# Research on Pseudorandom Number Generator Based on Several New Types of Piecewise Chaotic Maps

Authors: Hongyan Zang; Yue Yuan; Xinyuan Wei
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper proposes three types of one-dimensional piecewise chaotic maps and two types of symmetrical piecewise chaotic maps and presents five theorems. Furthermore, some examples that satisfy the theorems are constructed, and an analysis and model of the dynamic properties are discussed. The construction methods proposed in this paper have a certain generality and provide a theoretical basis for constructing a new discrete chaotic system. In addition, this paper designs a pseudorandom number generator based on piecewise chaotic map and studies its application in cryptography. Performance evaluation shows that the generator can generate high quality random sequences efficiently.


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## Body

## 1. Introduction

Chaos, as a nonlinear deterministic dynamic system with external complex phenomena due to its inherent randomness, has been widely considered and intensively studied. In the 1870s, the mathematician Li and his tutor Yorke put forward the definition of chaos in the famous article “Period Three Contains Chaos” [1], that is, the Li–Yorke chaos discrimination theorem. Then, a chaotic discrimination theorem, namely, Marotto’s theorem, that is more suitable for high-dimensional discrete dynamic systems, was given by Marotto [2]. These two theorems provide an important theoretical basis on which later scholars have studied one-dimensional discrete chaotic systems [2–4]. Since 1989, chaotic systems have been widely used in the field of secure communication due to their sensitivity to the initial values or parameters, ergodicity, and randomness-like properties [5–8]. After constructing a new chaotic system, we can use it to design a new chaotic pseudorandom number generator (CPRNG). Therefore, it is of great theoretical and practical significance to study and construct a new chaotic system.In many studies on one-dimensional discrete chaotic systems, piecewise chaotic mapping has gradually attracted scholars’ attention. Among them, piecewise linear chaotic mapping has a simpler form in chaotic systems, and because of its relatively good uniformity, it is easy to implement by fixed point algorithms with limited digital precision, which is convenient for its application in the fields of cryptography and communication [9]. As a typical one-dimensional piecewise linear discrete system, the tent map is a basic example for the promotion and application of chaotic theory systems. Many scholars have constructed other chaotic systems based on the tent map. In [10], a new one-dimensional piecewise chaotic map was constructed by combining a tent map with a logistic map. In [11], a class of oblique tent maps was improved, and it was proved that the chaotic system has excellent dynamic key space and practicability, making it more suitable for secure communication and other fields [12], based on the deformation of a tent map, provides piecewise linear chaotic mapping, and uses the period three theorem and topological conjugation theory to construct quadratic polynomial chaotic mapping and realize homogenization of a chaotic sequence.For the study of piecewise nonlinear mapping, most approaches involve theoretical analyses of a system’s own dynamic characteristics and periodic phenomena [13, 14] or piecewise transformations based on known chaotic mapping. In [15], a class of one-dimensional piecewise nonlinear discrete dynamic systems under modulo operation is constructed, and the improved Marotto’s theorem is used to give a general theory of the chaotic behavior of piecewise nonlinear functions with nonzero origin. Reference [16] built a piecewise nonlinear chaotic mapping system based on logistic mapping so that the system parameters have a larger value range and better uniformity. Reference [17] discussed the branch of chaotic attractors in piecewise smooth one-dimensional mapping with a large number of switching manifolds based on different nonlinear smoothing models and applied it in the field of electronic science.In the field of cryptography, the research of pseudorandom number generator based on chaotic system mainly focuses on the following aspects: proposing new chaotic system and designing controller to realize chaotic synchronization [18] and improving the existing chaotic system to enhance its complexity and make it have greater Lyapunov [19, 20]. Some mathematical methods are used to improve the random performance of pseudorandom number generator [21], the software and hardware implementation of pseudorandom number generator [22], and the encryption scheme and cryptosystem based on chaotic pseudorandom number generator [23, 24]. The one-dimensional discrete chaotic system has the advantages of simple structure and easy realization. Therefore, it is an important content to construct a large number of general one-dimensional discrete chaotic systems.The structure of this article is as follows. Section2 constructs several types of general one-dimensional discrete piecewise maps, and based on the Li–Yorke discriminant theorem and Marotto’s theorem, sufficient conditions for chaotic mapping are given. One-dimensional discrete piecewise chaotic nonlinear mapping and a numerical simulation are performed. Section 3 gives two types of segmented chaotic mapping models with symmetry, and based on the proposed models, three examples of chaotic mapping satisfying the conditions are given. In Section 4, a new pseudorandom number generator based on piecewise chaotic map is designed, and the randomness and key sensitivity of the generator are analyzed. Finally, we conclude the full text in Section 5.


