ”structural flaws”. in related literature) and based on q-Power Decisional× Second, [10] shows efficient implementing the random Diffie-Hellman and q-Strong Diffie-Hellman assump- oracle is impossible. Later, [35] finds that the random- tions. Unfortunately, these two assumptions are not stan- oracle instantiations proposed by Bellare and Rogaway dard assumptions used in cryptography and seem signif- from 1993 and 1996, and the ones implicit in IEEE icantly stronger than DDH, DQR and so on. Motivated P1363 and PKCS standards are weaker than a random by basing OT on weaker complexity assumption, [28] oracle. What is worse, [35] shows that how the hash n presents a protocol for OTh using a blind identity-based function defects deadly damages the securities of the encryption which is based on decisional bilinear Diffie- cryptographic schemes presented in [4], [5]. Therefore, in Hellman (DBDH) assumption. Using cut-choose tech- this paper, we only consider the schemes which are fully- nique, [36] later presents two efficient protocols for fully- simulatable and without turning to a random oracle. 2 simulatable OT1 respectively based on DDH assumption To our best knowledge, only [6] and [28] respectively n and DNR assumption, where the DDH-based protocol present such fully-simulatable protocols for OTh . How- is the most efficient one among these fully-simulatable ever, the assumptions the former uses are not standard works. and the latter uses is too expensive. Therefore, a well- The protocols mentioned above are proved their se- motivated problem is to find a protocol or framework curities in the plain stand-alone model which not nec- for efficient, fully-simulatable, secure against malicious n essarily allows concurrent composition with other arbi- adversaries OTh under weaker complexity assumptions. trary malicious protocols. [48] overcomes this weakness and further the research by presenting a framework 1.2 Our Contribution under common reference string (CRS) model for fully- 2 In this paper, we present a framework for efficient, simulatable, universally composable OT1 and instan- tiating the framework respectively under DDH, DQR fully-simulatable, secure against non-adaptive malicious n and worst-case lattice assumption. It is notable that adversaries OTh whose security is proven under stand conditioning on a trusted CRS is available, the DDH- model (i.e., without turning to a random oracle). To our n based instantiation of the framework is the most efficient best knowledge, this is the first framework for such OTh .
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