Solvability and Nilpotency of Lie Algebras in Cryptography and Steganography

Abstract

This paper investigates the role of solvable and nilpotent Lie algebras in the domains of cryptography and steganography, emphasizing their potential in enhancing security protocols and covert communication methods. In the context of cryptography, we explore their application in public-key infrastructure, secure data verification, and the resolution of commutator-based problems that underpin data protection strategies. In steganography, we examine how the algebraic properties of solvable Lie algebras can be leveraged to embed confidential messages within multimedia content, such as images and video, thereby reinforcing secure communication in dynamic environments. We introduce a key exchange protocol founded on the structural properties of solvable Lie algebras, offering an alternative to traditional number-theoretic approaches. The proposed Lie Exponential Diffie–Hellman Problem (LEDHP) introduces a novel cryptographic challenge based on Lie group structures, offering enhanced security through the complexity of non-commutative algebraic operations. The protocol utilizes the non-commutative nature of Lie brackets and the computational difficulty of certain algebraic problems to ensure secure key agreement between parties. A detailed security analysis is provided, including resistance to classical attacks and discussion of post-quantum considerations. The algebraic complexity inherent to solvable Lie algebras presents promising potential for developing cryptographic protocols resilient to quantum adversaries, positioning these mathematical structures as candidates for future-proof security systems. Additionally, we propose a method for secure message embedding using the Lie algebra in combination with frame deformation techniques in animated objects, offering a novel approach to steganography in motion-based media.

1. Introduction

Mathematical structures, such as Lie algebras, and their solvable variants play a crucial role in many areas of modern mathematics and theoretical computer science. Solvable Lie algebras, characterized by the gradual vanishing of their commutator series, represent a significant tool in group theory, Lie algebra theory, and applied mathematics. In recent decades, their applications in cryptography and steganography have become the focus of intensive research, as these algebras offer robust and efficient methods for securing data and enabling covert communication. Cryptography, the field concerned with protecting information through encoding and decoding, employs various algebraic structures in the design of security protocols. In this context, solvable Lie algebras can be utilized to construct public-key systems, establish secure communication channels, and verify data authenticity. Their mathematical properties, particularly those involving commutator relations and nonlinear functions, enable the development of advanced techniques that hinder attacks on security systems. Steganography, which deals with hidden communication methods, leverages solvable Lie algebras to protect covert messages embedded in multimedia data. The goal in this domain is to develop techniques for the undetectable exchange of information. Solvable Lie algebras offer capabilities for efficiently encoding and concealing data within dynamic media, such as images, videos, and 3D graphics, facilitating the development of sophisticated steganographic methods resistant to detection and analysis.

This paper provides an in-depth investigation of the mathematical foundations of solvable Lie algebras, analyzes their properties in the context of security, and explores practical applications in both cryptography and steganography. We focus on theoretical aspects of commutator relations and exponential functions that can be employed for generating secure keys and protecting data in hidden communications. Furthermore, we examine concrete examples of implementing solvable Lie algebras in security protocols and steganographic systems, evaluating their benefits as well as potential limitations and challenges. Ultimately, the aim of this research is to demonstrate how solvable Lie algebras can enhance existing security systems by offering a new dimension of data protection and covert communication in the digital age.

The following structure of the paper will be presented. Section 2 presents a review of related works, outlining previous research involving algebraic methods in cryptography and steganography. Section 3 introduces the cryptographic potential of solvable Lie algebras, detailing their use in constructing secure key exchange protocols. Section 4 explores the application of solvable Lie algebras in steganography, focusing on embedding confidential data within multimedia content. Section 5 discusses the role of nilpotent Lie algebras in cryptography, emphasizing their algebraic properties for developing secure encryption mechanisms. Also, this section examines how nilpotent Lie algebras can enhance steganographic techniques. In this section, we also show the testing of the proposed steganographic method, evaluating its effectiveness and robustness. Section 6 concludes the study by summarizing the main contributions and proposing directions for future research.

2. Related Works

In [1], the authors present a novel data-hiding method using multiple stego-images, based on a modified least significant bit (LSB) matching technique that improves both embedding capacity (EC) and image quality. The main advantage of this approach is that each original pixel generates four new pixels, each containing hidden secret data, which are then adjusted to enhance the quality of the stego-images. Consequently, four distinct stego-images are produced, with each embedding one bit per pixel. This method shows strong resistance to steganalysis. In [2], the authors proposed an enhanced reversible data-hiding (RDH) technique employing a shadow image, which is a copy of the cover image. By applying simple addition and subtraction operations to the shadow image pixels, the method achieves higher capacity, improved stego-image quality, and increased ASA. During embedding, three shadow images of the cover image are created and then adjusted based on XOR properties of the least significant bit planes. This technique demonstrates significant robustness against various steganalysis attacks, including regular and singular analyses, pixel difference histogram analysis, and bit pair analysis, while also effectively addressing the boundary overflow problem.

The steganographic method described in [3] belongs to key generation techniques rather than traditional approaches like bit injection or substitution. It generates a hidden message using the data carrier and a complex stego-key, without modifying or adding anything to the data carrier itself. Furthermore, the authors introduced a steganographic approach based on the decomposition of Catalan numbers, which also keeps the data carrier completely unchanged and produces the hidden message solely from the key. As a result, the original file remains intact and cannot be compared to others due to the absence of any modifications. In the paper [4], the authors described the concept of a new data-hiding technique for steganography in RGB images where a secret message is embedded in the blue layer of specific bytes. For increasing security, bytes are chosen randomly using a random square Hermitian positive definite matrix $N_{8\times 8}$, which is at the same time the key and stego-key.

Table 1. The advantages/disadvantages of the proposed method compared to the previous methods.

Stability and Robustness
1	Numerical Stability	The implementation of Lie algebra in Numerical Stability improves algorithm robustness by preserving geometric structures and minimizing error accumulation.
2	Robust Signal Processing	The use of Lie algebra in Robust Signal Processing enhances system resilience by enabling structure-preserving transformations that maintain performance under uncertainties and noise.
Optimality in Estimation
3	Minimum Variance	Previous methods [1,2]
Positive Definiteness in Metrics
4	Distance Measures	The application of Lie algebra in Distance Measures enables accurate and consistent computation of distances on nonlinear manifolds by leveraging the underlying group structure.
Flexibility in Handling Ill-posed Problems
5	Weight Adjustment	Previous method [4]
Improved Solutions for Underdetermined Systems
6	Handling Underdetermined Systems	Previous methods [1,3]
Robustness Against Perturbations
7	Reduced Sensitivity	The use of Lie algebra in Reduced Sensitivity enables more efficient control of nonlinear systems by reducing sensitivity to parameter variations through structural symmetry analysis.

outlines the benefits of this new approach compared to earlier methods.

