Theorem to Generate Independently and Uniformly Distributed Chaotic Key Stream via Topologically Conjugated Maps of Tent Map

Abstract

This paper proposes a theorem to generate chaotic key stream from topologically conjugated maps of Tent Map. In this theorem, the condition for topological conjugation between Tent Map and a class of chaotic maps is first determined. Then, the chaotic attractor of the maps is divided into 2^𝑛 unequal subintervals, the chaotic orbit is sampled once in 𝑛 time iteration, and, finally, the independently and uniformly distributed 2^𝑛 phase key stream is obtained. The theoretical and numerical analyses show that the chaotic key stream generated by the proposed theorem successfully is independent and uniform, has a certain complex degree close to the maximum approximate entropy for 2ⁿ phase key stream, and satisfies the randomness requirement defined in NIST SP800-22. This theorem can be used in fields such as cryptography and numerical simulation.

1. Introduction

Random number generation is very important in cryptography, especially in key construction. In symmetric and asymmetric cryptosystems, the random number is the primary resource for key generation. Further, random number generators (RNGs) are used to create challenges, nonce, padding bytes, and blinding values in many cryptographic protocols.

There are two basic types of generators that produce random sequences: true random number generators (TRNGs) [1] and pseudorandom number generators (PRNGs) [2–5]. PRNGs are generally faster than TRNGs, and, therefore, PRNGs are preferable in applications requiring a large number of random numbers.

Chaotic system, characterized by sensitive dependence on initial conditions, similarity to random behavior, continuous broad-band power spectrum, inherent determinism, and simplicity of realization, can be potentially exploited for PRNGs [6, 7]. Most applications of chaotic maps use a one- or multidimensional chaotic map as a PRNG to produce a binary stream, which is then XORed with the plaintext to produce the cipher-text [8–11]. Thus, the statistical property—independently and identically distributed (IID)—of chaotic key stream plays an important role in avoiding cipher-text attacking.

Tent Map is described by the uniform probability law, and, hence, the statistical independence of successive binary symbols is the main problem to be addressed. Some studies focus on experiments that determine the selection of a suitable Tent Map parameter for different applications where the statistical independence is of interest. The relationship between the Tent Map parameter and the statistical independence decision is indicated in [12]. Complying with the fair coin tossing model in [13], the threshold is given according to the Tent Map parameter to obtain a statistically independent key stream. Other studies have focused on theoretical proofs. Logistic Map is topologically conjugated with Tent Map [14], and, therefore, Hu et al. proposed a method for generating chaotic key stream based on Logistic Map [15]. This key stream is proved to be independent and uniform. AdrianLuca and Vlad provided another method for generating IID samples using Logistic Map by combining discrete noisy channel with the test of independence in contingency tables [16]. Until now, the research for generating IID key stream by topological conjugation has been limited to Logistic Map. However, in our study, we determine a more general condition, under which a class of chaotic systems can produce the IID key stream. Meanwhile, the proof in [14] can be considered as the example for the theorem proposed in this paper.

2. Theorem for Generation of IID Key Streams

In this section, we specify the conditions and process required for a class of topologically conjugated mapping systems of Tent Map to generate the IID key stream.

Definition 2.1 (see [17]). For two one-dimensional maps,we have 𝑥_𝑘+1=𝑔𝑥_𝑘𝑦,𝑥∈𝐼⊂𝑅,_𝑘+1=𝑔𝑦_𝑘,𝑥∈𝐽⊂𝑅. (2.1) If there exists a continuous and invertible map ℎ∶𝐼→𝐽, such that 𝑦_𝑘=ℎ(𝑥_𝑘), making ℎ⁻¹∘𝑓∘ℎ(𝑥_𝑘)=𝑔(𝑥_𝑘) and ℎ∘𝑔∘ℎ⁻¹(𝑦_𝑘)=𝑓(𝑦_𝑘), then 𝑓 and 𝑔 are said to be topologically conjugated via ℎ.

Lemma 2.2 (see [17]). If 𝑓 and 𝑔 are topologically conjugated via ℎ, then 𝑓^𝑛 and 𝑔^𝑛 are topologically conjugated via ℎ.

