We study the performance of polarizing codes over a degraded symmetric wiretap channel under a total variation distance (TVD) secrecy constraint. We show that the leakage can be bounded by the sum of the TVDs of the bit-channels corresponding to the confidential and frozen bits. In the asymptotic regime, this gives a new criterion to design wiretap codes with vanishing TVD leakage. In finite blocklength, it allows us to compute lower bounds for the secrecy rate of different families of polarizing wiretap codes over a binary erasure wiretap channel.

To be presented at ISIT 2025

We study the fundamental limits of covert communications over general memoryless additive-noise channels. We assume that the legitimate receiver and the eavesdropper share the same channel and therefore see the same outputs. Under mild integrability assumptions, we find a general upper bound on the square-root scaling constant, which only involves the variance of the logarithm of the probability density function of the noise. Furthermore, we show that, under some additional assumptions, this upper bound is tight. We also provide upper bounds on the length of the secret key required to achieve the optimal scaling.

We extend previous results on covert communication over the additive white Gaussian noise channel to two other types of additive noise channels. The first is the Gaussian channel with memory, where the noise sequence is a Gaussian vector with an arbitrary invertible covariance matrix. We show that the fundamental limit for covert communication over such a channel is the same as over the channel with white, i.e., memoryless, Gaussian noise. The second type of channel we consider is one with memoryless generalized Gaussian noise. For such a channel we prove a general upper bound on the dominant term in the maximum number of nats that can be covertly communicated over n channel uses. When the shape parameter p of the generalized Gaussian noise distribution is in the interval (0,1], we also prove a matching lower bound.

We propose a lattice-based scheme for secret key generation from Gaussian sources in the presence of an eavesdropper, and show that it achieves the strong secret key capacity in the case of degraded source models, as well as the optimal secret key / public communication rate trade-off. The key ingredients of our scheme are the use of the modulo lattice operation to extract the channel intrinsic randomness, based on the notion of flatness factor, together with a randomized lattice quantization technique to quantize the continuous source. Compared to previous works, we introduce two new notions of flatness factor based on$L^{1}$distance and KL divergence, respectively, which might be of independent interest. We prove the existence of secrecy-good lattices under$L^{1}$distance and KL divergence, whose$L^{1}$and KL flatness factors vanish for volume-to- noise ratios up to$2\pi e$. This improves upon the volume-to- noise ratio threshold$2\pi $of the$L^{\infty }$flatness factor.

In this paper we consider the diversity-multiplexing gain tradeoff (DMT) of so-called minimum delay asymmetric space-time codes for the$n \times m$MIMO channel. Such codes correspond to lattices in$M_{n}(\mathbb {C})$with dimension smaller than$2n^{2}$. Currently, very little is known about their DMT, except in the case$m=1$, corresponding to the multiple input single output (MISO) channel. Further, apart from the MISO case, no DMT optimal asymmetric codes are known. We first discuss previous criteria used to analyze the DMT of space-time codes and comment on why these methods fail when applied to asymmetric codes. We then consider two special classes of asymmetric codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also show that lattice codes based on$\mathbb {Q}$-central division algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all$\mathbb {Q}$-central division algebra codes that are based on standard embeddings. While the$\mathbb {Q}$-central division algebra based codes achieve the largest possible DMT of a code restricted to either real or quaternion space, they still fall short of the optimal DMT apart from the MISO case.

We consider a key encapsulation mechanism (KEM) based on Module-LWE where reconciliation is performed on the 8-dimensional lattice$E_{8}$, which admits a fast CVP algorithm. Our scheme generates 256 bits of key and requires 3 or 4 bits of reconciliation per dimension. We show that it can outperform Kyber in terms of the modulus$q$with comparable error probability and similar requirements in terms of bandwidth. We prove that our protocol is IND-CPA secure and improves the security level of Kyber by 7.3%.

We consider a semi-deterministic wiretap channel where the main channel is noiseless and the eavesdropper’s channel is a binary erasure channel (BEC). We provide a lower bound for the achievable secrecy rates of polar and Reed-Muller codes, and compare it to the second order coding rate. To the best of our knowledge, this is the first work which demonstrates the secrecy performance of polar and Reed-Muller codes in short blocklengths. The results show that under a total variation secrecy metric, Reed-Muller codes can achieve secrecy rates very close to the second order approximation rate. On the other hand, we observe a significant gap between the lower bound for the achievable rates of polar codes and the the second order approximation rate for short blocklengths.