The solvability of Lie algebras represents significant potential in the field of cryptography, especially because it allows the construction of secure mathematical models based on their internal structure. Solvable Lie algebras, i.e., those in which the series of commutators becomes trivial after a finite number of steps, find application in the theory of algebraic groups as well as in the construction of cryptographic systems. A characteristic feature of these algebras is that the series of commutators, after a finite number of steps, becomes trivial, making them suitable for use in the theory of algebraic groups as well as in specific cryptographic applications. That is, there exists $k\in N$ such that

$$g\left(k\right)=0, \mathrm{where} g(i+1)=\left[g\right(i),g(i\left)\right] \mathrm{and} g\left(0\right)=g.$$

Due to their structured non-commutativity and available representation theory, solvable Lie algebra provides a framework for the construction of cryptographic protocols that rely on complex systems based on groups or lattices. They are particularly useful in non-commutative cryptographic primitives, where standard assumptions like the hardness of the conjugacy search problem (CSP) or the decomposition problem are interpreted in the context of Lie algebra [5,6]. Solvable Lie algebras can serve as a foundation for constructing cryptographic schemes, particularly in the context of public-key cryptography and schemes that rely on algebraic structures. Their application is because certain problems related to their structure represent significant mathematical complexity, which directly contributes to the security of the system.

3. Cryptographic Potential of Solvable Lie Algebras

In many cryptographic systems based on Lie algebras, problems involving commutators are naturally complex and hard to solve. In the case of a solvable Lie algebra, the complexity of commutator expressions can be sufficient to serve as the foundation for security protocols. These systems can use solvable algebras for key generation or data verification, further complicating attacks based on the analysis of algebraic properties. This complexity is analogous to the difficulty of factorization in RSA or solving the discrete logarithm problem [7]. Solvable algebras, through control over the depth of the commutator series, enable the design of protocols with adaptable algebraic complexity.

Structural similarities between solvable Lie algebras and algebraic groups make them highly suitable for creating public-key cryptosystems. Key generation and identity verification in such systems can be based on the difficulty of solving certain algebraic problems within the structure of solvable Lie algebras. For example, it is possible to construct digital signatures or encryption systems that use these algebras for generating secure keys and verifying messages. The exponential function defined on Lie algebras, often through matrix exponentiation or the Baker–Campbell–Hausdorff formula, allows for the construction of analogs of classical public-key systems. Key exchange is based on non-commutative “exponentiation” and the difficulty of its inverse problem, known as the Lie exponential Diffie–Hellman problem (LEDHP) [8]. Thanks to the specific structure of the commutator series, solvable Lie algebras enable the construction of systems with precisely controlled commutative behavior. This feature provides additional protection against attacks that rely on simple linear or commutative properties of algebraic structures. Such controlled non-commutativity supports the development of cryptographic techniques that are resistant to common analysis methods involving commutators and derivatives. Control over algebraic properties in solvable Lie algebras allows the construction of cryptosystems resistant to attacks using linear algebra methods. As these algebras can be decomposed into lower-dimensional subalgebras with well-defined commutator structures, they offer flexible platforms for setting up high-complexity problems [9]. Solvable algebraic structures have also been proposed for secure data representation and steganography, where information is embedded in images or signals through algebraic transformations resistant to statistical detection [10].

3.1. Key Exchange Protocol Based on Solvable Lie Algebras

Below, we present a key exchange protocol that uses solvable Lie algebras over a finite field ${\mathbb{F}}_q$ based on the assumption of the difficulty of the LEDHP problem. Let g be a finite-dimensional, solvable Lie algebra over ${\mathbb{F}}_q$ with a known basis $\{e_1,\dots ,e_n\},$ and let the Lie bracket $[x,y]$ be bilinear and non-commutative. The exponential function is defined (formally or through matrices) as:

$$exp (x)=\sum _{k=0}^{\infty }{\displaystyle \frac{x^k}{k!}}$$

In practice, this series is truncated or replaced with the shortened BCH formula.

Protocol Steps:

• Generation of Public and Private Keys: Alice and Bob agree on a common solvable Lie algebra, including the basis and relevant commutator relations. Each party selects a private element within the Lie algebra to generate their private key. Let Alice and Bob generate elements $x_A$ and $x_B$ within the Lie algebra.
• Exchange of Public Keys: Alice uses her private element $x_A$ to compute her public key $y_A=e^{x_A}.$ Similarly, Bob computes his public key $y_B=e^{x_B}.$
• Key Combination: Alice sends her public key $y_A$ to Bob, and Bob sends $y_B$ to Alice. Using the received public key and her private element, Alice computes the shared secret key $K_A=y_B^{x_A} .$ Similarly, Bob computes $K_B=y_A^{x_B}$ If the commutator properties of the algebra ensure that $y_B^{x_A}{=y}_A^{x_B}$ both parties obtain the same shared key: ${ K}_A=K_B.$
• Verification and Use of the Shared Key: The shared secret key can now be used for encrypting and decrypting messages between Alice and Bob. However, a fundamental security analysis of this approach is required to determine whether the underlying problem is computationally hard in the context of Lie algebras, like the discrete logarithm problem in classical cryptographic systems.

3.2. The Lie Exponential Diffie–Hellman Problem (LEDHP)

Let g be a finite-dimensional, solvable Lie algebra over a finite field ${\mathbb{F}}_q,$ and let $exp :g\to G$ be an exponential map onto a (matrix) group G. Given elements $Y_A=exp \left(x_A\right), Y_B=\exp \left(x_B\right)$ for unknown $x_A,x_B\in g,$ compute

$$\exp \left(\left[x_A,x_B\right]\right)\mathrm{or} K=\exp \left(x_A+x_B\right)$$

depending on the scheme design.

This is the non-abelian analog of the classical Diffie–Hellman problem but with additional hardness stemming from the non-commutative nature of the Lie bracket. Since $[x_A,x_B]\ne 0,$ recovering $x_A$ or $x_B$ from $exp \left(x_A\right)$ is nontrivial. The LEDHP is believed to be hard under the following conditions:

• g is chosen such that inversion of the exponential map is computationally infeasible;
• The commutator depth is high;
• Lie algebra is non-nilpotent (but solvable), ensuring a rich non-abelian structure.

