Hidden in Plain Numbers: A New Steganographic Approach

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Author: Denis Avetisyan

Researchers are exploring a unique method of concealing data within numerical sequences, leveraging the mathematical properties of semigroups.

This paper details a steganographic technique using the gap distribution and Frobenius number of symmetric numerical semigroups to embed hidden information.

Concealing information within seemingly innocuous data streams remains a central challenge in secure communication. This paper, ‘Steganographic information hiding via symmetric numerical semigroups’, introduces a novel approach leveraging the arithmetic properties of numerical semigroups to embed hidden data within the distribution of gaps between representable integers. By utilizing symmetric semigroups, the scheme achieves a balanced gap structure, rendering encoded values statistically indistinguishable from random noise, and relies on the computational hardness of numerical semigroup membership inference for security. Could this number-theoretic primitive offer a viable path towards post-quantum resilient information hiding and covert communication channels?

Unveiling Hidden Structures: The Foundation for Secure Communication

Current cryptographic systems frequently depend on the computational difficulty of certain mathematical problems, such as factoring large numbers or solving discrete logarithms. While effective today, the relentless advancement of computing power – including the development of quantum computers – poses a significant threat to these methods. As computational capabilities increase, algorithms once considered unbreakable become increasingly susceptible to attack, potentially compromising sensitive data. This vulnerability stems from the fact that these systems rely on complexity as a security measure; a sufficiently powerful computer, given enough time, can eventually overcome that complexity. Consequently, researchers are actively exploring alternative cryptographic approaches, including those leveraging the properties of mathematical structures like numerical semigroups, which offer the potential for security based on inherent properties rather than computational difficulty.

Numerical semigroups present an intriguing alternative to conventional cryptographic methods by leveraging the mathematical properties of their internal structure. These semigroups, constructed from sets of non-negative integers closed under addition, inherently possess ‘gaps’ – numbers that cannot be expressed as a non-negative integer combination of the semigroup’s generators. This characteristic allows for the subtle embedding of information within the seemingly innocuous arrangement of representable and unrepresentable integers. Unlike traditional cryptography, which relies on computational complexity, information hidden within a numerical semigroup’s gaps is protected by the semigroup’s inherent mathematical properties, offering a potential resilience against advances in computing power. The challenge lies in efficiently encoding and decoding data within this ‘gap distribution’, but the theoretical framework suggests a path toward a more robust and mathematically grounded approach to information security.

Numerical semigroups establish a compelling method for information concealment through their unique mathematical structure. These semigroups, built upon non-negative integer combinations of a finite set of generators, inherently possess ‘gaps’ – positive integers that cannot be expressed as such a combination. It is within these gaps, and the precise arrangement of representable numbers, that data can be subtly encoded. The seemingly abstract properties of these additive structures allow for the creation of a hidden informational space, where the presence or absence of a number within the semigroup’s generated sequence acts as a binary signal. This approach differs significantly from traditional cryptographic methods, as security isn’t derived from computational complexity, but rather from the difficulty of discerning meaningful patterns within the semigroup’s inherent structure – effectively transforming mathematical limitations into a protective barrier for sensitive data.

The Frobenius number, representing the largest integer unattainable through non-negative integer combinations of a given set of numbers, fundamentally limits the scope of representable data within a numerical semigroup. This value doesn’t simply define an upper bound, but actively sculpts the ‘hidden space’ available for information encoding. For a numerical semigroup generated by three variables – denoted as a_1, a_2, and a_3 – Davison established a critical lower bound: the Frobenius number is guaranteed to be greater than or equal to 3\sqrt{a_1a_2a_3}. This mathematical constraint dictates a minimum ‘gap’ within the representable integers, influencing the density and subtlety with which information can be embedded and, crucially, providing a quantifiable measure of the security afforded by this information-hiding technique. A larger Frobenius number expands this ‘hidden space’ and enhances security, while its relationship to the generating variables offers a pathway to designing semigroups with tailored cryptographic properties.

Hiding Within the Structure: A Steganographic Protocol

The steganographic protocol utilizes numerical semigroups – sets of non-negative integers that are closed under addition and contain 0 – as the foundational structure for data concealment. Information is not embedded within the elements of the semigroup, but rather within the ‘gaps’ – the positive integers not present in the semigroup. A numerical semigroup S of n is a subset of the non-negative integers that satisfies 0 \in S, and for all a, b \in S, a + b \in S. The gaps of S are the positive integers not in S. By carefully constructing the semigroup, specific gaps can be designated to represent bits or other data, effectively hiding a message within the inherent structure of the numerical semigroup itself. The properties of semigroups, particularly their predictable gap distribution, facilitate both encoding and decoding of the concealed information.