Lemma 2.3 (see [17]). If 𝑓 and 𝑔 are topologically conjugated via ℎ, and 𝜌_𝑔 is the probability density function of 𝑔, then the probability density function of 𝑓 is 𝜌_𝑓(𝑥)=𝜌_𝑔ℎ⁻¹(𝑥)𝑑ℎ⁻¹(𝑥).𝑑𝑥 (2.2)

Theorem 2.4. For tent map, we have𝑥_𝑘+1=𝑔𝑥_𝑘=2𝑥_𝑘,0≤𝑥_𝑘≤122−2𝑥_𝑘,12≤𝑥_𝑘≤1 (2.3) when 𝑚𝑎=−4, and 𝑚,𝑎∈𝑅, 𝑓𝑥_𝑘=𝑚𝑥²_𝑘+4𝑥_𝑘,𝑥_𝑘∈[min{0,𝑎},max{0,𝑎}], (2.4) and 𝑔(𝑥_𝑘) are topologically conjugated via ℎ𝑥_𝑘=𝑎sin²𝜋𝑥_𝑘2,𝑎≠0. (2.5)
For the map 𝑓(𝑥_𝑘)=𝑚𝑥²_𝑘+4𝑥_𝑘 defined by (2.4), we choose 𝑛 as sampling step, that is, 𝑥_𝑘+1=𝑓^𝑛(𝑥_𝑘), and can generate an independent uniformly distributed chaotic key stream {𝑠_𝑖}^∞₀ by the following process.(1)If 𝑎>0, the chaotic attractor domain [0,𝑎] is partitioned into 𝑁=2^𝑛 subdomains 𝜏_𝑖=[𝑡_𝑖,𝑡_𝑖+1), where 𝑡_𝑖=ℎ(𝑖/𝑁),𝑖=0,1,…,𝑁−1.(2)If 𝑎<0, the chaotic attractor domain [𝑎,0] is partitioned into 𝑁=2^𝑛 subdomains 𝜏_𝑖=[𝑡_𝑖,𝑡_𝑖+1), where 𝑡_𝑖=ℎ((𝑁−𝑖)/𝑁),𝑖=0,1,…,𝑁−1.(3)Then, chaotic key stream {𝑠_𝑖}^∞₀ is defined as: if 𝑥_𝑘∈𝜏_𝑖, then 𝑠_𝑘=𝑖.