The security of the scheme relies on the intractability of LEDHP. The following assumptions underpin its security:

• Non-invertibility of $exp—$The exponential map in non-abelian contexts (especially over finite fields) is nontrivial to invert, especially when approximated via BCH series.
• Non-commutativity—Commutators inject nonlinearity, preventing simple reductions to known linear problems.
• Structural Obscurity—Solvable Lie algebras can be chosen to have large, derived length and intricate bracket structures, making algebraic attacks impractical.

Quantum algorithms such as Shor’s are effective primarily in abelian or group settings (e.g., integer factorization, discrete logarithms). The LEDHP setting lacks:

• Group homomorphism from $exp \left(x\right)$ to an abelian group;
• Efficient Fourier sampling strategies applicable to nonlinear Lie algebras;
• Quantum circuits for solving non-commutative exponential problems.

Hence, current quantum algorithms do not offer a clear advantage over classical methods in solving LEDHP, though further research is required. This positions solvable Lie algebra-based cryptography as a potential post-quantum approach. An example is presented below.

Let ${\mathbb{F}}_q={\mathbb{F}}_{13}$ (the finite field with 13 elements). Consider the 2-dimensional solvable Lie algebra:

$$g=spa{n}_{{\mathbb{F}}_{13}}\left\{X,Y\right\},\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}[X,Y]=Y$$

This is a subalgebra of $g{l}_2\left({\mathbb{F}}_{13}\right).$ Define:

$$X=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right) and Y=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$$

We verify:

$$\left[X,Y\right]=XY-YX=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)-\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)-\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

We define our Lie algebra as follows:

$$\left[X,Y\right]=H≔\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

Let

$$g=spa{n}_{{\mathbb{F}}_{13}}\left\{X,Y\right\},$$

with the Lie bracket defined as $[X,Y]=H.$

We now construct a key exchange scheme based on truncated matrix exponentiation, where participants choose private linear combinations of X and Y, compute the exponential map truncated at second order (as in the BCH formula), and derive public matrices to exchange.

Step 1: Generation of Private and Public Keys

Participant A selects the private element:

$$x_A=5X+3Y=\left(\begin{array}{cc}0& 5\\ 3& 0\end{array}\right).$$

Its square is

$$x_A^2={\left(\begin{array}{cc}0& 5\\ 3& 0\end{array}\right)}^2=\left(\begin{array}{cc}15& 0\\ 0& 15\end{array}\right)\equiv \left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right) mod 13.$$

Using the truncated BCH formula,

$$exp(x_A)\approx I+x_A+{\displaystyle \frac{1}{2}}{x}_A^2.$$

In ${\mathbb{F}}_{13}$, since $2^{-1}\equiv 7,$ we compute:

$$exp\left(x_A\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\left(\begin{array}{cc}0& 5\\ 3& 0\end{array}\right)+7\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right)=\left(\begin{array}{cc}2& 5\\ 3& 2\end{array}\right).$$

Thus, A’s public key is

$$Y_A=exp (x_A)=\left(\begin{array}{cc}2& 5\\ 3& 2\end{array}\right)$$

Participant B selects:

$$x_B=6X+7Y=\left(\begin{array}{cc}0& 6\\ 0& 7\end{array}\right),\hspace{1em}\hspace{1em}{x}_B^2=\left(\begin{array}{cc}42& 0\\ 0& 42\end{array}\right)\equiv \left(\begin{array}{cc}3& 0\\ 0& 3\end{array}\right).$$

Hence,

$$exp(x_B)\equiv \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\left(\begin{array}{cc}0& 6\\ 7& 0\end{array}\right)+7\left(\begin{array}{cc}3& 0\\ 0& 3\end{array}\right)=\left(\begin{array}{cc}9& 6\\ 7& 9\end{array}\right)$$

B’s public key is

$$Y_B=exp(x_B)=\left(\begin{array}{cc}9& 6\\ 7& 9\end{array}\right).$$

Step 2: Shared Key Derivation

To derive a shared secret, one computes:

$$x=x_A+x_B=11X+10Y=\left(\begin{array}{cc}0& 11\\ 10& 0\end{array}\right),$$

$$x^2=\left(\begin{array}{cc}110& 0\\ 0& 110\end{array}\right)\equiv \left(\begin{array}{cc}6& 0\\ 0& 6\end{array}\right).$$

Then,

$$exp (x)\approx I+x+{\displaystyle \frac{1}{2}}{x}^2\equiv \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\left(\begin{array}{cc}0& 11\\ 10& 0\end{array}\right)+7\left(\begin{array}{cc}6& 0\\ 0& 6\end{array}\right)=\left(\begin{array}{cc}3& 11\\ 10& 3\end{array}\right).$$

Thus, the shared key is

$$K=exp (x)=\left(\begin{array}{cc}3& 11\\ 10& 3\end{array}\right).$$

The security of this protocol relies on the difficulty of recovering the private linear combinations $x_A$ or $x_B$ from their exponentiated forms over ${\mathbb{F}}_{13}$, given only the publicly exchanged matrices. This form of discrete exponential obfuscation is analogous in spirit to the classic DHKE scheme but operates in the framework of matrix Lie algebras, as seen in works such as [11,12].

The solvability of Lie algebras holds significant potential in cryptographic applications due to the ability of these structures to support the construction of secure mathematical frameworks. Solvable Lie algebras, defined by the property that their commutator series eventually vanishes, are frequently employed in the theory of algebraic groups and their cryptographic applications. A deep understanding of the solvability of Lie algebras and their deployment in the context of security systems can lead to the design of efficient and secure cryptographic protocols. Solvable Lie algebras can be used to develop cryptographic schemes, particularly in public-key cryptography and algebra-based encryption systems. Their application can ensure security by leveraging the mathematical difficulty of solving problems grounded in their structure. In many Lie algebra-based cryptographic systems, solving commutator-related problems is inherently complex. For solvable Lie algebras, the commutator problem may be sufficiently hard to serve as a basis for security protocols. Such systems can utilize solvable algebras for key generation or data verification, thus complicating attacks based on the analysis of algebraic structures. The structural similarities between solvable Lie algebras and algebraic groups make them particularly suitable for the design of public key systems. Key generation and identity verification in these systems can be based on the hardness of solving algebraic problems within solvable Lie algebras. For instance, one possible approach involves designing digital signatures or encryption systems that rely on solvable Lie algebras to generate secure keys and verify messages.

Given their specific commutator series, solvable Lie algebras allow for the construction of systems with controlled commutative behavior. This feature helps defend against attacks that exploit simple linear or commutative properties. Such characteristics support the development of cryptographic techniques that resist common analytical methods involving commutators and derivations [13,14,15,16].

