(Xxxxx and Muramatsu Sample Clauses

(Xxxxx and Muramatsu. [34]) The error probability averaged over the random assemble is bounded by Pr{(Un, X˜n) ∈/ Tn} and ϵn decay exponentially. log n X˜n|Xn (x˜n|xn) ≥ I(X˜; U|X) − γ , 1 PX˜n|U nXn (x˜n|un, xn) , P t=1 where Un ∼ Qn PUt with PUt = PU for t ∈ [1 : n]. ECn [Pr{gn (X˜n) n ψn(φn (gn (X˜n)), Y n)}] Next, we determine the codebook, and the enrollment (en- coding) and authentication (decoding) procedures. Random Code Generation: Generate exp{n(I(X˜; U ) + 2γ)} ≤ e−γn + EC [Pr{(gn(X˜n), Y n) ∈/ An}]. (50) ∈ T Q X → Q ⊂ U i.i.d. sequences of u˜n from PU and denote the set of these se- quences as n. Let gn : ˜n n n be a mapping of the measurement x˜n into u˜n. The mapping rule is that it searches u˜n such that (u˜n, x˜n) n. In case there are multiple such u˜n, the encoder picks one at random. On the contrary, if there does not exist such a sequence, u˜n is chosen. Now prepare Using Lemma 1, the right-hand side of (50) converges to zero exponentially. The proof of Lemma 2 is seen in the argument of evaluating the error probability for the Xxxxx- Xxx problem with general sources in [34]. Next, we define a security measure MJ = e nRJ bins. Assign each sequence u˜n ∈ Qn to one µn = Σ PZn (zn ) PSJ|Zn=zn,Cn − PS˜PJ|Zn=zn,Cn , of MJ bins according to a uniform distribution on Jn. This zn∈Zn − where PA PB and PS˜ denote the variational distance between probability distributions PA and PB, and the uniform distribution on the set Sn, respectively. Lemma 3. (Xxxxxxxx and Xxxxxx [18, Lemma 12]) An