Modular Gap Partitioning operates by strategically selecting gaps within a numerical semigroup based on their residue classes modulo a chosen integer. This process enables the embedding of a message by associating specific residue classes with the presence or absence of a gap. Critically, even when restricting the selection of gaps to only those belonging to designated residue classes, the asymptotic gap density remains at 1/2. This consistent density is a key characteristic, ensuring that the modification of the semigroup structure caused by message embedding does not introduce detectable statistical anomalies. The mathematical principle relies on the predictable distribution of gaps within the semigroup, allowing for reliable encoding and decoding of the embedded message.

Symmetric Numerical Semigroups are crucial to the protocol’s functionality because they provide a mathematically defined structure that guarantees both message recoverability and a degree of security. These semigroups are constructed such that for every generator g, its complement N - g is also a generator, where N represents the numerical semigroup’s Frobenius number. This symmetry allows the receiver to uniquely determine the generating sequence used for encoding, even with incomplete information, facilitating message decryption. Furthermore, the symmetric property complicates cryptanalysis, as any attempt to deduce the message must account for the dual nature of the generating set, increasing the computational difficulty for potential adversaries. The use of these semigroups avoids the need for a shared secret key, relying instead on the mathematical properties of the structure itself for security.

The construction of numerical semigroups for this steganographic protocol heavily relies on the selection of appropriate generating sequences; specifically, ‘Telescopic Generating Sequences’ are employed due to their predictable and controllable structure. These sequences, defined as \{a, a+d, a+2d, ..., a+nd\} where ‘a’ is the initial value, ‘d’ is the common difference, and ‘n’ is the degree, facilitate the creation of semigroups with precisely defined gap distributions. The predictable nature of these sequences allows for the controlled introduction of gaps – the differences between consecutive semigroup elements – which are then utilized for message embedding. By manipulating the parameters ‘a’, ‘d’, and ‘n’, the protocol can engineer semigroups with specific gap characteristics, ensuring both the capacity and recoverability of the encoded message. The telescopic property ensures that the semigroup’s structure is readily analyzable, a crucial requirement for the decoding process.

Efficient Gap Computation: A Graph-Theoretic Approach

The computation of gaps within a numerical semigroup – the non-negative integers not representable as a sum of the semigroup’s generators – presents a computational challenge that scales with the size of the semigroup. Traditional methods typically involve iterating through potential values and testing for representability, resulting in a time complexity that is often exponential or factorial with respect to the largest generator. As the size of the largest generator, and therefore the potential range of missing values, increases, the number of computations required to determine all gaps grows rapidly. This is especially pronounced for large semigroups where the number of generators and their values are substantial, rendering brute-force approaches impractical for real-time applications or large-scale datasets. The problem isn’t simply the size of the semigroup itself, but the combinatorial explosion of possible sums that must be checked against the potential gaps.

The Residue Graph is a directed graph constructed to optimize the calculation of minimal representable values within a numerical semigroup. Nodes in the graph represent residue classes modulo the smallest generator, g, of the semigroup. Edges connect residue classes i and j if j can be obtained by adding a multiple of a generator g_k to i. The weight of each edge corresponds to the coefficient used in this addition. By representing the semigroup’s structure as a graph, the minimal value representable in each residue class is determined by finding the shortest path from the node representing the class to a node representing zero, effectively minimizing the number of generator additions needed.

Dijkstra’s Algorithm is applied to the constructed Residue Graph to determine the minimal representable value for each residue class modulo the smallest generator of the numerical semigroup. The algorithm functions by iteratively exploring nodes in the graph, assigning a distance representing the minimum number of steps required to reach that node from the starting point (representing zero). This process effectively calculates the smallest non-negative integer expressible as a sum of multiples of the semigroup’s generators within each residue class. The resulting minimal values directly correspond to the gaps in the semigroup, and their identification facilitates the embedding of messages by mapping data to these gaps.

Traditional methods for gap computation in numerical semigroups typically involve iterative checks of potential representable values, resulting in a time complexity proportional to the size of the largest gap. The graph-based approach, utilizing a Residue Graph and Dijkstra’s Algorithm, reduces this computational burden by transforming the gap computation problem into a shortest-path search. This allows for a complexity dependent on the number of nodes and edges in the Residue Graph – a structure significantly smaller than the range of potential representable values – and the efficiency of Dijkstra’s Algorithm, typically O(E + V \log V), where E is the number of edges and V is the number of vertices. Empirical results demonstrate a substantial decrease in computation time, particularly for large semigroups where traditional methods become intractable.

