H − A Sample Clauses

H − A. Since A knows the random values of the corrupted players, it learns, from ows (1b), the values Zi, and thus hi = 2(Zi) for i = 1, . . . , n 1, before it plays on behalf of [n/2♩+1. Even if it cannot modify anymore zn in ow (1b) (already committed to in cn), it can choose Zn to bias the value hn, and thus the nal key, that is de xxx as sk = H0(ssid', h1, . . . , hn): this is introduced in ow (2a). Such an inconsistent value is possible since no honest player can verify the value of Zn: this is the Di e-Xxxxxxx value between two corrupted players. A More generally, if controls enough players so that each honest player is between two corrupted players, then it can learn, from ows (1b), the values Xi that will be sent in ows (2b). If two corrupted players are neighbors, they can send a value Xi of their choice, since it comes from a Di e-Xxxxxxx value between these two corrupted players. In the attack above, the adversary could learn all the hi early enough, so that its control on hn could bias the key. If it can control an hi, without knowing the other values, there is still enough entropy in the key derivation: the nal key is uniformly distributed: contributiveness. ^ We now prove the (n/2, n)-contributiveness, using the above intuition. We need to construct, for any real-world adversary A (interacting with real parties running the protocol), an ideal-world adversary S (interacting with dummy parties and the functionality sF XX XXX) such that no environment Z can distinguish between an execution with A in the real world and S in the ideal world with non-negligible probability. S A We incrementally de ne a sequence of games starting from the one describing a real execution of the protocol in the real world, and ending up with game G8 which we prove to be indistinguishable with respect to the ideal experiment. The key point will be G7. G0 is the real-world game. In G1, we start by simulating the encryption, decryption and hash queries, canceling some unlikely events (such as collisions). Granted the ideal tweakable cipher model (see details in Appendix C), we can extract the passwords used by for players corrupted from the beginning of the session. G2 and G3 allow to be sure that the authenticators for non-corrupted players are always oracle-generated. In G4, we show how to deal with the simulation of the rst ows. In G5, we deal with only oracle-generated ows. In G6, we deal with (possibly) non-oracle-generated ows from round 2. G7 is the crucial game, where w...