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Proof of achievability From the set of BECs {W (i)} for ∀i∈ [ 1,N], the size of each information set |A_{L}(i)|= lfloor L 1-epsilon _{m_{i}})rfL 1-epsilonm {dot {i}}), the unit of which is in bits per L channel uses.

Proof The achievability follows from Theorem 1.

Since achievability directly follows from Theorem 3, we proceed to the converse part.

Proof: The achievability part follows from C 1 str in Theorem 6 by evaluating (38 - 41 38 - 41ero mean jointly Gaussian channel inputs X1, X2 and X3.

The achievability follows directly from Theorem 1 by taking U = Y since Y = f XD, Xr, SD), letting XR be independent of Ur and Xr considering the fact that Yr is only determined by XR and SR, and by setting xr = gr ur, sD), xD = gd y, ur, sD).

Proof: Achievability: The proof follows from the region of Theorem 1.

Achievability of these rates follows from [8, Proposition 2].

This fact is readily obtained from the alternative achievability proof given in Section 4. We note that [25, 26] consider the problem key generation from common randomness over wiretap channels and exploit a Wyner-Ziv coding scheme to limit the amount of information conveyed from the source to the destination via the wiretap channel.

It is noted that in this case, the common message can be ignored, i.e., T = ∅ and R0 = 0. Proof: Achievability: Considering (35 - 37 35 - 37roof follows from theorem 3. Converse: Consider a ( 2 n R 0, 2 n R 1, 2 n R 2, 2 n R 3, n ) code with an average error proofbility ofollows → 0. Defrom Theoremlowing RV for i = 1,..., n: T n = M 0 (42).

(On a 10-point scale, the average achievability rating increased by more than half a point, from 5.75 to 6.39).

Here we outline an achievability argument based on the consideration of a conceptual wiretap channel from the destination back to the source and eavesdropper suggested in [13, Theorem 3].