Pseudorandom Number Generator (PRNG) Design Using Hyper ‐ Chaotic Modified Robust Logistic Map (HC ‐ MRLM)

: Robust chaotic systems, due to their inherent properties of mixing, ergodicity, and larger chaotic parameter space, constitute a perfect candidate for cryptography. This paper reports a novel method to generate random numbers using modified robust logistic map (MRLM). The non ‐ smooth probability distribution function of robust logistic map (RLM) trajectories gives an un ‐ even binary distribution in randomness test. To overcome this disadvantage in RLM, control of chaos (CoC) is proposed for smooth probability distribution function of RLM. For testing the proposed design, cryptographic random numbers generated by MRLM were vetted with National Institute of Standards and Technology statistical test suite (NIST 800 ‐ 22). The results showed that proposed MRLM generates cryptographically secure random numbers (CSPRNG).


Introduction
Random number sequences have immense impact on numerous applications, such as signal processing [1,2], stochastic simulations [3,4], spread spectrums [5,6], gaming [7][8][9], statistics [10], captcha [11,12], machine learning [13,14], and cryptography etc. The random number generators that pertain to excellent statistical properties are considered critical for robust cryptographic applications [15]. The random numbers are categorized into two types: (1) true random number generator (TRNG), and (2) pseudorandom number generator (PRNG). TRNGs are generally based on physical and natural phenomenon and are non-deterministic, such as quantum random process, photon noise, frequency jitter in the oscillator, thermal noise, human brain signals, and free-running oscillator [16][17][18]. The generated sequences can be passed through the sampling and post-processing techniques to enhance the randomness. TRNG properties reveal that the truly generated random sequences must be non-reproducible, unpredictable, and statistically unbiased [19]. On the other hand, PRNGs are based on mathematical functions stem from initial condition to generate deterministic sequences over a long period. These PRNGs possess good statistical properties, such as repeatability, reproducibility, and fast execution time. A special type of PRNG, desirable for cryptography, is cryptographically secure PRNG (CSPRNG). A CSPRNG is unpredictable and generate CSPRNG. The larger parameter space of robust chaotic maps can be beneficial to fulfill cryptographic mechanisms. In this paper, we used a robust chaotic map with positive Lyapunov exponent, however the distribution of this map is non-uniform. The non-uniform distribution of the opted map is the cause of the failure of NIST 800-22 tests that require equal probability of '0' and '1' in a binary sequence. Hence, the control of chaos is proposed which styles the probability of 0's and 1's equally likely. The uniform region of output is selected to produce CSPRNG by applying the scaling operation followed by modulo shifting operation. The threshold value is chosen carefully and considered critical in mapping the output value to either zero or one. The probability for the occurrence of each value of the set {0,1} remains equally likely. To validate the proposed PRNG design, we apply the NIST 800-22 test suite, which passes all the tests successfully. Hence, the proposed CSPRNG can be used effectively in different cryptographic applications including key generation, image encryption, and watermarking. This paper is organized into following sections. Section 2 discusses the robust logistic map's bifurcation diagram, Lyapunov exponent, and ergodic behavior. The proposed scheme of PRNG design using RLM and NIST 800-22 tests evaluation is presented in Section 3. In Section 4, the performance analysis of proposed CSPRNG is explained for correlation, key-space and key sensitivity analysis. Finally, the conclusion of this paper is provided in Section 5.

The Robust Chaotic Logistic Map
The chaotic logistic map is one-dimensional. One-dimensional maps are considered simple because they are based on single mathematical equation. The chaotic logistic map, due to its simple structure, is generally studied to generate a PRNG for cryptographic services [42][43][44][45]. Equation (1) defines the one-dimensional logistic map.

Ergodicity
Ergodicity is an essential and closely related to the mixing property of chaotic system. In an ergodic system, orbit of every initial point from its definition interval leads to cover the complete phase space after a large number of iterations. For example, if one picks arbitrary initial condition and another number in phase space then trajectory of eventually visit during the course of iterations sometime or the other. The orbits having nature of visiting complete phase are called ergodic. Hence when iterated, the ergodic orbits are mixed throughout the whole interval.
By iterating Equation (2), the orbits of different arbitrary chosen inputs provide ergodic behavior and cover the complete phase space (0,1) when γ ≥ 4 ( Figure 4). It has good mixing properties when γ ∈

Bifurcation Diagram
The bifurcation diagram shows the mapping of orbits. It gives an inherent behavior of the chaotic system with the change of control parameters. The logistic map and its variants demonstrate period doubling bifurcation as shown in Figure 1 and Figure 6. With the change in control parameter, chaotic obits switch their behavior to stable/unstable that results in period doubling bifurcation. All stable orbits mean they are covering the complete phase space, and hence the system is chaotic. The bifurcation diagram of RLM in Figure 6 shows chaotic behavior at γ ≥ 4 and this range keep growing till γ = 31.

