On the eavesdropper's correct decision in Gaussian and fading wiretap channels using lattice codes

Ernvall-Hytönen, Anne-Maria; Hollanti, Camilla

doi:10.1109/itw.2011.6089380

Cited by 9 publications

(12 citation statements)

References 12 publications

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“…2) In fact, when the dimension n is fixed the secrecy gain is totally determined by the kissing number A 2 . The lattice with the best secrecy gain (in boldface) is the one with the smallest kissing number, which can also be seen directly from (15). This agrees with the observation in [10] that the best secrecy gain is achieved by extremal lattices, for being extremal in this special case is equivalent to having A 2 = 0.…”

Section: Unimodular Lattices In Small Dimensionssupporting

confidence: 87%

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A Classification of Unimodular Lattice Wiretap Codes in Small Dimensions

Lin

Oggier

2013

IEEE Trans. Inform. Theory

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Lattice coding over a Gaussian wiretap channel, where an eavesdropper listens to transmissions between a transmitter and a legitimate receiver, is considered. A new lattice invariant called the secrecy gain [1] is used as a code design criterion for wiretap lattice codes since it was shown to characterize the confusion that a chosen lattice can cause at the eavesdropper: the higher the secrecy gain of the lattice, the more confusion. In this paper, a formula for the secrecy gain of unimodular lattices is derived. Secrecy gains of extremal odd unimodular lattices as well as unimodular lattices in dimension n, 16 ≤ n ≤ 23 are computed, covering the 4 extremal odd unimodular lattices and all the 111 non-extremal unimodular lattices (both odd and even) providing thus a classification of the best wiretap lattice codes coming from unimodular lattices in dimension n, 8 < n ≤ 23. Finally, to permit lattice encoding via Construction A, the corresponding error correction codes are determined.

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Section: Unimodular Lattices In Small Dimensionssupporting

confidence: 87%

“…This conjecture was recently proven by A.-M. Ernvall-Hytönen [14], [15] for a special class of lattices called extremal even unimodular lattices. The idea of the proof is to write the secrecy function of a lattice Λ as a function of the quantity…”

Section: B Previous Resultsmentioning

confidence: 81%

A Classification of Unimodular Lattice Wiretap Codes in Small Dimensions

Lin

Oggier

2013

IEEE Trans. Inform. Theory

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“…Here we still use χ Λ to denote the weak secrecy gain. This conjecture was recently proven by A.-M. Ernvall-Hytönen [15] for a special class of lattices called extremal even unimodular lattices and by F. Lin and F. Oggier [11] for unimodular lattices in dimension n, 8 < n ≤ 23. In this paper, we will compute the weak secrecy gain of known even 2and 3-modular lattices, and claim that their secrecy gains are larger than that of unimodular lattices in the same dimension, since the secrecy gain is by definition the maximum of the secrecy function.…”

Section: Preliminaries and Previous Resultsmentioning

confidence: 89%

Secrecy gain of Gaussian wiretap codes from 2- and 3-modular lattices

Lin

Oggier

2012

2012 IEEE International Symposium on Information Theory Proceedings

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Lattice coding over a Gaussian wiretap channel is considered with respect to a lattice invariant called the secrecy gain, which was introduced in [1] to characterize the confusion that a chosen lattice can cause at the eavesdropper: the higher the secrecy gain of the lattice, the more confusion. In this paper, secrecy gains of several 2-and 3-modular lattices are computed. Most are shown to have a secrecy gain larger than the best unimodular lattices can achieve.

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“…June 13, 2018 DRAFT where the equality in (a) holds iff MF is also uniform, and (b) is due to the chain rule. 13 The symmetrization of a non-binary channel is similar to that of a binary channel as shown in Lemma 8. When X and X are both non-binary, X ⊕ X denotes the result of the exclusive or (xor) operation of the binary expressions of X and X.June 13, 2018 DRAFT…”

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confidence: 94%

Achieving Secrecy Capacity of the Gaussian Wiretap Channel With Polar Lattices

Liu

Yan

Ling

2018

IEEE Trans. Inform. Theory

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In this work, an explicit scheme of wiretap coding based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. Firstly, polar lattices are used to construct secrecy-good lattices for the mod-Λs Gaussian wiretap channel. Then we propose an explicit shaping scheme to remove this mod-Λs front end and extend polar lattices to the genuine Gaussian wiretap channel. The shaping technique is based on the lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. By employing the asymmetric polar coding technique, we construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. As a result, the encoding complexity for the sender and the decoding complexity for the legitimate receiver are both O(N log N log(log N )). The proposed scheme is proven to be semantically secure.In simple terms, the secrecy capacity is defined as the maximum achievable rate under both the reliability and strong secrecy conditions. When W and V are both symmetric, and W is degraded with respect to V , the secrecy capacity is given by C(V ) − C(W ) [3], where C(·) denotes the channel capacity. DRAFT Polar codes [7] have shown their great potential in solving the wiretap coding problem. The polar coding scheme proposed in [8], combined with the block Markov coding technique [9], was proved to achieve the strong secrecy capacity when W and V are both binary-input symmetric channels, and W is degraded with respect to V . More recently, polar wiretap coding has been extended to general wiretap channels (not necessarily degraded or symmetric) in [10] and [11]. For continuous channels such as the GWC, there also has been notable progress in wiretap lattice coding. On the theoretical aspect, the existence of lattice codes achieving the secrecy capacity to within 1 2 nat under the strong secrecy as well as semantic security criterion was demonstrated in [6]. On the practical aspect, wiretap lattice codes were proposed in [12] and [13] to maximize the eavesdropper's decoding error probability. June 13, 2018 DRAFT 3 A. Our contribution Polar lattices, the counterpart of polar codes in the Euclidean space, have already been proved to be additive white Gaussian noise (AWGN)-good [14] and further to achieve the AWGN channel capacity with lattice Gaussian shaping [15] 1 . Motivated by [8], we will propose polar lattices to achieve both strong secrecy and reliability over the mod-Λ s GWC. Conceptually, this polar lattice structure can be regarded as a secrecy-good lattice Λ e nested within an AWGN-good lattice Λ b (Λ e ⊂ Λ b ). Further, we will propose a Gaussian shaping scheme over Λ b and Λ e , using the multilevel asymmetric polar coding technique. As a result, we will accomplish the design of an explicit lattice coding scheme which achieves the secrecy capacity of the GWC with semantic security.• The first technical contribution of this paper is the explicit construction of secrecy-good polar latt...

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On the eavesdropper's correct decision in Gaussian and fading wiretap channels using lattice codes

Cited by 9 publications

References 12 publications

A Classification of Unimodular Lattice Wiretap Codes in Small Dimensions

A Classification of Unimodular Lattice Wiretap Codes in Small Dimensions

Secrecy gain of Gaussian wiretap codes from 2- and 3-modular lattices

Achieving Secrecy Capacity of the Gaussian Wiretap Channel With Polar Lattices

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