Introduction

The increasing use of multimedia transmission is apparent in various communication domains, such as naval operations, mobile picture messaging applications, biological information dissemination, imaging for medical use, scientific observations, and others1,2. The rapid advancement in computer and web technologies has enabled accelerated transmission across various networks. Therefore, it is crucial to establish a secure system for communication to achieve reliable distribution of multimedia content. Cryptography involves converting plain text into indistinguishable forms called cipher text3,4. It includes two main methods: a conventional approach based on mathematical concepts such as number theory and the concept of algebra and a nontraditional approach derived from the concepts of dynamical systems. Fractals offer notable advantages in encryption and decryption procedures, encompassing enhanced security, heightened key sensitivity, efficient data encryption, resilience against attacks, and flexibility5. The utilization of fractals contributes to the development of secure and efficient encryption and decryption algorithms. At the same time, the theory of dynamical systems serves as a conceptual framework for leveraging fractals within cryptographic systems. In 1976, Diffie–Hellman presented the first instance in the world of cryptography, a public key exchange protocol5,6. The innovative concept entailed computing a shared key utilizing prime numbers within a designated key size. Following that, a protocol exchange key that is dependent on the Mandelbrot and Julia sets was introduced by M. Alia and associates. They proceeded to conduct a comparative analysis using the Diffie–Hellman protocol. In this context, the encryption process utilizes intricate fractal geometry, specifically emphasizing the efficiency of its generation. Before the emergence of public key algorithms incorporating fractals, the author thoroughly examined different algorithms, including DES, RSA, and ECDH, and their respective applications in sharing keys, data encryption, and digital signatures7,8. Fractals, which are non-regular geometric shapes that display consistent levels of irregularity at all scales, were initially not entirely utilized in cryptographic systems.

An Endeavour was undertaken to encrypt a message using random numbers, and the Mandelbrot set was fractal. While the encryption process was successful, the decoding phase presented difficulties due to the requirement of a flawless decoder that effectively prevents the repetition of numbers. In addition to the Mandelbrot set, various other fractal functions have been utilized to develop secure cryptosystems. In this context, a cryptosystem was developed to use an asymmetric cryptography strategy to encrypt photos over the internet9. It utilizes the complex mathematical structure and fractals’ deterministic character to provide a novel public-key cryptosystem using the Iterated Function System (IFS). This paper presents a cryptosystem that utilizes the Noor iterated fractal function to generate a shared key. The fractal function operates within a feedback system, and including one- and two-stage looping functions increases the complexity of the key generation process9,10,11. The proposed cryptosystem establishes a safe communication space by establishing a common secret key between the two parties without any network exchange. In addition, the paper analyses the system’s performance in terms of key space and sensitivity. The primary contributions are as follows:

  • A novel cryptosystem is designed by integrating fractal functions and Noor iteration to achieve robust security and low latency encryption. The chaotic properties of fractals enhance key sensitivity, while Noor iteration provides speed.

  • The encryption and decryption algorithms are specified, leveraging the generated fractal keys and Noor iteration values for secure text transmission.

  • Comprehensive security analyses are conducted, including assessments of key space, sensitivity, differential cryptanalysis, and statistical properties.

  • Performance evaluations measure the encryption/decryption speeds, reliability, and resilience against attacks. Results demonstrate the efficiency gains of the approach.

The structure of this paper is as follows. “Background” Section provides the basic facts about fractals, the Mandelbrot set, the Julia set, and the Noor orbit. “Fractal Generation Process” Section discusses the generation process and the proposed algorithm’s flowchart. “Experimental Setup” Section presents the computational complexity of the proposed algorithm. “Result and Discussion” Section presents the results and discussion, followed by the conclusion in “Conclusions” Section.

Background

The encryption and decryption of data are accomplished with the use of a single key when a symmetric key technique is employed. This key serves both purposes of encrypting and decrypting the data. In most cases, the key must be sent across the communication network, which requires the negotiation of cryptosystem security12. A technique that involves the production of a secret key on the end of the recipient is conceivable and might help lessen the susceptibility to key interception. The connectivity of fractal functions, namely Julia’s set and Mandelbrot’s set, gains relevance in this context because it permits the development of a safe secret key on both ends, preventing unauthorized entities from compromising key secrecy13. The Mandelbrot set, where each point is associated with a corresponding Julia set see Fig. 1, the Noor iterative process also generates a related Julia set, referred to as the Noor-based Julia set. The Noor iteration incorporates multiple intermediate computations at each step, resulting in behavior that is highly complex, dynamic, and sensitive to initial conditions. This behavior is we used to generate a key, and because of this sensitivity, the generated keys also show highly sensitive behavior. for visual diagrams is see Fig. 2a,b and pseudocode shown in Detailed Algorithm.