Lyapunov Exponent
The Lyapunov exponent is a quantitative measure of chaos and orbital divergence, by which the chaotic orbit of the logistic map is verified. A positive value of Lyapunov exponent indicates chaotic behavior and orbital divergence. The Lyapunov exponents of RLM for γ = 0 to 16 is shown in Figure 7. The Lyapunov exponent for RLM is positive for γ ≥ 4.

Proposed Methodology
The objective of this study is to investigate RLM for CSPRNG. Initially, we show that NIST 800-22 test suite PRNG for RLM is unsecure. Therefore, in this study, the design of CSPRNG is presented. The NIST 800-22 test of PRNG using CSPRNG validates the effectiveness of proposed methodology. The next subsection will cover both methodologies in detail.

Parameter Selection
The initial conditions and control parameters are chosen arbitrarily for the maps. Small perturbation on initial conditions and control parameters gives us completely different trajectories that are used to measure the key sensitivity and key space analysis. Small perturbations on control value investigate the behavior of bifurcations of the maps as shown in Figure 6. The logistic map bifurcations diagram in Figure 1 show that all trajectories are stable and covers complete phase space when γ = 4. Based on the results in Figure 6, RLM is chaotic and covers the complete phase space, and all trajectories are stable for γ ∈ [4,31].

PRNG Using RLM
In this study, PRNs generated using RLM are verified using NIST test suit. The PRNGs are crucially important, because the achieved randomness directly affects the security of encrypted applications. PRNGs having uniform probability distribution of binary sequences are desirable in cryptography. The chaotic trajectories generated using RLM having infinite real number values in range ∈ 0,1 . However, Figure 8 validated the non-uniform distribution of RLM trajectories. Therefore, efficient mapping is required from real to binary domain that gives random bits stream of '0' and '1' with uniform probability distribution. One of the methods is to threshold the real domain to generate binary sequence. We choose a median value as a threshold Ʈ = 0.5. The performance of the proposed RNG is evaluated with NIST-800-22 statistical test suite [46], which includes 15 different tests. A bit stream of length 1 million (1M) is required for NIST-800-22 statistical tests. The flow graph of unsecure random bit generator (RBG) is shown in Figure 9, and the steps for generating a random bit stream using RLM PRNG are as follows: Step-1: Choose an arbitrary chosen initial condition ∈ (0,1) and control parameter γ ∈ [4,31] as an input to iterate RLM for generating output ∈ [0,1].
Step-3: For real to binary domain mapping, thresholding is applied to the floating-point numbers.
Threshold value of Ʈ = 0.5 is chosen.
Step-4: Each generated output of RLM is checked where it falls. The binary value of '1' is chosen if ≥ Ʈ or '0' otherwise.
Step-5: Stop iterating RLM once the bit stream of 1M is generated.
The generated bit stream using above mention steps is tested using NIST 800-22 test. Based on the results in Table 1, standard NIST test on RLM's PRNG is failed. This is because the non-uniform probability distribution nature of RLM trajectories. Thus, PRNGs using RLM are not suitable to be used in cryptography. The next subsection presents the proposed CSPRNG design using modified RLM.

Modified RLM (MRLM) Aided by CoC
PRNG to be used in any cryptographic application requires passing all the NIST tests. The Lyapunov exponent of RLM is positive but NIST test is failed. During mapping from real to decimal domain, maybe the inherent randomness of chaotic map is reduced. Therefore, in this study CoC is proposed where small perturbations are applied to the input parameters to generate RLM chaotic trajectories with uniform probability distribution function. For the CoC of RLM, the output region 0.1 0.6 is desirable where the output of RLM is uniform as shown in Figure 10. The flow graph of proposed CSPRNG using Modified RLM (MRLM) is give in Figure 11; Figure 12. The pseudocode of the proposed method is also presented in Algorithm 1. The detail of steps involved in generating CSPRNG using proposed MRLM is given below: Step-1: Choose an arbitrary initial condition ∈ 0,1 and control parameter ∈ 4,31 .
Step-4: Apply CoC to choose the output region within the range ∈ 0.1, 0.6 by carefully small perturbations are applied to the input to filter out the desired output sequence.
Step-7: Apply threshold to the resultant floating values such that each random value is mapped to '1' if Ʈ , else it is mapped to '0'. In this regard, choose a threshold value such that the probability distribution of 0's and 1's is almost equal. In our case, we choose Ʈ 0.5 for CoC to generate cryptographically secure pseudorandom numbers.
Step-8: Stop iterating the map when 1M bits stream is generated.
The generated bits stream is then tested using NIST suit. Based on the results in Table 2, the proposed MRLM passed all the NIST tests with chosen 31 and 0.78.

Performance Analysis
The proposed method is tested for the well-established performance parameters of correlation, key space and key sensitivity analysis. The details of these properties are explained in the next subsection.