## 2. Three Types of One-Dimensional Discrete Piecewise Chaotic Maps

First, we introduce the Li–Yorke chaos discrimination theorem, which is expressed as follows.Lemma 1 (see [1]).
LetJ be an interval and let f:J⟶J be continuous. If there is a point a∈J for which the points b=fa, c=fb, and d=fc satisfy d≤a<b<c (or d≥a>b>c), then it is a chaotic map in the sense of Li–Yorke.
In addition, we introduce other related theories. LetBrx∗ be a closed ball with point x∗ as the center and radius r, if the fixed point x∗ of the differentiable map g in Rn satisfies the following two conditions:(1)
There is a real numberr>0, such that the modulus of all the eigenvalues of the Jacobian matrix Dgx of any point x in Brx∗ is greater than 1.(2)
There is a pointx0≠x∗ and a natural number m≥2 in Brx∗ such that gmx0=x∗, and point x0 satisfies detDgmx0≠0.Then, the fixed pointx∗ is a regressive repulsor of the mapping g [3].Lemma 2 (Marotto’s theorem [2]).
If then-dimensional map g:Rn⟶Rn has a regressive repellent, then the map g has chaotic behavior in the Li–Yorke sense.Next, based on the above two lemmas, several kinds of one-dimensional discrete piecewise chaotic maps are given.