Fig. 1
figure 1

Mandelbrot Set and related images of Julia Sets.

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Fig. 2
figure 2

Encryption (a) and decryption (b) mechanisms use public keys generated by the sender and the receiver.

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Fractals

Fractals exhibit self-similarity, meaning they have infinite levels of detail; no matter how much you zoom in, the pattern remains the same. This property, called scale invariance, allows fractals to maintain their structure at any level of magnification, making them ideal for modeling the complex shapes found in nature. Nature encompasses numerous phenomena that exhibit intricate and asymmetrical characteristics, rendering their representation through traditional geometric principles an insurmountable challenge. Mathematical chaos and fractals offer invaluable frameworks to explore and comprehend such intricate phenomena. A fractal can be defined as a collection that manifests self-similarity across various scales14,15,16. This is described as a “rough or fragmented geometric shape that can be divided into parts, each exhibiting, to a certain degree, a reduced-size replica of the entire structure.” Research on fractals constitutes a relatively recent field of study with broad applicability across scientific and technological domains9. The pioneering work of Mandelbrot has served as a catalyst and inspiration for advancements in this area. The Mandelbrot set and Julia set to have a connection that may be defined as The Julia set can be thought of as a big book with an endless number of pages; each page has a particular picture matching a certain Julia set13. On the other hand, the Mandelbrot set can be viewed as a single page inside this book, serving as a short directory that offers a complete overview of the full collection of Julia sets (see Fig. 1).

There are four main ways to produce fractals: random, escape-time, iterated function, and strange attractors fractals. These approaches are considered more advanced for studying fractals compared to the traditional Peano–Picard iterative method. This new iterative method is seen as superior and produces iterated function system fractals, strange attractors and escape-time fractals. Iteration methods help achieves the self-similarity seen in fractals, usually employing two different kinds of feedback devices: one-step and two-step. Both kinds operate through iterative processes6,14. One-step feedback devices make use of Iterations of Peano-Picard. Fractal theory was dependent on one-step feedback for a very long period. And Picard orbits were used to investigate fractal models prior to the development of two-step feedback circuits. The result of two-step feedback is calculated using two input numbers, producing a new number. Fibonacci numbers, for instance, are produced by a two-step feedback machine, leading to more advanced fractals8,17. This was a cutting-edge method for computing, visualizing, and analysing fractal structures. Many people believe that nature’s geometry is based on simple shapes like lines, circles, ellipses, polygons, spheres, and quadratic surfaces. For example, car tires are circular, the solar system orbits the sun in an elliptical path, and poles are cylindrical. But have we ever considered the shape of a mountain? Can we describe the structure of animals and plants? How can the network of veins that supply blood be described using traditional geometry? Many natural objects are so complex and irregular that classical geometry cannot model them effectively. To analyze these complex forms, fractals and mathematical chaos are useful tools14,18.

The current fascination with Julia sets and associated mathematics began in the 1920s alongside Gaston Julia. Born in 1893, Gaston Julia showed extraordinary talent in the all subjects from a young age but was particularly enthusiastic about maths. He graduated in 1914, nonetheless, he had to stop working when he was summoned to serve in World War I. On January 25, 1915, he was severely injured in battle, leaving him with a facial injury that required him to live the remainder of his life wearing a mask8,14. Despite this setback, Julia quickly resumed his studies after the war. In 1918, he gained significant recognition with their publication. He introduced the modern concept of the Julia set, bringing him instant renown within the field of mathematics.

A fascination with Julia sets grew rapidly throughout the ensuing 10 years, with several eminent mathematicians researching them. Despite the lack of computing power, the first person to approximate a picture of a Julia set was Harald Cramer. But the challenge of producing intricate pictures without computers led to a decline in interest. Julia sets gained attention again in the 1970s when Benoit Mandelbrot began studying iterations. With better computing technology available by then, Pictures with even greater detail might be created9,18. Julia sets continue to be an active area of research, focusing on their intricate structure and calculating their "fractal dimension." Recent studies have also explored the relationship in hyperbolic geometry between Julia sets and Kleinian group limit sets.

