Application of high performance one-dimensional chaotic map in key expansion algorithm

In this paper, we present a key expansion algorithm based on a high-performance one-dimensional chaotic map. Traditional one-dimensional chaotic maps exhibit several limitations, prompting us to construct a new map that overcomes these shortcomings. By analyzing the structural characteristics of classic ID chaotic maps, we propose a high-performance ID map that outperforms multidimensional maps introduced by numerous researchers in recent years. In block cryptosystems, the security of round keys is of utmost importance. To ensure the generation of secure round keys, a sufficiently robust key expansion algorithm is required. The security of round keys is assessed based on statistical independence and sensitivity to the initial key. Leveraging the properties of our constructed high-performance chaotic map, we introduce a chaotic key expansion algorithm. Our experimental results validate the robust security of our proposed key expansion algorithm, demonstrating its resilience against various attacks. The algorithm exhibits strong statistical independence and sensitivity to the initial key, further strengthening the security of the generated round keys.


I. INTRODUCTION
With the rapid advancement of information technology, the significance of information security has garnered increasing attention, consequently driving the progress of cryptography.Among various encryption techniques, block ciphers hold a crucial position.In 2000, the Advanced Encryption Standard (AES) emerged as a pivotal block cipher.The AES encryption algorithm consists of multiple encryption rounds, where each round involves XOR operations between round keys and encryption blocks [1].
Chaos is renowned for its sensitivity to initial values and the unpredictable nature of the sequences it generates [2,3].In recent years, chaotic maps have found extensive applications in the realm of encryption [4][5][6][7][8][9][10][11][12].However, traditional lowdimensional chaotic systems exhibit certain shortcomings in cryptographic applications, such as discontinuous chaotic intervals and predictable chaotic signals.To address these issues, researchers have proposed high-dimensional chaotic maps [13][14][15][16].Although higher dimensions result in more complex mapping forms, they also increase computational requirements.Consequently, we draw inspiration from these developments and aim to design a high-performance onedimensional chaotic map with a simple structure.
In a block cipher algorithm, apart from round key addition, all other steps do not utilize keys.This implies that an attacker could calculate the inverse without possessing the key, underscoring the pivotal role of the round key in ensuring the security of the block cipher.The round key is derived from a key expansion algorithm, thus emphasizing the significance of devising a secure key expansion algorithm.Upon analyzing the key expansion algorithm employed by AES, we observe that it undergoes a reversible serial transformation process.If the round key for any round is known, one can deduce the round key for other rounds or even the initial key.This substantially diminishes the security of block ciphers and exposes vulnerabilities to side-channel attacks and other forms of intrusion.Inspired by the successful applications of chaotic maps in various cryptographic domains, we propose leveraging chaotic maps to generate a more secure key expansion algorithm.
In this study, we introduce a high-performance 1D chaotic map tailored for our key expansion algorithm, drawing insights from the analysis of various classical 1D chaotic map structures.By examining the nonlinear dynamics inherent in our mapping, we showcase its superiority over alternative maps.Subsequently, we put forth a chaotic key expansion algorithm built upon this chaotic map, accompanied by a thorough security analysis.
The subsequent sections of this paper are organized as follows: Section II provides an analysis of several wellestablished classical 1D chaotic maps.In Section III, we present our high-performance 1D chaotic map, along with an examination of its Lyapunov Exponent and K-Entropy.Section IV introduces our chaotic key expansion algorithm, while addressing its security considerations.Finally, Section V concludes this paper, summarizing the key findings and contributions.

A. Logistic map
The Logistic map represents a classical 1D chaotic map, which can be mathematically expressed by Eq. ( 1) [17].

C. Sine map
In addition, we introduce another classical onedimensional map, known as the Sine map.The Sine map can be mathematically represented by Eq. ( 3) [19].( 1) sin( ( )) where [ 4,4] x − is the state variable and [0, 4] r  is the control parameter.The bifurcation diagram of the Sine map is depicted in Fig. 3. Upon examining the limitations of the previously introduced classical 1D chaotic maps, it becomes evident that they share common weaknesses, such as discontinuities in the chaotic interval.These maps may exhibit non-chaotic phenomena, including fixed points, at certain control parameter values.Consequently, the predictability of chaotic signals restricts their applicability in cryptography.Our objective is to overcome these limitations without increasing the dimensionality of the map.
We embark on improving the structure of the existing 1D map by leveraging the inherent characteristics of nonlinear components.Chaos, as a typical nonlinear phenomenon in iterative maps, necessitates the presence of nonlinear components that eliminate the superposition effect.The sine map, a classical nonlinear component, effectively constrains the range of state variables.Additionally, the tangent map displays exceptional sensitivity to changes in initial values within specific ranges, making it an ideal candidate for constructing chaotic maps.
By amalgamating these existing structures and incorporating both the sine map and the tangent map, we propose a high-performance 1D chaotic map, mathematically expressed by Eq. (4).

