A Generalized Stability Theorem for Discrete-Time Nonautonomous Chaos System with Applications

Firstly, this study introduces a definition of generalized stability (GST) in discrete-time nonautonomous chaos system (DNCS), which is an extension for chaos generalized synchronization. Secondly, a constructive theorem of DNCS has been proposed. As an example, a GST DNCS is constructed based on a novel 4-dimensional discrete chaotic map. Numerical simulations show that the dynamic behaviors of this map have chaotic attractor characteristics. As one application, we design a chaotic pseudorandom number generator (CPRNG) based on the GST DNCS. We use the SP800-22 test suite to test the randomness of four 100-key streams consisting of 1,000,000 bits generated by the CPRNG, the RC4 algorithm, the ZUC algorithm, and a 6-dimensional CGSbased CPRNG, respectively. The numerical results show that the randomness performances of the two CPRNGs are promising. In addition, theoretically the key space of the CPRNG is larger than 2. As another application, this study designs a stream avalanche encryption scheme (SAES) in RGB image encryption.The results show that the GST DNCS is able to generate the avalanche effects which are similar to those generated via ideal CPRNGs.


Introduction
Chaos, characterized by its deterministic, unpredictable features and extremely sensitive dependence on initial conditions, stems from nonlinear systems (e.g., see [1][2][3]).During the last three decades, chaos theory has been developed to model complex nature and social phenomena by using quite simple mathematical models.Thus it has captured much attention of the scientific community for predicting the behavior of systems in the real world.
Nonautonomous discrete systems were introduced in [4]; as we can see, they also appear connected to some nonautonomous difference equations (e.g., see [5,6]).A lot of natural questions concerned the nonautonomous dynamical systems; therefore the research of the nonautonomous system is recently very intensive (e.g., see [7][8][9]).However, there is no much research on discrete nonautonomous chaotic systems.
Mathematically chaos synchronization (CS) means that the trajectories of two different chaotic systems exhibit identical phenomena with time evolution.Synchronization phenomenon is a kind of typical collective behaviors that could be found in many physical, biological, and engineering systems (e.g., see [10][11][12][13][14][15][16]).
Chaos generalized synchronization (CGS) means that, with time evolution, the trajectories of two different chaotic systems tend to become identical with respect to a transformation in a specific domain.Therefore CGS has more general meaning than CS.The study of CGS has also attracted much attention (e.g., see [17][18][19][20][21][22][23][24][25]).
Generally speaking, there are different methods such that two systems achieve generalized synchronization such as design control laws to force coupled systems to satisfy a prescribed functional relation [26][27][28].In series papers [29][30][31][32], we have studied the general representations of two systems to achieve GS.
This paper extends the concept of the generalized synchronization [38] to generalized stability (GST) for discretetime nonautonomous chaos system (DNCS), and then we propose a corresponding GST theorem.Using the GST theorem helps design a novel nonautonomous chaotic discrete 2 Mathematical Problems in Engineering system and construct a chaos-based pseudorandom number generator (CPRNG).We test the randomness of the CPRNG, the RC4 algorithm, the ZUC algorithm [39], and a 6-dimensional CGS-based CPRNG2 [40] by the SP800-22 test suite of the INST [41], respectively.At last, as an application, by using the CPRNG and the SAES a RGB image has been encrypted in communication.
The rest of this paper is organized as follows.Section 2 introduces the definition and the theorem of GST.Section 3 presents a novel 4-dimensional nonautonomous chaotic discrete system and an 8-dimensional GST system and simulates the dynamic behaviors of the GST system.Section 4 designs the CPRNG and makes the statistic tests for the CPRNG, the RC4 algorithm, the ZUC algorithm, and the 6-dimensional CGS-based CPRNG2, respectively.An image encryption example of the CPRNG and the SAES is introduced in Section 5. Finally, Section 6 presents some concluding remarks.

Definition and Theorem of GST
A point of view states that two events with relationship of cause and effect might be described via CGS for two systems.Motivated by CGS, let us introduce the concept of GST.Definition 1.Consider two systems where If there exists a transformation  : R  × Z + → R  , where for ∀ > 0, there exist  1 > 0 and  2 > 0 such that all trajectories of (1) and (2) with initial conditions (X(0), Then the systems in (1) and ( 2) are said to be in GST with respect to the transformation (X(), ).System (1) is called the driving system; system (2) is said to be the driven system.
In order to construct the novel DNCS with the GST property, we present the following.Theorem 2. Let X, Y, X  , (X, ), and (Y, X, ) be defined by (3) Then Therefore, two dynamic systems (1) and ( 2) are in GST via the transformation , if and only if the function (X  , Y, ) makes the zero solution of the error equation ( 9) stable.This completes the proof.
which is an invertible matrix.Define a transformation  : R 4 × Z + → R 4 as follows: where Step 3. First, we need the following lemma.
In fact, define a positive definite function and then Therefore, ΔV (2,1) (e, ) is a negative semidefinite function.
According to Lemma 3,(19) is zero solution stable.Hence we conclude, from Theorem 2, that the constructed dynamic systems (12) and (17) are in GST with respect to transformation (15) for any initial value (X(0), Now we choose (22) as the initial condition: The first 5000 iterations of variable components of Y are obtained as shown in Figure 3.The evolution of state variables, − 1 (), − 2 (), − 3 (), and − 4 (), is shown in Figure 4.The simulation result shows that the system has chaotic attractor characteristics.And Figure 5 shows that X and Y are rapidly in GST with respect to transformation (X, ) = X + ().
In summary, both theory and numerical simulation shows that the constructed dynamic systems ( 12) and ( 17) are in GST.

