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International Journal of Advances in Electronics and , ISSN: 2393-2835 Volume-5, Issue-7, Jul.-2018 http://iraj.in PSEUDO RANDOM NUMBER GENERATION BASED ON

1BHAVYA GARG, 2PRAGYA GOEL, 3SRISHTI GUPTA, 4UTKARSHA GOSWAMI

1,2,3,4 Indira Gandhi Delhi Technical University for Women, Kashmere Gate, Delhi-110006, India E-mail: [email protected], [email protected], [email protected], [email protected]

Abstract - In this paper, we attempt to survey the work done in the past on finding new ways to obtain Pseudo Random Number Generators using Chaos theory. Chaotic behaviour is a characteristic of nonlinear wherein simple rules can give rise to complex behaviours that show apparent . Thus, to cater to the various use of random , we summarise the past work done in this as well as compare the various techniques proposed to highlight key features that have proved to be positive and further, providing scope of working with them to obtain a much effective Pseudo Random Number Generator.

Keywords - Random number; Chaos; Security; PRNG

I. INTRODUCTION II. RANDOM NUMBER

Random Numbers have always fascinated A random number belongs to a sequence of numbers mathematicians and scientists because of their natural generated randomly such that a large set of such occurrence, and because of their use in wide variety numbers represents a uniform underlying distribution. of applications including Security, Sampling and Almost always, the numbers generated are Computer Simulation. Random Number Generation is independent thus making it impossible to predict the thus studied extensively and various techniques have next number from any of the previously generated been put forward in generating random numbers. random numbers. Such numbers are called pseudo random numbers. Random numbers find their Random number generators are generally categorized applications in various fields like gaming, sampling, as true random number generators (TRNG) and gambling and security. A good random number must pseudo random number generators (PRNG). TRNG possess the two important properties: Uniformity and are obtained from natural processes like atmospheric Independence. noise, but these are not efficient enough to be harnessed effectively for practical purposes. PRNG III. CHAOS THEORY on the other hand provides results that are good enough for all practical purposes. Chaos theory is an emerging field of that deals with nonlinear dynamical systems(A Techniques used to devise pseudo random number that changes over time and does not show a linear generators (PRNGs) mostly fall in some form of dependence on its components). It exhibits the mathematical distributions or linear congruential property of being extremely sensitive to the initial relations. Chaos based dynamical systems because of conditions i.e a slight change can produce an their apparent randomness are being studied lately to effectively large difference in the behavior. It must be devise PRNGs. topologically (meaning that is non-constant and will evolve over time), and it must have dense This random behaviour in chaotic systems owes to its periodic orbits, which means that every point in space properties, namely, , extreme sensitivity to can be arbitrarily approached. Despite their initial conditions and system . Chaos is deterministic simplicity, over time these systems can observed as a subtle behaviour of nonlinear systems produce totally unpredictable and wildly divergent wherein the randomness introduced results from the (chaotic) behavior i.e such systems seem to be sensitivity to initial conditions on defining predictable in the beginning and later they appear to deterministic processes. be random.

Therefore, in this paper we plan to study previous IV. RANDOM NUMBER AND CHAOS work done to generate pseudo random numbers using THEORY chaos theory and combine our findings to reveal and observations that can be used in this field As software generated random sequences are not in the to either improve on an existing truly random, classically chaotic systems can be technique or to devise a new that exploits viable alternatives. Chaos theory plays an active role chaotic systems to generate pseudo random numbers. in the improvement of the quality of PRNGs. The

