## The Secrecy Capacity Region of the Gaussian MIMO Multi-Receiver Wiretap Channel (2009)

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Citations: | 31 - 18 self |

### BibTeX

@MISC{Ekrem09thesecrecy,

author = {Ersen Ekrem and Sennur Ulukus},

title = {The Secrecy Capacity Region of the Gaussian MIMO Multi-Receiver Wiretap Channel},

year = {2009}

}

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### Abstract

In this paper, we consider the Gaussian multiple-input multiple-output (MIMO) multi-receiver wiretap channel in which a transmitter wants to have confidential communication with an arbitrary number of users in the presence of an external eavesdropper. We derive the secrecy capacity region of this channel for the most general case. We first show that even for the single-input single-output (SISO) case, existing converse techniques for the Gaussian scalar broadcast channel cannot be extended to this secrecy context, to emphasize the need for a new proof technique. Our new proof technique makes use of the relationships between the minimum-mean-square-error and the mutual information, and equivalently, the relationships between the Fisher information and the differential entropy. Using the intuition gained from the converse proof of the SISO channel, we first prove the secrecy capacity region of the degraded MIMO channel, in which all receivers have the same number of antennas, and the noise covariance matrices can be arranged according to a positive semi-definite order. We then generalize this result to the aligned case, in which all receivers have the same number of antennas, however there is no order among the noise covariance matrices. We accomplish this task by using the channel enhancement technique. Finally, we find the secrecy capacity region of the general MIMO channel by using some limiting arguments on the secrecy capacity region of the aligned MIMO channel. We show that the capacity achieving coding scheme is a variant of dirty-paper coding with Gaussian signals.

### Citations

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Citation Context ...2 − K1) ] 1dt (135) where ⊙ denotes the Schur (Hadamard) product, and 1 = [1 . . . 1] ⊤ with appropriate size. Since the Schur product of two positive semi-definite matrices is positive semi-definite =-=[40]-=-, the integrand is non-negative implying the non-negativity of the integral. � In light of this lemma, using (133) in (130), we get α ≤ 1 ∫ ΣZ [ J(X + N2) 2 Σ2 ] −1dΣN + ΣN − Σ2 (136) = 1 2 log |J(X +... |

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Citation Context ... is kept totally ignorant of this part. Later, Csiszar and Korner considered the general wiretap channel, where there is no presumed degradation order between the legitimate user and the eavesdropper =-=[2]-=-. They found the capacity-equivocation rate region of this general, not necessarily degraded, wiretap channel. Manuscript received March 18, 2009; revised September 08, 2010; accepted September 22, 20... |

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Citation Context ...an signals. ∗ This work was supported by NSF Grants CCF 04-47613, CCF 05-14846, CNS 07-16311 and CCF 0729127. 11 Introduction Information theoretic secrecy was initiated by Wyner in his seminal work =-=[1]-=-. Wyner considered a degraded wiretap channel, where the eavesdropper gets a degraded version of the legitimate receiver’s observation. For this degraded model, he found the capacity-equivocation rate... |

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Citation Context ... the achievability of the secrecy rates in Theorem 6 by extending Marton’s achievable scheme for broadcast channels [41] to multi-receiver wiretap channels. For that purpose, we will use Theorem 1 of =-=[42]-=-, where the authors provided an achievable region for Gaussian vector broadcast channels using Marton’s achievable scheme in [41]. While using this result, we will combine it with a stochastic encodin... |

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Citation Context ...ver wiretap channel. Our approach will be to first find the secrecy capacity region of the degraded case, then to generalize this result to the aligned case by using the channel enhancement technique =-=[29]-=-. Once we obtain the secrecy capacity region of the aligned case, we use this result to find the secrecy capacity region of the most general case by some limiting arguments as in [29,30]. Thus, the ma... |

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Citation Context ...rovide a converse proof for the Gaussian scalar multi-receiver wiretap channel. We explicitly show that the main ingredient of these two converses in [22], [23], which is the entropy-power inequality =-=[24]-=-–[26], 1 is insufficient to conclude a converse for the secrecy capacity region. The second reason for the separate presentation is to present the main ingredients of the technique that we will use to... |

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Citation Context ...for the Gaussian scalar multi-receiver wiretap channel. The first one uses the connection between the minimum-mean-square-error (MMSE) and the mutual information along with the properties of the MMSE =-=[26,27]-=-. In additive Gaussian channels, the Fisher information, another important quantity in estimation theory, and the MMSE have a complementary relationship in the sense that one of them determines the ot... |

