A Modernistic Approach for Chaotic Based Pseudo Random Number Generator Secured with Gene Dominance

Sådhanå (2021) 46:8 Ó Indian Academy of Sciences

https://doi.org/10.1007/s12046-020-01537-5Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

A modernistic approach for chaotic based pseudo random number generator secured with gene dominance

SATHYA KRISHNAMOORTHI1,* , PREMALATHA JAYAPAUL2 and VANI RAJASEKAR3

1 Department of Computer Technology-UG, Kongu Engineering College, Perundurai, India 2 Department of Information Technology, Kongu Engineering College, Perundurai, India 3 Department of Computer Science and Engineering, Kongu Engineering College, Perundurai, India e-mail: [email protected]; [email protected]; [email protected]

MS received 18 March 2020; revised 25 September 2020; accepted 17 November 2020

Abstract. Random numbers play a key role in diverse ﬁelds of cryptography, stochastic simulations, gaming, etc. Random numbers used in cryptography must satisfy additional properties of forward secrecy. Chaotic systems have been a potential source of random number generators. Both lower (One-dimension) and higher (two, three-dimension) chaotic systems are popularized in the generation of random bit sequences. Higher-order chaotic systems have a higher resistance to attacks owing to multiple dimensional outputs. Logistic map initially designed in one-dimension has been extended to two- dimensions to improve security. This paper proposes to use the concept of biological Gene Dominance to further improvise the randomness of 2D Logistic map. The sequences X and Y are considered to be parent genes that determine the value of parameter ‘r’ for the next iteration. The scatter plot of the proposed 2D Logistic Map with Gene Dominance (2DLMGD) shows almost uniform distribution of points in the region. The generated sequences are statistically tested using NIST SP 800-22 test suite and the results show that all sequences pass the tests. The random sequences are analysed for key sensitivity, information entropy, linear complexity, correlation to verify their conformity for use in cryptographic applications.

Keywords. Random number generator; chaotic systems; 2D logistic map; gene dominance; statistical test for randomness; security analysis.