Pseudo Random Number Generation based on Chaos Theory

53 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-5, Issue-7, Jul.-2018 http://iraj.in advantage of using chaos in this field lies in its chaotic signal. It used matrix method to synthesize disordered behavior and its unpredictability. Chaotic chaotic signals. A chaotic piecewise linear one- behavior depends upon the called as dimensional map as a PRNG was proposed by “seed” or “key”. In a Pseudo Random Number Kocarev and Stojanovski et al. A theoretical analysis Generator (PRNG), the initial conditions are fixed suggesting that chaotic maps have perfect from difference equations. cryptographic properties including those of good correlation properties and long cycle length was done V. BACKGROUND WORK by Li et al.They also pointed out that bit streams generated through a single chaotic system are A lot of research has gone into chaos and randomness potentially insecure as the output may leak some theory.Chaos illustrates the rather striking fact that information about the chaotic system. Thus a new complex behaviour arises from simple rules when PRNG based on a couple of piecewise linear chaotic nonlinearities are present. One tries to apply this maps was developed with independently iterated bit notion to generate pseudo-random number generators streams. They also justified their theoretical claims (PRNGs) because random numbers are widely used through a few numerical experimentations on the not only in and Monte Carlo proposed pseudo random bit generator. applications but also in less obvious applications as quoted in Pecora et al., L’Ecuyer, Kocarev & Parlitz, Kocarev and Jakimoski in 2003, chaotic maps were Hidalgo et al. , Fernandez et al. from 1990-2003. used to devise new ways to generate PRNG In 2004 Since simple chaotic maps are capable of generating Fu et al proposed a chaos-based random number stochastic-like signals, implementations based on generator using piecewise chaotic map.A one-way chaotic systems are usually less involved than those coupled chaotic map lattice was used by Huaping et based on more complex (Stojanovski & al for a new PRNG. They used a combination of Kocarev 2001; Kocarev & Jakimoski 2003). chaotic and conventional method on spatiotemporal chaotic system to show the effectiveness of random Liu had proposed a new way to generate pseudo- number thus generated. random number generator (PRNG) based on modification of on which further, a This pseudo-random number generator can be used as was designed. A chaotic random an ideal synchronous and self-synchronizing stream number generator was developed by Wang et al. cipher for secure communications. In 2005 Li et al which was realized by an analog circuit. In 2006, z- designed and analysed a random number generator logistic map based PRNG was given by Wang et al. based on a piecewise-linear map. A new pseudo- in which the binary sequence through the chaotic random number generator (PRNG) based on modified was realized under finite computing precision. logistic map was proposed by Liu and a design of a González et al. (2005) used statistical chaotic stream cipher using it was also suggested. measures to calculate the effectiveness of Further, a chaotic random number generator was randomisation achieved by modus operandi on developed by Wang et al and realized it by an analog Lorenz Equations. This was originally proposed by circuit. López-Ruiz et al. (1995). Rosso et al. (2007a) showed that as chaotic systems are deterministic in , its In 2006, Wang et al proposed a pseudo-random associations with can be exploited to number generator based on z-logistic map, where the obtain effective randomness. binary sequence through the chaotic orbit was realized under finite computing precision. In 2007, Though, a mix of two chaotic signals didn’t show Ergun and Ozogur demonstrated that stroboscopic positive results. Use of first order nonlinear Poincare map of a non-autonomous chaotic electronic differential equations such as Lorenz equations that circuit can be used to generate bit streams which shows chaotic behaviour were then thought of to be passed the four basic tests of FIPS-140-2 and NIST used to generate PRNG in 1982 by Oishi and tests suite.Lately, Hu et al proposed a true random Inoue.Here, they were able to generate a uniform number generator (which generates a 256-bit random random number generator with an arbitrary number by computer mouse movement), where the Kolmogorov . Later a PRNG was developed authors used three chaos based approaches namely: with second-order digital filter on a hardware by Lin discretized 2D chaotic map permutation, and Chua in 1993. spatiotemporal chaos and MASK algorithm to eliminate the effect of similar mouse movement Further, logistic map was extensively studied and patterns. The results have been tested through NIST various ways to generate pseudo random number tests suite. Patidar et al devised a PRNG based on the were proposed with varying levels of success in chaotic and tested the same with NIST achieving randomness. First was in 1996, a logistic as well as DIEHARD test suites and none of the tests map based PRNG proposed by Andrecut. Kolesov et had failed for both of these test suites. al in 2001 designed a digital PRNG using discrete

Pseudo Random Number Generation based on Chaos Theory

54 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-5, Issue-7, Jul.-2018 http://iraj.in VI. GAP ANALYSIS

