Interdisciplinary vision
Visual imagination
Artistry
Imagine the architectural marvel of Cologne Cathedral: every spire points precisely to the heavens, every line serving a specific purpose. This is exactly how future wireless communication systems work – they create invisible architectural forms from radio waves, where each beam must find its intended recipient amidst the chaos of the electromagnetic space. But what happens when an uninvited guest, an eavesdropper, lurks in this space, ready to steal our secrets?
Today, we dive into the mathematical symphony of secure communication, where each note is a bit of information, each chord a directed beam, and the entire composition plays on the frequencies of sixth-generation mobile networks.
The Architecture of an Invisible World
Sixth-generation communication systems promise us data transfer speeds that surpass everything we’ve known before. These systems operate on millimeter waves – tiny particles of the electromagnetic spectrum, so small they can be likened to threads in an intricate tapestry. Yet, these threads are fickle: they easily break when they encounter obstacles and quickly fade over distance.
To compensate for these shortcomings, engineers create directional beams, like spotlights in a theater that illuminate a specific area of the stage. The base station must first find the recipient in this electromagnetic theater by directing a beam precisely towards them. This process is called beam acquisition – an elegant dance of discovery in the space of radio waves.
But imagine that in the theater's audience, there isn't just a legitimate viewer but also a secret observer trying to eavesdrop on the performance. How can we ensure that a secret message reaches only its intended recipient?
The Geometry of a Secret
At the heart of our research lies an elegant mathematical model: the binary-indicated beam channel. This is a simplified yet profoundly meaningful reflection of real directional communication systems. Imagine a compass rose, but instead of wind directions, we have directions for transmitting radio signals.
Our system consists of three participants in this electromagnetic drama: the base station (transmitter), the legitimate receiver, and a secret eavesdropper. The base station does not know the exact location of the receivers and must explore the space by sending signals in different directions.
A key feature of our model is block memory. This means that the positions of the participants remain unchanged for short periods – as if they were frozen in specific poses during one musical piece but could move at the start of the next. Mathematically, this is reflected by the transmission directions remaining constant within a symbol block but changing independently between blocks.
Each block contains L channel uses, like L notes in a musical phrase. At each moment, the base station sends a signal X, which can be represented as a vector of zeros and ones, where a one indicates an active transmission direction.
The Mathematics of Feedback
One of the key elements of our system is feedback – an echo that returns to the transmitter from the receivers. This is as if a musician in a concert hall could hear not only their own playing but also the sound's reflection from the walls, which helps them understand the acoustic properties of the space.
In our case, the base station receives backscattered signals from both the legitimate user and the eavesdropper. These signals arrive with a single delay, like an echo heard a moment after a word is spoken. Such information is invaluable for refining location estimates of the receivers.
Mathematically, the system is described by the product of probability distributions that reflect all interdependencies between signals, states, and feedback. This is a complex, multi-layered structure where each element influences its neighbors, creating a network of dependencies akin to a lace pattern.
The Limits of Secrecy
At the core of our research lies a fundamental question: what is the maximum rate at which information can be transmitted so that it reliably reaches the legitimate receiver but remains completely inaccessible to the eavesdropper? This is the secret capacity – the mathematical equivalent of the art of being loud for your friends and silent for your enemies simultaneously.
Imagine a waterfall: the water (information) must flow at the highest possible speed to a pond (the legitimate receiver), yet not a single drop should fall into a nearby ravine (to the eavesdropper). The speed of this flow is our sought-after value.
For channels without block memory (when L = 1), the situation is critical. In this case, the system cannot ensure the secret transmission of information – the difference between the information available to the legitimate receiver and the eavesdropper turns out to be zero. This is like trying to whisper in a room with perfect acoustics; every sound is equally audible to everyone present.
However, when block memory increases (L ≥ 2), the possibility for secure communication emerges. Mathematically, this is expressed through complex expressions involving entropy functions and depending on constraints on transmission power.
Upper Bound: The Theoretical Limit
The upper bound of secret capacity represents a theoretical maximum – an unreachable but important benchmark, like the horizon for a traveler. It is based on the capacity of the main channel between the base station and the legitimate receiver, without accounting for the eavesdropper's presence.
Mathematically, this boundary is expressed as the sum of entropy terms, where each term corresponds to a specific use of the channel in a block. The formula includes a sequence of values c_j that reflect the optimal strategy for distributing transmission energy:
The first value c_1 is the minimum between half of the total number of possible directions K and the power constraint B. Subsequent values are calculated iteratively, taking into account the already «spent» directions.
This boundary shows us the ceiling of performance – just as the height of a cathedral vault determines the maximum volume the space can hold.
Lower Bound: The Achievable Reality
The lower bound represents what we can reliably achieve with specific transmission strategies. It is based on a fundamental principle of information theory: the secret rate cannot exceed the difference between the information reliably delivered to the legitimate user and the information received by the eavesdropper.
This boundary is achieved through a complex coding strategy that uses two key tools: random coding and binning. Random coding helps ensure transmission reliability, while binning ensures secrecy by dividing messages into groups in a way that the eavesdropper cannot determine which specific message was transmitted.
The Joint Communication and Sensing Algorithm
The heart of the practical implementation is the JCAS (Joint Communication and Adaptive Sensing) algorithm – an elegant solution that simultaneously performs two tasks: transmits information and explores the surrounding environment. It’s like a dancer who both performs choreography and studies the stage, adapting their movements to its features.
