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## Providing Secrecy With Structured Codes: Tools and Applications to Two-User Gaussian Channels (2009)

Citations: | 44 - 17 self |

### Citations

12170 |
Elements of Information Theory
- Cover, Thomas
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Citation Context ...,M can be reliably decoded from Y N1 . The decoding procedure is given in Appendix B. Let Pe denote the probability of decoding error which converges to 0 as N goes to ∞ 4. Then, by Fano’s inequality =-=[38]-=-, we have 1 N H ( uN1,M|Y N1 ) (39) ≤ 1 N ( 1 + Pe H (uN1,M) ) = 1 N + Pe M∑ i=1 Ri . (40) Therefore lim N→∞ 1 N I (uN1,M; Y N1 ) (41) = lim N→∞ 1 N (H (uN1,M) − H ( uN1,M|Y N1 ) ) (42) ≥ M∑ i=1 Ri − ... |

1485 |
Digital Communications
- Proakis
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(Show Context)
Citation Context ...tructure on the set from which the codewords are sampled can be helpful in proving certain information theoretical results [17]. This could be as simple as sampling codewords from a QAM constellation =-=[18]-=-. In [17], a lattice is used, which can be viewed as a constellation defined over N channel uses. This structured random code approach [17] is useful in multi-terminal problems: the structure of these... |

1190 | Communication Theory of Secrecy Systems - Shannon - 1949 |

1024 | Writing on dirty paper - Costa - 1983 |

740 |
Broadcast Channels with Confidential Messages
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Citation Context ...d by the eavesdropper (Eve) is a degraded version of the signal observed by the receiver, the long secret keys needed to achieve secrecy per Shannon’s notion are not necessary [2]. Csiszár and Körner =-=[3]-=- extended Wyner’s setting to the general discrete memoryless wiretap channel and established its secrecy capacity. Manuscript received July 28, 2009; revised December 31, 2011 and June 29, 2013; accep... |

559 | The wiretap channel - Wyner - 1975 |

419 | Interference Alignment and Degrees of Freedom of the K-User Interference Channel - Cadambe, Jafar - 2008 |

359 | The Gaussian wiretap channel - Leung-Yan-Cheong, Hellman - 1978 |

301 |
The wire-tap channel,” Bell Syst
- Wyner
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Citation Context ...lar, it was shown that it is possible that the eavesdropper gains no information regarding the secret message having intercepted the cryptogram, albeit at the expense of very long keys [1]. Wyner, in =-=[2]-=-, established that, if the signal received by the eavesdropper (Eve) is a degraded version of the signal observed by the receiver, the long secret keys needed to achieve secrecy per Shannon’s notion a... |

233 | Secure transmission with multiple antennas—part II: The MIMOME wiretap channel
- Khisti, Wornell
- 2010
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Citation Context ...channels with additive Gaussian noise. The maximum reliable transmission rate with secrecy was identified for some of these models including the Gaussian wiretap channel [4], the MIMO wiretap channel =-=[5]-=-, [6] and the MIMO Gaussian broadcast channel with confidential messages [7], [8]. On the other hand, secrecy capacity regions for models with multiple transmitters remain in general as open problems ... |

174 | Joint physical layer coding and network coding for bi-directional relaying
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Citation Context ...gn unwanted interference, for example, in Gaussian interference channels with more than two users [19]–[22]. Additionally, it renders the analysis of some network topologies feasible: for example, in =-=[23]-=-, [24], using structured codes allows the relaying scheme to be equivalent to a modulo sum operation, making it easy to trace the signal over a multi-hop relay network. A natural question therefore is... |

167 |
Communication theory of secrecy systems,” Bell Sys
- Shannon
- 1949
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Citation Context ...ed. Index Terms— Information theoretic secrecy, lattice codes, cooperative jamming, Gaussian wiretap channels. I. INTRODUCTION THE notion of information theoretic secrecy was firstproposed by Shannon =-=[1]-=- whereby a message transmitted to a receiver is guaranteed to be kept secret from an eavesdropper, irrespective of the computational power the eavesdropper possesses. In particular, it was shown that ... |