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Citation Context ...ignals. Index Terms—Gaussian MIMO broadcast channel, information theoretic security, secrecy capacity region. I. INTRODUCTION I NFORMATION theoretic secrecy was initiated by Wyner in his seminal work =-=[1]-=-. Wyner considered a degraded wiretap channel, where the eavesdropper gets a degraded version of the legitimate receiver’s observation. For this degraded model, he found the capacity-equivocation rate... |

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Citation Context ...o provide a converse proof for the Gaussian scalar multi-receiver wiretap channel. We explicitly show that the main ingredient of these two converses in [22,23], which is the entropy-power inequality =-=[24,25]-=-, is not sufficient to conclude a converse for the secrecy capacity region. The second reason for the separate presentation is to present the main ingredients of the technique that we will use to prov... |

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Citation Context ...easons for the presentation of the scalar case separately. The first one is to show that, existing converse techniques for the Gaussian scalar broadcast channel, i.e., the converse proofs of Bergmans =-=[22]-=- and El Gamal [23], cannot be extended in a straightforward manner to provide a converse proof for the Gaussian scalar multi-receiver wiretap channel. We explicitly show that the main ingredient of th... |

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Citation Context ...rrelated random variables with well-defined densities. Then, we have J(X|U) ≥ 1 Var(X|U) (61) with equality if (U, X) is jointly Gaussian. We now provide the conditional form of the de Bruin identity =-=[24,25]-=-. The vector generalization of this lemma will be provided in Lemma 16 in Section 5.4, and hence, its proof is omitted here. Lemma 3 Let X, U be arbitrarily correlated random variables with finite sec... |

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Citation Context ...11750 In recent years, information theoretic secrecy has gathered a renewed interest, where most of the attention has been devoted to the multi-user extensions of the wiretap channel, see for example =-=[3]-=-–[21]. One natural extension of the wiretap channel to the multi-user setting is the problem of secure broadcasting.In this case, there is one transmitter which wants to communicate with several legit... |

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Citation Context ... secrecy capacity region of the general MIMO channel. The single user version of the Gaussian MIMO multi-receiver wiretap channel we study here, i.e., the Gaussian MIMO wiretap channel, was solved by =-=[32,33]-=- for the general case and by [34] for the 2-2-1 case. Their common proof technique was to derive a Sato-type outer bound on the secrecy capacity, and then to tighten this outer bound by searching over... |

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49 | Relay channels with confidential messages - Oohama |

44 | Gradient of mutual information in linear vector Gaussian channels
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Citation Context ... be extended to our secrecy context, 3even for the degraded scalar case. Thus, we need a new technique which we construct by using the Fisher information matrix and the generalized de Bruin identity =-=[31]-=-. After we obtain the secrecy capacity region of the degraded MIMO channel, we adapt the channel enhancement technique to our setting to find the secrecy capacity region of the aligned MIMO channel. T... |

40 | The secrecy capacity of the MIMO wiretap channel
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Citation Context ... secrecy capacity region of the general MIMO channel. The single user version of the Gaussian MIMO multi-receiver wiretap channel we study here, i.e., the Gaussian MIMO wiretap channel, was solved by =-=[32,33]-=- for the general case and by [34] for the 2-2-1 case. Their common proof technique was to derive a Sato-type outer bound on the secrecy capacity, and then to tighten this outer bound by searching over... |

40 | Towards the secrecy capacity of the Gaussian MIMO wire-tap channel: The 2-2-1 channel
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Citation Context ... MIMO channel. The single user version of the Gaussian MIMO multi-receiver wiretap channel we study here, i.e., the Gaussian MIMO wiretap channel, was solved by [35], [36] for the general case and by =-=[37]-=- for the 2-2-1 case. Their common proof technique was to derive a Sato-type outer bound on the secrecy capacity, and then to tighten this outer bound by searching over all possible correlation structu... |

36 | The discrete memoryless multiple access channel with confidential messages - Liu, Maric, et al. - 2006 |

32 |
Information theory and the central limit theorem
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Citation Context ...e basic definitions. The unconditional versions of the following definition and the upcoming results regarding the Fisher information can be found in standard detection-estimation texts; to note one, =-=[36]-=- is a good reference for a detailed treatment of the subject. Definition 1 Let X, U be arbitrarily correlated random variables with well-defined densities, and f(x|u) be the corresponding conditional ... |