Randomness Test.
The NIST SP800-22 test suite [41] consists of 15 statistical tests (see the first column in Table 2) that were developed to test the randomness of binary sequences produced by either hardware or software based cryptographic random or pseudorandom number generators [41].Each statistical test is formulated to test a specific null hypothesis ( 0 ): the sequence being tested is random.A significance level () can be chosen for the tests.If  value ≥, then the null hypothesis is accepted; that is, the sequence appears to be random.Typically,  is chosen in the range [0.001, 0.01].NIST SP800-22 test suite is stricter than the FIPS140-2 test suite NIST [43].A binary sequence which has passed all tests of FIPS140-2 test suite may not pass all tests in the NIST SP800-22 test suite.
In order to test the pseudorandomness of the CPRNG, we transform the "16-bit" stream defined by (25) to the {0, 1} bit stream as follows.
Construct a transform  2 : {0, 1, . . ., 2 16 − 1} → {0, 1}, which is defined by  "zeros(1, )" is a zero row vector of dimension .Consequently, the RC4 algorithm-based PRNG is designed.Then, the SP800-22 test suite is used to test 100 keystreams randomly generated by the RC4 PRNG.The statistical test results are listed in the 3rd column of Tables 1 and 2. Now, we compare all test results shown in Tables 1 and 2. It can be clearly observed, from the statistical properties of the pseudorandomness of the sequences generated via the four PRNGs, that all the CPRNGs have satisfactory, indeed very promising, randomness properties.

Key Space.
The key set parameters of CPRNG includes the initial condition X(0), Y(0), and the matrix  = ( , ).It can be proved that if the perturbation matrix Δ = ( , ) satisfies | , | < 0.07419, the matrix  + Δ is still invertible.Therefore the CPRNGs have 4 + 4 + 16 key parameters denoted by Let the key set be perturbed by First, we compare the difference between the keystream  =  1 (S) including 10 6 codes' length generated by key set (30) and keystreams    s generated by the perturbed key  3. Observe that the average percentage of different codes is 49.995% and the mean correlation coefficient is 2.8295 × 10 −7 .The results may suggest that the keystream  has no significant correlations with the perturbed keystreams    s and the streams    s.Furthermore, the key space of the CPRNG is larger than 10 14×24 > 2 1116 , which will help to increase security.

Simulations on Avalanche Image Encryption
This study designs a stream encryption scheme with avalanche effect.

Conclusions
The main results of this paper are concluded as follows.
(1) This study first introduces the definition of GST in DNCS, which is an extension for chaos generalized synchronization in DNCS, and then proposes a constructive GST theorem, which provides a general representation for GST DNCS.
(2) As an example, a novel 4-dimensional nonautonomous discrete chaotic map and an 8-dimensional GST system are constructed, whose trajectories display chaotic attractor characteristics.
(4) The key space of the CPRNG is larger than 2 1116 , which is large enough against brute-force attacks.
(5) An image encryption example on the CPRNG with the SAES is given.It shows that the CPRNG is able to generate significant avalanche effects.The numerical simulations also show that the CPRNG is a qualified candidate for SAES.
Furthermore, to prevent attacks, one can consider a "onetime-pad" scheme: let  be a set in the seed space (initial conditions) of the CPRNG, and assume that Alice and Bob share a one-to-one map  :  → .Before each communication, Alice randomly selects an element  ∈  and sends it to Bob.Then, they both use () as the seed for onetime encryption.
In summary, based on generalized synchronization methods, GST and corresponding theorem are introduced.They have been successfully used to design CPRNG with sound randomness for practical applications.It is expected that the GST theory deserves further detailed investigations on its theoretical analyses and numerical simulations.

Table 1 :
The calculated mean -values of SP800-22 statistical tests for the 100 binary sequences with length 10 6 produced by the CPRNG proposed in this paper, the RC4 PRNG, the ZUC algorithm and the CPRNG2, respectively.Select a significance level  = 0.01.

Table 2 :
The acceptance rates of SP800-22 statistical tests for the 100 binary sequences with length 10 6 produced by the CPRNG proposed in this paper, the RC4 PRNG, the ZUC algorithm, and the CPRNG2, respectively.Here select a significance level  = 0.01.

Table 3 :
The statistic data for the percentages of the codes of the keystream variations between  and    s, as well as  and    s.

Table 3 .
Observe that the average percentage of different codes is 50.001% and the mean correlation coefficient is 0.00071329.It is very closed to the ideal different value 50% and ideal correlation coefficient 0.Then, let us compare the same keystream  with the 100 keystreams    s generated by the function of Matlab command randi([0 1], 1, 10 6 ).The comparison results are shown in fourth column in Table