### 2.1. Construction of One-Dimensional Discrete Piecewise Chaotic Map

Theorem 1.
Letf be a continuously differentiable strictly monotonically increasing function on the closed interval 0,1, let a be a real number on the open interval (0, 1), and define a function g1 of the following form:(1)g1x=fx/a−f0f1−f0,0≤x≤a,f1−x/1−a−f0f1−f0,a<x≤1.If the functiong1 satisfies g1′x>1 in the interval 0,a∪a,1, then g1 is a chaotic map in the sense of Li–Yorke.Proof of Theorem 1.
The functionf is continuous in the interval 0,1, and g1 is a continuous function in J:0,1⟶J:0,1. Therefore, it is only necessary to prove that there are four points satisfying the conditions of the Li–Yorke theorem:(2)g1y3=y4≤y1<g1y1=y2<y3=g1y2.
LetFx=g1x−x, then F′x=g1′x−1, and in the interval 0,a, F′x=g1′x−1≥0 can be obtained from g1′x>1.
Therefore, in the interval0,a, Fx=g1x−x>F0=0, that is, g1x>x.
g1 is a unimodal function, and its function image structure is shown in Figure 1.
Takey2=a, y3=g1y2=g1a=1, and y4=g1y3=g11=0. Obviously there is(3)y4<y2<y3.
Because ofg10=0<y2=a<1=g1y2, from the continuity of g1 and the intermediate value theorem ∃y1∈0,y2, if y2=g1y1, then(4)y4<y1<y2.
Combining equations (3) and (4), there are four points satisfying the conditions of the Li–Yorke theorem:(5)g1y3=y4≤y1<g1y1=y2<y3=g1y2.
In conclusion, the functiong1 satisfies the Li–Yorke discriminant theorem; thus, it is a chaotic map in the sense of Li–Yorke.
Analogous to the piecewise chaotic map constructed by Theorem1, another one-dimensional discrete piecewise chaotic map is constructed below.Figure 1
Rough function diagram ofg1.Theorem 2.
Letf be a continuously differentiable strictly monotonically increasing function on the closed interval 0,1 and a be a real number on the open interval (0, 1), and define a function g2 of the following form:(6)g2x=1−fx/a−f0f1−f0,0≤x≤a,1−f1−x/1−a−f0f1−f0,a<x≤1.If the functiong2 satisfies g2′x>1 in the interval 0,a∪a,1, then g2 is a chaotic map in the sense of Li–Yorke.The proof process is similar. The function image structure ofg2 is shown in Figure 2.Next, we discuss the relationship between Theorems1 and 2. Let formula (1) be Ax and formula (3) be Bx; we can know when a=0.5 has Ax=1−Bx; Ax and Bx are symmetric piecewise mappings.Based on this, we can find a homeomorphic mappingCx:Cx=1−x, having(7)C∘A=C1−Bx=1−1+Bx=Bx,B∘C=BCx=B1−x,and because we know that Bx function is obviously symmetric with respect to x=0.5, then there is Bx=B1−x on 0,1, so C∘A=B∘C holds. Then, there is a topological conjugate relationship between the mappings of Theorems 1 and 2. Furthermore, from the fact that homeomorphic mapping Cx is a linear mapping, we can see that Theorems 1 and 2 satisfy affine conjugation and therefore have the same dynamic behavior.Figure 2
Rough function diagram ofg2.Theorem 3.
Letf be a continuously differentiable strictly monotonically increasing function on the closed interval 0,1 and a be a real number on the open interval (0, 1), and define a function g3 of the following form:(8)g3x=fx/a−f0f1−f0,0≤x<a,1−f1−x/1−a−f0f1−f0,a≤x≤1.If the functiong3 satisfies g3′x>1 in the interval 0,a∪a,1, then g3 is a chaotic map in the sense of Li–Yorke.Proof of Theorem 3.
According to the conditions, the function image structure of functiong3 is shown in Figure 3.
According to the theorem,g30=0; thus, x∗=0 is the fixed point of g3x. Since the derivative of function g3 on [0, 1] is greater than 1, x∗=0 is a repulsive fixed point.(1)
Consider the auxiliary functionh1x=g3x, with x∈a,1:Choosex1∈a,1 that satisfies hx1=g3x1=x∗=0 to obtain x1=h−1x∗=g3−1x∗=a.(2)
Consider the auxiliary functionh2x=g3x−x1, x∈0,a:
Becauselimx⟶a−h2x=limx⟶a−g3x=1>a=x1, there is δ0>0, and for ∀x∈a−δ0,a, there is always h2x=g3x>x1.
Takex¯∈a−δ0,a; then, hx¯=g3x¯−x1>0, and h0=g30−x1=−x1<0.
According to the one-dimensional intermediate value theorem, there isx0∈0,x¯ such that hx0=g3x0−x1=0, that is, g3x0=x1. Then,(9)g3x0=x1⟶g3g3x0=g3x1=x∗⟶g32x0=x∗.
In conclusion, the fixed pointx∗=0 of g3x satisfies the following conditions:(1)
Taker=x1 and satisfy x0∈0,x¯∈0,x1 such that Brx∗ becomes a closed ball including x∗. Its Jacobian matrix is Dgx=g3′x>1. Thus, for any x∈Brx∗, the modulus of the eigenvalue of Dgx is greater than 1.(2)
There exists a point0<x0<1 and a natural number m=2 in Brx∗ such that g3mx0=x∗, and x0 is nondegenerate, that is, it satisfies(10)detDgmx0=detg3x1⋅detg3x0=g3′x1⋅g3′x0≠0.
Therefore, the pointx∗ is a regressive repulsor of map g3 and thus is a chaotic map in the sense of Li–Yorke.
The proof is over. Next, we verify the above three theorems by a numerical simulation.Figure 3
Rough function diagram ofg3.