31 | Shitz), “A note on secrecy capacity of the multi-antenna wiretap channel
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Citation Context ...e proof for the degraded MIMO channel. We will then use the channel enhancement technique to extend our converse proof to the aligned MIMO channel. Finally, we will use some limiting arguments, as in =-=[29,30]-=-, to come up with a converse proof for the most general MIMO channel. 5 Degraded Gaussian MIMO Multi-receiver Wiretap Channel In this section, we establish the secrecy capacity region of the degraded ... |

31 | An extremal inequality motivated by multiterminal information theoretic problems
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Citation Context ...n the additive Gaussian channel, how the Fisher information matrix behaves under the addition of two independent random vectors is crucial. Regarding this, we have the following lemma which is due to =-=[39]-=-. Lemma 5 ([39]) Let U be a random vector with differentiable density, and let ΣU ≻ 0 be its covariance matrix. Moreover, let V be another random vector with differentiable density, and be independent... |

30 | Secure broadcasting over fading channels
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Citation Context ...st channels, which suggests that we might be able to find the secrecy capacity region for some special classes of multi-receiver wiretap channels. This suggestion has been taken into consideration in =-=[8]-=-–[11]. In particular, in [9]–[11], the degraded multi-receiver wiretap channel is considered, where there is a certain degradation order among the legitimate users and the eavesdropper. The correspond... |

30 |
A new entropy power inequality
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Citation Context ...entropy-power inequality refers to the original form of this inequality which was proposed by Shannon [24], but not the subsequent variants of this inequality such as Costa’s entropy-power inequality =-=[27]-=-. Indeed, using Costa’s entropy-power inequality, it is possible to provide a converse proof for the secrecy capacity region of the Gaussian scalar multireceiver wiretap channel [28]. (41) Combining (... |

24 | The relay channel with a wiretapper - Yuksel, Erkip - 2007 |

24 | Secrecy in cooperative relay broadcast channels - Ekrem, Ulukus - 2011 |

23 |
On the capacity of channels with additive non-Gaussian noise
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Citation Context ...f Theorem 2 for To prove that (91) and (92) give the secrecy capacity region, we need the results of some intermediate optimization problems. The first one is the so-called worst additive noise lemma =-=[41]-=-, [42]. Lemma 4 ([42, Lemma II.2]): Let be a Gaussian random vector with covariance matrix , and be a positive semidefinite matrix. Consider the following optimization problem: (93) where and are inde... |

22 | On the secrecy of multiple access wiretap channel - Ekrem, Ulukus - 2008 |

22 | Secrecy capacity of a class of broadcast channels with an eavesdropper
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Citation Context ...a certain degradation order among the legitimate users and the eavesdropper. The corresponding secrecy capacity region is derived for the two-user case in [9], and for an arbitrary number of users in =-=[10,11]-=-. The importance of this class lies in the fact that the Gaussian scalar multi-receiver wiretap channel belongs to this class. In this work, we start with the Gaussian scalar multi-receiver wiretap ch... |

16 | Cooperation with an untrusted relay: A secrecy perspective - He, Yener - 2010 |

15 | Secrecy rate region of the broadcast channel with an eavesdropper
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Citation Context ...iretap channel is considered, where there is a certain degradation order among the legitimate users and the eavesdropper. The corresponding secrecy capacity region is derived for the two-user case in =-=[9]-=-, and for an arbitrary number of users in [10,11]. The importance of this class lies in the fact that the Gaussian scalar multi-receiver wiretap channel belongs to this class. In this work, we start w... |

13 | Secrecy capacity region of a multi-antenna Gaussian broadcast channel with confidential messages - Liu, Poor |

13 | Confidential messages to a cooperative relay - Bloch, Thangaraj - 2008 |

13 | Information theoretic proofs of entropy power inequalities. Submitted to
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Citation Context ...hannels, the Fisher information, another important quantity in estimation theory, and the MMSE have a complementary relationship in the sense that one of them determines the other one, and vice versa =-=[28]-=-. Thus, the converse proof relying on the MMSE has a counterpart which replaces the Fisher information with the MMSE in the corresponding converse proof. Hence, the second converse uses the connection... |

12 | Multiple access channels with generalized feedback and confidential messages
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(Show Context)
Citation Context ...0 In recent years, information theoretic secrecy has gathered a renewed interest, where most of the attention has been devoted to the multi-user extensions of the wiretap channel, see for example [3]–=-=[21]-=-. One natural extension of the wiretap channel to the multi-user setting is the problem of secure broadcasting.In this case, there is one transmitter which wants to communicate with several legitimate... |