4. Solvability of Lie Algebras in Steganography

The solvability of Lie algebras can contribute to enhancing the security of hidden messages through non-obvious algebraic operations that can be utilized for encoding and decoding data. This approach may prove useful in applications requiring high levels of confidentiality, such as military communication, digital protection of intellectual property, or communication in high-risk environments. Solvability is a mathematical property of Lie algebras based on solving logarithmic problems within their structure. These properties can be employed to perform seamless algebraic manipulations on concealed data that are difficult to detect or reconstruct without access to specific parameters, such as a private key. A message can be encoded into elements of Lie algebra using operations such as commutation, exponentiation, or additive manipulations, exploiting the solvable structure. A steganographic protocol based on solvable Lie algebras can resemble a key exchange protocol; however, instead of exchanging secret keys, this method embeds messages within the structure of the Lie algebra (see

Table 2. A steganographic protocol based on solvable Lie algebras.

Creation of a Hidden Key	Alice and Bob agree upon a common Lie algebra and its solvable structure, along with the parameters for commutator relations. Alice selects a secret element within the Lie algebra, denoted by $x_A$, which will be used for message encoding.
Message Encoding	Alice uses her secret element $x_A$ to encode the message. For instance, the message can be transformed into a vector or a matrix, which is then modified by applying commutator operations and exponentiation, leveraging the solvability of the Lie algebra. The message may be represented as an element M in the Lie algebra, embedded into the algebraic structure such that the public element $y_A=e^{x_A}$ conceals the original content.
Concealed Message in the Output Value	The result of the encoding process is an element y in the Lie algebra, representing the encrypted message. Since the message is encoded through internal operations to the Lie algebra, further transformations may be performed to ensure that the concealed message is not easily detectable, even when the output data are observed.
Message Decoding	Bob, who possesses the private key $x_B$, can use his secret to reconstruct the original message from the element y. The decoding process involves solving a logarithmic problem within the Lie algebra, which is computationally hard without knowledge of the private key.

).

We utilize a solvable Lie algebra to hide a message within 3D graphical objects. The goal is to develop a system that enables the secure embedding of textual messages into 3D models (e.g., digital object images), such that the presence of the message is not evident, even under analysis of the graphical object. This system may employ a solvable Lie algebra whose elements represent groups of operations (e.g., rotation, translation, or scaling) over 3D objects. Algebraic processes used to manipulate object geometry are imperceptible to the human eye but allow for message decoding if the specific key for each operation is known. For example, consider the Lie algebra $so\left(3\right)$, which describes rotations in three dimensions. The algebra $so\left(3\right)$ is characterized by the non-commutativity of rotations, which means these operations can be used to manipulate the positions of points on a 3D object, embedding information within the rotational transformations [17]. Bob, who possesses the private key, utilizes it to reverse the transformations and recover the original message. If Bob knows which rotational parameters correspond to which ASCII values, he can accurately reconstruct the hidden message. The receiver (Bob), who knows the embedding method (including the value of α\alphaα and the axis used), extracts the rotation matrices from the 3D object’s transformation history or through a controlled sampling of geometric points. Using inverse trigonometric functions, Bob recovers the angles ${\theta }_i$ from each matrix. For instance, from a z-axis rotation matrix:

$${\theta }_i=arccos \left(R_{11}\right)=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n} \left(R_{21}\right)$$

Bob calculates:

$$c_i={\theta }_{i}i/\alpha$$

which rounds the result off to the nearest integer to obtain the ASCII code. The codes are then mapped back to characters to reconstruct the original message.

This extended example offers a more precise description of the process, demonstrating how transformations are encoded and subsequently reversed into a concrete textual format. By leveraging the structure of Lie groups and their associated algebras, it becomes possible to implement secure and mathematically robust steganographic schemes that integrate well with complex media, such as 3D graphics or geometric models. This approach not only enhances the confidentiality of the embedded information but also makes unauthorized detection or decoding highly challenging due to the algebraic complexity and the use of computationally hard inverse problems.

4.1. Secure Message Embedding Using Lie Algebra $s{l}_2\left(R\right)$ and Frame Deformation in Animated Objects

Beyond static 3D geometry, Lie algebras can also be applied in the temporal domain, especially for animated or dynamic geometric objects. In this example, we utilized the Lie algebra $s{l}_2\left(R\right)$ to encode and embed messages via controlled frame deformations over time. The algebra $s{l}_2\left(R\right)$ consists of all $2\times 2$ real matrices with trace zero:

$${sl}_2\left(R\right)=\left\{A\in M_{2\times 2}\left.\left(R\right)\right|Tr\left(A\right)=0\right\}$$

A typical basis of $s{l}_2\left(R\right)$ is:

$$H=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right], E=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right], F=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right].$$

The exponential map $exp:s{l}_2\left(R\right)\to S{L}_2\left(R\right)$ allows us to move from the algebra to group elements (i.e., linear transformations acting on vector spaces, animation frames, or key points).

Encoding Process:

Take a message “YES”, and encode it to ASCII:

• Y → 89
• E → 69
• S→ 83

Each code $c_i$ is mapped to a matrix $A_i\in s{l}_2\left(R\right)$ via:

$$A_i={\lambda }_{i}H+{\mu }_{i}E+{\nu }_{i}F\mathrm{where} {\lambda }_i+{\mu }_i+{\nu }_i=c_i$$

The parameters ${\lambda }_i,{\mu }_i,{\nu }_i$ are chosen deterministically or pseudorandomly (shared secret). The animated object (e.g., a rotating cube or a deformable mesh) consists of frames ${\mathbb{F}}_t,$ each corresponding to a transformation at time t. For the selected frames, the transformation matrix is adjusted:

$$T_t=exp \left(ϵA_i\right)$$

where ϵ is a small perturbation factor to ensure minimal visual difference, and $A_i\in s{l}_2\left(R\right)$ encodes the message. The transformation $T_t$ is applied to vertices, joints, or camera movement. Since these are part of a time-based animation, the changes can be imperceptible yet mathematically consistent.

Decoding Process:

• Frame Sampling:

The receiver samples the selected frames and recovers the transformation matrices $T_t.$

2.

Logarithmic Inversion:

For each matrix $T_t\in S{L}_2\left(R\right)$, compute:

$$A_i=log \left(T_t\right)$$

to recover the original element in $s{l}_2\left(R\right).$

3.

Parameter Extraction:

From $A_i={\lambda }_{i}H+{\mu }_{i}E+{\nu }_{i}F$, extract coefficients, then compute:

$${\lambda }_i+{\mu }_i+{\nu }_i=c_i$$

Rounding it off to the nearest integer and decoding it as ASCII.

