Science and Information Conference 2015 July 28-30, 2015 | London, UK
A Cost-Effective True Random Bit Generator Using a Pair of Robust Signum-Based Chaotic Maps Wimol San-Um, Patinya Ketthong,Winai Chankasame and Jeerana Noymanee Intelligent Electronic Systems (IES) Research Laboratory Master Program of Engineering Technology Faculty of Engineering, Thai-Nichi Institute of Technology (TNI) Patthanakarn 37, Suanlaung, Bangkok, Thailand
[email protected] and
[email protected] Abstract—this paper presents a cost-effective random-bit generator through a newly proposed inverted signum-based piecewise-linear chaotic map, which provides not only robust chaos against parameter changes but also symmetric bifurcation for zero-thresholding for digital random-bit generation. Chaos dynamics are described in terms of equilibria and Jacobian analysis, bifurcation diagram, Lyapunov exponent, time-and frequency domain signal, and cobweb plots. NIST standard tests suite have been realized for statistical analysis of randomness of binary sequence, and the sufficient length of 1,000,000 bits successfully passed all NIST standard tests. Experimental results of digital random-bit sequences on have been performed using a cost effective Arduino with Atmel SAM3X8E ARM Cortex-M3 CPU. The proposed random-bit generator offers a potential alternative in compact and robust random bit sequence for applications in computer information security. Keywords—True Random Bit Generator; Robust; SignumBased Chaotic Maps
I.
INTRODUCTION
Information security has remarkably become a crucial issue under consideration for both research and practical applications due to rapid advancements of Information and Communication Technology (ICT). Cryptography has consequently been exploited as a solution to information security where a TrueRandom-Bit (TRB) generator is typically utilized not only in confidential key generation, but also in some computation algorithms. The TRB is typically defined as an algorithm that enables the generation of digital bit sequences with randomness properties. Existing hardware-based TRBs were generally implemented by random physical phenomenon such as the amplification of direct resistor noises [1] or jitter noises of digital clock signals [2]. Despite the fact that such limitations can be conquered through proper custom circuits, randomness extraction is still a challenging topic in the designs based on such devices. Recently, chaotic systems have been of great attention due to various potential applications. Chaotic systems have been characterized as a system that offers a sensitive dependence on initial conditions, i.e. a small perturbation ultimately results in dramatic change insystem states [3]. The use of chaotic signals as sources of randomness in hardware TRBs has been
suggested, including, for example, discrete-time chaotic maps [4], switched-capacitor chaotic circuit [5], and double-scroll. In addition, various chaotic systems are employed in chaotic ciphers. The logistic map, the Hénon map and the piecewise linear chaotic map (PWLCM), for example, are used in stream ciphers[1,4–6], whereas the logistic map, the PWLCM, the Baker map and the Cat map are used for block ciphers [3,7– 12]. Typically, chaotic dynamical systems exhibit two types of chaotic attractors, i.e. a fragile chaos (the attractors disappear with perturbations of a parameter or coexist with other attractors), and a robust chaos, defined by the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space [13]. Therefore, robust chaos is relatively important in the implementation of random bit generator. Small changes or errors in hardware implementation cannot be obstacles for random bit generators. This paper therefore presents a new random-bit generator. The random signal source is a new signum-based piecewiselinear chaotic map, which provides not only robust chaos against parameter changes but also symmetric bifurcation for zero-thresholding for digital random-bit generation. Chaos dynamics are described in terms of equilibria and Jacobian analysis, bifurcation diagram, Lyapunov exponent, time-and frequency domain signal, and cobweb plots. Autocorrelation, histogram, and NIST standard tests suite have been realized for statistical analysis of randomness of binary sequence, and the sufficient length of 1,000,000 bits successfully passed all NIST [14] standard tests. Experimental results of digital random-bit sequences on have been performed using a cost effective Arduino with Atmel SAM3X8E ARM Cortex-M3 CPU. The proposed random-bit generator offers a potential alternative in compact and robust random bit sequence for applications in computer information security. This paper is organized as follows; Section 2 describes the existing chaotic systems. Section 3 proposes a pair of chaotic maps for use as a source of randomness. In addition, the statistical analyses are also investigated through the standard NIST tests. Finally, experimental results are demonstrated in both time and frequency domains.
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Science and Information Conference 2015 July 28-30, 2015 | London, UK
Fig. 1. Bifurcation diagrams and Lyapunov exponents; (a) Standard cubic map, (b) Modified standard cubic map using hyperbolic tangent function
II.
