A Bidirectional Generalized Synchronization Theorem-Based Chaotic Pseudorandom Number Generator

In order to design good pseudorandom number generator, using a bidirectional generalized synchronization theorem for discrete chaos system, this paper introduces a new 5-dimensional bidirectional generalized chaos synchronization system (BGCSDS), whose prototype is a novel chaotic system. Numerical simulation showed that two pair variables of the BGCSDS achieve generalized chaos synchronization via a transform H. A chaos-based pseudorandom number generator (CPNG) was designed by the new BGCSDS. Using the FIPS-140-2 tests issued by the National Institute of Standard and Technology (NIST) verified the randomness of the 1000 binary number sequences generated via the CPNG and the RC4 algorithm respectively. The results showed all the tested sequences passed the FIPS140-2 tests. The confidence interval analysis showed the statistical properties of the randomness of the sequences generated via the CPNG and the RC4 algorithm do not have significant differences. So, the CPNG is suitable to be used in the information security filed.


Introduction
As a nonlinear dynamics phenomenon, chaos has many properties to be worthwhile use, such as pseudo-random characteristics, the unpredictability of the orbit, and the extreme sensitivity of the initial state and so on [1].The feature of chaotic systems which makes them suitable for generating pseudo-random sequence is important.One way is to use a single chaotic map, such as Tent map [2], Henon map [3] and so on [4]- [5].Another way is to extend or compound some common chaotic maps, such as Logistic map [6]- [7], Henon map [8].By summarizing the literatures, it is found that the extension methods are simple addition, improved and coupled mostly.Using the existing chaotic theorem to extend the chaotic maps is still in some sense.
Based on a bidirectional generalized synchronization theorem for discrete chaos system in [15], a novel BGCSDS was introduced.By a transformation from the real set to the integer set, a chaos-based pseudo-random number generator (CPNG) was designed.The key set was initial condition and system parameters of the BGCSDS.
Let the key set be perturbed randomly by  for 1000 times where -16 -5 10 < Δ <10 .We verify the randomness of the binary number sequences by the FIPS 140-2 tests, and compare the correlation coefficients and the percentages of the different bits between two different key steams.The outputs of the CPNG are passed the tests.Comparing with the confidence intervals between the CPNG and the RC4 algorithm, it shows that the randomness of the sequences generated via the two ways do not have significant differences.

Algorithm
To design the CPNG based on GCS for bidirectional discrete system, some basic concepts are introduced. where If there exists a transformation  : n m H R R and a subset such that all trajectories of (1), (2) with intial conditions  ( (0), (0)) , then the systems in (1) and ( 2) are said to be in GS with respect to the transformation H.

Research Method
In this section, a novel discrete chaotic system could be constructed using the Theorem 1, and then a new CPNG based on the BGCSDS is designed.The last procedure is to verify the randomness of binary sequence generated via the CPNG.All the simulations are implemented using Matlab 7.0.

A New BGCS Discrete System
Suppose the discrete chaotic system has the form [16]: Its largest Lyapunov exponent is 0.964763, which shows the system is chaotic.Let the chaotic system (5) be amended as follow: ( 1) ( ) where Based on the Theorem 1, the system ( 5) is extended to 5-dimensional system, where   ( ), ( ), ( ) ( ), ( ) where Y k H X k makes the error equation (4) zero asymptotically stable.

Chaotic Pseudo-random Number Generator
In order to transform the streams of the BGCSDS into key streams with integers . Let the transform T be defined by: where Then the binary sequence ( ) s k can be obtained by transferring ( ( )) T X k into binary codes: Hence, we obtain a chaotic pseudo-random number generator (CPNG).

Results and Discussion
The chaotic trajectories of the BGCSDS and the characteristics of the pseudorandom sequence generated via the formula (10) are simulated and analyzed in this section.

Chaotic trajectories of The BGCSDS
In system (6), we select the following parameters and initial conditions (0) (0.5,0.5,0.5 a a a a a a .By calculating the Lyapunov exponents of the 5-dimensional system, two positive Lyapunov exponents 0.69507 and 0.48318 can be obtained.It shows the system (6) is chaotic.Then the trajectories of system ( 6) and ( 8) are shown in Figure 1.From (d), it can be seen that the two pair of variables are in GCS with respect to H, so it is in line with expectations.