Due to the solvable nature of the employed Lie algebra, recovering the encoded message without access to the rotational parameters (i.e., the secret key) becomes extremely difficult. Furthermore, the subtle alterations to the 3D object remain imperceptible to the human eye, making it unlikely that an attacker could detect the presence of hidden data. Even if one were to analyze the object for rotational anomalies, the complexity of solving logarithmic problems in Lie algebras poses an additional barrier to successful extraction without full knowledge of the secret parameters. The solvability of Lie algebras in steganographic schemes offers substantial potential for secure hidden communication. The utilization of algebraic structures enables the execution of complex operations that are mathematically protected, thereby increasing overall security and minimizing the risk of detection. While this methodology is not yet widely adopted, it holds promise for the development of new steganographic systems that meet high security standards. Traditional steganographic techniques, such as the least significant bit (LSB) method, rely on modifying the least significant bits of image or audio files. Although straightforward to implement, such methods are vulnerable to statistical analysis and spectral detection techniques. In contrast, the application of Lie algebras and Lie groups in steganography offers several distinct advantages:

• Encoding data through exotic, nonlinear transformations within Lie groups makes detection and reverse engineering significantly more difficult.
• The exponential and logarithmic functions associated with Lie algebras introduce additional layers of complexity that hinder analytical decryption.
• Lie groups are frequently utilized in cryptographic frameworks involving differential operators, enabling dynamic encoding of data through continuous transformations.

4.2. Exponential Map of a Lie Algebra and the Logarithmic Problem

One of the key tools in the application of Lie algebras to steganography is the exponential map:

$$exp:g\to G$$

which maps each element X from a Lie algebra g to an element of the corresponding Lie group G, defined by the series:

$$\exp \left(X\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{x^k}{k!}}$$

In the opposite direction, the logarithmic function enables the reconstruction of a Lie algebra element from a Lie group element:

$$\log \left(A\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{(-1{)}^{k+1}}{k}}(A-I{)}^k.$$

These properties allow for the design of cryptographic functions based on the computational hardness of the logarithmic problem in Lie groups, which significantly increases the difficulty of attacking hidden data. Suppose we wish to hide a textual message using the group of rotation matrices from $S{L}_2\left(R\right)$, each character can be represented by its ASCII code and embedded via a rotation matrix:

$$R\left(\theta \right)=\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]$$

The procedure involves the following steps:

• Encoding

  ○

  Each ASCII code $c_i$ is converted into an angle ${\theta }_i=c_i\cdot \alpha ,$ where α is a fixed scaling factor.

  ○

  The message is represented as a sequence of matrices $R\left({\theta }_i\right)$.

• Embedding into a Carrier Medium

  ○

  These matrices are used to modify an image or another medium through geometric transformations that are not easily detectable.

• Decoding

  ○

  The matrices are extracted and analyzed using the inverse of the exponential map:

  $${\theta }_i=arccos \left(R_{11}\right),$$

  ○

  The angle ${\theta }_i$ is then divided by α to recover the original ASCII code.

  ○

  Finally, the ASCII codes are mapped back to characters to reconstruct the original text.

This extended example offers a more precise description of the process, demonstrating how transformations are encoded and subsequently reversed into a concrete textual format. By leveraging the structure of Lie groups and their associated algebras, it becomes possible to implement secure and mathematically robust steganographic systems [18]. The exponential map from the Lie algebra $so\left(3\right)$ to the Lie group $SO\left(3\right)$ plays a crucial role in the theory of rotations, particularly when manipulating objects in three-dimensional space. One of the most important tools for calculating rotations is Rodrigues’ formula, which provides a simple way to determine the rotation matrix as a function of the angle θ and the vector u, which represents the axis of rotation. The formula can be written as follows:

$$exp \left(\theta \widehat{u}\right)=I+\sin \left(\theta \right)\widehat{u}+(1-\mathit{cos}\left(\theta \right)){\widehat{u}}^2$$

where $\widehat{u}\in so\left(3\right)$ is the skew-symmetric matrix corresponding to a unit vector $u\in {\mathbb{R}}^3$, representing the axis of rotation. The matrix $\widehat{u}$ for the vector $u=(u_x,u_y,u_z)$ has the form:

$$\widehat{u}=\left(\begin{array}{ccc}0& -u_z& u_y\\ u_z& 0& {-u}_x\\ -u_y& u_x& 0\end{array}\right)$$

Rodrigues’ formula connects the Lie algebra and the rotation group $SO\left(3\right),$ enabling easy translation between infinitesimal and finite rotations. By using this formula, it is possible to efficiently handle rotations in a computationally effective manner while maintaining the geometric clarity of the transformation. This connection between algebraic and geometric representations of rotations is of paramount importance for many practical applications in science and engineering, including the study of rigid body motion and rotations in three-dimensional space, as well as in the design and operation of robotic systems and virtual environments [19].

4.3. Message Encoding Using $SO\left(3\right)$

Input:

• $v \in {\mathbb{R}}^3$ (carrier signal, i.e., the 3D vector representing the original structure)
• m (the message, numerically encoded)

Output:

• v_rot (the rotated vector with the encoded message)

Steps:

• Map the message m to an angle θ:

  ○

  Convert the message into an angle by scaling it:

  $$\theta ={\displaystyle \frac{m\pi }{128}}$$

  where m is the numerical representation of the message, and π is a constant.

• Choose a unit axis for rotation:

  ○

  Select a unit vector $u=(u_1,u_2,u_3)\in {\mathbb{R}}^3$ representing the axis of rotation. This axis should be a unit vector, meaning $u_{12}+u_{22}+u_{32}=1$

• Construct the skew-symmetric matrix $\widehat{u}.$

  ○

  The skew-symmetric matrix $\widehat{u}$ corresponding to the vector u is given by

  $$\widehat{u}=\left[\begin{array}{ccc}0& -u_z& u_y\\ u_z& 0& {-u}_x\\ -u_y& u_x& 0\end{array}\right]$$

  This matrix represents the cross-product operator for the vector u.
• Compute the rotation matrix R:

  ○

  Using the exponential map of $SO\left(3\right),$ the rotation matrix R is given by

  $$R=exp (\theta \widehat{u})=I+sin (\theta )\widehat{u}+(1-cos (\theta )){\widehat{u}}^2$$

  where $exp \left(\theta \widehat{u}\right)$ denotes the matrix exponential, I is the identity matrix, and ${\widehat{u}}^2$ is the square of the skew-symmetric matrix $\widehat{u}$ [20].