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( 1) sin(tan( ( ( ) ))) where is the state variable and 4 [0,3 10 ] r  is the control parameter.We have named this new chaotic map the 1D-sin-tan-quadratic chaotic map (1D-STQCM).The bifurcation diagram of the 1D-STQCM is presented in Fig. 4, demonstrating its ability to exhibit chaos across a significantly wider parameter range.In the subsequent analysis, we delve into the dynamical performance of the 1D-STQCM, evaluating its key characteristics and properties.

A. Lyapunov Exponent
The Lyapunov Exponent serves as a crucial index for describing the stability and chaotic properties of dynamical systems.It quantifies the rate at which adjacent orbits in phase space diverge, thus capturing the system's sensitive dependence.This measure is commonly employed to analyze nonlinear dynamical systems, particularly those exhibiting chaotic behavior [20].Chaotic systems display a high degree of sensitivity to initial conditions, where small perturbations can lead to significant deviations in system behavior.The Lyapunov Exponent effectively captures this sensitive dependence and provides insights into the stability and chaotic nature of the system.
Typically, the Lyapunov Exponent is expressed as a real number or a set of real numbers.Each index corresponds to a specific direction within the system, describing the rate of separation along that particular direction.A positive Lyapunov Exponent indicates exponential divergence between adjacent orbits, indicating chaotic behavior.Conversely, a negative Lyapunov Exponent suggests exponential convergence between adjacent orbits, indicating stability.A Lyapunov Exponent of zero signifies linear separation or convergence, indicating bounded behavior within the system.
It is worth noting that the Lyapunov Exponent is a statistical measure that is not sensitive to the specific evolution path of the system.It is typically obtained by calculating an average value and can be estimated using numerical simulation or mathematical analysis methods.In other words, a larger positive Lyapunov Exponent indicates a more pronounced chaotic performance.Reference [21] provides a method for computing the Lyapunov Exponent, expressed by Eq (5).
where LE denotes the Lyapunov Exponent and () i fx is the time series of length n generated by the chaotic system.The curve illustrating the Lyapunov Exponent of the 1D-STQCM in relation to the control parameter is presented in Fig. 5. Furthermore, a comparison between the maximum Lyapunov Exponent of the 1D-STQCM and other chaotic maps is presented in Table 1.Notably, despite being one-dimensional, the dynamic performance of the 1D-STQCM surpasses that of more recent three-dimensional maps, as evident from the table.[22] 15.2462 0.8862 EQM [23] 16.6540 1.4343 3D-ICQM [24] 17.1231 0.6893 3D-ECM [25] 16.9166 0.9238 1D-STQCM 17.9990 1.4440

B. K-Entropy
In discrete chaotic systems, K-Entropy serves as a vital concept for measuring the complexity and uncertainty inherent in the system.It stems from the notion of entropy, which is a fundamental concept in information theory used to quantify the uncertainty of random variables.In the context of discrete random variables, entropy describes the average amount of information present.In discrete chaotic systems, K-Entropy extends the concept of entropy to discrete variable sequences.It characterizes the rate at which information grows during the dynamic evolution of discrete chaotic systems.
The calculation method for K-Entropy involves dividing the state sequence of the system into different subsequences of length K .Subsequently, the entropy of each subsequence is computed, and the average of these entropies is determined.This average reflects the overall rate of information growth within the system.The value of K-Entropy is typically directly linked to the complexity and chaotic nature of the system.In a simple periodic system, the K-Entropy value may be low as the entropy of the sequence tends to remain stable.However, in a chaotic system, characterized by high sensitivity and uncertainty, the K-Entropy value tends to be higher due to the rapid increase in sequence entropy.
Reference [26] provides a method for calculating K-Entropy, as expressed in Eq. ( 6) 1 2 1 2 0 0 , ,..., 1 lim lim lim ( , ,..., ) ln( ( , ,..., )) where n is the embedding dimension,  denotes the time delay, p is the joint probability.The K-Entropy of the 1D-STQCM is illustrated in Fig. 6, revealing that with the appropriate selection of control parameters, our map exhibits a significantly high K-Entropy.This characteristic renders the generated chaotic sequence highly unpredictable.The superiority of the 1D-STQCM in terms of K-Entropy is also demonstrated in Table 1, further highlighting its advantages over other chaotic maps.

A. Proposed key expansion algorithm
Our proposed key expansion algorithm targets the AES encryption algorithm structure with a key length of 128 bits, which encrypts the block for 10 rounds, so our key expansion algorithm will produce 10 round keys with a length of 128 bits.For other structures of block ciphers, the corresponding key expansion algorithm can be obtained by making simple changes.Before presenting the algorithm, we first make some notational conventions.We agree that a word is 4 bytes, that is, 32 bits, and use the array  Given an initial key, the result of a key expansion is shown in Table 2. Then we analyze the security of the proposed key expansion algorithm.

