Privacy Amplification. Comparing the security condition given in Definition 1.8 with Xxxxxxx’x criterion of secu- rity (perfect security) given in Definition 1.6 implies that a secure communication is the sense of Definition 1.8 is not necessarily a secure communication in the sense of Defini- tion 1.6. That is, Definition 1.8 requires only Xxx’s ratio of information about the message to be negligible. Hence, she can gain a possibly considerable amount of information about the message. Similarly, Condition 1.18b guarantees that only Xxx’s ratio of information about the key is negligible but not her information about any single bit of the key. On the other hand, a pair of agreed keys according to Definition 1.11 guarantees only a secure communication in the sense of Definition 1.8 according to Lemma 1.2. Conclusively, Definition 1.8 and Definition 1.11 are too weak to provide privacy require- ments [7,8] for the entire message and the entire key, respectively. Thus, the AS conditions in these definitions are referred to as the weak sense of security in the literature [3, 41]. Also, the achievable key rate and key capacity introduced in Definition 1.11 are known as the weak sense of achievable key rate and that of key capacity, respectively 19. New definition for the achievable key rate is given below to assure that the entire key is secure. Following [3], we first combine the security and the randomness conditions of Definition 1.11 into a security index S(n) , log(|K|) — H(K|z, e, P ) , (1.30)
Privacy Amplification. After information reconciliation, Xxxxx and Xxx both know x0 but cannot use it as key, since Xxx has some information about it (in fact, Xxx knows some positions of x0 with certainty in our setting). Xxxxx and Xxx rectify this situation in the next step, called privacy amplification. The simple idea is that Xxxxx and Xxx can apply a strong extractor: Xxxxx chooses a seed uniformly at random and sends it to Bob. Then, they both apply the extractor to x0. Since for Eve x0 has large min-entropy, this gives a bit string which is close to uniform with respect to Xxx’s information.
Privacy Amplification. The Alices apply a privacy amplification protocol to generate the final key systems XX0 XX0 · · · XXX . The winning condition for the M -partite parity-CHSH game is [RMW18] a1,j ⊕ a2,j = x1,j ∧ x2,j ⊕ Mi=3 ai,j ! , (2) M 4 where a1,j, . . . , xX,x, x0,x, and x2,j are realizations of the random variables specified in round j of the RMW18 Protocol. The winning probability for an arbitrary classical strategy is PCHSH = 3 . The Xxxx inequality corresponding to the classical–quantum threshold in the tripartite case is [HKB20] ν = O1O+O3 − O0O− ≤ 1, (3) i where O± = (O0 ± O1)/2, i refers to the ith party, and O0 and O1 are observables corresponding to the inputs 0 and 1, respectively, and are defined in [HKB20].
Privacy Amplification. [9] The transmitter and the legiti- mate receiver publicly agree on a deterministic function they apply to their common sequence to generate a secret key. In this work, we focus primarily on the first phase of the key- distillation process and we investigate the optimal transmission strategy to adopt when the terminals in the network deploy multiple antennas. of messages over the public channel. A secret-key rate is defined as the ratio between the number of key bits k obtained at the end of a key-distillation strategy and the number of noisy channel uses n required to obtain it. A secret-key rate R = k/n is achievable if there exists a secret-key distillation strategy such that, • on denoting the secret key distilled at the transmitter and that at the legitimate receiver by K and Kˆ, the error probability is zero asymptotically, that is: public communication channel W nR R + YnR B lim n→∞ P hK =/ Kˆi = 0; (3) A XnT HE HR + ZnE E W • the mutual information between the secret key and the eavesdropper observations is arbitrarily low (strong se- crecy constraint [11]); that is, if we denote the messages sent on the public two-way channel by the the random variable F and we collect the outputs of the eavesdropper nE E