• Rotate the vector:

  ○

  Apply the rotation to the input vector v to obtain the encoded message:

  $$v_{rot}=R\cdot v$$

  where $v_{rot}$ is the rotated vector, incorporating the hidden message.

• Output:

  ○

  The final encoded vector $v_{rot}$ is the output, representing the original structure with the encoded message embedded in its geometric transformation.

This method of encoding information using rotations in $SO\left(3\right)$ is a straightforward yet effective approach to embedding messages within 3D geometry, leveraging the rotational symmetries of space. Rodrigues’ formula and the exponential map provide a natural way to encode and decode the information smoothly, without visually distorting the underlying structure.

4.4. Evaluation of the Heisenberg Encoding Scheme: Quality and Capacity

To assess the performance of the Heisenberg-based encoding method, we consider several criteria:

• PSNR (Peak Signal-to-Noise Ratio): This is a standard measure used to evaluate the quality of images. By comparing the original image with the encoded one, we can determine how much distortion has occurred. A higher PSNR value generally indicates that the image quality is well preserved.
• Hausdorff Distance: In the case of 3D models, the Hausdorff distance is used to measure geometric distortion. This metric calculates the maximum distance between two sets, helping us understand how accurately the encoded model retains its original shape and structure.
• Payload Capacity: This refers to the amount of data that can be hidden within the encoded signal. It is quantified in bits per transformation or per vector, providing a measure of how much information can be securely embedded without introducing noticeable distortion or degradation in quality.
• Robustness Against Detection: It is crucial to test the robustness of the encoding method against potential detection methods. This can be performed through statistical analysis to identify any patterns or anomalies, and by applying geometric perturbations to check how resistant the encoded data are to modifications or attacks.

The subsequent analysis, presented in

Table 3. Embedding capacity, SSIM, execution time, and detection rate.

Sample	Image Type	Image Resolution	Message Length (in Characters)	Embedding Capacity (Bytes)	SSIM (Range 0–1)	ET (in Seconds)
315	JPEG	2560 × 1440	1178	88,473,600	0.82-Good	2.74
298	JPEG	1920 × 1080	1331	49,766,400	0.79-Good	2.03
245	JPEG	1280 × 720	1312	22,118,400	0.73-Good	1.77
290	JPEG	1024 × 768	1252	18,874,368	0.79-Good	1.69
250	JPEG	800 × 600	842	11,520,000	0.57-Acceptable	1.41
305	PNG	1920 × 1080	1256	49,766,400	0.81-Good	2.23
218	PNG	1280 × 720	1089	22,118,400	0.78-Good	1.14
247	PNG	1024 × 768	1087	18,874,368	0.64-Good	1.23
277	PNG	800 × 600	778	11,520,000	0.55-Acceptable	1.33
231	PNG	640 × 480	732	7,372,800	0.49-Poor	1.02
310	BMP	2560 × 1440	1478	88,473,600	0.97-Good	2.18
310	BMP	1920 × 1080	1479	49,766,400	0.77-Good	2.14
305	BMP	1280 × 720	858	22,118,400	0.71-Good	2.02
309	BMP	1024 × 768	847	18,874,368	0.72-Good	1.12
254	BMP	800 × 600	985	11,520,000	0.58-Acceptable	1.38

, was conducted using the criteria of Image Type (carrier of the secret data), Image Resolution, and Secret Message Length. The results were then evaluated in terms of embedding capacity (bytes), Structural Similarity Index (SSIM), execution time (ET), and detection rate (ER).

As highlighted in previous work [20], the Heisenberg algebra, with its nontrivial center, offers a natural mechanism for securely embedding data. The algebra’s structure allows for efficient data encoding while minimizing the risk of detection. For a complete evaluation of the Heisenberg encoding method, it is essential to consider how each of these factors—quality, payload capacity, and resistance to detection—contributes to the overall effectiveness of the scheme. Quantum computing poses a potential threat to many classical cryptographic methods, as quantum algorithms—such as Shor’s algorithm—allow for the efficient resolution of problems that are considered hard on classical computers, such as integer factorization and discrete logarithm problems [20,21]. As a result, there is a need to explore methods for improving the resistance of solvable Lie algebras to quantum attacks. The following outlines key aspects of enhancing the security of Lie algebra-based cryptographic schemes against quantum computers. While many cryptographic schemes are based on problems involving Lie algebras, these schemes can be easily compromised using quantum computers. The discrete logarithm problem in Lie groups is a fundamental issue in several cryptographic systems based on Lie algebras, and quantum computers can efficiently solve this problem using Shor’s algorithm [22]. Specifically, methods based on Lie algebras are not robust in terms of security when subject to quantum attacks, as quantum algorithms can quickly solve the discrete logarithm problem [23]. To increase the quantum resistance of solvable Lie algebras, several techniques can be applied:

• By introducing quantum-resistant operators that cannot be easily solved with quantum algorithms, the complexity of solving problems can be increased. These modified operations may involve changes to commutator relations or redefinition of basic algebraic structures [24].
• Combining classical Lie algebra-based methods with quantum-resistant schemes, such as lattice-based cryptography, can create more robust cryptographic systems [25]. This approach allows secure data protection even in a post-quantum era.
• Non-commutative algebraic structures often provide greater resistance to quantum attacks. Using solvable Lie algebras with non-commutative structures can make quantum attacks much more complex. Non-commutative problems enable the creation of cryptographic systems that are harder to decipher with quantum computers.
• Integrating hash functions that are secure in the context of quantum computing can further strengthen cryptographic systems based on Lie algebras. These hash functions can be used in key generation and data protection [26].
• Quantum computers tend to solve lower-dimensional problems more quickly. Increasing the dimensionality in solvable Lie algebras can enhance problem complexity, making it harder for quantum computers to extract useful information.

5. Nilpotency of Lie Algebras in Cryptography and Steganography

The concept of nilpotency in Lie algebras, as an algebraic property that can be utilized in cryptography, offers intriguing opportunities for the design of novel cryptographic systems. A nilpotent Lie algebra is defined as an algebra for which there exists a positive integer n such that any nested Lie bracket (commutator) of n elements vanishes. This structural property can be leveraged in the development of secure mechanisms. Nilpotency can prove beneficial in various aspects of cryptographic systems. The key features that make nilpotency particularly valuable for security purposes include the following:

• In systems based on Lie algebras, nilpotency can hinder adversaries from reconstructing the original data when commutator-based operations are employed. In cases where commutation involves multiple layers before annihilation, the reverse engineering process becomes computationally infeasible without knowledge of the correct key.
• Nilpotent Lie algebras can be used to develop new encryption methods that take advantage of the algebraic properties of commutators, offering enhanced security in data encryption while reducing vulnerability to cryptanalytic attacks.
• Nilpotent Lie algebras enable the construction of complex commutation patterns for encryption, resulting in high-level security. Repeated commutators can serve as the basis for generating key functions whose patterns are extremely difficult to identify without possessing the appropriate secret key.
• These algebras can also be utilized in digital signature algorithms. By constructing digital signatures based on operations in nilpotent Lie algebras, the resulting schemes can offer increased security, as attackers would need to resolve complex commutator equations to reconstruct valid signatures.