The mathematician and philosopher Leibniz studied recursive self-similarity in the seventeenth century, which is when the mathematics behind fractals first emerged. Karl Weierstrass presented a function with a graph in 1872 that is today known as fractal, It is never differentiable but is continuous everywhere. Helge von Koch, dissatisfied In 1904 with Weierstrass’s abstract explanation, offering a more geometric representation known today as the Koch curve. In 1915, Waclaw Sierpinski developed his triangle and carpet. In 1918, Bertrand Russell acknowledged a “supreme beauty” in the evolving field of fractal mathematics9,14. In 1938, Paul Pierre Levy extended the concept of self-similar curves with his paper on curves and surfaces that are similar to their whole form, describing what is now referred to as the Levy C curve. Additionally, Georg Cantor proposed real line subsets with distinctive characteristics, known today as Cantor sets, which are also considered fractals. Henri Poincare, Felix Klein, Pierre Fatou, and Gaston Julia explored iterated functions in the complex plane in the late 19th and early twentieth centuries. In the 1960s, Benoit Mandelbrot explored self-similarity in papers such as "How Long Is the Coast of Britain?" and "Statistical Self-Similarity and Fractional Dimension," building on the earlier work of Lewis Fry Richardson. The term “fractal” was finally used by Mandelbrot in 1975 to characterize an object whose topological dimension is less than its Hausdorff–Besicovitch dimension6,14.

Fractals dimensions and chaos

There are many ways to define dimension that result in non-integer or fractal dimensions. These types of dimensions are particularly useful for describing fractal objects. Here, we will focus on the similarity dimension, which is used to construct regular fractal objects. The concept of dimension is closely related to scaling. While objects in classical Euclidean geometry are those that have whole-number dimensions, Fractal geometry is the study of objects that have dimensions that are not integers9,14,18. Euclidean geometry describes shapes like lines, ellipses, and circles. Fractal geometry, on the other hand, is based on algorithms sets of instructions for creating fractals. The world around us is made up of objects with whole-number dimensions: points (dimension zero), lines and arcs (one dimension), planar forms, such as squares and circles (two dimensions), and solid items such as cubes and spheres (three dimensions)14. But a lot of things in nature are better explained using dimensions that fall between whole digits. For instance, a straight line’s measurement is exactly one, a curve with fractals has a measurement ranging from one to two, based on how much area it fills as It turns and twists. A fractal is closer when it fills a plane more. Its dimension is to two. Similarly, a wavy fractal surface has a measurement ranging from two to three. For instance, a landscape with fractals with small bumps would have a dimension nearer two, but a landscape with uneven surfaces and Numerous moderately large hills would have a dimension nearer three6,9.

A century ago, if someone had discussed the idea of chaotic dynamics, he would never take serious for talking gibberish. Because that time the belief that each and everything be predicted exactly without any mistake because according to classical mechanic the unpredictability could only originate from outside influences. In 1963 a researcher Lorenz, which simplified the complex equation Navier Stokes, by the use of that time advanced device computer into three differential equations called by the name of its inventor Lorenz equations he studied the behavior of that equation and found that they are very different and shows completely chaotic behavior. After some years ago a group of mathematician restudies the Lorenz research and discovered that deterministic, or totally predictable, systems are capable of exhibiting chaotic behavior called “deterministic chaos.” Mathematical chaos and fractals are similar to instruments for understanding several equations pertaining to dynamics, which is connected to phenomena. Furthermore, fractal refers to a certain static geometric arrangement6.

Chaotic systems can also be divided into two types: conservative and dissipative. Conservative systems include those found in astronomical systems, also called “Physicist’s chaos,” where the state of space density remains constant but gradually changes form. Engineer’s chaos is the focus, in these systems, with time, the phase space density decreases, and they have “Strange Attractors.” Moreover, dissipative systems are separated into discrete and continuous. Continuous systems are, like analogue type systems which work over the time, while discrete we also called the digital systems are operate at definite time gaps14,17. There are two types of nonlinear chaotic systems: autonomous and non-autonomous. An external input signal is present in the non-autonomous systems that means they are depend on time. These systems often have a periodic forcing signal on the left side of their equations. Autonomous systems, on the other hand, do not have an external input and do not depend on time. The Lorenz equation serves as an illustration of an independent system.

Although there are numerous uses for chaos theory, we’ll focus on one here: information protection is a measure issue. We speak to this as cryptography. It’s a means of protecting confidential information. Information is mixed up with some unrelated data in cryptography by using intricate arithmetic, including chaotic systems. This makes it tough for unauthorized peoples. The protected information can only be unlocked and viewed by authorized peoples who possess the appropriate key. Consequently, although chaos seems random, it really helps to safeguard our digital secrets. This is just one instance of how, with the right application, something that appears unkempt may be quite helpful. An ever-changing chaotic system is described by equations that include variables and constants that show the system’s “state” over time in phase space9,19. The system’s path, or orbit, shows that the changes as time goes by, like the speed of a planet, with quantifiable conditions like speed and increase of rate. In this case, the emphasis is on the voltage and current variations affect electrical circuits, starting from a first condition and creating disordered signals. A mathematical concept is used to describe the system’s states in some case two or in some case three dimensions is called the phase space. Even though there is considerable disagreement over who originated the concept. Boltzmann or Liouville and Gibbs (1902) were the ones who initially present it. The Irish mathematician Sir William Edwin Hamilton, and Carl Jacobi contributed to the development of this concept.

