Random Sequence Generator Based on Nonlinear Function Ioan VASILACHE Military Equipment and Technologies Research Agency Bucharest, Romania Dan GRECU Military Equipment and Technologies Research Agency Bucharest, Romania Constantin GROZEA Military Equipment and Technologies Research Agency Bucharest, Romania Bogdan CRISTEA Military Equipment and Technologies Research Agency Bucharest, Romania

ABSTRACT In this article we present a pseudorandom binary sequence generator based on a nonlinear function witch exhibits in certain conditions a chaotic behavior. The pseudorandom generator was implemented on an 8 bit microcontroller and the generated bit stream was tested with a battery of randomness tests. Hence we are able to directly demonstrate that our generator has good statistical properties and can be used in cryptographic applications such as stream ciphers. Keywords: pseudorandom generator, nonlinear function, chaos, randomness tests, stream cipher. 1. INTRODUCTION Among the most promising applications of chaotic systems is their use in the field of chaotic encryption where the utilization of nonlinearities and the forcing of the dynamical system to a chaotic state will fulfill in our opinion the basic cryptographic requirements. Due to the nonlinear mechanisms that lead to a chaotic behavior, this one is too difficult to predict by analytical methods without the secret key (initial conditions and/or parameters) being known. Nevertheless when such chaotic systems are implemented on binary machines a major problem is the usage of floating point numbers because chaos has only meaning on real numbers. Depending upon

the precision of the numerical implementation the chaotic generator has a pseudorandom nature due to the finite precision representation. As a result, in order to ensure good statistical properties for our generator, special considerations must be taken into account. 2. THE PSEUDORANDOM GENERATOR Our goal was to design a nonconventional pseudorandom generator suited for cryptographic applications such as stream ciphers. Stream ciphers work very well for real time data such as voice and video, where only small pieces of data are known at a time. The simplest example for a stream cipher is when a bit of data is “xor-ed” with another bit of a stream generated by a pseudorandom generator. To decrypt the message the same pseudorandom bit stream is generated and the inverse operation (another “xor”) is done. x0

x0’

z-1

z-1

PWAM

0 x1

-1

x2

M

Figure 1 – Chaotic pseudorandom generator

x3

We have considered a chaotic pseudorandom generator (figure 1) with the main part represented by a discrete time chaotic map known as “Piecewise Affine Markov Map” (PWAM) [1]. The chaotic map is defined by the following equation:  B( D − x ) , x < D fM1( x) =  ,  B( x − 2 D ) , x ≥ D

(1)

where B=3, D=1, x0=0.1. The above equation is defined on the real number domain so that the initial condition (x0) and the output variable (x1) belong to the interval (-3,3). Having in mind the fact that the above chaotic map must be implemented on a finite precision machine we represent the real numbers x0, x1 in floating point single precision arithmetic (sign 1 bit, exponent 8 bits, mantissa 23 bits). The second block stands for selection logic so that only the values that lie in the subinterval (-1,0) are selected. This must be done in order to select a subinterval with a good distribution of generated values by the PWAM, because some of the chaotic properties are lost after imposing the precision of the numerical representation. M is a mapping block described by the equation: x − a  x 3 = 255 ⋅ 2 , (2) b − a   where [ ] stands for integer part function; a=-1, b=0; x0’, x3∈{0,1,…,255} are integer numbers represented on 8 bits, x0’=255. Thus each of the single precision real numbers (x2) is transformed into an 8 bit integer in order to generate the pseudorandom bit stream. The last transformation is applied on 8 bits once to ensure good statistical properties of the generated bit stream. The encryption key is represented in our implementation by the initial condition of PWAM x0 (32 bits) and the initial condition of the final “xor” operation x0’ (8 bits) that is a 40 bits key length is used. The key length can be easily increased by several methods for example by increasing the precision of the numerical representation and/or by breaking the initial key into several sub keys that are used sequentially. 3. THE DEVELOPMENT MICROSYSTEM AND RANDOMNESS TESTS The chaotic pseudorandom generator described above was implemented on an 8 bit microcontroller and the generated bit stream was tested with a battery of randomness tests.