1. Introduction impractical for many applications. This lead to the popularization of the use of deterministic algorithms to A sequence of random numbers or random bits plays an produce random-like (pseudo) number sequences with the imperative role in assorted scientific fields of Monte Carlo above-mentioned properties [10–12]. Implementations of Simulations [1, 2], cryptography [3–5], stochastic simula- these deterministic algorithms are effective and platform- tions, gaming [6]. independent. Researchers are then challenged with the Random numbers with a higher degree of disorder find question of choosing good quality seeds and various tech- helpful in generation of keys for symmetric cryptography, niques were proposed to effectively garner the seeds for initialization vector and salt in hashing, the nonce in good results [13–17]. challenge-response. Random number generators are Pseudorandom numbers are generated in many ways for required to possess the following properties: use in various applications. Cryptographic algorithms need random numbers produced from a secure source. It can 1. Uniform distribution of all possible numbers in the either be from a hardware system or a software system. keyspace. Most of the hardware components utilize natural random- 2. The present number generated does not depend on the ness inherent with them and securely process them to past generated sequence. produce a random sequence of bits. Such an example would To produce random number sequence a source of entropy be the use of ring oscillators, Field Programmable Gate is needed. Entropy needs to be truly random like atmo- Array (FPGA) implementations. They were not found to spheric noise, thermal noise and the like [7–9]. Imple- efficient for fast cryptographic applications. mentations of such true random number generators are Researchers started focusing on software-based random number generation. Many ideas were proposed using mathematical systems like de Burjin sequences, Linear *For correspondence Congruential Generators, Mersenne Twister, etc. These 8 Page 2 of 12 Sådhanå (2021) 46:8 systems possess the vulnerability of being more dimension to make the system more complex. 2D Logistic predictable even when a single element in the sequence is map have the parameter ‘r’, that determines the behaviour exposed. Later advancements developed to harness the of the chaotic system. Instead of using single ‘r’ value, this random behaviour of chaotic systems. paper proposes to have two ‘r’ values and at any iteration, Dynamic systems exhibiting chaotic behaviour and being the ‘r’ value to be used is determined by gene dominance. reactive to any slight variations in initial parameters make them a potential candidate for Pseudo Random Number Generators (PRNG). Researchers from around the world 2. Genes and alleles devised the exploitation of various chaotic systems like Logistic map [18], Lorentz chaotic system [19–22], skew Every living organism is made up of cells. Cells carry the tent map [23, 24], saw tooth map [25], quantum chaotic genetic material in the form of chromosomes. The number map [26–28], Henon map [29], zigzag map [30] for pro- of chromosomes in a cell is specific to a species. For an duction of random numbers and in the encryption of digital instance, human beings have 23 pairs of chromosomes in a images. cell that determine every minuscule characteristic of an Of all the chaotic maps in use, Logistic maps were most individual. In the case of eukaryotic amphimixis, an prevalent owing to their simple formulation defined in one organism begins from a single cell ‘zygote’ resulting from dimension [31]. In later stages, a range of solutions was the fertilization of male and female gametes. Gametes are proposed to overcome the downside of the Logistic map haploid cells with half number (n/2) of chromosomes. that are cryptanalyzed in [32–35]. Haploid cells are produced in the process of Mitosis cell In [36]Liet al proposed a reseeding technique in the use division. During fertilization, the chromosomes pair up of the Logistic map to extend the periodicity of the gen- from each parent gamete. erator. The reseeding technique causes small perturbations The chromosomes contain Deoxyribo Nucleic Acid periodically to eliminate the low period and improve the (DNA) that encodes the gene traits for all characteristics. randomness. They also optimized the period selection to For each trait, the zygote carries two genes each from two reseed the generator and their hardware implementation parents. The trait to be expressed is determined by genetic showed a higher throughput of 250 Mbit/s. dominance. A gene can be dominant or recessive. The Chen et al [37] propositioned the design of PRNG based presence of dominant and recessive alleles determines the on digitalization of modified nonlinear Logistic map. They character to be expressed. reduced the computation cost with one multiplication Punnett Square is a helpful tool to show all possible cases operation to compute the orbit. To increase the length of of parents being dominant or recessive. From figure 1 ‘A’ is cycle, the output sequences are scrambled by noise from an a dominant gene, ‘a’ is a recessive gene. Whenever a external PRNG namely Linear Feedback Shift Register dominant gene ‘A’ is present only its character is expres- (LFSR). The generated random bit sequence passed the sed. When both genes are recessive, the recessive gene tests of NIST SP800-22. character is expressed. Murillo-Escobar et al [38] improved the performance of From figure 1, four possible offspring can be inferred as the Logistic map by including a multiplication and modulo 1 operation to the modified logistic map. The histograms AA – Expression of Dominant Allele. and Lyapunov exponents of modified Logistic map have Aa – Expression of Dominant Allele. higher results compared to conventional logistic map. Their results had considerable improvement in terms of statistical tests, security analysis. Multimodal maps were introduced in [39] by Garcıá- Parent ♂ (Male) Martıńez et al to generate key streams for digital image genotype encryption. This technique produced probabilistic encryption in which encrypting the same image twice produced A a two different cipher texts. Liu et al [40] constructed a non-stationary time-varied A AA Aa Logistic map by continuously varying the parameter ‘r’ of the chaotic system. This non-stationary logistic map can resist the attack of phase space reconstruction. The value of ‘r’ is driven randomly from the interval [3.5699, 4]. a aA aa In this paper, we propose to use the natural phenomena (Female)

of gene dominance in the Logistic Map to improvise the ♀ randomness. Two-dimensional (2D) Logistic Map has been long used in producing random numbers. 2D Logistic map Figure 1. Punnett square. is an extension of traditional Logistic map into a second Sådhanå (2021) 46:8 Page 3 of 12 8

