Conference Paper

Providing secrecy with lattice codes

Electr. Eng. Dept., Pennsylvania State Univ., University Park, PA
DOI: 10.1109/ALLERTON.2008.4797696 Conference: Communication, Control, and Computing, 2008 46th Annual Allerton Conference on
Source: IEEE Xplore


Recent results have shown that lattice codes can be used to construct good channel codes, source codes and physical layer network codes for Gaussian channels. On the other hand, for Gaussian channels with secrecy constraints, efforts to date rely on random codes. In this work, we provide a tool to bridge these two areas so that the secrecy rate can be computed when lattice codes are used. In particular, we address the problem of bounding equivocation rates under nonlinear modulus operation that is present in lattice encoders/decoders. The technique is then demonstrated in two Gaussian channel examples: (1) a Gaussian wiretap channel with a cooperative jammer, and (2) a multi-hop line network from a source to a destination with untrusted intermediate relay nodes from whom the information needs to be kept secret. In both cases, lattice codes are used to facilitate cooperative jamming. In the second case, interestingly, we demonstrate that a non-vanishing positive secrecy rate is achievable regardless of the number of hops.

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    • "We also achieve a slightly higher secrecy rate when we simplify our model to the model used for example in [27]. A drawback of our scheme is the weak secrecy criterion instead of strong or even perfect secrecy. "
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    ABSTRACT: We investigate the problem of secure communications in a Gaussian multi-way relay channel applying the compute-and-forward scheme using nested lattice codes. All nodes employ half-duplex operation and can exchange confidential messages only via an untrusted relay. The relay is assumed to be honest but curious, i.e., an eavesdropper that conforms to the system rules and applies the intended relaying scheme. We start with the general case of the single-input multiple-output (SIMO) L-user multi-way relay channel and provide an achievable secrecy rate region under a weak secrecy criterion. We show that the securely achievable sum rate is equivalent to the difference between the computation rate and the multiple access channel (MAC) capacity. Particularly, we show that all nodes must encode their messages such that the common computation rate tuple falls outside the MAC capacity region of the relay. We provide results for the single-input single-output (SISO) and the multiple-input single-input (MISO) L-user multi-way relay channel as well as the two-way relay channel. We discuss these results and show the dependency between channel realization and achievable secrecy rate. We further compare our result to available results in the literature for different schemes and show that the proposed scheme operates close to the compute-and-forward rate without secrecy.
    Full-text · Article · Jun 2014 · IEEE Transactions on Information Forensics and Security
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    • "In [10] the existence of wiretap lattice codes (based on the ensemble of random lattice codes) achieving the secrecy capacity under the weak secrecy criterion was demonstrated. Finally, we note that the secrecy capacity of the continuous mod-lattice channel with feedback was studied in [11], and that standard lattices codes for the Gaussian channel [12] were used to provide weak/strong secrecy in the settings of cooperative jamming and interference channels in [13] [14] [15]. "
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    ABSTRACT: We prove that nested lattice codes can achieve semantic security and strong secrecy over the Gaussian wiretap channel. The key tool in our proof is the flatness factor which characterizes the convergence of the conditional output distributions corresponding to different messages and leads to an upper bound on the information leakage. We not only show the existence of lattice codes that are good for secrecy, but also propose the {flatness factor} as a new design criterion. Both the modulo-lattice Gaussian channel and the genuine Gaussian channel are considered. In the latter case, we propose a new secrecy coding scheme based on the discrete Gaussian distribution over a lattice, which achieves the secrecy capacity to within a half nat under mild conditions. No \textit{a priori} distribution of the message is assumed, and no dither is used in our proposed schemes.
    Preview · Article · Oct 2012 · IEEE Transactions on Information Theory
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    • "Lattice codes have been proposed for Gaussian wiretap channels in [14]. Security for a network with several two-way relays arranged in a line with cooperative jamming was considered in [9], where a lattice-based scheme was proposed. In all of the above works, weak informationtheoretic security (mutual information rate to eavesdropper tends to zero) has been used as a secrecy metric. "
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    ABSTRACT: Bidirectional relaying, where a relay helps two user nodes to exchange messages has been an active area of recent research. In the compute-and-forward strategy for bidirectional relaying, the relay computes a function of the two messages using the naturally-occurring sum of symbols simultaneously transmitted by user nodes in a Gaussian Multiple Access Channel (MAC), and the computed function value is forwarded to the user nodes in an ensuing broadcast phase. In this work, we consider the Gaussian MAC in bidirectional relaying with the messages taking values in a finite Abelian group and the relay computing the sum within the group under an additional secrecy constraint for protection against a honest but curious relay. The secrecy constraint is that, while the relay should decode the group sum, the individual message of each user should be perfectly secure from the relay. We exploit the symbol addition that occurs in a Gaussian MAC to design explicit modulations at the user nodes that achieve independence between the received symbols at the relay and each of the two individual transmitted messages under an average transmit power constraint. We provide a lattice coding strategy for reliable computation of the group sum at the relay with perfect secrecy, and study rate versus average power trade-offs in the large-dimension regime. Our results for secure compute-and-forward are significant because we achieve perfect security with finite average transmit power, and this has been done using a novel approach involving Fourier-analytic tools.
    Full-text · Article · Jun 2012 · IEEE Transactions on Information Theory
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