Chaos theory is an attractive field in which we study the systems that are very sensitive to their first state. Small adjustments made in the beginning might have unexpected effects. The chaos theory helps to explain the irregular behaviors of nature, such as the weather, mountains, and organisms like the DNA and brain. Though it’s not, chaos appears random and unpredictable. In reality, it abides by a few stringent guidelines derived from mathematical formulas. The researchers may examine and comprehend these equations14. Therefore, despite their chaotic appearance, chaotic systems actually have an underlying order. Researchers can use mathematical methods to examine this underlying structure. Computer graphics are an important research tool for chaos.

Applications of fractals

Fractals have various practical applications across different fields as follows.

  • Smart cities contain fractals Fractals, which are patterns that repeat at different scales, have many real-world uses in different areas. One interesting example is how cities grow. When cities get bigger, they often do so in a way that looks like fractals. This means that as a city expands, it creates patterns that look similar whether you’re looking at a small neighborhood. As cities grow, they often take in nearby smaller towns and villages. This process can create a city layout that looks kind of messy or random at first glance. But actually, these “fractal cities” can work really well14. These naturally growing city patterns can form networks of roads, buildings, and neighborhoods that work efficiently. Sometimes, they might even work better than cities that were carefully planned out from the start.

  • Fractals are in medicines Fractals, which are repeating patterns found in nature, are useful in medicine, especially for finding diseases like cancer. This use of fractals in medicine is called "fractal medicine." In a healthy body, blood vessels grow in patterns that look like fractals. They branch out in similar ways whether you’re looking at small or large areas. This is normal and helps our bodies work well. But when cancer grows, it doesn’t follow these normal fractal patterns14,20. Cancer cells grow in unusual ways that don’t match the usual fractal patterns of healthy tissue. Because of this difference, doctors can use their knowledge of fractals to spot cancer more easily. They can look for areas where the usual fractal patterns are broken or different. This method helps doctors find health problems more accurately. It’s a clever way of using math patterns to improve medical care

  • Fractals are technologies Fractals, which are patterns that repeat at different sizes, are very useful in technology, especially for working with images and making antennas. When working with the images, fractals help save space by the use of compression techniques. They enable us to resize images to smaller files without sacrificing quality. This is called fractal image coding (FIC). It’s also excellent for creating 3D models with less data. Additionally, fractals are employed to create various antenna designs that can catch higher frequencies7. Fractals’ recurring patterns like Hilbert curve complement radio waves perfectly. This enables engineers to create compact, efficient antennas that can receive more signals14.

  • Technology in fractals Fractals have a big power on the area of art. Artists are inspired by the beautiful self-similar patterns, which are both simple and complicated at the same time. Fractals are remarkable because they feature patterns that repeat again and over, growing smaller each time in depth eye-catching results. The spectators are fascinated by the intriguing forms and designs created by this9. This concept intrigues artists, who incorporate it into their creations. Artists employ fractals in a variety of ways. They could use computer tools to produce digital fractal art, paint patterns resembling fractals, or even sculpt 3D objects inspired by fractal shapes, such as cups and T-sarts, which can be both enigmatic and beautiful.

The mandelbrot set definition

The collection of complex numbers is called the Mandelbrot set see Fig. 1 and is defined as the set of all complex numbers denoted by for which the orbit of the point 0 inside the repetition of a particularly complex function f(z) = z^2 + c remains bounded14. Formally, the definition of the Mandelbrot set M is as follows:

$${\text{M}} = \left\{ {{\text{c}} \in {\text{C}}:\left\{ {f^{ \wedge } {\text{n(0)}}} \right\}\;{\text{is}}\;{\text{bounded}}\;{\text{as}}\;{\text{n}}\;{\text{approaches}}\;{\text{infinity}}} \right\}$$

where C is the complex plane, f^n(z) denotes the nth iterate of f(z), and 0 is the complex number 0 + 0i. The Mandelbrot set’s boundary is structured like a fractal and highly complex and intricate set10,14. Analyzing the Mandelbrot set and its properties has been a rich and exciting field of study in mathematics and resulting in many significant findings and insights in the field.

Julia Set

The Julia set see Fig. 1 is a group of intricate figures defined as the boundary of the group of all iterations of a given complex function f(z) under a given initial condition c. Formally, J(f), the Julia set associated with an intricate function f(z), and a complex number cis characterized as follows:

$${\text{J}}({\text{f}}) = \left\{ {{\text{z}} \in {\text{C}}:\left\{ {f^{ \wedge } {\text{n(z)}}} \right\}\;{\text{is}}\;{\text{bounded}}\;{\text{as}}\;{\text{n}}\;{\text{approaches}}\;{\text{infinity}}} \right\}$$

where C is the complex plane, f^n(z) denotes the nth iterate of f(z), where z is a complex number13,14. The set J(f)'s border is known as the Julia set, which has a fractal-like structure that is highly complex and has intricate details and patterns.