Suppose Alice and Bob wish to establish a shared secret key using a key exchange protocol based on a nilpotent Lie algebra g. This protocol relies on specific commutator operations derived from the structure of g, which is assumed to be nilpotent.

• Alice and Bob independently generate key pairs by applying a series of commutator operations within g. These nested operations yield a shared secret that can be used for encryption and decryption.
• Utilizing an algorithm defined over the nilpotent algebra, Alice and Bob exchange partial key data. An adversary attempting to intercept the key would be forced to analyze the intricate commutator relationships in the algebra, a task that becomes infeasible without full knowledge of the exact operations and parameters involved.
• Even with access to numerous observed commutators exchanged between Alice and Bob, an attacker would be unable to reconstruct the original secret keys without knowledge of the underlying structure and parameters of the nilpotent Lie algebra, thus ensuring robust cryptographic security.

Quantum cryptography employs principles from quantum mechanics to design systems that surpass classical approaches in security. However, it can also be enhanced by algebraic tools such as nilpotent Lie algebras. In this context, nilpotent algebras can be used in quantum key distribution (QKD) protocols, where security depends on the algebraic structure of operations executed by the communicating parties and the potential eavesdropper [27,28]. In such a protocol, Alice and Bob aim to establish a shared secret using quantum cryptography enhanced with operations from a nilpotent Lie algebra. The goal is to ensure that, even if a quantum adversary gains access to the transmitted quantum states, they cannot reconstruct the key due to the inherent complexity of the algebraic operations.

Protocol Steps:

• Alice and Bob generate quantum states (qubits) that correspond to specific elements of a nilpotent Lie algebra. Each qubit is associated with a Lie algebra element.
• A sequence of commutator operations typical of nilpotent Lie algebras is applied. Each operation corresponds to an interaction among quantum states. Due to nilpotency, after a certain depth of iterated commutators, the result converges to zero, providing a natural security layer.
• Quantum information (e.g., quantum states) is transmitted through quantum channels. Even if an attacker intercepts the data, the layered algebraic structure of the commutators prevents successful reverse engineering without exact knowledge of the algebra and its operations.
• Because the operations are grounded in nilpotent Lie algebra structures, an attacker intercepting the information cannot recover the original keys without knowledge of the precise algebraic parameters. The underlying security premise is that reconstructing the system’s key or structure from partial quantum data is highly complex—even in the quantum setting.

The nilpotency of Lie algebras offers significant potential in cryptography, particularly for the design of secure encryption schemes, digital signature algorithms, and key exchange protocols. The utilization of commutator-based operations within nilpotent structures provides a solid foundation for cryptographic methods that are inherently resistant to attack in the absence of full algebraic knowledge. Their application may lead to the development of highly secure systems where deducing unknown keys or commutator patterns is computationally intractable.

Visualizations of rotation matrices that encode the message “HELLO” using the exponential map and ASCII codes. Each matrix represents one character, converted into an angle through a scaling factor, and then into a rotation matrix from the group $SO\left(2\right)\subset S{L}_2\left(R\right).$ Nilpotent Lie algebras, as a specific class of algebraic structures with the property of limited commutation, can be used in steganography for hiding information through carefully structured mathematical operations [29,30]. Here, we will consider the conceptual and practical applications of nilpotency in Lie algebras within the context of steganographic systems. Nilpotent Lie algebras provide a foundation for constructing complex yet controlled mathematical transformations, which are essential for secure and concealed data encoding. Nilpotent Lie algebras enable the construction of transformation sequences that gradually become “silent” (i.e., harder to distinguish) as the iterative process approaches zero. This property can be exploited for encoding data in multidimensional signals, where the structures of hidden data are masked in the final iterations of the commutator chain (see

Table 4. Nilpotency of Lie algebras in cryptography and steganography.

Encoding the Message	○ The message (text, image, or any binary string) is mapped to elements of the nilpotent Lie algebra g, where each component of the message corresponds to a specific generator of the algebra. ○ For instance, the textual character “A” could be encoded in the generator $X\in g,$ while the character “B” could be encoded in $Y\in g.$
Hiding Operations	○ Using commutators $[X,Y], [X,[X,Y\left]\right], \dots ,$ the information is iteratively spread throughout the signal structure. ○ The final commutators, which approach zero due to nilpotency, serve to mask the key data, making it nearly imperceptible in the final signal.
Application to Signal	○ The encoded data are implemented in audio, video, or digital images through imperceptible changes in frequencies, colors, or pixels.
Decoding	○ With a key known to the recipient, the message is decoded by reversing the iterative process through algebra.

).

The Heisenberg Lie algebra plays a critical role in various fields, including steganography, cryptography, and signal processing. It is a prime example of a nilpotent Lie algebra, where the commutators of the algebra eventually vanish after a finite number of steps. This property makes it particularly useful in hiding data within multidimensional signals such as images and audio recordings. The Heisenberg algebra ${\eta }_n$ has generators $X_1,\dots ,X_n,Y_1,\dots ,Y_n,Z$ with relations:

$$\left[X_i,Y_j\right]={\delta }_{ij}Z, \left[X_i,Z\right]=0, [Y_i,Z]=0.$$

Here, the commutator $[X_i,Y_j]$ generates the central element Z, and the nilpotency of the algebra arises from the fact that higher-order commutators involving $X_i,Y_i$ and Z vanish quickly. This structure allows for iterative reductions, which are crucial for secure data encoding.

Encoding Process:

• Input Data: The message to be hidden is mapped to a series of values associated with the generators $X_i,Y_j.$
• Encoding Through Commutators: Using $\left[X_i,Y_j\right],$ information is encrypted in the central element Z, which can then be modulated onto a digital signal.
• Application to Signal: Hidden information is incorporated into image pixels or time intervals of an audio recording through the values of the central element.

