International Association
for Cryptologic Research

<p> Oblivious Pseudorandom Functions (OPRFs) allow a client to evaluate a pseudorandom function (PRF) on her secret input based on a key that is held by a server. In the process, the client only learns the PRF output but not the key, while the server neither learns the input nor the output of the client. The arguably most popular OPRF is due to Naor, Pinkas and Reingold (Eurocrypt 2009). It is based on an Oblivious Exponentiation by the server, with passive security under the Decisional Diffie-Hellman assumption. In this work, we strengthen the security guarantees of the NPR OPRF by protecting it against active attacks of the server. We have implemented our solution and report on the performance. Our main result is a new batch OPRF protocol which is secure against maliciously corrupted servers, but is essentially as efficient as the semi-honest solution. More precisely, the computation (and communication) overhead is a multiplicative factor $o(1)$ as the batch size increases. The obvious solution using zero-knowledge proofs would have a constant factor overhead at best, which can be too expensive for certain deployments. Our protocol relies on a novel version of the DDH problem, which we call the Oblivious Exponentiation Problem (OEP), and we give evidence for its hardness in the Generic Group model. We also present a variant of our maliciously secure protocol that does not rely on the OEP but nevertheless only has overhead $o(1)$ over the known semi-honest protocol. Moreover, we show that our techniques can also be used to efficiently protect threshold blind BLS signing and threshold ElGamal decryption against malicious attackers. </p>

Recent targeted attacks have increased significantly in sophistication, undermining the fundamental assumptions on which most cryptographic primitives rely for security. For instance, attackers launching an Advanced Persistent Threat (APT) can steal full cryptographic keys, violating the very secrecy of “secret” keys that cryptographers assume in designing secure protocols. In this article, we introduce a game-theoretic framework for modeling various computer security scenarios prevalent today, including targeted attacks. We are particularly interested in situations in which an attacker periodically compromises a system or critical resource completely, learns all its secret information and is not immediately detected by the system owner or defender. We propose a two-player game between an attacker and defender called FlipIt or The Game of “Stealthy Takeover.” In FlipIt, players compete to control a shared resource. Unlike most existing games, FlipIt allows players to move at any given time, taking control of the resource. The identity of the player controlling the resource, however, is not revealed until a player actually moves. To move, a player pays a certain move cost. The objective of each player is to control the resource a large fraction of time, while minimizing his total move cost. FlipIt provides a simple and elegant framework in which we can formally reason about the interaction between attackers and defenders in practical scenarios. In this article, we restrict ourselves to games in which one of the players (the defender) plays with a renewal strategy, one in which the intervals between consecutive moves are chosen independently and uniformly at random from a fixed probability distribution. We consider attacker strategies ranging in increasing sophistication from simple periodic strategies (with moves spaced at equal time intervals) to more complex adaptive strategies, in which moves are determined based on feedback received during the game. For different classes of strategies employed by the attacker, we determine strongly dominant strategies for both players (when they exist), strategies that achieve higher benefit than all other strategies in a particular class. When strongly dominant strategies do not exist, our goal is to characterize the residual game consisting of strategies that are not strongly dominated by other strategies. We also prove equivalence or strict inclusion of certain classes of strategies under different conditions. Our analysis of different FlipIt variants teaches cryptographers, system designers, and the community at large some valuable lessons: 1.Systems should be designed under the assumption of repeated total compromise, including theft of cryptographic keys. FlipIt provides guidance on how to implement a cost-effective defensive strategy.2.Aggressive play by one player can motivate the opponent to drop out of the game (essentially not to play at all). Therefore, moving fast is a good defensive strategy, but it can only be implemented if move costs are low. We believe that virtualization has a huge potential in this respect.3.Close monitoring of one’s resources is beneficial in detecting potential attacks faster, gaining insight into attacker’s strategies, and scheduling defensive moves more effectively. Interestingly, FlipIt finds applications in other security realms besides modeling of targeted attacks. Examples include cryptographic key rotation, password changing policies, refreshing virtual machines, and cloud auditing.