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IEEE Journal on Selected Areas in Information Theory

Covert Capacity-Key Tradeoff Over Discrete Memoryless Networks

Abdelaziz Bounhar, Mireille Sarkiss, Michèle Wigger

Published Jan 1, 2026• 0 citations
Publisher

Abstract

This paper characterizes the covert capacity-key tradeoffs for discrete memoryless channels (DMC) and discrete memoryless multiple-access channels (DMMAC). The focus is on channels where communication rates are measured as the number of transmitted bits divided by the square-root of the blocklength and by the square-root of the covertness constraint. Previous results had determined the largest achievable covert rates for these channels, as well as the minimum key rates required to achieve these largest covert rates. In this work, we additionally determine the minimum key rates required at reduced covert rates. Stated differently, we determine the set of all covert rates that are achievable for a given set of key rates. This is termed thecovert capacity-key tradeoff. Our results on the covert capacity-key tradeoffs over DMCs and DMMACs show that for small key rates and when the adversary observes the inputs through a better channel than the legitimate receiver, binary signalling at all transmitters is optimal even for larger input alphabets and the capacity-key tradeoff grows linearly. Moreover, in this regime the covert capacity-key tradeoff of DMMACs is square implying that each of the transmitters can simultaneously achieve its own largest covert rate depending only on its own available key rate, irrespective of the other transmitter. For larger key rates or when the adversary is not uniformly stronger than the legitimate receiver over all inputs, the covert capacity-key tradeoff region of DMMACs is non-square and a tradeoff arises between the rates of the various transmitters.