The algorithm works as follows: at the beginning of each block, the system initializes a set of possible transmission directions. As feedback is received, this set adaptively shrinks. If feedback indicates that the signal was received (Y = 1), the system knows it has «felt out» the correct direction and can refine it. If the signal did not pass through (Y = 0), the corresponding directions are excluded from consideration.
A particularly interesting point: the algorithm uses a «divide and conquer» strategy. After finding the correct direction, the system begins to split the set of possible positions in half, sending signals into each half with equal probability. This maximizes the information transfer rate, like a binary search in the space of directions.
It is important to note that the algorithm uses feedback only from the legitimate receiver, ignoring signals from the eavesdropper. This is not an accident but a deliberate choice: using information from the eavesdropper could help them unlock secrets.
Performance Analysis
The difference between the upper and lower bounds shows us how far the proposed algorithm is from the theoretical optimum. This difference equals the amount of information the eavesdropper can extract from their observations.
Mathematically, it is expressed through the mutual information between the input signals and the eavesdropper's outputs, conditioned on the state of their channel. Calculating this value requires a careful analysis of all possible feedback scenarios and corresponding probabilities.
A key observation: the difference approaches zero as the block length L increases. This means that with growing L, the proposed algorithm approaches the theoretical performance limit. This is intuitively clear: more channel uses in a block provide more opportunities for precise tuning and optimization of the strategy.
Numerical Results and Visualization
Numerical experiments conducted with K = 32 possible quantized directions illustrate the theoretical predictions. The graphs show several important patterns:
First, both the upper and lower bounds increase with the power constraint B. This is natural: more energy allows for the transmission of more information. The growth resembles a logarithmic function – fast at the beginning and slowing down at larger values of B.
Second, both bounds increase with the block length L. This confirms the importance of block memory for ensuring secrecy. Systems with L = 1 cannot provide secure transmission, but even with L = 2, possibilities for protected communication appear.
Third, and this is especially important, the gap between the bounds narrows as L increases. For L = 2, the difference is a noticeable amount, but at L = 8 or L = 16, the bounds almost converge. This demonstrates the effectiveness of the proposed JCAS algorithm when working with long blocks.
Fundamental Trade-offs
Our research reveals several fundamental trade-offs in directional communication systems. The first trade-off is between transmission speed and secrecy. To ensure perfect secrecy, the system must «sacrifice» part of its capacity. This sacrifice is quantitatively expressed through the difference between the main channel capacity and the secret capacity.
The second trade-off is between sensing and data transmission. The system must spend part of its resources exploring the environment and determining the location of receivers. The more accurate this sensing is, the more efficient the subsequent data transmission, but the fewer resources remain for the actual transmission.
The third trade-off is related to block memory. Long blocks provide better performance but require the assumption of a stationary environment for the duration of the block. In real systems, receivers can move, which limits the practical length of blocks.
Connection to Physical Principles
Behind the mathematical derivations lie deep physical principles. The directionality of transmission in the millimeter wave range is a consequence of the wave properties of electromagnetic radiation. Short waves focus better but are less effective at bending around obstacles. This creates a natural defense: a signal directed toward a legitimate receiver is physically more difficult to intercept from other directions.
Block memory reflects the quasi-static nature of radio channels. In reality, channel coefficients change over time scales related to the speed of object movement and Doppler shifts. The assumption of constancy within a block is a mathematical idealization of this physical phenomenon.
Feedback is of fundamental importance because it breaks the symmetry between the legitimate receiver and the eavesdropper. In classical encryption models, this symmetry is broken through secret keys. In our case, the asymmetry is created through physical interaction with the legitimate receiver.
Practical Applications and Future Directions
The results of our research have direct application in developing 6G communication systems. Joint communication and sensing algorithms are becoming critically important in systems with massive antenna arrays, where the number of possible transmission directions is in the hundreds or thousands.
In automotive applications, where vehicles constantly exchange information about location and intentions, ensuring secrecy becomes a matter of safety. Our methods can help protect this crucial information from malicious actors.
In Internet of Things networks, where a multitude of devices compete for spectrum access, directional transmission can not only improve spectral efficiency but also provide inherent security through spatial separation.
Future research can develop in several directions. The first is to generalize to multi-user scenarios where multiple legitimate receivers and eavesdroppers are simultaneously present in the system. The second is to account for more realistic channel models with fading and multipath propagation. The third is the development of adaptive algorithms that can dynamically select the block length based on changing environmental conditions.
Theoretical Observations
From a theoretical perspective, our work demonstrates the power of the information-theoretic approach to analyzing physical communication systems. The concept of secret capacity, introduced by Wyner in 1975, finds a new application here in the context of modern directional systems.
The observation about the role of feedback in providing secrecy is particularly important. In classical models, feedback does not increase the capacity of memoryless channels. However, in our case, where memory is related to the spatial structure of signals, feedback becomes a critical resource.
The mathematical structure of our bounds opens up an interesting geometric interpretation. The upper bound corresponds to the information capacity of an «unfolded» channel, where all spatial degrees of freedom are used optimally. The lower bound reflects the performance of a real algorithm that must navigate a space of uncertainty.
As we conclude our journey through the mathematical landscapes of secure communication, we see how abstract theoretical concepts find physical embodiment in future technologies. Each formula tells a story about the balance between openness and secrecy, between efficiency and safety. In this balance lies the key to creating communication systems worthy of humanity's digital future.
Mathematics once again proves its ability to find order in the seeming chaos of electromagnetic space, transforming the problem of secure communication into an elegant symphony of numbers, where each note sounds precisely in the right place and at the right time.