165 | The secrecy capacity of the MIMO wiretap channel
- Oggier, Hassibi
- 2011
(Show Context)
Citation Context ...els with additive Gaussian noise. The maximum reliable transmission rate with secrecy was identified for some of these models including the Gaussian wiretap channel [4], the MIMO wiretap channel [5], =-=[6]-=- and the MIMO Gaussian broadcast channel with confidential messages [7], [8]. On the other hand, secrecy capacity regions for models with multiple transmitters remain in general as open problems excep... |

159 | Discrete memoryless interference and broadcast channels with confidential messages: Secrecy rate regions
- Liu, Maric, et al.
(Show Context)
Citation Context ...onfuse the eavesdropper while not causing excessive harm to the intended receiver [12], [25]. This strategy has been used in a number of channel models to improve secrecy rates; see [12], [13], [16], =-=[26]-=-–[28] for example. In this work, we focus on the simplest Gaussian channel model where such a strategy is known to be useful. The model consists of a Gaussian wiretap channel and a 0018-9448 © 2014 IE... |

156 | Guaranteeing secrecy using artificial noise - Goel, Negi - 2008 |

134 | The approximate capacity of the many-to-one and one-to-many Gaussian interference channels - Bresler, Parekh, et al. - 2010 |

123 | Lattices which are good for (almost) everything - Erez, Litsyn, et al. - 2005 |

114 |
Achieving 1/2 log (1+SNR) on the AWGN channel with lattice encoding and decoding
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- 2004
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Citation Context ...ignal X N transmitted over N channel uses from a nested lattice codebook is given by X N = (uN + d N ) mod c. (12) Here uN is the lattice point chosen from ∩ V(c), and d N is the dithering vector =-=[32]-=-. Remark 2: Conventionally, d N is defined as a continuous random vector which is uniformly distributed over V(c) [32]. This so called dithering vector is used to facilitate the analysis of the proba... |

109 | The Gaussian multiple access wire-tap channel
- Tekin, Yener
(Show Context)
Citation Context ...regions for models with multiple transmitters remain in general as open problems except for some special cases, e.g., sum secrecy capacity for a degraded Gaussian multiple access wiretap channel [9], =-=[10]-=-. Upper bounds, lower bounds and some asymptotic results on the secrecy capacity exist, see for example [11]–[16]. To prove achievability, Shannon’s random coding argument is used in these works, in w... |

93 | Averaging bounds for lattices and linear codes
- Loeliger
- 1997
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Citation Context ...ayer. Here, for each layer, a nested lattice code is used instead. As a result, the corresponding decoding algorithm and the error probability analysis are different. Reference [37] uses results from =-=[39]-=-. The rate derivation in our work uses results from reference [32]. A consequence of Corollary 2 is as follows: Corollary 3: For √ ab, such that 2 √ ab is not an integer and 1/ √ ab is not an intege... |

88 | The General Gaussian Multiple Access and Two-Way Wire-Tap Channels: Achievable Rates and Cooperative Jamming
- Tekin, Yener
- 2008
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Citation Context ...tegy in secure communication, where the legitimate transmitters introduce judicious interference into the channel to confuse the eavesdropper while not causing excessive harm to the intended receiver =-=[12]-=-, [25]. This strategy has been used in a number of channel models to improve secrecy rates; see [12], [13], [16], [26]–[28] for example. In this work, we focus on the simplest Gaussian channel model w... |

80 | An extremal inequality motivated by multi terminal information theoretic problems - Liu, Viswanath - 2007 |