B. Independence of the round key
To verify the independence of the round key, we need to calculate the number of bit change rate (NBCR).The ideal value of NBCR is 50%, meaning that the round key is independent [27].To compute NBCR we first compute the Hamming distance, since NBCR is equal to the Hamming distance of two sequences divided by the bit length of the sequence.The Hamming distance between two sequences is defined as different bits in binary.We generated 10,000 round keys from an initial key according to our algorithm, counted the Hamming distance between these round keys and the 88 initial key, and drew the histogram as shown in Fig. 7.The Hamming distance divided by the key length, that is, divided by 128 bits, yields the NBCR.Naturally we can plot the NBCR distribution of 10000 round keys as shown in Fig. 8.It can be seen that the NBCR of round keys is close to the ideal value of 50%, which indicates that round keys are independent.

C. Strict avalanche criterion
After testing the independence of the round key, we also need to test the sensitivity of the key expansion algorithm to the initial key, which manifests as a strict avalanche effect.A strict avalanche effect means that any bit of the initial key is reversed, with a 50% probability for every bit of the round key [28].We reverse the 1 bit of the initial key in Table 2, apply the key expansion algorithm to obtain 10 round keys, and for each round, calculate the Hamming distance between the corresponding round keys, and the results are shown in Table 3.It can be seen that the Hamming distance between each pair of corresponding round keys is about half of the key length, indicating that our key expansion algorithm satisfies the strict avalanche effect, thus proving the sensitivity to the initial key.

D. Security analysis
In The combination of confusion and diffusion is considered a fundamental element in achieving the security of cryptographic algorithms.The effects of confusion and diffusion work together to reinforce each other, making the security of cryptographic algorithms more robust.In key expansion algorithms, diffusion can be understood as the idea that small changes in the initial key can be spread out and mixed in the round key by distributing each bit of information in the initial key to as many positions as possible.Based on previous experiments, our key expansion algorithm is sensitive to the initial key and satisfies the diffusion effect.Confusion, on the other hand, can be understood as creating a highly complex and unpredictable relationship between the initial key and the round key, by making the relationship between the initial key and the round key confusing and complicated.To implement the confusion effect, we use highly unpredictable chaotic mapping in our key extension algorithm.
A side-channel attack [29] is a method employed in cryptanalysis that utilizes the physical information leakage arising from the execution of encryption operations by an encryption device, rather than directly attempting to crack the encryption algorithm, in order to obtain sensitive information.The fundamental concept behind a side-channel attack is that the internal state of the encryption device exhibits various physical characteristics, such as changes in power consumption and electromagnetic radiation, during the execution of encryption operations.These physical characteristics are correlated with the device's internal operation process and data, which can be monitored and recorded by specialized devices or sensors.By collecting a significant amount of side-channel data, an attacker can employ statistical analysis, pattern recognition, and other techniques to infer the round key utilized by the encryption algorithm.
Based on our previous analysis, the keys generated by our key expansion algorithm are independent, and even if an attacker manages to obtain a round key, they cannot deduce the initial key.Therefore, our key expansion algorithm effectively withstands side-channel attacks.
Differential attack is another commonly used method in cryptanalysis.It aims to obtain key information, such as the key or plaintext, by analyzing the output differences of cryptographic algorithms when subjected to different input differences [30].The fundamental concept behind a differential attack is to select a pair of input plaintexts with a small difference between them and observe the resulting output difference during the algorithm's execution.By repeating this process multiple times, collecting many differential pairs, and performing counting and analysis, an attacker can infer certain bits of the key or the internal state of the algorithm.
The key expansion algorithm satisfies the strict avalanche effect, and the change of initial key does not cause the characteristic difference of round key, which can effectively resist differential attacks.

V. CONCLUSION
In this study, we have proposed a high-performance 1D chaotic map, named 1D-STQCM.The Lyapunov Exponent and K-Entropy tests conducted on 1D-STQCM have demonstrated its robust performance.Furthermore, our key expansion algorithm generates round keys that exhibit independence from the initial key and sensitivity to changes in the initial key.These characteristics address the limitations found in many existing key expansion algorithms, the AES key expansion algorithm, and provide effective resistance against side-channel attacks and differential attacks.In the future, the application of 1D-STQCM can be extended to various domains, such as information encryption, random number generation, and the construction of strong S-boxes.The versatility and security properties of 1D-STQCM make it a promising tool for enhancing security measures in these areas.

[ 2
, 2] x − is the state variable and [0, 2] r  is the control parameter.The bifurcation diagram of the Quadratic map is depicted in Fig. 2.

Fig. 2
Fig. 2 Bifurcation diagram of the Quadratic map

Fig. 3
Fig. 3 Bifurcation diagram of the Sine map

IK
11 keys including the initial key and the 10 round keys of 44 words.Both initial and round keys are expressed in hexadecimal.Our proposed key expansion algorithm is denoted by Algorithm 1.

Fig. 7
Fig. 7 Distribution of Hamming distance between 10000 round keys and the initial key.

Table 1 A
comparison of the maximum K-Entropy and maximum Lyapunov Exponents of 1D-STQCM with other maps.

Table 2
An instance of key expansion using our algorithm.

Table 3
Hamming distance between corresponding round keys.