Key Space Analysis
The key space of secret keys or seed values must be larger than 2100 to resist against the brute force attack [47]. The proposed CSPRNG using MRLM is based on the two parameters of initial condition 0,1 and control parameter of 3.9,31 . The precision of double floating-point is 10 −16 stated in the IEEE floating-point standard [48]. Therefore, can be any of 10 16 values. Similarly, can be any value in the range 31 3.9 10 2.71 10 . Therefore, the key-space of proposed PRNG is 2.71 10 10 2 . The given range is used to compare the key space of proposed method with recently proposed methods is given in Table 3. Based on the results, the proposed scheme has a larger key space compared to recently proposed schemes. The proposed method satisfies the key space requirement and resists the brute force attack. Behnia et al. [50] ∈ 0,1 ∈ 0,1 1 0 10 10 10 2

Correlation Analysis
The correlation coefficient is used to measure the dependence and statistical relationship among random variables. The correlation coefficient between the two sequences is given as: , , where and are two random variable sequences ∑ , ∑ .The MRLM is iterated 1000 times by ignoring initial transient. To measure correlation using (2) and (3), small perturbations of ∆ are applied to control parameter to generate MRLM trajectories. The chosen value of ∆ is required that generate uncorrelated MRLM trajectories.
The correlation coefficient falls in the range ∈ [−0.0875,0.0915] as shown in Figure 13. In other words, there is no correlation between the generated sequences.

Key Sensitivity Analysis
Essentially, a cryptographic algorithm is required to be highly sensitive to small change in the key [47]. In this study, to measure key sensitivity between the keys, arbitrary keys of length 128-bits are generated with the change of one bit at least significant bit positions. A method for initial condition mapping to generate CSPRNG is given in Figure 14. Key sensitivity analysis measures small change in keys can generate highly uncorrelated CSPRNGs. The steps involved for initial condition mapping is given below: Step-1: Choose an arbitrary 128-bit hexadecimal key Step-2: Split the key into sixteen bytes. To spread the effect of small change at least significant bit over complete key and CSPRNGs, sixteen bytes are XORed together to generate an 8-bit number.
Step-3: Convert the 8-bit binary number into decimal.
CSPRNGs are generated by iterating MRLM with varying mapped and arbitrary chosen fixed control parameter 31. Arbitrary chosen keys with Single bit change at least significant bit are given in Table 4, generate highly CSPRNGs of 1000 bits are given in. Based on results in Figure  15, CSPRNGs are apparently uncorrelated and highly sensitive to small change in the key. The correlation between the generated sequences falls in the range ∈ (−0.0245, 0.0138).

CoC on Chaotic Map
Essentially, the probability distribution function of any given chaotic map describes its behavior. Smooth probability distribution function is required in chaotic map to generate CSPRNG. Therefore, in Table 5 four different chaotic maps such as circle [51], iterative [52], tent [53], and singer [54] maps are chosen with non-smooth probability distribution function. NIST STS test is performed on these maps with and without applying proposed CoC. It is apparent in Table 4 that typical chaotic maps used for comparison pass all NIST STS tests using the proposed CoC.
In any given chaotic map, changing the control parameters beyond the given limit move the fix points to infinity hence chaotic behavior is vanished. Therefore, modifications are proposed in maps to keep the chaotic fix points stable and within the range ∈ 0, 1 [1,53]. Chaotic maps are based on mathematical equations. Modified chaotic maps for larger parameter space having variable internal region that is shrink and expand with the change in control parameters. The change in internal region is described mathematically based on chaotic map's equation aided scaling and modulo operations. In [53] author modified chaotic tent map (MCTM) that enlarge the parameter space. The internal region is mathematically derived and aided by modulo and scaling operations. In [1] chaotic logistic map's parameter space is expanded with derived internal region. The internal region required deriving carefully for chaotic behavior and entirely based on map's equation. To analyze internal region selection, two different use cases are presented. Case 1 is measuring the NIST STS of MCTM and RLM with their respective internal region equations. Case 2 is measuring NIST STS by swapping the internal region equations. Based on the results in Table 6, it is evident that the wrongly derived internal region entails a failed NIST STS test.

Conclusions
In this study, modifying RLM using the proposed CoC generates cryptographically secure pseudorandom numbers. The RLM has large chaotic parameter space and possesses a positive Lyapunov exponent, but it has a non-uniform probability distribution of trajectories that makes it undesirable for cryptography. Hence, CoC in RLM gives a uniform distribution of the output sequence. The CSPRNG proposed design is vetted using NIST 800-22 tests. The correlation, key-sensitivity and key-space statistical analysis show that large parameter space of RLM gives sufficiently large key length and key space to resist all known attacks. The proposed modified RLM (MRLM) has the potential to be used in various cryptographic applications including image, telemedicine, electronic payment, computation, text source, personal information, biometrics, and military among others.