10 | On secure broadcasting
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Citation Context ...a certain degradation order among the legitimate users and the eavesdropper. The corresponding secrecy capacity region is derived for the two-user case in [9], and for an arbitrary number of users in =-=[10,11]-=-. The importance of this class lies in the fact that the Gaussian scalar multi-receiver wiretap channel belongs to this class. In this work, we start with the Gaussian scalar multi-receiver wiretap ch... |

9 | Effects of cooperation on the secrecy of multiple access channels with generalized feedback - Ekrem, Ulukus |

9 |
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Citation Context ... Π is the set of all possible one-to-one permutations on {1, . . ., K}. We will show the achievability of the secrecy rates in Theorem 6 by extending Marton’s achievable scheme for broadcast channels =-=[41]-=- to multi-receiver wiretap channels. For that purpose, we will use Theorem 1 of [42], where the authors provided an achievable region for Gaussian vector broadcast channels using Marton’s achievable s... |

9 | Shitz), “A vector generalization of Costa’s entropy-power inequality with applications
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Citation Context ...e entropy-power inequality [24]–[26] to prove the secrecy capacity region of the Gaussian scalar multi-receiver wiretap channel can be alleviated by using Costa’s entropy-power inequality as shown in =-=[28]-=-. 0018-9448/$26.00 © 2011 IEEE2084 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 4, APRIL 2011 the Gaussian scalar multi-receiver wiretap channel. The first one uses the connection between th... |

7 |
Estimation of non-Gaussian random variables in Gaussian noise
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Citation Context ...for the Gaussian scalar multi-receiver wiretap channel. The first one uses the connection between the minimum-mean-square-error (MMSE) and the mutual information along with the properties of the MMSE =-=[26,27]-=-. In additive Gaussian channels, the Fisher information, another important quantity in estimation theory, and the MMSE have a complementary relationship in the sense that one of them determines the ot... |

6 | The capacity region of the degraded multiple-input multiple-output compound broadcast channel
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(Show Context)
Citation Context ..., a Bergmans type converse does not work. Thus, we do not expect that the channel enhancement technique will be sufficient to extend our converse proof from the SISO case to the MIMO case, similar to =-=[40]-=-, where the channel enhancement technique alone was not sufficient for the extension of a converse proof technique from the scalar Gaussian case to the vector Gaussian case. Consequently, we will not ... |

3 |
EE478 Multiple user information theory. Lecture notes
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Citation Context ...sentation of the scalar case separately. The first one is to show that, existing converse techniques for the Gaussian scalar broadcast channel, i.e., the converse proofs of Bergmans [22] and El Gamal =-=[23]-=-, cannot be extended in a straightforward manner to provide a converse proof for the Gaussian scalar multi-receiver wiretap channel. We explicitly show that the main ingredient of these two converses ... |

3 |
The worst additive noise constraint under a covariance constraint
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Citation Context ...rem 3 for K = 2 To prove that (83) and (84) give the secrecy capacity region, we need the results of some intermediate optimization problems. The first one is the so-called worst additive noise lemma =-=[37,38]-=-. Lemma 4 Let N be a Gaussian random vector with covariance matrix Σ, and KX be a positive semi-definite matrix. Consider the following optimization problem, min p(x) I(N;N + X) s.t. Cov(X) = KX (85) ... |

3 |
Gamal, EE478 Multiple User Information Theory, lecture notes
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(Show Context)
Citation Context ...sentation of the scalar case separately. The first one is to show that, existing converse techniques for the Gaussian scalar broadcast channel, i.e., the converse proofs of Bergmans [22] and El Gamal =-=[23]-=-, cannot be extended in a straightforward manner to provide a converse proof for the Gaussian scalar multi-receiver wiretap channel. We explicitly show that the main ingredient of these two converses ... |

1 |
On secure broadcasting,” presented at the
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Citation Context ...a certain degradation order among the legitimate users and the eavesdropper. The corresponding secrecy capacity region is derived for the two-user case in [9], and for an arbitrary number of users in =-=[10]-=-, [11]. The importance of this class lies in the fact that the Gaussian scalar multi-receiver wiretap channel belongs to this class. In this work, we start with the Gaussian scalar multi-receiver wire... |

1 | The relay channel with a wire-tapper,” presented at the 41st Annu - Yuksel, Erkip - 2007 |