76 | On the Degrees-of-Freedom of the K-User Gaussian Interference Channel,” submitted to
- Etkin, Ordentlich
- 2008
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Citation Context ... [17] is useful in multi-terminal problems: the structure of these codes makes it possible to align unwanted interference, for example, in Gaussian interference channels with more than two users [19]–=-=[22]-=-. Additionally, it renders the analysis of some network topologies feasible: for example, in [23], [24], using structured codes allows the relaying scheme to be equivalent to a modulo sum operation, m... |

74 | Interference Alignment on the Deterministic Channel and Application to Fully Connected Gaussian Interference Networks
- Cadambe, Jafar, et al.
- 2009
(Show Context)
Citation Context ...roach [17] is useful in multi-terminal problems: the structure of these codes makes it possible to align unwanted interference, for example, in Gaussian interference channels with more than two users =-=[19]-=-–[22]. Additionally, it renders the analysis of some network topologies feasible: for example, in [23], [24], using structured codes allows the relaying scheme to be equivalent to a modulo sum operati... |

68 | The secrecy capacity region of the gaussian mimo multi-receiver wiretap channel
- Ekrem, Ulukus
(Show Context)
Citation Context ...h secrecy was identified for some of these models including the Gaussian wiretap channel [4], the MIMO wiretap channel [5], [6] and the MIMO Gaussian broadcast channel with confidential messages [7], =-=[8]-=-. On the other hand, secrecy capacity regions for models with multiple transmitters remain in general as open problems except for some special cases, e.g., sum secrecy capacity for a degraded Gaussian... |

65 | Interference alignment with asymmetric complex signaling - settling the Host-Madsen-Nosratinia conjecture - Cadambe, Jafar, et al. - 2010 |

62 |
Multiple-access channels with confidential messages
- Liang, Poor
(Show Context)
Citation Context ...es, e.g., sum secrecy capacity for a degraded Gaussian multiple access wiretap channel [9], [10]. Upper bounds, lower bounds and some asymptotic results on the secrecy capacity exist, see for example =-=[11]-=-–[16]. To prove achievability, Shannon’s random coding argument is used in these works, in which the codewords are i.i.d. sequences sampled from a distribution defined over the channel inputs. On the ... |

62 | Capacity bounds for two-way relay channel
- Nam, Chung, et al.
- 2008
(Show Context)
Citation Context ...anted interference, for example, in Gaussian interference channels with more than two users [19]–[22]. Additionally, it renders the analysis of some network topologies feasible: for example, in [23], =-=[24]-=-, using structured codes allows the relaying scheme to be equivalent to a modulo sum operation, making it easy to trace the signal over a multi-hop relay network. A natural question therefore is wheth... |

58 | Multiple-input multiple-output Gaussian broadcast channels with confidential messages
- Liu, Liu, et al.
- 2009
(Show Context)
Citation Context ...e with secrecy was identified for some of these models including the Gaussian wiretap channel [4], the MIMO wiretap channel [5], [6] and the MIMO Gaussian broadcast channel with confidential messages =-=[7]-=-, [8]. On the other hand, secrecy capacity regions for models with multiple transmitters remain in general as open problems except for some special cases, e.g., sum secrecy capacity for a degraded Gau... |

57 | The wiretap channel with feedback: Encryption over the channel
- Lai, Gamal, et al.
(Show Context)
Citation Context ... N1 and d N2 are known by both the transmitters and the receivers. Since ∩V(c) is an Abelian group, when uN2 is independent from uN1 , and u N 2 is uniformly distributed over ∩V(c), we have [33], =-=[34]-=-: I (uN1 ; uN1 ± uN2 mod c) = 0. (24) Applying it to (23), we find that (18) is upper bounded by H (T ) ≤ N. (25) Equations (19)-(25) imply 1 N I ( uN1 ; X N1 ± X N2 ) ≤ 1. (26) Applying this result ... |

52 | The case for structured random codes in network communication theorems
- Nazer, Gastpar
- 2007
(Show Context)
Citation Context ... over the channel inputs. On the other hand, it is known that introducing a structure on the set from which the codewords are sampled can be helpful in proving certain information theoretical results =-=[17]-=-. This could be as simple as sampling codewords from a QAM constellation [18]. In [17], a lattice is used, which can be viewed as a constellation defined over N channel uses. This structured random co... |