The Equilibrium Analysis
In a randomly generated N-bit sequence, we would expect approximately half of the bits in the sequence to be ones and approximately half to be zeros.Here, we denote the number of 0 and 1 as 0 N and 1 N respectively.The equilibrium analysis checks whether the number of ones in the sequences is significantly different from / 2 N , which also is named frequency test.We choose , the sequence passes the frequency test.That is, the sequence generated by (10) has sound equilibrium.

The Correlation Analysis
The correlation analysis are main analysis of auto-correlation function and crosscorrelation function.The auto-correlation function can be calculated by the following formula: which investigates the predictability of the sequence.If the auto-correlation coefficient is 0, the sequence is unpredictable and random.The auto-correlation function of ideal pseudo random sequence is  .While the cross-correlation function can be defined by: The cross correlation function describes the correlation between the two pseudorandom sequences.The number is more close to 0, the difference degree is greater.In formula (12), { } i s and ' { } i s represent two different sequences respectively, while mean s represents the mean of the sequence.Simulating the correlation characteristics of the sequence generated via (10), the results are shown in Figure 2. It can be seen that the auto-correlation is of  -like and the cross correlation is sound.

FIPS 140-2 Test
At present, there are some representative randomness testing standards, such as FIPS 140-2 test, SP800-22 test, and Marsaglia's Diehard battery test.Generally speaking, the pseudo-random sequences which pass these tests are of good randomness.In this paper, FIPS 140-2 is used to verify the randomness of the binary sequence.The test consists of four subtests.Each test needs 20000 binary {0, 1} codes.The test is passed if the tested values are fallen into the required intervals listed in required space in Tables 2 and 3, in which MT, PT and LT represent the Monobit test, Poker test, Longest Run test, respectively; LR stands for the length of the run.Define the significant level  =0.00001 , and prove the runs of the above 256 binary sequences have the normal distribution property.So the confidence intervals of the runs can be calculated by the following formula.
The RC4 algorithm is widely used in popular protocols.Although some defects are found in the key, the binary stream generated by the RC4 has a good random performance.Comparing the confident intervals of 1000 key streams generated by the CPNG and the RC4 algorithm respectively, the results are also shown in Tables 2 and 3.It follows that the confidence interval of the 1000 key streams generated by the two methods are of similar size.Based on the above analysis, it can be seen that the randomness of the sequences generated via the CPNG and the RC4 algorithm do not have significant differences.Besides, we compare the results of the statistical tests with the results described in [6].In [6], a pseudorandom binary sequence generator was proposed based on a combination of two logistic maps.As a result it turned out that over 95% of the sequences passed the test of FIPS 140-2.For the binary number sequences generated via the CPNG, it turned out all the tested sequences passed the test.It shows that the proposed method to extend chaotic maps is effective.

Conclusion
This paper presents a novel 5-dimensional bidirectional discrete chaotic system with the GS property.The trajectories of the novel BGCSDS has two positive Lyapunov exponets.The simulations show that the dynamics of the BGCSDS appear obviously chaotic characteristics.Based on the new BGCSDS, a CPNG is designed by a transform T. The pseudorandom number sequences generated by the CPNG via different keys are different at mean value 49.9961%, and have mean correlation coefficient 0.00569.The values are very close to the ideal value 50% and 0. The confidence interval analysis of FIPS 140-2 test showed the sequences generated via the CPNG pass the FIPS 140-2 test, and do not have significant differences with the sequences generated via the RC4 algorithm.It can be expected that the CPNG are promising for information security encryption.


ISSN: 1693-6930 TELKOMNIKA Vol.11, No. 2, June 2013: 409 -416 410 chaotic synchronization (GCS) is one of the focal research topics in CS, which provides a new tool for constructing secure communication systems can be gotten by the GCS of the variables   1 2 3

Table 1 .
The computed mean of the percentages of the different bits is about 49.9961%, which is very close to the ideal value 50%.While the mean of the correlation coefficients is 0.00569.The above results imply that two different key streams are almost completely independent.Consequently we can assume that the key space of the CPNG is large than

Table 1 .
The percents of the variations and the correlation coefficients among 1000 group key streams

Table 2 .
The MT, PT and LT results of 1000 key streams generated by the CPNG and the RC4 algorithm

Table 3 .
The run test results of 1000 key streams generated by the CPNG and the RC4 algorithm