We consider a network of two nodes separated by a noisy channel with two-sided state information, in which the input and output signals have to be coordinated with the source and its reconstruction. In the case of non-causal encoding and decoding, we propose a joint source-channel coding scheme and we develop inner and outer bounds for the strong coordination region. While the inner and outer bounds do not match in general, we provide a complete characterization of the strong coordination region in three particular cases: i) when the channel is perfect; ii) when the decoder is lossless; and iii) when the random variables of the channel are independent from the random variables of the source. Through the study of these special cases, we prove that the separation principle does not hold for the joint source-channel strong coordination. Finally, in the absence of state information, we show that polar codes achieve a subset of the best known inner bound for the strong coordination region, therefore offering a constructive alternative to random binning and coding proofs.

In this paper we consider space-time codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also point out that lattice codes based on Q-central division algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all Q-central division algebra codes that are based on standard embeddings.

Despite several works on secrecy coding for fading and MIMO wiretap channels from an error probability perspective, the construction of information-theoretically secure codes over such channels remains an open problem. In this paper, we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information. Our channel model includes static channels, i.i.d. block fading channels, and ergodic stationary fading with fast decay of large deviations for the eavesdropper's channel. We extend the flatness factor criterion from the Gaussian wiretap channel to fading and MIMO wiretap channels, and establish a simple design criterion where the normalized product distance/minimum determinant of the lattice and its dual should be maximized simultaneously. Moreover, we propose concrete lattice codes satisfying this design criterion, which are built from algebraic number fields with constant root discriminant in the single-antenna case, and from division algebras centered at such number fields in the multipleantenna case. The proposed lattice codes achieve strong secrecy and semantic security for all rates Rb- Ce- κ, where Cband Ceare Bob and Eve's channel capacities, respectively, and κ is an explicit constant gap. Furthermore, these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models. Finally, we consider a compound wiretap model with a more restricted uncertainty set, and show that rates Rb- C̅e- κ are achievable, where C̅bis a lower bound for Bob's capacity and C̅eis an upper bound for Eve's capacity for all the channels in the set.

We develop a random binning scheme for strong coordination in a network of two nodes separated by a noisy channel, in which the input and output signals have to be coordinated with the source and its reconstruction. In the case of non-causal encoding and decoding, we propose a joint source-channel coding scheme and develop inner and outer bounds for the strong coordination region. While the set of achievable target distributions is the same as for empirical coordination, we characterize the rate of common randomness required for strong coordination.

This paper addresses the question of achieving capacity with lattice codes in multi-antenna block fading channels when the number of fading blocks tends to infinity. A design criterion based on the normalized minimum determinant is proposed for division algebra multi-block space-time codes over fading channels; this plays a similar role to the Hermite invariant for Gaussian channels. Under maximum likelihood decoding, it is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for deterministic channels and ergodic fading channels. Moreover, if the number of receive antennas is greater than or equal to the number of transmit antennas, the same constant gap is achieved under naive lattice decoding as well. In the case of independent identically distributed Rayleigh fading, the error probability vanishes exponentially fast. In contrast to the standard approach in the literature, which employs random lattice ensembles, the existence results in this paper are derived from the number theory. First, the gap to capacity is shown to depend on the discriminant of the chosen division algebra; then, class field theory is applied to build families of algebras with small discriminants. The key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants.

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.

International audience

We develop a polar coding scheme for empirical coordination in a two-node network with a noisy link in which the input and output signals have to be coordinated with the source and the reconstruction. In the case of non-causal encoding and decoding, we show that polar codes achieve the best known inner bound for the empirical coordination region, provided that a vanishing rate of common randomness is available. This scheme provides a constructive alternative to random binning and coding proofs.

This work addresses the question of achieving capacity with lattice codes in multi-antenna block fading channels when the number of fading blocks tends to infinity. In contrast to the standard approach in the literature which employs random lattice ensembles, the existence results in this paper are derived from number theory. It is shown that a multiblock construction based on division algebras achieves rates within a constant gap from block fading capacity both under maximum likelihood decoding and naive lattice decoding. First the gap to capacity is shown to depend on the discriminant of the chosen division algebra; then class field theory is applied to build families of algebras with small discriminants. The key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants.

This paper shows that a family of number field lattice codes simultaneously achieves a constant gap to capacity in Rayleigh fast fading and Gaussian channels. The key property in the proof is the existence of infinite towers of Hilbert class fields with bounded root discriminant. The gap to capacity of the proposed lattice codes is determined by the root discriminant. The comparison between the Gaussian and fading case reveals that in Rayleigh fading channels the normalized minimum product distance plays an analogous role to the Hermite invariant in Gaussian channels.