50 | Cooperation with an Untrusted Relay: A Secrecy Perspective. Submitted to - He, Yener - 2008 |

45 | A layered lattice coding scheme for a class of three user Gaussian interference channels,” ArXiv pre-print cs.IT/0809.4316
- Sridharan, Jafarian, et al.
- 2008
(Show Context)
Citation Context ...AND YENER: PROVIDING SECRECY WITH STRUCTURED CODES 2125 Proof: Compared to the channel model in (8), the added complexity here is that node D1 is interfered by node S2. We use a layered coding scheme =-=[37]-=- to eliminate this interference. This scheme involves technical details related to how nested lattice codes are decoded at the receiver. For clarity, we keep these details in Appendix B while providin... |

43 | Capacity of symmetric k-user gaussian very strong interference channels.” [Online]. Available: http://arxiv.org/abs/0808.2314 - Sridharan, Jafarian, et al. |

40 | On the secrecy of multiple access wiretap channel
- Ekrem, Ulukus
- 2008
(Show Context)
Citation Context ...ON THEORY, VOL. 60, NO. 4, APRIL 2014 cooperative jammer. This model can also be viewed as a special case of a number of two-user Gaussian channel models considered in previous work [12], [16], [26], =-=[29]-=-. Hence improving the achievable secrecy rate for this model implies that the achievable rates for all these models can be improved as well. Previously, this model was studied in [27] with the optimal... |

30 | The Gaussian Wiretap Channel With a Helping Interferer
- Tang, Liu, et al.
- 2008
(Show Context)
Citation Context ...12], [16], [26], [29]. Hence improving the achievable secrecy rate for this model implies that the achievable rates for all these models can be improved as well. Previously, this model was studied in =-=[27]-=- with the optimal transmission power control strategy, where both the cooperative jammer and the sender of the message use codewords sampled from a Gaussian distribution. It was found that the secrecy... |

29 | Two-hop secure communication using an untrusted relay - He, Yener - 2009 |

29 |
On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets Wiener
- Forney
- 2004
(Show Context)
Citation Context ...ince d N1 and d N2 are known by both the transmitters and the receivers. Since ∩V(c) is an Abelian group, when uN2 is independent from uN1 , and u N 2 is uniformly distributed over ∩V(c), we have =-=[33]-=-, [34]: I (uN1 ; uN1 ± uN2 mod c) = 0. (24) Applying it to (23), we find that (18) is upper bounded by H (T ) ≤ N. (25) Equations (19)-(25) imply 1 N I ( uN1 ; X N1 ± X N2 ) ≤ 1. (26) Applying this r... |

25 | On the secrecy capacity of arbitrary wiretap channel - Bloch, Laneman - 2008 |

24 |
Gamal. Cooperation for Secrecy: The Relay-Eavesdropper Channel
- Lai, El
- 2008
(Show Context)
Citation Context ...channel to confuse the eavesdropper while not causing excessive harm to the intended receiver [12], [25]. This strategy has been used in a number of channel models to improve secrecy rates; see [12], =-=[13]-=-, [16], [26]–[28] for example. In this work, we focus on the simplest Gaussian channel model where such a strategy is known to be useful. The model consists of a Gaussian wiretap channel and a 0018-94... |

23 | Compute-and-forward: Harnessing interference with structured codes - Nazer, Gastpar - 2008 |

23 | K-user interference channels: Achievable secrecy rate and degrees of freedom
- He, Yener
- 2009
(Show Context)
Citation Context ...2,t ) + q√bZ N1 (77) which is identical to (64) with i replaced by i − 1. This will be used by D1 when decoding lower layers. We next determine the power Pi and the rate Ri of each layer. As in [37], =-=[42]-=-, we let the right hand side of (68) equal the right hand side of (76): Pi γ 2 Pi + Ai = 1 + γ 2 Pi Ai . (78) It is easy to check that, with α = 1−2γ 2+ √ 1−4γ 2 2γ 4 , (78) has the following solution... |