Noor iteration

The Noor iteration method is an advanced iterative scheme in fixed-point theory, which extends traditional methods such as Picard, Mann, and Ishikawa iterations10. Mathematically, it incorporates three recursive steps in its update process, allowing greater control over convergence behavior and dynamical complexity.

Definition

(Noor Orbit). Let’s examine a series \(\left\{ {x_{n} } \right\}\) of repeats for the starting point \(x_{0} \in X\) such that,

$$\left\{ \begin{gathered} x_{n + 1} :x_{n + 1} = \left( {1 - \alpha n} \right)x_{n} + \alpha_{n} Ty_{n} \hfill \\ y_{n} = \left( {1 - \beta_{n} } \right)x_{n} + \beta_{n} Tz_{n} \hfill \\ z_{z} = \left( {1 - \gamma } \right)x_{n} + \gamma_{n} Tx_{n} n = 0,1,2,3,... \hfill \\ \end{gathered} \right\}$$
(1)

where \(\alpha_{n} ,\beta_{n} ,\gamma_{n} \in \left[ {0,1} \right]\) and \(\left\{ {\alpha_{n} } \right\},\left\{ {\beta_{n} } \right\},\left\{ {\gamma_{n} } \right\}\) are the sequences convergent away from \(0\). Noor orbit refers to the iterative sequence seen above. It is denoted by \(NO\). This is determined by five tuples \(\left( {T,x_{0} ,\alpha_{n} ,\beta_{n} ,\gamma_{n} } \right)\). \({\alpha }_{n},{\beta }_{n},{\gamma }_{n}\) are control parameters that introduce nonlinearity, depth, and flexibility into the iterative path. This tri-level recursion structure contributes significantly to the dynamic behavior of the resulting system, especially when applied to generate fractals. the Noor iteration provides a mathematical advantage by generating highly sensitive and complex fractal structures, particularly in the construction of Noor-based Julia sets14. This sensitivity is beneficial for cryptographic key generation, as small changes in initial parameters produce vastly different outputs, increasing unpredictability and security.

Fractal generation process

The iterative formula is used in the fundamental process of creating fractals:\(z_{n + 1} \leftarrow f(z_{n} )\) where \(z_{0}\) is the starting point of \(z,\) and \(z_{i}\) value of the complicated number \(z\) at the \(i^{th}\) iteration. Mandelbrot’s self-squared function, for instance, generates fractals.\(f(z) = z^{2} + c\), both the complex values, z and c. We purpose the use of transformation function \(\overline{z}^{2} + c\) for producing fractal images in relation to the Noor orbit where the complex quantities are denoted by z and c.\(n\) are a real number14,19,21.

Encryption and decryption mechanism

Encryption and decryption are methods used to keep data safe when it is sent over a network. These processes ensure that the information remains secure during transmission. Cryptography. In different format of data type is supported such as audio, videos and other multimedia formats are encrypted before sending through the network. While in transit, the encrypted data appears random and meaningless. Ultimately, cryptography aims to protect information from unauthorized access through transmission, ensuring it reaches the recipient securely. Once received, decryption is used to convert the data back to its original form. For example, when someone sends a “secret” message, encryption is used. The original data, known as plain-text, is changed into arbitrary bits, referred to as cipher-text, with a key and a process. Depending on the key, the encryption algorithm may produce different results each time. The cipher-text is then sent over a communication channel. On the receiving end, the same key and process are used to change the cipher-text back into the original plain-text. In encryption, the two main terms are plain-text (the original message) and cipher-text (the encrypted form). The key and algorithm convert the plain-text into cipher-text, and the recipient uses the same key to restore it to its original form.

Cryptography

Cryptography is both an art and a science used to protect confidential and important information from hackers by using various techniques and algorithms for encryption and decryption, following specific rules and methods. The purpose of cryptography is to prevent unauthorized access to information. Data shared over a network can be stolen in different forms, such as text messages, images, audio files, and videos, which are combined during communication or stored on devices. The process of securing this information is called encryption, and the reverse process, where the original information is recovered, is called decryption.