Proof. (1)  𝑓(𝑥) Is Surjective
When 𝑚𝑎=−4,𝑓(𝑥_𝑘)=𝑚𝑥²_𝑘+4𝑥_𝑘=(−4/𝑎)𝑥²_𝑘+4𝑥_𝑘.  If 𝑎>0, it can be concluded that when 𝑥_𝑘∈[0,𝑎],𝑓(𝑥_𝑘)∈[0,𝑎].  If 𝑎<0, it can be concluded that when 𝑥_𝑘∈[𝑎,0],𝑓(𝑥_𝑘)∈[𝑎,0].
Thus, 𝑓(𝑥) is surjective in its definition domain [min{0,𝑎},max{0,𝑎}].
(2) Property of Topological Conjugation
If 𝑓 and 𝑔 are topologically conjugated via ℎ, then, according to Definition 2.1, ℎ⁻¹∘𝑓∘ℎ𝑥_𝑘=𝑔𝑥_𝑘,thatis,𝑓∘ℎ𝑥_𝑘=ℎ∘𝑔𝑥_𝑘. (2.6)
For Tent Map (2.3) and transformer (2.5), ℎ∘𝑔𝑥_𝑘=𝑎sin²𝜋2𝑥_𝑘2=𝑎sin²𝜋𝑥_𝑘,0≤𝑥_𝑘≤12𝑎sin²𝜋2−2𝑥_𝑘2=𝑎sin²𝜋−𝜋𝑥_𝑘=𝑎sin²𝜋𝑥_𝑘,12≤𝑥_𝑘≤1=𝑎sin²𝜋𝑥_𝑘,0≤𝑥≤1=ℎ2𝑥_𝑘. (2.7) When 𝑚𝑎=−4, for map (2.4), 𝑓𝑥_𝑘=−4𝑎𝑥²_𝑘+4𝑥_𝑘=4𝑎𝑥_𝑘𝑎−𝑥_𝑘. (2.8) Hence, 𝑓∘ℎ𝑥_𝑘=4𝑎⋅𝑎sin²𝜋𝑥_𝑘2𝑎−𝑎sin²𝜋𝑥_𝑘2=4𝑎sin²𝜋𝑥_𝑘2⋅cos²𝜋𝑥_𝑘2=𝑎2sin𝜋𝑥_𝑘2⋅cos𝜋𝑥_𝑘2²=𝑎sin²𝜋𝑥_𝑘. (2.9) Thus, 𝑓∘ℎ(𝑥_𝑘)=ℎ∘𝑔(𝑥_𝑘).
Hence, 𝑓 and 𝑔 are topologically conjugated via ℎ.
(3)  𝑓(𝑥_𝑘) Is Chaotic
To assure the key stream {𝑠_𝑖}^∞₀ produced by 𝑓(𝑥_𝑘) is chaotic pseudorandom number with the potential characters proposed in paragraph 3 in Section 1, we prove the 𝑓(𝑥_𝑘) is chaotic now.
As the reference [18–20] proposed, the Lyapunov exponents of corresponding orbits of two conjugated interval maps are the same. According to the computing method of Lyapunov exponent 𝜆𝑔,𝑥=lim_𝑛→∞1𝑛^𝑛_𝑖=1ln𝑔^′𝑥_𝑖, (2.10) if 𝜆(𝑔,𝑥) is the Lyapunov exponent of the orbit of 𝑔 through 𝑥,𝜆(𝑓,𝑦) is the Lyapunov exponent of the orbit of 𝑓 through 𝑦=ℎ(𝑥), and 𝑓∘ℎ(𝑥_𝑘)=ℎ∘𝑔(𝑥_𝑘) is the conjugation, then 𝜆(𝑔,𝑥)=𝜆(𝑓,𝑦) [18]. And the largest 𝜆(𝑔,𝑥) with respect to changes of 𝛿𝑥₀ is independent of 𝑥₀ [20].
More precisely, for the map (2.4) 𝑓(𝑦_𝑘)=𝑚𝑦²_𝑘+4𝑦_𝑘 established by Theorem 2.4, Tent Map defined by (2.3), and 𝑦_𝑘=ℎ(𝑥_𝑘)=𝑎sin²(𝜋𝑥_𝑘/2) defined by (2.5), we have 𝑓∘ℎ(𝑥_𝑘)=ℎ∘𝑔(𝑥_𝑘), and 𝑓^′(ℎ(𝑥_𝑘))⋅ℎ^′(𝑥_𝑘)=ℎ^′(𝑔(𝑥_𝑘))⋅𝑔^′(𝑥_𝑘), that is, 𝑓^′(𝑦_𝑘)⋅ℎ^′(𝑥_𝑘)=ℎ^′(𝑔(𝑥_𝑘))⋅𝑔^′(𝑥_𝑘), thus, 𝑓^′𝑦_𝑘=ℎ^′𝑔𝑥_𝑘ℎ^′𝑥_𝑘⋅𝑔^′𝑥_𝑘=(𝜋𝑎/2)sin2𝜋𝑥_𝑘(𝜋𝑎/2)sin𝜋𝑥_𝑘⋅𝑔^′𝑥_𝑘=2cos𝜋𝑥_𝑘⋅𝑔^′𝑥_𝑘. (2.11) Also, for 𝑥_𝑘 in Tent Map, we have cos(𝜋𝑥_𝑘)=cos(𝜋2𝑥_𝑘−1), then 