EXISTING RELATED CHAOTIC MAPS
Chaotic systems are deterministic nonlinear systems. In other words, chaotic systems are governed by exact mathematical form with chances to provide stochastic behaviors. Typically, chaotic systems are sensitive to both initial conditions and control parameters. There are several advantages for utilizing chaotic maps as random bit generation syste. First, a chaotic system provides random behaviors which resemble real souce of randomness through simple devices such as microcontroller. Second, a chaotic system provides deterministic behaviors. Unlike those conventional statistical systems based on stochastic variances, all kinds of random behaviors can be controlled exactly by known system parameters. Last, a chaotic system naturally provides transient behaviors that simulate real-time generation of random signal in time-domain which is important in random bit generator. It can be considered that chaotic systems are relatively useful as a source of randomness as its function can be control by system parameter while stochastic system cannot be controlled. The proposed signum-based piecewise-linear chaotic map is based on the well-known cubic map expressed as
xn +1 = λxn − xn3
(1)
where Ȝ (0, 3) is a bifurcation parameter. It can be considered from (1) that the function exploits an N-shape nonlinearity for generating chaos. One possible modification of the nonlinear term xn3 in (1) is the use of hyperbolic tangent function that closely resembles the N-shape nonlinearity, i.e.
xn+1 = λxn − tanh(βxn )
(2)
Where the additional control parameter ȕ is included for adjusting the slope of the N-shape nonlinearity. As for purposes of comparison, preliminary investigations of the three chaotic maps are performed by bifurcation diagrams and Lyapunov exponent (LE) as for qualitative and quantitative measures of chaos, respectively. The bifurcation diagram indicates possible long-term values, involving fixed points or periodic orbits, of a system as a function of a bifurcation parameter. The stable solution is represented by a straight line while the unstable solutions are generally represented by dotted lines, showing thick regions. On the other hand, the LE is defined as a quantity that characterizes the rate of separation of infinitesimally close trajectories and is expressed as
1 n→∞ N
LE = lim
N
¦ log n =1
2
dX n+1 dX n
(3)
where N is the number of iterations. Typically, the positive LE indicates chaotic behaviors of dynamical systems and the larger value of LE results in higher degree of chaoticity. Figs. 4(a) and 4(b) show the bifurcation diagrams and Lyapunov exponents of a standard cubic map, and a modified standard cubic map using hyperbolic tangent function. It is apparent that there are some periodic window appearing in the bifurcation diagrams and some points on LE spectrum are lower than zero. Such characteristic is unprofitable as the mobile nodes are not truly random. As for an improvement in terms of bifurcationsymmetry and smooth chaos over parameter space, this paper presents an alternative of the nonlinear terms xn3 and tanh(ȕxn) by using the hard-switching signum function given by [13]
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Number of Values of Xmax
Number of Values of Xmax
Lyapunov Exponent (LE)
Lyapunov Exponent (LE)
Science and Information Conference 2015 July 28-30, 2015 | London, UK
Fig. 2. Cobweb plots and waveforms in time-domain; (a) Case 1: xn+1=Įxn-sign(xn), (b) Case 2: xn+1=-Įxn+sign(xn)
xn+1 = αxn − sign( xn )
(4)
Where Į is a bifurcation parameter. It can be seen that the system is simple with a single linear term Įxn and a nonlinear term sign (xn). However, the polarity of each term may have an effect on chaotic dynamics through locations of equilibria. III.
PROPOSED INVERTER PAIR OF SIGNUM-BASED CHAOTIC MAP
This paper further generalizes Equation (4) in [12] given by
xn+1 = ±αxn B sign( xn )
(5)
In other words, Eq. (5) can be denoted into two cases based on the signs of each terms as follows;
Case1: xn+1 = αxn − sign( xn )
(6)
Case2: xn+1 = −αxn + sign( xn )
(7)
It is apparent that both cases in (6) and (7) exhibit the same shapes of Bifurcation diagrams and Lyapunov exponent values. This means that the chaotic map offers robust chaos over the region [1, 2]. Robust chaos is defned by the absenc of periodic windows and coexisting attractors in some neighborhood of the parameter spaces [14]. Such robust chaos features are important for true randomness in node mobility models. In contrast, the two cases cases in (6) and (7) exhibit different Cobweb plots and the waveforms in time-domain are laso completely different. Such features can be described by stability analysis that aims to solve for dynamics behaviors of system equilibrium. The Jacobian can be found through the absolute value of the first derivative as
J ( xn ) =
d ( xn+1 ) d (±αxn B sign( xn )) = dx dx
(8)
Based upon the identity sign(x) = 2H(x)-1 where H(x) is a Heaviside step function. The Jacobian in (8) can be written and provides the solution as follows;
J ( xn ) = ±
d (αxn ) d (2 H ( x) − 1) B = ± α B 2δ ( x) dx dx
(9)
Typically, the chaotic map becomes unstable in the case where J(xn)>1. The system fixed points can be calculated by substituting xn into xn+1 in (5) and omit the subscript n for simplicity. As a result, the equation becomes
± αx B sign( x) = x
(10)
Solving Eq. (10) yields the fixed points for each case. For case 1 in (6), the three fixed point (x1*, x2*, x3*) are given by
1 x1* = 0 x2* = + 1 x* = − α −1 3 α −1
(11)
On the other hand, the three fixed point (x1**, x2**, x3**) of (7) can also be described as