Techniques used in cryptography fall into two categories: Secret Key Cryptography comes first, also known as Symmetric Key Cryptography, and Public Key Cryptography comes in second. also called Asymmetric Key Cryptography. The primary goal of cryptography is to provide solutions and security for users. Today, it plays a crucial role in ensuring the security of communication networks. Fractals such as Julia and Mandelbrot sets, especially when generated using advanced iterations like Noor iteration, possess chaotic behavior, sensitive dependence on initial conditions, and nonlinear dynamics. These characteristics are desirable in cryptography for the same reasons: Fractal-based systems inherently produce outputs with high entropy due to their complex geometrical structures and non-repeating patterns. Even a minute variation in the initial fractal parameters (e.g., real or imaginary parts of complex constants) leads to drastically different outputs15. The iterative nature of fractals, especially when enhanced with multi-level schemes like Noor iteration, enables the generation of a vast and dynamic key space. In contrast, fractal-based systems generate complex, aperiodic sequences that offer better resistance to linear cryptanalysis, differential cryptanalysis, and frequency analysis. By integrating fractals into the cryptographic process—particularly through composite functions and iterative schemes we enhance the non-linearity, confusion, and diffusion characteristics that are critical for robust encryption. This justifies the use of fractals not only as a mathematical novelty but as a core security feature of the proposed hybrid cryptosystem.

Symmetric algorithm

Using symmetric keys in encryption, for encryption and decryption, the same key is utilized. The main advantage of this method is that it requires less processing power and operates rapidly when encrypting data. Symmetric keys algorithm is divided into two categories: stream ciphers and block cyphers. In block ciphers, all the data is divided into chunks or blocks, and a key is used to encrypt each block.

Asymmetric algorithms

The asymmetric algorithms are not work like symmetric algorithms in the symmetric encryption, the encryption and decryption both processes share the same type of key. This method’s primary benefit is that encrypts information is rapidly and with minimal use of processing power use. Block ciphers and stream ciphers are the two modes available for this kind of encryption. Data is split up into sections or blocks in block ciphers, and each block is encrypted with a different key. Data is divided into bits, such as 1111100011, and then randomly assigned before encryption rules are applied in stream ciphers.

In contrast, asymmetric key encryption employs distinct keys for both encryption and decryption. There are two keys: one public and the other private. "Public key encryption" is another name for this technique. Secure communication is ensured, for instance, if we encrypt data using the public key and only the private key may decrypt it. In an alternative scenario, the system uses the private key to confirm the sender’s identity when we use it to lock or decrypt data. Public key encryption is important in particular because it allows safe data transfer between users who do not already share a private key.

Process flow of proposed algorithm

The security of sensitive data transmitted over networks is an increasingly critical priority as cyber threats evolve. Attackers rapidly develop advanced techniques to intercept confidential information and break encryption systems. However, traditional cryptographic approaches often lack the combination of strong security and real-time performance needed to protect against these emerging risks. The proposed encryption method is symmetric in nature. This is because both the sender and receiver independently generate the same private key using identical fractal parameters and the Noor iterative procedure. The security of the system relies on the synchronized generation of the dynamic key, rather than the exchange of keys over a communication channel. In symmetric cryptography, the same key is used for both encryption and decryption, and that is precisely the mechanism implemented in our system19. The Noor-based fractal generation algorithm ensures that the key remains difficult to reconstruct or predict without knowledge of the initial parameters and iterative function.

This research is motivated by the need for robust encryption schemes optimized for low-latency operations. By leveraging the mathematical properties of fractal functions and chaotic systems, there is potential to achieve greater key randomness and algorithmic complexity than conventional ciphers. Integrating fractal cryptography with fast iterative methods like Noor iteration can enhance speed without sacrificing security. The proposed method distinguishes itself from other fractal-based cryptosystems, including those based on Mandelbrot or Iterated Function Systems (IFS), through the integration of the Noor iteration scheme. Unlike classical fractal generation methods, the Noor iteration introduces additional control parameters and intermediate computation stages in each step, allowing for greater flexibility and sensitivity in the fractal behavior. This iterative refinement leads to the generation of complex, dynamic, and highly sensitive Noor-based Julia sets, which serve as the foundation for private key generation in our cryptosystem22. The increased complexity and unpredictability of these fractals offer enhanced resistance against brute-force and pattern-based cryptanalytic attacks, thus providing a significant security advantage. Furthermore, the Noor-based approach enables multi-layered key derivation and supports adaptive cryptographic constructs that are difficult to replicate using traditional Mandelbrot or IFS-based methods.

We aim to design an encryption scheme that can provide provably secure and efficient cryptography to safeguard data in motion17,23. With data breaches extracting sensitive information at scale, securing communications with robust and lightweight cryptosystems is an increasingly vital countermeasure. The techniques developed through this research seek to directly address these motivations and the growing need for resilient real-time encryption in the face of escalating cyber threats.

