RESULTADOS A PARTIR DE ENCUESTA TIPO MIXTA APLICADA A

Here, we state a result that guarantees that part of W1remains (weakly) secret

from Receiver 2. We first define a new version of the capacity region _{R that} takes secrecy into account.

Definition 6.6 The equivocation-capacity region_Re of a given IMA chan-

nel is the set of all triples (R1, R2, Re) such that for any ǫ > 0, there exists a

block length T and a (T, ǫ)-code for which we have 1

TH(Wi)≥ Ri− ǫ, i = 1, 2

and _T1H(W1|Y2)≥ Re− ǫ, where Y2 is the received sequence at Receiver 2.

Definition 6.7 For a given distribution pQ pX1|Q pX2|Q pU1|X1,Q, define

Re,i(Q, X1, X2, U1) to be the set of all triples (R1, R2, Re) of non-negative real

numbers satisfying the 6 constraints of Definition 6.4 and the additional con- straint

Re ≤ min



R1, I(X1; Y1|U1, X2, Q), I(X1, X2; Y1|U1, Q)− R2

− minI(X1; Y2|U1, X2, Q), I(X1, X2; Y2|U1, Q)− R2

. (6.32)

In addition, define

Re,i , ∪Q,X1,X2,U1Re,i(Q, X1, X2, U1),

Theorem 6.4 For any IMA channel given by a set of alphabets and a distri-

bution pY1,Y2|X1,X2, we have

Re,i⊆ Re.

To prove this theorem, we show the existence of a scheme that has the following characteristics. Transmitter 1 uses the same structured code as in the proof of Theorem 6.1, but the private index j is split into two sub-indices jn and js. We are able to show that there exists a code for which jn is not

necessarily secret, but js is kept secret from Receiver 2. The bound on the

equivocation is derived in a similar way as in Proposition 6.1. A detailed proof is given in Appendix D.

Note that Theorem 6.4 provides weak secrecy from Receiver 2. If a certain (R1, R2, Re) lies in Re, then the results of Maurer and Wolf, explained in

Appendix E, can be used to show that we can achieve strong secrecy at a rate arbitrarily close to Re.

Discussion and Future

Work

7

In this thesis, we show that it is possible to guarantee a computable lower bound on the strong perfect secrecy capacity for wireless relay networks with an arbitrary acyclic topology. Our results are for the well-motivated model of Gaussian signal interaction, as well as for the simpler deterministic interaction model. We also provide an upper bound on the perfect secrecy capacity for arbitrary wireless networks. Unfortunately, it seems difficult to characterize the gap between the upper and lower bounds.

The Gaussian result should be viewed as a first step towards an approx- imate characterization of the perfect secrecy capacity of arbitrary networks. Future efforts should go into finding an upper bound that can be related to the lower bound expression, as well as decreasing the subtractive constants α, β and γ in Theorem 3.2.

The result for deterministic signal interaction can provide valuable insights into possible coding schemes. In addition, it proves to be more easy to handle. Therefore, we believe that it is important to pursue the problem under this model as well. The main aim would again be to find an upper bound on the secrecy capacity that can be related to the lower bound. Towards this aim, one can also try to improve the lower bound by considering additional coding techniques like decode-and-forward and destructive interference. Our results for the fan network show that for special network topologies with discrete memoryless channels, one can obtain characterizations of the perfect secrecy capacity. To find these characterizations, we used intuitions and techniques from our results for general networks.

Feedback from the destination to the source is a promising feature, but studying it for arbitrary wireless networks seems very challenging. Our results for the line network are an encouraging first step. A valuable extension would be a non-trivial upper bound on the secret key capacity for this small network.

During my doctoral studies, I have worked on several other problems that are not contained in this thesis. The most important results are the following: We considered the problem of lossy compression of a Gaussian source that is to be reproduced by two independent decoders, each having access to a different Gaussian side-information source that is correlated with the data source. We found the rate-distortion region for the case when the side-information sources are physically degraded. In more recent work, we studied lossy compression of a discrete source with erased side-information. We found that this type of setup shares many properties with the Gaussian case, and all our Gaussian results were repeated for this setup. These results can be found in 1. and 8. (in the CV at the end of the thesis).

In joint work with Vasudevan (7. in the CV), we considered a source coding situation where two encoders, observing correlated sources, can collaborate by means of a rate-limited link from one to the other. The aim is to jointly describe the sources to a decoder. We provided an inner bound on the rate- distortion region and showed tightness for two special cases.

Recently, we presented a result obtained in collaboration with Vasudevan and Vojnovi´c (see 4. in the CV at the of this thesis). In this contribution, we studied the algorithmic problem of distributed binary consensus in a complete graph. We showed that when each node can use 3 states for memory and signaling, the reliability of the best consensus algorithm improves dramatically as compared to the case when the nodes have binary memory and binary messages. We also showed that the convergence is as fast as for the case without limits on the state.

Together with Rethnakaran Pulikkoonattu, we developed a software called Xitip, which is a C-version of the “Information-Theoretic Inequality Prover” (ITIP) by Yeung and Yan [50]. Our version of this software is faster, more easily portable, and it includes a graphical user interface. Further references about this work are given in the CV at the end of the thesis.

Appendix for Chapter 3

A