𝜆𝑓,𝑦=lim_𝑛→∞1𝑛^𝑛_𝑖=1ln𝑓^′𝑦_𝑖=lim_𝑛→∞1𝑛^𝑛_𝑖=1ln2cos𝜋𝑥_𝑘⋅𝑔^′𝑥_𝑘=lim_𝑛→∞1𝑛^𝑛_𝑖=1ln2cos𝜋𝑥_𝑘+𝜆𝑔,𝑥=lim_𝑛→∞1𝑛ln2cos𝜋𝑥₁⋅2cos𝜋𝑥₂⋯2cos𝜋𝑥_𝑛+𝜆𝑔,𝑥=lim_𝑛→∞1𝑛lnsin𝜋𝑥₁2cos𝜋𝑥₁⋅2cos𝜋𝑥₂⋯2cos𝜋𝑥_𝑛sin𝜋𝑥₁+𝜆𝑔,𝑥=lim_𝑛→∞1𝑛lnsin𝜋𝑥₁2cos𝜋𝑥₁⋅2cos𝜋2𝑥₁⋯2cos𝜋2^𝑛𝑥₁sin𝜋𝑥₁+𝜆𝑔,𝑥=lim_𝑛→∞1𝑛lnsin𝜋2𝑥₁⋅2cos𝜋2𝑥₁⋯2cos𝜋2^𝑛𝑥₁sin𝜋𝑥₁+𝜆𝑔,𝑥=lim_𝑛→∞1𝑛lnsin𝜋2^𝑛𝑥₁sin𝜋𝑥₁+𝜆𝑔,𝑥=lim_𝑛→∞1𝑛lnsin𝜋2^𝑛𝑥₁−lim_𝑛→∞1𝑛lnsin𝜋𝑥₁+𝜆𝑔,𝑥≤lim_𝑛→∞1𝑛ln𝑛−0+𝜆𝑔,𝑥=𝜆𝑔,𝑥=ln2. (2.12)
Hence, the largest Lyapunov exponent of 𝑓(𝑥) is ln2, and the map (2.4) is a chaotic system.
(4) Chaotic Key Stream Is Uniformly Distributed
From Lemma 2.3, the probability density function of 𝑓 is 𝜌_𝑓=𝜌_𝑔ℎ⁻¹(𝑥)𝑑ℎ⁻¹(𝑥).𝑑𝑥 (2.13)
Therefore, the probability of 𝑓 in the domain [𝑡_𝑖,𝑡_𝑖+1) is ^𝑡𝑖+1_𝑡𝑖𝑑ℎ⁻¹(𝑥)𝑑𝑥𝑑𝑥=_ℎ()^{ℎ((𝑖+1)/𝑁)}_𝑖/𝑁𝑑ℎ⁻¹(𝑥)=ℎ⁻¹(𝑥)_ℎ()^{ℎ((𝑖+1)/𝑁)}_𝑖/𝑁=ℎ⁻¹ℎ𝑖+1𝑁−ℎ⁻¹ℎ𝑖𝑁=𝑖+1𝑁−𝑖𝑁=1𝑁. (2.14)
Thus, the value of chaos has the same probability in each domain, and, hence, chaotic key stream is uniformly distributed.
(5)  𝑠_𝑘 Is Independent of 𝑠_𝑘+1 (Property of Independence)
In this part, we will show that the value of 𝑠_𝑘+1 depends on the subdomains 𝜏^𝑗_𝑖=[𝑡^𝑖_𝑗,𝑡^𝑖_𝑗+1),  𝑗=0,1,…,𝑁−1,  𝑡^𝑖_𝑗=ℎ((1/𝑁)(𝑖+𝑗/𝑁)), where 𝑥_𝑘 is uniformly distributed according to the proof (4), and is independent of the 𝜏_𝑖 to which 𝑥_𝑘 belongs.
For simplicity, we will only consider the condition 𝑎>0. According to Definition 2.1, Lemma 2.2, and the sampling way 𝑥_𝑘+1=𝑓^𝑛(𝑥_𝑘), we get that, for 𝜃∈[0,1], 𝑥_𝑘=ℎ(𝜃)=𝑎sin²(𝜋𝜃/2) is increasing and has a one-to-one correspondence with 𝜃, and 𝑥_𝑘∈𝜏^𝑖_𝑗⇔𝜃∈[(1/𝑁)(𝑖+𝑗/𝑁),(1/𝑁)(𝑖+(𝑗+1)/𝑁)), then 𝑥_𝑘+1=ℎ(2^𝑛𝜃)=𝑎sin²(𝜋2^𝑛𝜃/2)∈ℎ([𝑖+𝑗/𝑁,𝑖+(𝑗+1)/𝑁)). For ℎ is an even function with period 2, then 𝑥_𝑘+1𝑗∈ℎ𝑖+𝑁,𝑖+𝑗+1𝑁=ℎ𝑗𝑁,ℎ𝑗+1𝑁ℎ,𝑖iseven𝑁−𝑗−1𝑁,ℎ𝑁−𝑗𝑁,𝑖isodd. (2.15)
Thus, when 𝑠_𝑘=𝑖, 𝑠_𝑘+1=𝑗,𝑖iseven𝑁−𝑗−1,𝑖isodd, (2.16)
that is, 𝑠_𝑘+1 is independent of 𝑠_𝑘.
Considering (1) to (5), the theorem proposes a method to establish a class of topologically conjugated maps of Tent Map and can generate independently and uniformly distributed chaotic key stream.