Nilpotent Lie algebras, especially the Heisenberg algebra, offer significant advantages in steganographic applications. The commutator structure of these algebras provides a controlled environment in which data can be securely encoded into signals, while the nilpotency ensures that the operations are limited, easy to implement, and easily reversible for data extraction. In steganography, this controlled commutation makes it difficult for an attacker to detect the hidden information without knowledge of the algebraic structure used. This makes Heisenberg algebra a promising tool for creating robust and secure steganographic systems. The use of Heisenberg algebras in cryptographic and steganographic systems is well documented in the literature. For example, the works of [31,32,33] demonstrate various applications of the Heisenberg algebra in information security, highlighting its practical utility and theoretical foundations. These studies lay the groundwork for the current application of the Heisenberg algebra in steganographic systems, where its properties are leveraged to secure data encoding and to ensure robustness against attacks. The parameters used in our construction are derived from a finite-dimensional representation of the Heisenberg Lie algebra over the finite field ${\mathbb{F}}_{13}$. Specifically, we selected the canonical basis elements X, Y, and their Lie bracket $[X,Y]=H$ were

$$X=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),Y=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),H=[X,Y]=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

This choice satisfies the defining relation $[X,Y]=H,$ while maintaining matrix simplicity that enables efficient computation in cryptographic protocols. These choices are consistent with prior works on Lie algebra-based cryptography (see [34,35]) and the BCH formula approximation for matrix exponentiation in cryptographic settings ([36]). Steganography is used for covert communication, and the encoding and decoding processes often rely on transformations involving Lie algebras. Exponential and logarithmic maps are critical for these transformations. Below is an explanation of the encoding and decoding steps using exponential and logarithmic maps, which allow for secure and efficient data hiding. The exponential map enables the transformation of data from Lie algebra to Lie group, which allows for secure data encoding. The encoding process involves the following steps:

• Prepare the data: The initial data M is first represented as a vector in Lie algebra g.
• Map to the Lie group: The vector from the Lie algebra is transformed into a point in the Lie group G via the exponential map:

$$g=exp \left(x\right)$$

3.

Final Integration: Once the data are mapped to the Lie group, it can be used for data protection or further encryption.

The logarithmic map is a crucial tool for decoding an encoded message. It serves as the inverse of the exponential map used for encoding. This process is necessary for reversing the transformation applied during encoding. Below is a breakdown of how it works:

Start with the encoded message:

After the encoding process, we obtain a Lie group element $g\in G,$ which corresponds to the hidden message [37]. This element represents the encoded data.

Apply the logarithmic map:

To retrieve the original data, we apply the logarithmic map $log\left(g\right)$ to the group element g [38]:

$$x=log\left(g\right).$$

This map returns an element $x\in g,$ which is an element from the Lie algebra and corresponds to the original vector before encoding [39].

The result x is then interpreted as the original message. Additional processing might be necessary, such as scaling or mapping back to the original format of the data, depending on the encoding scheme used (see

Table 5. Sample to visualize the encoding and decoding process.

Encoding Process (Exponential Map)	$M\to Linear Transformation x\to exp \left(g\right)\to Embed Encoded Message$ In the encoding phase, the data M undergoes a linear transformation, after which the exponential map $exp\left(g\right)$ is applied to embed the encoded message.
Decoding Process	$Encoded Message\to Extractg\to log\left(x\right)\to Inverse Transformation M$ During decoding, the encoded message is first extracted to obtain the group element g, followed by the application of the logarithmic map $log\left(x\right).$

).

This step reverses the transformation, resulting in the recovery of the original data M. Several recent studies have explored advanced techniques for image encryption and privacy protection, including a high-quality reconstruction DCT compression approach combined with nonlinear dynamics and a 3D memristive cubic map using dual discrete memristors. The application of solvable Lie algebras in cryptography and steganography opens new opportunities for the development of security systems. However, these areas of research face numerous challenges. Solvable Lie algebras allow the development of advanced mathematical models that support complex security protocols, but they also raise questions about the efficiency of their implementations in real systems. In the context of cryptography, solvable Lie algebras enable secure information exchange, but they require high computational complexity, especially when used for generating public keys and verifying data. Although they offer robustness in terms of protection against attacks, their implementation can be demanding in terms of execution time and optimization. Moreover, while solvable Lie algebras are effective in protecting data from known attacks, further research is needed to determine their resilience against new types of threats, such as quantum computers, which pose a potential danger to many existing cryptographic methods.

Steganography uses solvable Lie algebras to hide information within multimedia data. Although these methods are highly effective in hiding data, there is a challenge in detecting their traces. Advanced analysis techniques, such as statistical analysis and the use of deep neural networks for recognizing hidden messages, can lead to the discovery of hidden information, making data protection even more critical. Additionally, the quality of multimedia data may be compromised using steganographic techniques, presenting a trade-off between security and visual quality. It is important to note that solvable Lie algebras are just one of many tools that can be used in security protocols and steganography. A combination of different mathematical approaches, such as group theory, coding theory, and quantum cryptography, could provide even more secure systems. Further research aimed at developing new methods that integrate solvable Lie algebras with other techniques, as well as optimizing existing protocols, is crucial for improving data security and protection in the digital world.

6. Conclusions

This study highlights the significant potential of solvable and nilpotent Lie algebras in advancing the fields of cryptography and steganography. By leveraging their unique algebraic structures, we demonstrated how these algebras can underpin secure key exchange protocols and enable robust message embedding techniques. The proposed cryptographic framework, based on the non-commutative properties of Lie brackets, not only resists classical attacks but also exhibits promising resilience against quantum threats. Furthermore, the integration of Lie algebraic methods with frame deformation in multimedia steganography introduces an innovative pathway for secure communication in dynamic environments. Overall, our findings underscore the relevance of Lie algebras as a foundation for developing next-generation security systems in both static and motion-based digital communication.

Future work may focus on several key directions. First, further exploration of different classes of Lie algebras, such as graded or filiform algebras, could uncover additional cryptographic primitives with enhanced security properties. Second, implementing the proposed protocols in real-world systems and evaluating their performance and efficiency in practical settings would provide valuable insights into their applicability. Third, combining Lie algebra-based methods with machine learning techniques could lead to adaptive steganographic systems capable of resisting evolving detection algorithms. Lastly, investigating the integration of Lie algebraic schemes into blockchain architectures may open new avenues for decentralized and quantum-resistant security solutions. Further research should focus on optimizing mathematical models, as well as exploring new applications in quantum cryptography and other advanced security technologies. Additionally, the development of hybrid approaches that combine different mathematical algebraic structures could lead to new, more secure methods for data protection and hidden communication. Ultimately, solvable Lie algebras represent a significant step forward in the field of mathematical security, but much more research and testing are needed to fully exploit their advantages in real-world applications.