The development microsystem has an 80C51 family microcontroller and a white noise generator. The white noise generator uses the signal obtained from a npn junction who is amplified and limited to 5 V. The signal obtained from noise generator is “read” by the microcontroller in order to obtain “the seed” for the pseudorandom generator who is software implemented in the microcontroller. The bit stream obtained from the microcontroller (as the generator output) is transferred through the serial interface to a IBM compatible personal computer and stored in binary files in order to apply the randomness tests.

IBM – PC or compatible

RS 232

80C51 FAMILY MICROCONTROLLER

White Noise

Figure 2 – The development microsystem To verify the randomness degree of the generated bit stream we have to find out how close or how far the statistical properties of the obtained bit stream are from the theoretical repartition laws (e.g. χ 2 distribution, series method, Kolmogorov). A battery of ten statistical tests [2] where used to appreciate the quality of the tested bit streams obtained with the above procedure: 1. The test no. 1: (“monobits”) 2. The test no. 2: (“dibits”) 3. The test no. 3: (“nibbles”) 4. The test no. 4: (“bytes”) 5. The test no. 5: (“classes”) 6. The test no. 6: (“poker”) 7. The test no. 7: ("binary position per tracks") 8. The test no. 8: (“serial correlations”) 9. The test no. 9: ("marks") 10. The test no. 10: ("gaps") Each of these tests classifies the tested bit stream by comparing the empirical statistical frequencies with the theoretical mean values. The deviation degree of the empirical results from the theoretical ones permits to classify the bit stream in one with very good statistical properties, with good statistical properties and one with poor statistical properties (the so called 1% and 5% threshold values).

4. EXPERIMENTAL RESULTS As an example we present here the results obtained with the above described battery of randomness tests for a bit stream of 80000 bits length.

Figure 6 – Classes test

Figure 3 – Monobits test

Figure 7 – Serial correlations tests

Figure 4 – Nibbles test

Figure 8 – Marks test

Figure 5 – Bytes test

We have done several statistical tests with different bit streams and our generator has passed all so that we have concluded that the bit stream generated with the above structure has very good

statistical properties and therefore can be used in a stream cipher.

Even if there is much work to do in this field of chaotic cryptographic algorithms we hope that with this article we have given some directions for future researches.

4. CONCLUSIONS We have shown that the proposed pseudorandom generator has good statistical properties and therefore can be used in cryptographic applications. Having in mind the properties of chaotic systems such as a random like nature, mixing properties, sensitivity to changes in initial conditions and parameters other possible applications could be realized: auto-synchronized cryptographic algorithms [3] or the implementation of nonlinear substitution boxes (S-boxes) using discretized versions of chaotic maps [4]. Is also worth to emphasize that chaos only is a necessary but not a sufficient property of encryption algorithms. For example our stream cipher has passed all statistical tests only after the final “xor” operation was introduced. So, our believe is that the best results will be obtained by mixing the classical encryption methods with chaotic ones.

5. REFERENCES [1] Delgado-Restituto, M., Rodriguez-Vasquez, A., “Piecewise Affine Markov Maps for Chaos Generation in Chaotic Communication”, ECCTD' 99, Stresa, Italy, 1999, pp. 1-5 [2] M. Radu, D. Grecu, “Programs for Testing Random Numerical Streams”, Scientific Bulletin of Politehnica University, Timisoara, 2000, pp. 1-2 [3] Marco Gotz, Kristina Kelber, Wolfgang Schwarz, “Discrete Time Chaotic Encryption Systems”, IEEE transactions on circuits and systems, vol. 44, no. 10, 1997, pp. 1-2 [4] Ljupco Kocarev, Goce Jakimoski, “Logistic map as a block encryption algorithm”, Physics Letters, October 2001, pp. 202-205