x1** = 0 x2** = + 1 x3** = − 1 α +1 α +1
(12)
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Fig. 3. Cobweb plots and waveforms in time-domain; (a) Case 1: xn+1=Įxn-sign(xn), (b) Case 2: xn+1=-Įxn+sign(xn)
The fixed point x0*=0 and x0**=0 in (11) and (12) cause the Jacobian in (9) becomes zero and the system consequently becomes stable as J(xn)> SET Initial condition X >> SET Parameter a >> LOOP >> Calculate X = a*X - sgn(X) >> IF X >= 0 THEN bI = 1 >> ELSE bI = 0 >> END IF >> END LOOP Fig. 4. Pseudo code for generating random bit from chaotic map Case 1
RANDOMNESS ANALYSIS
The the National Institute of Standards and Technology (NIST) has provided a statistical tests suite in order to evaluate the randomness of binary sequences. This paper generates chaotic signals by the proposed two cases of the signum-based chaotic maps for 1,000,000 iterations and simply proceed a comparison with zero, i.e. bit “1” for any values that greater than zero and bit “0” for any values that smaller than zero. Subsequently, the NIST test suite from a special publication 800-22rev1a [13] was realized using a typical 1,000,000 random bits. The test suite attempts to extract the presence of a pattern that indicates non-randomness of the sequences through probability methods described in terms of p-value. For each test methods, the p-value indicates the strength of evidence against perfect randomness hypothesis, i.e. a p-value greater than a typical confidence level of 0.01 implies that the sequence is considered to be random with a confidence level of 99%. Table 1 summaries NIST test results, indicating that the
Fig. 5. The quadri-shift register (QSR) based post-processing
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(a)
(b)
(d)
(c) Fig. 6. Chaotic waveforms in time domain, random bits,, and power spectrum
V.
PROPOSED RANDOM-BIT GENEERATOR
The proposed random bit generator iss based on the proposed chaotic maps in (6) and (7). The genneration system is programed on a cost effective Aduino with A Atmel SAM3X8E ARM Cortex-M3 CPU. Either system in (6)) and (7) can be used as random-bit generator as both system m offers smooth chaos and pass all NIST test. As the output iis symmetric, the signal can be quantized at zero while maintaining its randomness properties. This paper presentss a very simple generation as described in pseudo code in Fig.5. However, post-processing is still required to garuntee rrandomness. This work therefore realizes register (QSR) basedd post-processing with a particular structure of four shift registerrs A, B, C, and D as shown in Fig.4. This scheme is basicallyy based on XOR operations incorporating with linear shiftt registers. The principle is to perform XOR operation betweenn the bits coming from the TRB and those bits coming from the delayed bitstream memorized into a linear shift register. The operation is
also repeated several times and a few numbers of XORs are inserted between the different shifft registers to increase the complexity of each stage.
Fig. 7. The photograph of Aduino with Attmel SAM3X8E ARM Cortex-M3 CPU that used for generating chaotic signalss
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sufficient length of 1,000,000 bits successfully passed all NIST standard tests. Experimental results of digital random-bit sequences on have been performed using a cost effective Arduino with Atmel SAM3X8E ARM Cortex-M3 CPU. The proposed random-bit generator offers a potential alternative in compact and robust random bit sequence for applications in computer information security. ACKNOWLEDGMENTS The authors are grateful to Research and Academic Services Division of Thai-Nichi Institute of Technology (TNI) for research fund. [1] Fig. 8. Random bit generation after post-processing using QSR
VI.
[2]
EXPERIMENTAL RESULTS
The experimental results are summarized in Fig. 6. Fig.6 (a) shows the chaotic waveforms as well as its quantized random bit of the chaotic map in Eq. (6). In addition to Fig.6 (a), Fig.6 (c) also shows the respective frequency spectrum of the chaotic signals. It is apparent that the spectrum is relatively flat containing random frequency and amplitude. In the similar manner, Figs.6 (b) and (d) also chaotic signals in time and frequency domain, respectively, of Eq.(7). Fig. 7 shows the utilized photograph of Arduino with Atmel SAM3X8E ARM Cortex-M3 CPU that used for generating chaotic signals. Fig.8 illustrates the random bit generation after post-processing using QSR technique. It is seen that the output bit are truly random.
[3] [4]
[5]
[6]
[7]
VII. DISCUSSION AND CONCLUSION Cryptography has extensively been utilized for to information security where a True-Random-Bit (TRB) generator is a crucial system not only in confidential key generation, but also in some computation algorithms. Conventional techniques are based on stochastic phenomena and relatively expensive such as random physical phenomenon such as the amplification of direct resistor noises. This paper presents a cost-effective random-bit generator through a newly proposed inverted signum-based piecewise-linear chaotic map, which provides not only robust chaos against parameter changes but also symmetric bifurcation for zero-thresholding for digital random-bit generation. Chaos dynamics of the signum-baed system have been described in terms of equilibria and Jacobian analysis, bifurcation diagram, Lyapunov exponent, time-and frequency domain signals, and cobweb plots. NIST standard tests suite have been realized for statistical analysis of randomness of binary sequence, and the
[8]
[9]
[10]
[11]
[12]
[13]
[14]
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