aA – Expression of Dominant Allele. map. The 2D Logistic Map with Gene Dominance aa - Expression of Recessive Allele. (2DLMGD) considers the X and Y sequence as two parents and their genes determine the next value to be expressed for When two different dominant genes are present, a con- ‘r’ parameter. dition called ‘co-dominance’ exists, where the two char- Since Cases 2 and 4 produce plots in a different pattern, acters are expressed equally. this paper proposes to use both values of r i.e. r1 and r2. However, to have a better outcome with improved randomness, the values of ‘r’ to be used is determined 3. Two-dimensional logistic map genetically. The point (X, Y) generated in each iteration genetically determine the next ‘r’ value. X and Y sequence One dimensional Logistic map is given by Eq. (1) where ‘r’ have the genes to determine the character ‘r’. The proposed is the parameter that determines the behaviour of a chaotic system is depicted in a block diagram in figure 2. system. When ‘r’ ranges in (3.5699, 4] the system exhibit Initially, X0 and Y0 are assigned a seed value (typically dynamic behaviour. can be a sensor data) represented in 32-bit using IEEE single-precision floating-point representation. Using a xnþ1 ¼ rxnðÞ1 xn ð1Þ single-precision floating-point is sufficient to generate The two-dimensional logistic map can be formulated as 16-bit output and saves processing time instead of using in Eqs. (2) and (3). 64-bit double precision. The Xn?1 and Yn?1 each of 32-bit are generated iteratively from 2D Logistic map. xnþ1 ¼ rðÞ3yn þ 1 xnðÞ1 xn ð2Þ The least significant 16 bits of only ‘Xn?1’ are considered as output. The ‘Yn?1’ value is kept secret without being ynþ1 ¼ rðÞ3xnþ1 þ 1 ynðÞ1 yn ð3Þ expressed in the output. This allows resistance from the With the variation in the value of parameter ‘r’, the known state attack. The least significant bits have higher dynamic system exhibits varying behaviour. randomness comparing to the most significant bits. Cases of ‘r’: Hiding the ‘Yn?1’ ensures that complete internal state (X ? ,Y? ,r) are unknown to an outsider in the form of 1. r e (-1,1), the system is unstable with all points n 1 n 1 output. overlapped on a few positions. Given that the least significant 16 bits (17th –32nd)go 2. r =1, the system is plotted into the spiral distribution of out as output, the genes to determine ‘r’ can be placed at points around a centre point. st th any position from 1 to 16 -bit position of X ? and Y ? 3. r e (1,1.11), the system is plotted onto polygon with n 1 n 1 since the gene for ‘r’ need to be hidden and not exposed in periodic orbit. the form of output. Our implementation used 16th bit as the 4. r e [1.11,1.19], the system exhibits complete chaotic gene for ‘r’. The gene dominance chart is given in table 1. behaviour. When both genes G and G are dominant, an average of r 5. r[1.19, the system becomes unstable with points plotted M F 1 and r is taken to guarantee the equal influence of both outside the region. 2 genes on the outcome.

4. Proposed system 4.1 2DLMGD algorithm Initial population: The 2D Logistic map has been evaluated under 5 cases as above and it can be seen that only case 4 exhibits chaotic Male Parent, X0=32 bit seed behaviour. Existing random number generators use a 2D Female Parent, Y0=32 bit seed Logistic map with case 4 scenario, where ‘r’ is ﬁxed to any one value in the range [1.11, 1.19]. Case 2 produces spiral Selection of Genes: th plot around a centre point that makes it infeasible for use in Male Gene, GM =16 bit of Xn?1 th random number generation. We propose to apply Gene Female Gene, GF =16 bit of Yn?1 dominance in genetically determining which value of ‘r’ can be used to produce the next iteration X and Y values. Parameters:

This technique will improvise the randomness and also r1=1 extends the period of the random sequence. r2 can be any value declared initially in the range [1.11, The 2D Logistic map has shorter periods for any value of 1.19] ‘r’ in [1.11, 1.19]. To further improve the randomization of orbits and to hide the internal state of the generator, we Determination of Offspring: propose to apply the gene dominance to the 2D logistic r = Gene_Dominance (GM,GF,r,r1,r2) 8 Page 4 of 12 Sådhanå (2021) 46:8

X n+1

Yn+1

17th to 32nd Random X 2D Logistic Map bit of X n+1 0 Bit Sequence Y0 16th bit of

Initial Seeds 16th bit of X n+1 r Y n+1 r

Gene r1 Dominance r2

Figure 2. Block diagram of 2DLMGD.

Table 1. Genetic dominance chart.

th th GM (16 bit of Xn?1) GF (16 bit of Yn?1) Outcome Gene result (‘r’ Value) 0 0 Both genes are Recessive No change in ‘r’ value 01GF is dominant r=r2 10GM is dominant r=r1 1 1 Both genes are dominant r= average of r1 and r2

The procedure to apply Gene Dominance is presented in 5. Implementation Algorithm 1. The genes GM and GF determine the value of parameter ’r’ for the next iteration. Initially ‘r’ is given the The Two-dimensional logistic map is implemented in average value of r1 and r2 MATLAB 2018b. The initial seed values x0 and y0 are taken as 0.78211 and 0.64582 respectively. Initially, the parameter ‘r’ is set to 1.0939. Figure 3(a) shows the scatter plot for the specified 2D Logistic Map. Algorithm 1. Procedure for Gene Dominance. To implement the proposed 2DLMGD, in our experiment Procedure Gene_Dominance(GM, GF, r, r1, r2) r1=1 and r2=1.1879 are considered as genes expressed by if GM and GF both are recessive: th 16 bit of Xn?1 and Yn?1 respectively. Initial ‘r’ value is set return r; ##No change in ‘r’ value to be 1.0939 (average of r and r ). Gene Dominance can else if GM is dominant and GF is recessive: 1 2 return r=r1; now be precisely defined as in table 2. G G else if M is recessive and F is dominant: Figure 3(b) shows the improved distribution of (Xn?1, return r=r2; Yn?1) points in the scatter plot of 2DLMGD than the con- else if GM and GF are both dominant: ventional 2D Logistic Map. The points Xn?1 and Yn?1 return r=(r1+r2)/2; ##Average of r1 and r2 end if generated are in the range [0, 1]. The points are converted end Gene_Dominance into 32-bit single-precision floating point. The output sequence is generated from the Least Significant 16 bits of Xn?1. Sådhanå (2021) 46:8 Page 5 of 12 8