Focus and Tests Performed, Technique Author, Year Objective Gap Analysis Strength Bit/Word Type Involved First attempt at Various modifications of Random deriving the parameters can be Number random Mathematical Stephen possible which can Cellular Generation behaviour from Tests, bit type Wolfram, 1986 potentially give better automaton used using cellular underlying PRNG results, hence not automata. cellular optimum. process A remapped version of the Deals with Only a few statistical logistic equation floating point Neal R tests are performed to Re-mapped is used to devise numbers with Statistical tests Wagner, 1992 check the effectiveness logistic equation a pseudo logistic of the generator random number equation. generator. Have used the Only a limited tests like simplest and distributed test, Using the logistic most moments calculation and Distributed, map in the S.C. Phatak Make a RNG transparent chi-squared test have Moments chaotic regime and S.Suresh based on system been performed. calculation, Chi- (logmap) for a Rao, 1995 logistic map. exhibiting Standard NIST Tests, squared Test pseudo random order to chaos TestU01 and Diehard number generator transition tests are missing. Analysis and application of a The paper involves a lot Chaos-Based application of chaotic Toni of mathematics and thus Random pairwise-linear Only piecewise-linear Stojanovski is difficult to understand. Number 1D map for mathematically one-dimensional and Ljupčo A lot of symbols have Generators— random proven (PL1D) map as Kocarev, 2001 unclear and/or Part I: Analysis number random number ambiguous meaning. generation generator (RNG) Analysis of Separates out It attempts to analyse previous few properties past researches in a More attempts at Kocarev, that might general way and only cryptography using chaos - Ljupco., 2001 work and the provides some hints at based(block algo) theory for pitfalls of the future work that could than RNG cryptographic past researches be done techniques Propose a modified Used a single version of the to Manually tested Wai-kitWong, chaotic More trade-off Results not tested and achieve trade-off time for Lap-piu Lee cryptographic between size of applications of the new between the 3 different and Kwok-wo method (of text and time model have not been spread of the parameters and 3 Wong random number for encryption mentioned distribution of different file sizes generator )of ciphertext and the iterating a encryption time logistic map Uses 7 Paper meets the aim Q.V. Lawande, A new parameters as specified.To deal with B. R. Ivan and encryption seed to develop overflow of time steps, Self devised. Word S. D. scheme based a strong text Lorentz equations same initial settings are PRNG Dhodapkar, on Lorenz based used every time which 2005 dynamics encryption could be changed scheme

Pseudo Random Number Generation based on Chaos Theory

55 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-5, Issue-7, Jul.-2018 http://iraj.in Pseudo random bit generator (PRBG) based on two chaotic A new proposed solution Patidar, Vinod, Completely logistic maps that increases the Krishan K. new approach running side-by- complexity of the NIST tests. Bit Two chaotic maps Sud, and and tested side and PRNG, but not PRNG side-by-side Narendra K. against NIST starting from implemented in any Pareek, 2009 tests random application independent initial conditions Combines ISAAC and XORshift Christophe A new chaotic generators Guyeux, pseudo-random It does not explore ISAAC and with chaotic Qianxue number different chaotic NIST and TestU01 XORshift iterations. It Wang, and generator strategies and iteration Batteries of tests. generators with possesses Jacques M. (PRNG) is functions chaotic iterations important Bahi, 2010 proposed properties of topological chaos. The proposed key agreement protocol supports The semigroup property Introduced an mutual of Chebyshev enhanced authentication, Utilizes the semi- Xingyuan polynomials holds only key agreement could resist group Wang *, for some values of protocol attacks like property of Jianfeng Zhao, variables depending on based on replay attack Chebyshev 2010 the arithmetic precision Chebyshev etc. polynomials used in implementing chaotic map. Unlike other the protocol. protocols, Synchronised clocks are not required. Random numbers A new Iteration generated have The claim of the paper Developed a new Mohamed Function a chaotic that the random numbers Measuring mathematical Nageb System (IFS) to behaviour. The generated can be used as sensitivity of the formula termed as Elsherbeny and generate a PRNG can thus key generator in iterated function to Iteration Function Mahmud Rahal Pseudo Random be used as a different cryptosystem Lyapunov System to , 2012 Number key generator algorithms has not been Exponent generate random Generator. in different shown. numbers cryptosystem algorithms. The algorithm Michael mixes two The paper is well written Francois, A New Pseudo- chaotic maps and considers all the NIST Statistical Thomas Random based on an factors that may affect Tests, Pearson's Grosges, Number initial vector. the randomness of the mixing of two correlation Dominique Generator based A chaotic sequence. The paper also chaotic maps Coefficient. Bit Barchiesi, on Two Chaotic function based describes how does the PRNG Robert Erra, Maps on linear algorithm counters the 2013 congruences is attacks against system. used to

Pseudo Random Number Generation based on Chaos Theory

56 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-5, Issue-7, Jul.-2018 http://iraj.in compute and index positions of permutations.