This paper considers shifted inverse determinant sums arising from the union bound of the pairwise error probability for space-time codes in multiple-antenna fading channels. Previous work by Vehkalahti et al. focused on the approximation of these sums for low multiplexing gains, providing a complete classification of the inverse determinant sums as a function of constellation size for the most well-known algebraic space-time codes. This work aims at building a general framework for the study of the shifted sums for all multiplexing gains. New bounds obtained using dyadic summing techniques suggest that the behavior of the shifted sums does characterize many properties of a lattice code such as the diversity-multiplexing gain trade-off, both under maximum-likelihood decoding and infinite lattice naive decoding. Moreover, these bounds allow to characterize the signal-to-noise ratio thresholds corresponding to different diversity gains.

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.

We propose a simple yet complete lattice-based scheme for secret key generation from Gaussian sources in the presence of an eavesdropper, and show that it achieves strong secret key rates up to 1/2 nat from the optimal in the case of “degraded” source models. The novel ingredient of our scheme is a lattice-hashing technique, based on the notions of flatness factor and channel intrinsic randomness. The proposed scheme does not require dithering.

This work considers normalized inverse determinant sums as a tool for analyzing the performance of division algebra based space-time codes for multiple antenna wireless systems. A general union bound based code design criterion is obtained as a main result. In our previous work, the behavior of inverse determinant sums was analyzed using point counting techniques for Lie groups; it was shown that the asymptotic growth exponents of these sums correctly describe the diversity-multiplexing gain trade-off of the space-time code for some multiplexing gain ranges. This paper focuses on the constant terms of the inverse determinant sums, which capture the coding gain behavior. Pursuing the Lie group approach, a tighter asymptotic bound is derived, allowing to compute the constant terms for several classes of space-time codes appearing in the literature. The resulting design criterion suggests that the performance of division algebra based codes depends on several fundamental algebraic invariants of the underlying algebra.

The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP) are the core algorithmic problems on Euclidean lattices. They are central to the applications of lattices in many problems of communications and cryptography. Kannan's embedding technique is a powerful technique for solving the approximate CVP; yet, its remarkable practical performance is not well understood. In this paper, the embedding technique is analyzed from a bounded distance decoding (BDD) viewpoint. We present two complementary analyses of the embedding technique: we establish a reduction from BDD to Hermite SVP (via unique SVP), which can be used along with any Hermite SVP solver (including, among others, the Lenstra, Lenstra and Lovász (LLL) algorithm), and show that, in the special case of LLL, it performs at least as well as Babai's nearest plane algorithm (LLL-aided successive interference cancellation). The former analysis helps us to explain the folklore practical observation that unique SVP is easier than standard approximate SVP. It is proven that when the LLL algorithm is employed, the embedding technique can solve the CVP provided that the noise norm is smaller than a decoding radius λ1/(2γ) , where λ1is the minimum distance of the lattice, and γ ≈O(2n/4). This substantially improves the previously best known correct decoding bound γ ≈O(2n) . Focusing on the applications of BDD to decoding of multiple-input multiple-output systems, we also prove that BDD of the regularized lattice is optimal in terms of the diversity-multiplexing gain tradeoff, and propose practical variants of embedding decoding which require no knowledge of the minimum distance of the lattice and/or further improve the error performance.

This work considers inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic space-time block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse determinant sums and the diversity-multiplexing gain tradeoff is investigated. It is proven that the growth of the inverse determinant sum of a division algebra-based space-time code is completely determined by the growth of the unit group. This reduces the inverse determinant sum analysis to studying certain asymptotic integrals in Lie groups. Using recent methods from ergodic theory, a complete classification of the inverse determinant sums of the most well-known algebraic space-time codes is provided. The approach reveals an interesting and tight relation between diversity-multiplexing gain tradeoff and point counting in Lie groups.

We consider a wiretap scenario where the main channel and eavesdropper's channel are modulo lattice Gaussian channels. We prove that nested lattice codes can achieve strong secrecy for this model, which gives considerable insights to tackle the genuine Gaussian wiretap channel. The key tool in our proof is an L1convergence result for the conditional output distributions corresponding to different messages, which follows from the properties of the lattice Gaussian measure. No constraint on the a priori distribution of the message is imposed, which means that the proposed scheme is actually semantically secure. We not only show the existence of lattice codes that are good for secrecy, but also propose the flatness factor as a design criterion.

Earlier it was proven by Vehkalahti and Lu how the unit group and diversity-multiplexing gain trade-off (DMT) of division algebra-based space-time codes are linked to each other through inverse determinant sums. This work explores this relation further, showing that indeed the density of unit group completely determines the growth of the inverse determinant sum. In particular, in the case of Q(i)-central division algebras, the lower bound obtained from the DMT and the upper bound derived from the growth rate of units coincide.