Figure 3. Scatter plot of (a) 2D logistic map and (b) 2DLMGD.

Table 2. Genetic dominance chart for our implementation.

th th Allele of ‘Xn?1’ (16 bit) Allele of ‘Yn?1’ (16 bit) Outcome Gene result (‘r’ Value)

0 0 None of r1 and r2 is expressed No change in ‘r’ value 01r2 is expressed r=1.1879 10r1 is expressed r=1 1 1 Both r1 and r2 are expressed r=1.0939 (average of r1 and r2)

6. Statistical testing of proposed random number proposed 2DLMGD is higher in these tests compared to generator other existing maps and passes all the 15 statistical tests.

The random sequence generated by the proposed gene 7. Security analysis dominance algorithm is tested using a NIST SP 800-22 test suite. 1000 sequences are generated and tested. Each sequence has 106 bits. 15 statistical tests are done to verify 7.1 Uniformity whether the sequences passed the test. Each of the tests The random bit sequence should possess the property of returns a P-value (Probability value) where P-value=1 uniformity which denotes the number of ones and zeros in implies the sequence is perfectly random. The Sequence the sequence must be balanced. When an unbalance arises passes the test when the P-value is greater than or equal to a due to increased ones or zeros the resulting sequence may signiﬁcance level (called ‘a’). Normally, ‘a’ range from not be said to be uniform. The 16-bit random sequence 0.001 to 0.01. generated must ideally have 8 ones and 8 zeros. However, it The proportion of sequences (out of 1000) passing the may vary slightly. An Equilibrium degree (e) is calculated test with a =0.01 are compared against the test results of for a sequence of N bits using Eq. (4). various implementations of Logistic Map in table 3. From jjS S table 3, it can be inferred that some of the maps have very e ¼ 0 1 ð4Þ low P-value in frequency, block-frequency, runs, longest N runs, FFT, universal, approximate entropy, random excur- In Eq. (3), ‘N’ is the total sequence length with S0 sions, serial test and linear complexity. The P-Value of number of Zeros and S1 number of Ones. 8 Page 6 of 12 Sådhanå (2021) 46:8

Table 3. NIST SP 800-22 statistical test results.

Logistic map with Non-stationary Digitalized PRNG from PRNG from mutual Proposed additional input logistic map modiﬁed logistic chaotic logistic coupled 2D logistic 2DLMGD [41] P-value [40] P-value map [37] P-value map [34] P-value map [42] P-value P-value Frequency 0.5171 0.6298 0.0061 0.6355 0.4216 0.3461 Block- 0.1586 0.5621 0.3371 0.4568 0.3324 0.6497 frequency Cumulative- 0.2147 0.7939 0.4626 0.2655 0.2149 0.2648 sums(forward) Cumulative- 0.5351 –– 0.5585 – 0.6587 sums(reverse) Runs 0.2753 0.2113 0.1283 0.9673 0.2487 0.8431 Longest-runs 0.6699 0.8906 0.8881 0.0123 0.2567 0.8712 Rank 0.1546 0.3274 0.4685 0.5749 0.1930 0.1045 FFT 0.7967 0.6073 0.1626 0.0001 0.4100 0.8421 Overlapping- 0.2757 0.4848 0.8043 0.3795 0.3177 0.4679 templates* Non-periodic- 0.2262 – 0.4970 0.3521 – 0.1246 templates Universal 0.0849 0.8795 0.3241 0.4038 0.4102 0.1002 Approximate 0.0147 0.5549 0.4846 0.9060 0.3497 0.0846 entropy Random- 0.7653 0.9742 0.1473 0.0158 0.3014 0.8318 excursions* Random- 0.2686 0.5033 0.0013 0.5707 0.2987 0.3479 excursions Variant* Serial 1 0.3613 0.9005 0.2881 0.9329 0.2419 0.4399 Serial 2 0.0365 –– – – 0.1484 Linear- 0.1478 0.6337 0.0083 0.1252 0.5167 0.3464 complexity