By applying an Applying an extra extra parameter so that To develop a parameter the key will be new super key is modified Paper meets the modified and will Rakesh Kumar chaotic RNG to provide proposed objective but provide better Rai and that generates better their usefulness in any encryption and Rakesh Kumar random encryption and application has not been decryption thus Prajapati, 2013 numbers by decryption and shown. increasing applying an thereby security of extra parameter. increase chaotic security. cryptography A random secret key is generated of fixed size.To estimate the A random key randomize Large key increases the Aruna Tomar based character to security level but as the and Sunita cryptographic generate the key size increases Malik, 2014 approach is crypto data, information decoding discussed. base value becomes more complex. analysis is performed and thus used in information security. Based on the initial conditions The implementations K. Generate generated by have not been applied to image Thamizhchelvy different types PRNG, fractal other systems like e- generation using and G.Geetha of using is generated banking and online chaos , 2014 chaos theory. and the novelty shopping etc. to credit card number system is proposed The paper proposes an adaptive chaos Applied based data Piyush Kumar The reason(s) for Cryptography encryption Shukla, Ankur choosing only 16 Based on the Using Chaos framework of Khare, parameters have not Correlation, UACI, standard logistic Function for secure Murtaza Abbas been mentioned. The NPCR, and NIST map and the Fast Digital communication Rizvi, Shalini algorithms, while being statistical tests. Bit Engel Continued Logic-Based which can be Stalin and compared, are not PRNG Fraction (ECF) Systems in further Sanjay Kumar, arranged in any map Ubiquitous explored. The 2015 particular order. Computing limitations of chaos based data encryption techniques

Pseudo Random Number Generation based on Chaos Theory

57 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-5, Issue-7, Jul.-2018 http://iraj.in have been explored and their applications presented.

Create a universal tool (design ) with There are limited Combines the modular design number of machine Dieharder, NIST Designing a benefits of hash and parameters. When buffer Statistical Tests chaos machine function and Maciej A. customizable space is large, the and TestU01. that is able to pseudo-random Czyzewski, parameters, machine can be used Period of 2^(8m) to implement function, forming 2016 that can be only as RNG, otherwise 2^(16m) where m cryptographic flexible one-way applied where it can be used as a fully is the space primitives push-pull randomness functioning chaos parameter interface. and machine. sensitiveness is needed

The random numbers generated Expensive hardware is To develop a Yu-Guang using QRW required for QRNG. novel PRNG Yang and have better Moreover application for based on NIST, 8 bit PRNG Qian-Qian statistical cryptographic purposes quantum zhao, 2016 complexity, has not been focussed random walks recurrence upon plots and non- periodicity.

ANALYSIS & CONCLUSION number using different versions of logistic maps and its various applications. This is done by The study of all the research papers mentioned above simultaneously running two chaotic logistics map, by helped us understand the relationship between chaos applying some extra parameters, etc. and randomness. Edward Lorenz summarised Chaos Theory as “Chaos: When the present determines the After this thorough literature survey, we did the gap future, but the approximate present does not analysis of the papers considering the various approximately determine the future”. This definition parameters i.e the strength of the paper, various tests of Chaos is exploited for security related applications performed to check the validation of the proposed like Pseudo Random Number Generation. algorithm, the type of PRNG generated (Bit/Word We conclude that Chaos Theory can be used as a type), the techniques involved, etc and then created a Pseudo Random Number Generator as it provides comparison table. This analysis highlighted the unpredictability, extreme sensitivity to initial states importance of different papers. With the help of the and good pseudo-randomness. Since the Chaos based comparison table now we have a concise and clear RNG techniques can be easily realised and require form representing the papers which gives an easy low memory, they can be considered ideal for use in understanding. Wifi, VANETs, etc. Chaotic maps increase the strength of the algorithm as compared to the cases Logistic map is the best example to show how a when no chaos is used. The confusion and diffusion simple recursive nonlinear dynamic equation is of chaos functions should be used for providing better sufficient enough to give rise to a complex, chaotic trade-offs between security and computational behaviour. We understand that the behaviour of the complexity.We did a detailed study of various equation is predictable until value of r is between 0 research papers to understand the previous works and 3.56995 and beyond this there is an onset of related to chaos theory and randomness. The study chaotic behaviour. Although there are certain values helped us to get the insights of chaos theory, its of r for which the equation behaves non-chaotically. various applications in the creation of fractal images, Beyond 4, the value leaves the [0,1] and in the field of cryptography and also the different diverges. In order to understand the chaotic behavior, approaches to the generation of Pseudo random we used the simplest logistic equation:

Pseudo Random Number Generation based on Chaos Theory

58 International Journal of Advances in Electronics and Computer Science, ISSN: 2393-2835 Volume-5, Issue-7, Jul.-2018 http://iraj.in F(x)= rx(1-x) [4] Stojanovski, Toni, and Ljupco Kocarev. "Chaos-based random number generators-part I: analysis [cryptography]." IEEE Transactions on Circuits and Systems I: Fundamental and by variating the values of r, we could clearly see Theory and Applications 48.3 (2001): 281-288. the difference in the exhibited behavior. When we [5] Kocarev, Ljupco. "Chaos-based cryptography: a brief used logistic equation to look at its behaviour, we overview." IEEE Circuits and Systems Magazine 1.3 (2001): found out that for r<3.6 the graph had just a few 6-21. [6] Wong, Wai-kit, Lap-piu Lee, and Kwok-wo Wong. "A points even when the number of iterations of modified chaotic cryptographic method." Communications recursive function were 500. and Multimedia Security Issues of the New Century. Springer US, 2001. 123-126. [7] Lawande, Q. V., B. R. Ivan, and S. D. Dhodapkar. "Chaos based cryptography: a new approach to secure communications." BARC newsletter 258.258 (2005). [8] Patidar, Vinod, Krishan K. Sud, and Narendra K. Pareek. "A pseudo random bit generator based on chaotic logistic map and its statistical testing." Informatica 33.4 (2009). [9] Guyeux, Christophe, Qianxue Wang, and Jacques M. Bahi. "A Pseudo Random Numbers Generator Based on Chaotic Iterations: Application to Watermarking." WISM. 2010. Figure 6.2 : Plot for r=3.2 with seed1 = 0.4 and seed2 = 0.41 [10] Wang, Xingyuan, and Jianfeng Zhao. "An improved key agreement protocol based on chaos." Communications in Nonlinear Science and Numerical Simulation 15.12 (2010): When the same was done for r>3.6, even the very 4052-4057. first 500 iterations gave a very chaotic graph. [11] Elsherbeny, Mohamed Nageb, and Mahmud Rahal. "Pseudo- Random number generator using deterministic chaotic system." International Journal of Scientific & Technology Research 1.9 (2012): 95-97. [12] Francois, Michael, et al. "A new pseudo-random number generator based on two chaotic maps." Informatica 24.2 (2013): 181-197. [13] Rakesh Kumar Rai and Rakesh Kumar Prajapati. “Creation of new chaotic super random number generators for improving the security of chaotic cryptography.” International Journal of Research and Technology, Vol 2, Issue 11 (2013) Figure 6.2 : Plot for r=3.7 with seed1 = 0.4 and seed2 = 0.41 [14] Aruna Tomar and Sunita Malik. “A random key based cryptographic approach.” International Journal of Computer REFERENCES Science and Mobile Computing (2014) [15] Thamizhchelvy, K., and G. Geetha. "A NOVEL APPROACH TO GENERATE FRACTAL IMAGES USING CHAOS [1] Wolfram, Stephen. "Random sequence generation by cellular THEORY." automata." Advances in 7.2 (1986): 123- [16] Shukla, Piyush Kumar, et al. "Applied cryptography using 169. chaos function for fast digital logic-based systems in [2] Wagner, Neal R. "The logistic lattice in random number ubiquitous computing." Entropy 17.3 (2015): 1387-1410. generation." PROCEEDINGS OF THE ANNUAL [17] Czyzewski, Maciej A. "Chaos Machine: Different Approach ALLERTON CONFERENCE ON COMMUNICATION to the Application and Significance of Numbers." IACR CONTROL AND COMPUTING. Vol. 30. UNIVERSITY Cryptology ePrint Archive 2016 (2016): 468. OF ILLINOIS, 1992. [18] Yang, Yu-Guang, and Qian-Qian Zhao. "Novel pseudo- [3] Phatak, S. C., and S. Suresh Rao. "Logistic map: A possible random number generator based on quantum random walks." random-number generator." E 51.4 (1995): Scientific reports 6 (2016). 3670.

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Pseudo Random Number Generation based on Chaos Theory

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