DOI
10.1109/jsait.2026.3696919

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This paper characterizes the covert capacity-key tradeoffs for discrete memoryless channels (DMC) and discrete memoryless multiple-access channels (DMMAC). The focus is on channels where communication rates are measured as the number of transmitted bits divided by the square-root of the blocklength and by the square-root of the covertness constraint. Previous results had determined the largest achievable covert rates for these channels, as well as the minimum key rates required to achieve these largest covert rates. In this work, we additionally determine the minimum key rates required at reduced covert rates. Stated differently, we determine the set of all covert rates that are achievable for a given set of key rates. This is termed thecovert capacity-key tradeoff. Our results on the covert capacity-key tradeoffs over DMCs and DMMACs show that for small key rates and when the adversary observes the inputs through a better channel than the legitimate receiver, binary signalling at all transmitters is optimal even for larger input alphabets and the capacity-key tradeoff grows linearly. Moreover, in this regime the covert capacity-key tradeoff of DMMACs is square implying that each of the transmitters can simultaneously achieve its own largest covert rate depending only on its own available key rate, irrespective of the other transmitter. For larger key rates or when the adversary is not uniformly stronger than the legitimate receiver over all inputs, the covert capacity-key tradeoff region of DMMACs is non-square and a tradeoff arises between the rates of the various transmitters.
This paper explores the tradeoff between covert communication capacity and secret key requirements over discrete memoryless channels (DMCs). We focus on settings where under a covertness constraint both communication and key rates are measured as the number of bits per square root of the block-length. While previous work has identified the maximum covert communication rates and the corresponding minimum key rates needed to achieve them, our study characterizes the minimum key rates necessary for all of desired covert communication rates. In equivalent terms, we determine, for any given key rate, the set of achievable covert rates. This relationship defines what we call the covert capacity-key tradeoff.Our analysis reveals several new insights. In scenarios where only small key rates are available and the adversary has a stronger channel than the intended receiver, binary signaling is optimal—regardless of the specific channel characteristics or input alphabets. In these cases, the covert capacity increases linearly with the available key rate. In other cases and for larger key rates, the covert capacity-key tradeoff grows sublinearly.We also extend our findings to multi-access channels (MACs) with binary inputs.
We consider a scenario in which two legitimate transmitters attempt to communicate with a legitimate receiver over a discrete memoryless Multiple-Access Channel (MAC), while escaping detection from an adversary who observes their communication through another discrete memoryless MAC. If the MAC to the legitimate receiver is “better” than the one to the adversary, in a sense that we make precise, then the legitimate users can reliably communicate on the order of √n bits per n channel uses with arbitrarily Low Probability of Detection (LPD) without using a secret key. We also identify the pre-constants of the scaling, which leads to a characterization of the covert capacity region.
In this paper, we study the conditions under which covert communication at positive rates is feasible over discrete memoryless classical, classical-quantum, and quantum channels. For classical point-to-point channels, we show that if the dimension of the channel input probability simplex exceeds that of the channel output probability simplex, equivalently, if the input alphabet has larger cardinality than the output alphabet, then positive covert rates are achievable for certain classes of Discrete Memoryless Channels (DMCs). We further show that allowing the innocent symbol (i.e., the symbol transmitted in the no-communication mode) to be chosen appropriately can enlarge the class of DMCs for which positive covert rates are achievable. We also study covert communication over classical Multiple-Access Channels (MACs), where the additional transmitter effectively enlarges the set of available channel input pairs, and we show that the conditions required to satisfy the covertness constraint are less restrictive for MACs than for point-to-point channels. We extend these results to classical-quantum DMCs by showing that if the dimension of the input probability simplex exceeds the affine dimension of the set of output states that can be induced at the channel output, then positive covert rates are achievable for certain classes of classical-quantum DMCs. Finally, for quantum channels, we show that if the innocent state (i.e., the state transmitted in the no-communication mode) is mixed, then the covertness constraint can always be satisfied by a non-trivial input ensemble. Consequently, positive covert rates are achievable whenever the legitimate receiver can distinguish at least two states in a suitable such ensemble.
We consider a scenario in which K transmitters attempt to communicate covert messages reliably to a legitimate receiver over a discrete memoryless multiple-access channel (MAC) while simultaneously escaping detection from an adversary who observes their communication through another discrete memoryless MAC. We assume that each transmitter may use a secret key that is shared only between itself and the legitimate receiver. We show that each of the K transmitters can transmit on the order of √n reliable and covert bits per n channel uses, exceeding which, the warden will be able to detect the communication. We identify the optimal pre-constants of the scaling, which leads to a complete characterization of the covert capacity region of the K-user binary-input MAC. We show that, asymptotically, all sum-rate constraints are inactive unlike the traditional MAC capacity region. We also characterize the channel conditions that have to be satisfied for the transmitters to operate without a secret key.
This paper considers the covert identification problem in which a sender aims to reliably convey an identification (ID) message to a set of receivers via a binary-input discrete memoryless channel (BDMC), and simultaneously to guarantee that the communication is covert with respect to a warden who monitors the communication via another independent BDMC. We prove a square-root law for the covert identification problem. This states that an ID message of size exp(exp(Θ(√{ n})) can be transmitted over n channel uses. We then characterize the exact pre-constant in the Θ(·) notation. This constant is referred to as the covert identification capacity. We show that it equals the recently developed covert capacity in the standard covert communication problem, and somewhat surprisingly, the covert identification capacity can be achieved without any shared key between the sender and receivers. The achievability proof relies on a random coding argument with pulse-position modulation (PPM), coupled with a second stage which performs code refinements. The converse proof relies on an expurgation argument as well as results for channel resolvability with stringent input constraints.
This paper establishes the fundamental limits of a multi-access system where multiple users communicate to a legitimate receiver in presence of an external warden. Only a specific subset of the users, called covert users, needs their communication to remain undetected to the warden, while the remaining non-covert users have no such constraint. The fundamental limits show a tradeoff between the different rates that are simultaneously achievable at the various users in function of the secret-key rates that the different users share with the legitimate receiver. Interestingly, the presence of the non-covert users can enhance the capacities of the covert users, especially under stringent secret-key budgets. Our findings underscore the essential requirement of employing a multiplexing (coded time-sharing) strategy to exhaust the fundamental region of all rates that are simultaneously achievable at the covert and the non-covert users. As a side-product of our results, we also establish the covert-capacity secret-key tradeoff for standard single-user and multi-access covert communication systems (without non-covert users), i.e., the largest covert rates that are achievable under given secret-key rate budgets. Previous works had only established the minimum secret-key rates required at largest covert rates.