Encryption process

Encryption is a protective measure that ensures that even if someone not authorized can obtain the information that has been encrypted see Fig. 2a, all they will see is a string of incomprehensible alphabetic and numeric letters. Encrypting data requires the utilization of a public key and a function infrastructure algorithm to be effective. Formulating a complicated public key, referred to as “c,” will be the first step in the proposed technique. An associated Julia set see Fig. 3 that connects to a Mandelbrot set that is as specified by the function “f(n)” will use this complicated public key as a parameter6,20,24. A random Julia set that is linked and within the Mandelbrot set is chosen due to the interrelated nature of the Julia sets contained within the Mandelbrot set. It is necessary to make use of the iteration’s fixed point for the function “f(n),” which is represented by the letter “d,” to initialize the private key. After that, the message is encrypted utilizing the private key see Table 1, and it is then transferred in the appropriate manner8.

Fig. 3
figure 3

Julia set for \(m = 2\), \(\beta = 0.2,\,\,\alpha = 0.2\,\,\gamma = 1\), \(c = - 0.{1458 } + \, 0.{\text{1968i}}\).

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Table 1 Description of a key exchange Procedure.
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Decryption process

Decryption entails returning the data to its original format to make it easier for the person who is supposed to receive it to understand. This procedure reflects the early stages of the encryption and decryption procedures, described in detail in the encryption phase at the sender’s end. By carrying out these procedures, the same private position will be produced on the receiver’s end, fulfilling the requirement that the key be included inside the cipher text see Fig. 2b.

The encryption and decryption processes mentioned in an algorithm (see Fig. 2):

Encryption process

Step 1: Sender generates a public key Pk using the Julia set fractal function see Fig. 3.

Step 2: Initialize the private key Pk’ with the help of the Noor iteration function F(n).

Step 3: Input the plaintext message Z(msg).

Step 4: Blend Z(msg) with the private key Pk’ to obtain the encrypted message ZP(msg).

Step 5: Store the encrypted ZP(msg) in an array and transmit it to the recipient.

Decryption process

Step 1: Recipient obtains the sender’s public key Pk.

Step 2: Initialize the private key Pk’ using the same Noor iteration function F(n).

Step 3: Input the received encrypted message ZP(msg).

Step 4: Iterate ZP(msg) with the private key Pk’ to obtain ZP’(msg).

Step 5: Output the decrypted plaintext message ZP’(msg).

Detailed algorithm

  1. 1.

    alpha: Weighting factor for updating complex numbers during iteration

  2. 2.

    beta: Weighting factor for updating complex numbers during iteration

  3. 3.

    gamma: Weighting factor for updating complex numbers during iteration

  4. 4.

    power: Exponent used in the computation

  5. 5.

    n: Number of iterations

  6. 6.

    z: Complex number initialized to (0,0)

  7. 7.

    c: Complex number used in the computation

  8. 8.

    t: Array storing time steps

  9. 9.

    x: Array storing complex numbers

  10. 10.

     x0: Initial complex number

  11. 11.

    private_key_sender: Private key generated by the sender

  12. 12.

    M: Message input by the user

  13. 13.

    encrypted message: Encrypted message

  14. 14.

    decrypted message: Decrypted message

  15. 15.

    private_key_receiver: Private key generated by the receiver

  16. 16.

    decr: Temporary variable for decryption

  17. 17.

    decrypted_char: Decrypted character

  18. 18.

    encr: Temporary variable for encryption

Algorithm

figure a

Computational analysis of proposed algorithm

The proposed algorithm uses the Noor fractal function to secure communication between the channels through the keys. Our study shows the algorithm’s efficient characteristics, time complexity, and optimizations. The complexity of the time depends on two phases: the generation of a private key and the encryption and decryption of the given message. The private key iterates loop n times, where n is the number of iterations; in each iteration, the complex number is updated by using the function and shows the time complexity O(n^2) or O(n^3), which depends upon the computation. The time complexity of the generation of the key is approximately O(n^3) to the iterative nature of the algorithm. The linear time complexity of the message is O(m), and m represents the message length.

Some adjustment increases the optimization potential and performance of algorithms, such as:

  1. 1.

    Parameters alpha, beta, gamma, and power increase the algorithm’s efficiency.

  2. 2.

    By applying the efficient data structures to intermediate results storages and optimizing the computational overhead.

Experimental setup

The proposed encryption technique is implemented in Python 3.8.5 and evaluated on a system with a 2.60 GHz Intel Core i7 processor and 16 GB of RAM operating Windows 10. The cryptosystem is tested on varying-sized plaintext inputs, from 256 bytes to 10 MB. The key size is configured to 128-bit for the primary experiments. The encryption and decryption runtimes are measured using the Python time module.