3. Examples of the Theorem 2.4

This section consists of the illustration of the deduction in [14], which is an example of the proposed theorem. To illustrate the effectiveness and feasibility of the proposed theorem, another example is provided to verify the chaotic property, independently and uniformly distributed property, and randomness of key streams. Due to the differences between theoretical values and calculated ones of a chaotic Logistic Map proposed in [21], all the examples are run with MATLAB (R2011a, v7.12) codes on the Window XP (32-bit) or Win7 (32-bit), and all the results are rounded into 4 digits after the decimal point.

Example 3.1. In Theorem 2.4, when 𝑎=1,  𝑚=−4, 𝑓𝑥_𝑘=−4𝑥²_𝑘+4𝑥_𝑘=4𝑥_𝑘1−𝑥_𝑘. (3.1)
This map is the typical Logistic Map 𝑓(𝑥)=𝑢𝑥(1−𝑥), when 𝑢=4. It is used in [14] for the generation of key stream. According to Theorem 2.4, when 𝑚=−4, the ℎ(𝑥)=sin²(𝜋𝑥_𝑘/2), then the proof of Deduction 1 in [14] can be considered as an example of the theorem proposed in this paper.

Example 3.2. In Theorem 2.4, we choose 𝑎=8,𝑚=−1/2, and another chaotic map is obtained as follows: 𝑓𝑥_𝑘1=−2𝑥²_𝑘+4𝑥_𝑘, (3.2) and the transformer is ℎ𝑥_𝑘=8sin²𝜋𝑥_𝑘2. (3.3) We choose 𝑛=8 and divide the domain [0,𝑎]=[0,8] into 2^𝑛=256 intervals, each intervals' beginning pot and ending pot is shown in Figure 1. Not similar to the Tent Map, the partition of the attractor domain for i.i.d key stream generation is symmetric and is not uniform equal.

Figure 1 

Partition of the attractor domain.

(1) Chaotic Property of Map (3.2)

For map (3.2), we set the initial condition 𝑥₀=0.2323, and the results of 1,000 iterations are shown in Figure 2. It is observed that the values are nonperiodic and are distributed in almost the entire domain.

Figure 2 

1,000 iterations of map (3.2).

(2) Independently and Uniformly Distributed Property of {𝑠_𝑖}^∞₀

(a)  𝜒² Test [22]

For the chaotic map (3.2) established by Theorem 2.4, we choose 𝑥₀=0.2323 and sampling step 𝑛=8. After first 5,000 iteration abandoned, we get the 8-phase key stream {𝑠_𝑖}^𝑀₀, where 𝑀 is the lengths of sequence.

We set significance level 𝛼=0.05, then for 𝑀 more than 100000, we get the same threshold 𝜒²_𝛼(𝑀)=293.2478, and the results of 𝜒² test are shown in Table 1, where all the results pass the test successfully.

Table 1 

Results of uniform distribution of key stream {𝑠_𝑖}^𝑀₀ by map (3.2).

Then we choose 100 random 𝑥₀ and set 𝑀 as 20,000 to do 𝜒² test. From the results in Figure 3, it can be observed that most of the tests passed and only 3 tests did not pass. As indicated in [12], the choice of initial value has an important impact on the statistical property of the chaotic system. Therefore, we can conclude that the key stream generated from Theorem 2.4 is uniformly distributed under the appropriate initial value.

