This paper considers physical layer security and the design of secure coset codes for wiretap channels, where information is not to be leaked to an eavesdropper having a degraded channel. The eavesdropper's bounds for correct decoding probability and information are first revisited, and a new variant of the information bound is derived. The new bound is valid for a general channel with any fading and Gaussian noise. From these bounds, it is explicit that both the information and probability are upper bounded by the average flatness factor, i.e., the expected theta function of the faded lattice related to the eavesdropper. Taking the minimization of the average flatness factor as a design criterion, simple geometric heuristics to minimize it in the low signal-to-noise ratio (SNR) regime in Gaussian and Rayleigh fast fading channels are motivated. It is concluded that in the Gaussian channel, the security boils down to the sphere packing density of the eavesdropper's lattice, whereas in the Rayleigh fading channel a full-diversity well-rounded lattice with a dense sphere packing will provide the best secrecy. The proposed criteria are backed up by extensive numerical experiments.
@cite_30: 
@cite_33: Coding for the Gaussian Wiretap Channel can be done with nested lattices. The fine lattice must be designed in the same way as lattices used on the regular Gaussian Channel while the coarse one must be a lattice whose theta series is minimized. We present a criterion of design of both the fine and coarse lattice when used on a Rayleigh fading wiretap channel
@cite_6: We propose a new scheme of wiretap lattice coding that achieves semantic security and strong secrecy over the Gaussian wiretap channel. The key tool in our security 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 introduce the notion of secrecy-good lattices, but also propose the flatness factor as a design criterion of such lattices. Both the modulo-lattice Gaussian channel and genuine Gaussian channel are considered. In the latter case, we propose a novel 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 a priori distribution of the message is assumed, and no dither is used in our proposed schemes.
@cite_3: We consider the Gaussian wiretap channel, where two legitimate players Alice and Bob communicate over an additive white Gaussian noise (AWGN) channel, while Eve is eavesdropping, also through an AWGN channel. We propose a coding strategy based on lattice coset encoding. We define the secrecy gain as a design criterion for wiretap lattice codes, expressed in terms of the lattice theta series, which characterizes Eve’s confusion as a function of the channel parameters. The secrecy gain is studied for even unimodular lattices, and an asymptotic analysis shows that it grows exponentially in the dimension of the lattice. Examples of wiretap lattice codes are given.
@cite_0: We consider the MIMO wiretap channel, where a legitimate transmitter (Alice) wishes to communicate a confidential message to the legitimate receiver (Bob) over a MIMO channel, while its messages are being eavesdropped by Eve through another MIMO channel. Supposing that Alice uses the nested lattice code proposed in [8], we characterize the expected value of amount of information leakage to Eve, and from minimizing this expected value, we derive a code design criterion for MIMO lattice wiretap codes. The special cases of block and fast fading channels are also studied. We also illustrate our lattice code design criterion using the Alamouti Code as the simplest MIMO code available in the literature.
@cite_27: 
@cite_34: We consider a fading wiretap channel model where the transmitter has only statistical channel state information, and the legitimate receiver and eavesdropper have perfect channel state information. We propose a sequence of non-random lattice codes which achieve strong secrecy and semantic security over ergodic fading channels. The construction is almost universal in the sense that it achieves the same constant gap to secrecy capacity over Gaussian and ergodic fading models.
The security of lattice coset codes can be quantized either by Eve's correct decision probability, or alternatively by the mutual information of the message and Eve's received signal. For the (AWGN) channel, upper bounds are known for both approaches @cite_3 @cite_27 . More importantly, both are increasing functions of the of the lattice @math , yielding its minimization as a design criterion. Sequences of lattice coset codes achieving security and reliability are also constructed in @cite_6 . For different fading channel models, various alternative design criteria based on probability and information bounds were derived in @cite_33 @cite_30 , and @cite_0 , respectively. Codes achieving security and reliability in the (MIMO) channel, and an information bound, were given in @cite_34 . However, to the best of our knowledge, practical low-dimensional code constructions are an open problem in the fading (SISO) channel.