FrobCrypt: Implementation and Future Trajectories

A fully functional implementation of the FrobCrypt steganographic protocol has been developed in Python, serving as a practical demonstration of its core principles. This implementation allows for the encoding and decoding of hidden messages within cover data, showcasing the protocol’s operational mechanics beyond theoretical description. By providing a tangible codebase, researchers and developers can directly examine the protocol’s functionality, experiment with its parameters, and assess its performance characteristics. The Python implementation not only validates the protocol’s feasibility but also establishes a foundation for further refinement and integration into secure communication systems, opening avenues for practical applications requiring covert data transmission.

The completed FrobCrypt implementation serves as compelling evidence of the protocol’s practical viability and efficiency in secure communication scenarios. Through a functional demonstration in Python, the system showcases a streamlined approach to steganography, minimizing computational overhead while maintaining a high degree of security. This efficiency is particularly relevant for resource-constrained environments or applications demanding real-time communication, such as secure messaging or data transmission over networks with limited bandwidth. Beyond theoretical security guarantees, the implementation’s performance suggests that FrobCrypt could be readily integrated into existing communication infrastructure, offering a robust and adaptable solution for protecting sensitive information against unauthorized access.

The core of FrobCrypt’s security lies in its ability to render the hidden message statistically indistinguishable from random noise. This is achieved through a carefully engineered distribution of ‘gaps’ – the intervals between embedded message bits within the cover text. By maintaining a balanced gap density of precisely 1/2, meaning each gap has an equal probability of occurring, the protocol ensures that the distribution of these gaps mirrors that of purely random data. This prevents an attacker from identifying patterns indicative of hidden information, even with advanced statistical analysis. P(gap = x) = 1/2 for all possible gap sizes, effectively camouflaging the message within the noise and bolstering the protocol’s resilience against detection.

Continued investigation into FrobCrypt’s security profile is crucial, with future work dedicated to rigorously assessing its resilience against known steganographic and cryptographic attacks, including those targeting statistical anomalies or exploiting potential weaknesses in the gap density function. Beyond security, exploring the protocol’s adaptability represents a significant avenue for advancement; this includes testing its performance across diverse communication channels – such as lossy media formats, bandwidth-constrained networks, or even quantum communication systems – and examining how parameter adjustments impact both robustness and data capacity. Such research will determine the protocol’s true practical limits and unlock its potential for broader implementation in secure data transmission scenarios.

The presented work leverages the inherent structure within numerical semigroups to conceal information, mirroring a fundamental principle of pattern recognition. The manipulation of gap distributions, as explored within the paper, demonstrates how seemingly innocuous numerical properties can be harnessed for a specific purpose. This echoes Albert Einstein’s observation: “The most incomprehensible thing about the world is that it is comprehensible.” The ability to encode data within the arithmetic characteristics of these semigroups reveals an underlying order, transforming what appears random into a system capable of communication. If a pattern cannot be reproduced or explained, it doesn’t exist.

Beyond the Hidden Message

The exploration of numerical semigroups as a vessel for steganography reveals a curious truth: information, like numbers themselves, often resides within structure. This work, viewed as a microscope focused on the gaps within these algebraic forms, has illuminated a pathway – but the landscape remains largely uncharted. The current method, while demonstrating feasibility, is limited by the density of information embeddable within a given semigroup. Future investigations must address this constraint, perhaps by exploring higher-dimensional generalizations or leveraging the Frobenius number itself as a carrier of data – a more ambitious, but potentially richer, avenue.

A compelling, if slightly unsettling, question arises: how robust is this hidden information against intentional disruption? The model, at present, appears vulnerable to even minor alterations in the generating set of the semigroup. Subsequent research should prioritize the development of error-correcting codes tailored to the specific structure of these algebraic objects, transforming this delicate hiding place into a more resilient archive.

Ultimately, this field, like the semigroups it examines, feels incomplete. The true potential lies not merely in concealing messages, but in understanding the very limits of information density within mathematical structures. It suggests a deeper inquiry: could the principles governing information hiding within semigroups illuminate analogous properties in other algebraic systems, or even within the natural world itself?

Original article: https://arxiv.org/pdf/2602.04052.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-02-05 12:04