20 | On Secure Signaling for the Gaussian Multiple Access Wire-tap Channel
- Tekin, Serbetli, et al.
- 2005
(Show Context)
Citation Context ...city regions for models with multiple transmitters remain in general as open problems except for some special cases, e.g., sum secrecy capacity for a degraded Gaussian multiple access wiretap channel =-=[9]-=-, [10]. Upper bounds, lower bounds and some asymptotic results on the secrecy capacity exist, see for example [11]–[16]. To prove achievability, Shannon’s random coding argument is used in these works... |

18 | A new outer bound for the Gaussian interference channel with confidential messages - He, Yener - 2009 |

15 |
Cooperation and information theoretic security in wireless networks
- He
- 2010
(Show Context)
Citation Context ...eger lattice coding scheme described in this section is superior when 2 √ ab are integers and √ ab ≥ 2. Remark 9: The smallest known upper bound on the number of secure degrees of freedom is found in =-=[31]-=- to be 2/3, which still has a nonzero gap from 1/2. VII. CHANNEL GAIN MISMATCH The coding schemes presented in previous sections require aligning lattice points at the eavesdropper, which relies on 21... |

14 | Providing Secrecy with Lattice Codes - He, Yener - 2008 |

13 | On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets Wiener - Jr - 2003 |

12 | Secrecy capacity region of a class of one-sided interference channel - Li, Yates, et al. |

9 | Interference Alignment for Secrecy. Submited to - Koyluoglu, El-Gamal, et al. - 2008 |

6 |
The multiple access wire-tap channel: wireless secrecy and cooperative jamming
- Tekin, Yener
- 2007
(Show Context)
Citation Context ...n secure communication, where the legitimate transmitters introduce judicious interference into the channel to confuse the eavesdropper while not causing excessive harm to the intended receiver [12], =-=[25]-=-. This strategy has been used in a number of channel models to improve secrecy rates; see [12], [13], [16], [26]–[28] for example. In this work, we focus on the simplest Gaussian channel model where s... |

4 | Cooperative binning and channel prefixing for secrecy in interference channels”, Submitted to
- Koyluoglu, El-Gamal
(Show Context)
Citation Context ....g., sum secrecy capacity for a degraded Gaussian multiple access wiretap channel [9], [10]. Upper bounds, lower bounds and some asymptotic results on the secrecy capacity exist, see for example [11]–=-=[16]-=-. To prove achievability, Shannon’s random coding argument is used in these works, in which the codewords are i.i.d. sequences sampled from a distribution defined over the channel inputs. On the other... |

3 | Interference channels with strong secrecy
- He, Yener
(Show Context)
Citation Context ...g Y1/ √ b with Ỹ1 and X1/ √ b with X̃1 lead to the channel expression used in [27]. 2The achievable rate in this work also holds for the secrecy constraint limn→∞ H (W1) = limn→∞ H ( W1|Y n2 ) . See =-=[30]-=- for the additional steps to prove achievability for this stronger secrecy constraint. HE AND YENER: PROVIDING SECRECY WITH STRUCTURED CODES 2123 Remark 1: In this paper, we focus our attention to rea... |

1 |
Lattice coding for strongly secure compute-and-forward in a bidirectional relay
- Yang, Piantanida, et al.
- 2013
(Show Context)
Citation Context ...ently [35] proposed a lattice Gaussian signaling scheme for the Gaussian wiretap channel in which the transmitted signals could also be sampled outside of the Voronoi region of c and it was shown in =-=[36]-=- that this new scheme eliminates this information leakage without introducing a wiretap code as an outer code. IV. ACHIEVABLE SECURE DEGREES OF FREEDOM WITH NESTED LATTICE CODES We next apply the pr... |