*Average of multiple tests values is taken

Ideally, when a sequence has equal ones and zeros, the Autocorrelation of a sequence S is given by Eq. (5) equilibrium degree e tends to be zero. Figure 4 shows the where A is the number of identical bit positions in S comparison of the Equilibrium distribution of 2-D Logistic and S", D is the number of mismatching bit positions in map without and with Gene Dominance (GD). Figure 5 SandS".Figure6 shows the Auto-correlation of shows the distribution of 0’s and 1’s count in each 2DLMGD with bits shifted by 0 to 250 bit positions on sequence. both sides.

7.2 Auto-correlation 7.3 Key sensitivity Correlation is the degree of similarity between two sequences. Auto-correlation measures the similarity To make the proposed PRNG suitable for cryptographic between a sequence and its shifted version. Auto-correla- applications, it must be highly sensitive to initial condi- tion is measured to ﬁnd any periodic patterns in a sequence. tions. Any slight variation in the initial parameters likes x0, If auto-correlation measures close to 1, the sequence has y0 and r will have a drastic variation in the resulting random repetitive patterns. To measure the auto-correlation (AC) of sequence. This property is required for a PRNG to with- a sequence with size N, it is shifted r-bits (S") and measured stand Differential Analysis attack. The difference between against the original sequence. two sequences generated with varying parameters can be measured by cross-correlation. For two sequences to be jjA D completely unrelated, cross-correlation must be close to 0. AC ¼ ð5Þ N When the correlation reaches 1, the sequences are Sådhanå (2021) 46:8 Page 7 of 12 8

Figure 4. Equilibrium distribution.

Figure 5. Distribution of 0’s and 1’s. 8 Page 8 of 12 Sådhanå (2021) 46:8

Figure 6. Auto-correlation of 2DLMGD.

completely similar. Sequences S1,S2,S3 are generated by various sequence lengths. From table 4 it is found that slightly varying the X0 and Y0 as entropy is close to 16 in all the iterations with varied sequence length. S1: X0=0.78211; Y0=0.64582 S2: X0=0.78211?0.00000000001; Y0=0.64582 S3: X0=0.78211; Y0=0.64582?0.00000000001 7.5 Histogram Figure 7a shows the cross-correlation of sequences A histogram is a graphical representation of the frequency S1and S2 and figure 7(b) shows the cross-correlation of of objects in a list. A PRNG with good uniformity must sequences S1and S3. generate all numbers in the keyspace with equal probability. The proposed system generates random numbers of 16 0 16 7.4 Entropy analysis bits. Hence the keyspace is from 0 to 65535(2 to 2 -1). The Histogram in figure 8 shows the almost equal distri- The entropy of a PRNG denotes the degree of unpre- bution of generated random numbers. dictability of its random sequence. A PRNG generating random numbers sequence each with ‘n’ bits have a key- n n space of 2 . Entropy for a sequence S containing 2 num- 7.6 Linear complexity bers each with n-bit representation is given by H(S) in Linear Complexity is the minimal length of a Linear Eq. (6), where p(si) is the probability of a random number si present in the sequence S. Feedback Shift register that can reproduce the generated random sequence. For a PRNG to be more unpredictable, 2Xn1 the length of LFSR must be higher. Ideally, a perfect PRNG ðÞ¼ ðÞ ðÞ ðÞ HS psi log2 psi 6 producing n-bit sequence can have a linear complexity of ¼ i 0 length n/2. Linear complexity of sequences generated by Ideal PRNG with good disorder degree must have the 2DLMGD is calculated by Berlekamp-Massey Algorithm. entropy close to ‘n’. The proposed PRNG generates 16-bit Figure 9 shows the linear complexity of varying sequence random numbers by extracting the Least Significant 16-bits lengths with almost all sequences having n/2 linear of X sequence, where n=16. The entropy is measured for complexity. Sådhanå (2021) 46:8 Page 9 of 12 8

Figure 7. Cross-correlation between (a) S1 and S2, (b) S1 and S3. Table 4. Entropy of binary sequences of different length.

Sequence length Entropy

2000 15.89732 4000 15.79462 8000 15.92324 10000 15.67812 20000 15.48922 40000 15.73226 80000 15.91245 8 Page 10 of 12 Sådhanå (2021) 46:8

Figure 8. Histogram of the proposed 2DLMGD.

Figure 9. Linear complexity of random sequences of different lengths. Sådhanå (2021) 46:8 Page 11 of 12 8

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