In this paper we consider the error performance analysis of lattice alignment for the many-to-one interference channel. An upper bound on the error probability for the first receiver when lattice codes are used is derived. More precisely, we consider the case of joint maximum-likelihood (ML) decoding of the desired signal and the sum of interfering signals, derive the union bound for the error probability in terms of the theta series of these lattices, and show that it is related to the flatness factor.

In lattice-coded multiple-input multiple-output (MIMO) systems, optimal decoding amounts to solving the closest vector problem (CVP). Embedding is a powerful technique for the approximate CVP, yet its remarkable performance is not well understood. In this paper, we analyze the embedding technique from a bounded distance decoding (BDD) viewpoint. 1/(2γ)-BDD is referred to as a decoder that finds the closest vector when the noise norm is smaller than λ1/(2γ), where λ1is the minimum distance of the lattice. We prove that the Lenstra, Lenstra and Lovász (LLL) algorithm can achieve 1/(2γ)-BDD for γ ≈ O(2n/4). This substantially improves the existing result γ = O(2n) for embedding decoding. We also prove that BDD of the regularized lattice is optimal in terms of the diversity-multiplexing gain tradeoff (DMT).

Multiple Input Double Output (MIDO) asymmetric space-time codes for 4 transmit antennas and 2 receive antennas can be employed in the downlink from base stations to portable devices. Previous MIDO code constructions with low Maximum Likelihood (ML) decoding complexity, full diversity and the non-vanishing determinant (NVD) property are mostly based on cyclic division algebras. In this paper, a new family of MIDO codes with the NVD property based on crossed-product algebras over ℚ is introduced. Fast decodability follows naturally from the structure of the codewords which consist of four generalized Alamouti blocks. The associated ML complexity order is the lowest known for full-rate MIDO codes (O(M10) instead of O(M16) with respect to the real constellation size M). Numerical simulations show that these codes have a performance from comparable up to 1dB gain compared to the best known MIDO code with the same complexity.

Lattice reduction algorithms, such as the LLL algorithm, have been proposed as preprocessing tools in order to enhance the performance of suboptimal receivers in MIMO communications. In this paper we introduce a new kind of lattice reduction-aided decoding technique, called augmented lattice reduction, which recovers the transmitted vector directly from the change of basis matrix, and therefore doesn't entail the computation of the pseudo-inverse of the channel matrix or its QR decomposition. We prove that augmented lattice reduction attains the maximum receive diversity order of the channel; simulation results evidence that it significantly outperforms LLL-SIC detection without entailing any additional complexity.

Lattice reduction algorithms, such as the Lenstra-Lenstra-Lovasz (LLL) algorithm, have been proposed as preprocessing tools in order to enhance the performance of suboptimal receivers in multiple-input multiple-output (MIMO) communications. A different approach, introduced by Kim and Park, allows to combine right preprocessing and detection in a single step by performing lattice reduction on an v{augmented channel matrix}. In this paper we propose an improvement of the augmented matrix approach which guarantees a better performance. We prove that our method attains the maximum receive diversity order of the channel. Simulation results evidence that it significantly outperforms LLL reduction followed by successive interference cancellation (SIC) while requiring a moderate increase in complexity. A theoretical bound on the complexity is also derived.

In this paper we introduce a new right preprocessing method for the decoding of 2 times 2 algebraic space-time codes, called algebraic reduction, which exploits the multiplicative structure of the code. The principle of the new reduction is to absorb part of the channel into the code, by approximating the channel matrix with an element of the maximal order of the code algebra. We prove that algebraic reduction attains the receive diversity when followed by a simple zero-forcing (ZF) detection. Simulation results for the golden code show that using minimum mean squared error generalized decision feedback equalization (MMSE-GDFE left preprocessing), algebraic reduction with simple ZF detection has a loss of only 3 dB with respect to optimal decoding.

In this paper, block-coded modulation is used to design a 2 times 2 multiple-input multiple-output (MIMO) space-time code for slow fading channels. The golden code is chosen as the inner code; the scheme is based on a set partitioning of the golden code using two-sided ideals whose norm is a power of two. In this case, a lower bound for the minimum determinant is given by the minimum Hamming distance. The description of the ring structure of the quotients suggests further optimization in order to improve the overall distribution of determinants. Simulation results show that the proposed schemes achieve a significant gain over the un-coded golden code.

We consider a block coded modulation scheme for a 2 times 2 MIMO system over slow fading channels, where the inner code is the Golden Code. The scheme is based on a set partitioning of the Golden Code using two-sided ideals. A lower bound for the minimum determinant is given by the minimum Hamming distance. Performance simulations show that our GCRS schemes achieve a significant gain over the uncoded Golden Code.