SECRET KEY AGREEMENT Sample Clauses

SECRET KEY AGREEMENT to the probability distribution P(x, y) is given by the mutual information I(X, Y). I(X, Y) = . . x∈X y∈Y P(x, y) log P(x, y) P1(y)P2(y)

SECRET KEY AGREEMENT. Thus we can introduce a model like shown in figure 1.2.

SECRET KEY AGREEMENT. That means that: (1.6) the correlation (mutual information) between N copies of Eve’s ran- dom variable, the communication C and the key S is arbitrary small, (1.7) the entropy of the secret key is arbitrary close to its maximum, namely an uniform distribution and (1.8) the rate is non-zero in the limit of large blocks. Based on this definition it is hard to find the secret-key rate for a given distribution. Hence it is very useful to establish more easily computable upper and lower bounds for this quantity. It seems quite reasonable that the rate at which Xxxxx and Xxx agree on a secret bit cannot be lower than their shared information degraded by the mutual information between Eve and one of them. In other words the secret-key rate must be larger than what Eve knows about one random variable of the honest parties.

SECRET KEY AGREEMENT. Another interesting question is what happens if Xxxxx or Xxx perform local maps PX|X and PY|Y . This can be described as the following: Let X, Y, Z, X and Y be random variables jointly distributed like PXYZXY = PXYZ · PX|X · PY |Y . Then we can state due to the fact that the secret-key rate is the maximum rate taken over all possible protocols between Xxxxx and Xxx that

SECRET KEY AGREEMENT lation protocol previously described. After this protocol, the probability that Xxx accepts a message correctly is given by the fidelity F N 2 2 N F = (δ + ǫ ) (1.19) and in the case of a false accepted message we obtain the disturbance D N . Σ N 2 2 D = 1 − (ǫ + δ ) (1.20) Moreover we can say that Xxx accepts a message in general with the probability paccept = F N + DN (1.21) and thus we can derive Bob’s overall error probability, getting: D N βN = F N + DN (1.22) We are now interested in the conditional probability γN that Eve decides the wrong message 0 (0) (1) δF · F2 (1 − δF ) · F2 (0) (1) δD · D2 (1 − δD) · D2 Table 1.3: Resulting distribution after receiving the broadcasted signal under the condition that Xxx accepts the correct one. Therefore we introduce the probability δF = P(Z = X|X = Y) that Eve makes a right decision given that Xxx accepts the correct one.

SECRET KEY AGREEMENT. 11 We can now conclude Eve’s error probability for the whole message as the following: γN =

SECRET KEY AGREEMENT. PZ1...Z mZw . Let ZV := (Zi : i ∈ V ) and Zn denote the n i.i.d. In the secure secret key agreement, each user tries to realizations of Zi. Each user has access to the corresponding random variable. Upon observing n i.i.d. realizations, the recover a common randomness that is kept secure from the wiretapper. Specifically, we say that (F(n), E(n), . . . , E(n)) terminals communicate interactively using their observations and possibly independent private randomness on the noiseless and authenticated channel. In other words, the communication made by an user in any round depends on all the previous rounds communication and user’s observations. Let F(n) de-