Figure 3 

Result of 𝜒2 tests with 100 random number 𝑥1.

(b) Approximate Entropy Analysis of {𝑠_𝑖}^∞₀

For the chaotic map (3.2) established by Theorem 2.4, we choose 𝑥₀=0.2323 and sampling step 𝑛=8. After first 5,000 iteration abandoned, we get the 8-phase key stream {𝑠_𝑖}^𝑀₀, where 𝑀 is the lengths of sequence.

According to the algorithm of approximate entropy in [23], we choose length of compared run lcr=2, filtering level 𝑟=0.25SD, where SD is the standard deviation of {𝑠_𝑖}^𝑀₀. For different 𝑀, we compute the approximate entropy of {𝑠_𝑖}^𝑀₀ as shown in Table 2, where all the approximate entropies are close to the largest approximate entropy ln8=2.0794 for 8-ary key stream, which is inferred in [24] and follows Theorem 2 in [23].

Table 2 

Approximate entropy analysis of {𝑠_𝑖}^𝑀₀ with different 𝑀.

Then we choose 100 random 𝑥₀ and set 𝑀 as 6,000 with the same lcr=2 and 𝑟=0.25SD to compute the approximate entropy. From the results in Figure 4, it can be observed that the mean approximate entropy of {𝑠_𝑖}^𝑀₀ is close to the maximum approximate entropy ln8=2.0794 which shows that the sequence produced by Theorem 2.4 has a certain complex degree.

Figure 4 

Result of approximate entropy with 100 random number 𝑥1.

(3) Randomness Test of Key Streams {𝑠_𝑖}^∞₀

In this part, we change {𝑠_𝑖}^∞₀ into binary sequences to carry out the test of NIST SP800-22 (April 2010), whose 15 tests depict the deviations of a binary sequence from randomness [25]. 𝑃 value, as the probability against the null hypothesis, represents the degree of randomness of the tested sequence. When we choose significance level 𝛼=0.01, if the value is bigger than 𝛼=0.01, it demonstrates that the sequence passes the test and could be considered as random. And the bigger the value is, the more random the sequences are.

First we use the chaotic map (3.2) established by Theorem 2.4 and choose 𝑥₀=0.2323, sampling step 𝑛=8. After first 5,000 iteration abandoned, we get the 8-phase key stream {𝑠_𝑖}^𝑀₀, where 𝑀 is the lengths of sequence and 𝑀=125000. Then we test the binary sequences {𝑏_𝑖}₀^1000000, which is changed from 8-phase sequence {𝑠_𝑖}^𝑀₀, with NIST SP800-22. The results are shown in Table 3, implying that the key streams produced by theorem can satisfy the random requirements in NIST SP800-22.

Table 3 

SP800-22 test’s results.

Then we choose 100 random 𝑥₀ and set 𝑀=125000 to do SP800-22 test, and Table 4 lists the mean 𝑃 value and the passing ratio. For the significance level 𝛼 set as 0.01, it means that 99% of test samples pass the tests if the random numbers are truly random. The acceptance region of the passing ratio is given by [𝑝−3𝑝(1−𝑝)/𝑙,𝑝+3𝑝(1−𝑝)/𝑙], where 𝑙 represents the number of the samples tested and 𝑝=1−𝛼 is the probability of passing each test [26]. For 𝑙=100 and 𝑝=0.99, we obtain the confidence interval [0.9602,1.0198], that is, [0.9602,1]. From Table 4, we can get that the computed passing ratio for each test lies inside the confidence interval.

Table 4 

Mean 𝑃 value and the passing ratio in SP800-22 tests with 100 random 𝑥₀.

4. Conclusion

In this paper, we propose a new theorem for a class of topologically conjugated maps of Tent Map to generate independently and uniformly distributed key streams. Two examples are provided to validate that the key stream generated by the proposed theorem is theoretically and experimentally provedto beindependently and uniformly distributed. We also conducted experiments for testing the randomness of these key streams, and all the key streams passed the NIST SP800-22 test. In future, this theorem could be applied to information security, numerical simulation, and other fields.