Result and discussion

The encrypted message has real and imaginary components, displaying recurring and different patterns see Fig. 4. It also gives a visual picture of the encrypted message’s distribution and behavior within a complicated region. As data points, the three-dimensional graphic depicts the real and imaginary components of the encrypted values see Fig. 5. The x-axis represents the index and location of each value, while the y-axis represents the real component of the values and complex numbers. Throughout the encryption process, the z-axis depicts the imaginary component of the encrypted values and complex numbers. The exact input values determine the arrangement and distribution of these dots within the chart, resulting in distinct features and visual characteristics that simulate the algorithm’s features and visual aspects.

Fig. 4
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3-D chart of the encrypted message.

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Fig. 5
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Chart of the encrypted message.

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Encrypting introduces observable fluctuations and patterns reflecting the message’s transformational effects. These variations include distinct patterns, gradients, and tendencies illustrative of the behavior observed during encryption. The x-axis denotes the index and position of each value within the message. The y-axis, on the other hand, displays the values derived from the encryption procedure, which entails appending the private key to each message character. This illustration aims to shed light on the modifications and transformations that occur within the encrypted message.

Key space

The phrase “key space” refers to the complete collection of all possible configurations a given cryptographic key could take. It is crucial to have a significant-sized key space capable of holding off brute-force assaults to maintain strong data security25,26,27. The approach under consideration generates a 128-bit encryption key. As a result, the total number of possible keys reaches the incredible value of 2^128, which is so high that any realistic attempt at decryption is almost impossible18,28. This vast area of the key space is the foundation of a trustworthy cryptosystem, strengthening its robustness and effectiveness in maintaining data security. Table 2 illustrates how, as key size increases, the fractal key space grows, indicating that many keys are conceivable. Additionally, exponential growth indicates that the number of keys increases considerably for every tiny increase in key size, ensuring security confirmation. Table 3 shows the difficulty level of the fractal-based protocol compared to traditional encryption methods in terms of hardness for attackers to find the correct key. The results indicate that in fractal-based encryption, larger key sizes lead to stronger security and better protection of sensitive data.

Table 2 Fractal key space.
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Table 3 Key space comparison.
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Key sensitivity analysis

A change in the key’s value affects the output because of the key’s sensitivity. Fractals and chaotic systems are very sensitive. The butterfly effect concept shows how a small shift in a single variable may have far-reaching consequences. Our proposed technique uses a 128-bit key to encrypt sensitive information. The encrypted message would look very different if we changed the bit value from 0.7778–0.0035i to 0.7788–0.0040i, or it might not change at all14. As a result, we may infer that fractal keys are extremely sensitive. The sensitivity of fractal keys in encryption is contingent upon various aspects, including length, rotation, complexity, external sources, and encryption algorithm strength, as illustrated in Table 4.

Table 4 Key sensitivity analysis.
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Comparative analysis with the existing techniques

The fractal-based iterative algorithm presents a novel and efficient method for message encryption and decryption through keys that show chaotic and self-similar behavior, enhancing security. The security strength also stems from the algorithm’s linear time complexity concerning message length, making it advantageous for real-time communication applications over networks. Comparative studies between traditional cryptographic algorithms and this fractal-based approach reveal that fractal keys offer higher entropy and resistance against brute-force attacks than deterministic algorithms. The algorithm’s chaotic nature contributes to the unpredictability of the encryption process, granting it an advantage over deterministic methods. With its linear time complexity, the algorithm facilitates secure message encryption and decryption, making it suitable for small-scale and large-scale communication systems. Longer keys provide more choices, which make it harder for adversaries to figure out the correct key13. Complicated fractals produce a variety of patterns that are challenging to interpret the results. The proposed algorithm has less chance of unwanted access and better protection than the strong algorithms. Compared to simpler iterative schemes like those used in Mandelbrot or IFS-based methods, Noor iteration offers:

  • Faster divergence and richer fractal boundaries, useful in key complexity,

  • Greater parameter space for fine-tuning, enhancing control over the fractal dynamics,

  • Better chaotic behavior, which is essential for confusion and diffusion properties in encryption.

This makes Noor iteration not only mathematically significant but also practically advantageous in developing robust, secure, and adaptive cryptographic systems.

Conclusions

This study presents an encryption method based on the Noor iterated function. This approach provides an incredibly difficult procedure for creating the encryption key utilizing the iterated Noor set. A secure cryptosystem results from the fact that the process of producing keys only needed a small number of isolated factors that were kept hidden on the website. The fractal function is impenetrable due to its chaotic nature and enormous encryption key to any type of threat. The performance analysis shows that the proposed method of key generation delectated the slight changes in key, and the result gives a secure and reliable cryptosystem to meet today’s challenging environment. The traditional cryptography method is no longer sufficient. Therefore, an effective yet straightforward encryption and decryption technique is essential for any kind of data encryption. When using fractals in cryptography, a greater range of key values can be obtained than traditional techniques that rely on the prime values for a particular key size.