### 2.2. Numerical Simulation of One-Dimensional Discrete Piecewise Chaotic Map

Substitutingfx=sinex−1 into Theorem 1, the image of function g1, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 4(a)–4(c). In Theorem 2, the image of function g2, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 4(d)–4(f). In Theorem 3, the image of function g3, the bifurcation graph of parameter a and the Lyapunov exponent graph are as shown in Figures 4(g)–4(i).Figure 4
(a–c) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by makingfx=sinex−1 in Theorem 1. (d–f) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=sinex−1 in Theorem 2. (g–i) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=sinex−1 in Theorem 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Substitutingfx=e0.01sinx into Theorem 1, the image of function g1, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 5(a)–5(c). In Theorem 2, the image of function g2, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 5(d)–5(f). In Theorem 3, the image of function g3, the bifurcation graph of parameter a and the Lyapunov exponent graph are as shown in Figures 5(g)–5(i).Figure 5
(a–c) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by makingfx=e0.01sinx in Theorem 1. (d–f) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=e0.01sinx in Theorem 2. (g–i) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=e0.01sinx in Theorem 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Substitutingfx=0.3xcosx+0.1ex into Theorem 1, the image of function g1, the bifurcation diagram of parameter a, and the Lyapunov exponent diagram are as shown in Figures 6(a)–6(c). In Theorem 2, the image of function g2, the bifurcation diagram of parameter a, and the Lyapunov exponent diagram are as shown in Figures 6(d)–6(f). In Theorem 3, the image of function g3, the bifurcation graph of parameter a and the Lyapunov exponent graph are as shown in Figures 6(g)–6(i).Figure 6
(a–c) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by makingfx=0.3xcosx+0.1ex in Theorem 1. (d–f) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=0.3xcosx+0.1ex in Theorem 2. (g–i) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=0.3xcosx+0.1ex in Theorem 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 2.1. Construction of One-Dimensional Discrete Piecewise Chaotic Map

Theorem 1.
Letf be a continuously differentiable strictly monotonically increasing function on the closed interval 0,1, let a be a real number on the open interval (0, 1), and define a function g1 of the following form:(1)g1x=fx/a−f0f1−f0,0≤x≤a,f1−x/1−a−f0f1−f0,a<x≤1.If the functiong1 satisfies g1′x>1 in the interval 0,a∪a,1, then g1 is a chaotic map in the sense of Li–Yorke.Proof of Theorem 1.
The functionf is continuous in the interval 0,1, and g1 is a continuous function in J:0,1⟶J:0,1. Therefore, it is only necessary to prove that there are four points satisfying the conditions of the Li–Yorke theorem:(2)g1y3=y4≤y1<g1y1=y2<y3=g1y2.
LetFx=g1x−x, then F′x=g1′x−1, and in the interval 0,a, F′x=g1′x−1≥0 can be obtained from g1′x>1.
Therefore, in the interval0,a, Fx=g1x−x>F0=0, that is, g1x>x.
g1 is a unimodal function, and its function image structure is shown in Figure 1.
Takey2=a, y3=g1y2=g1a=1, and y4=g1y3=g11=0. Obviously there is(3)y4<y2<y3.
Because ofg10=0<y2=a<1=g1y2, from the continuity of g1 and the intermediate value theorem ∃y1∈0,y2, if y2=g1y1, then(4)y4<y1<y2.
Combining equations (3) and (4), there are four points satisfying the conditions of the Li–Yorke theorem:(5)g1y3=y4≤y1<g1y1=y2<y3=g1y2.
In conclusion, the functiong1 satisfies the Li–Yorke discriminant theorem; thus, it is a chaotic map in the sense of Li–Yorke.
Analogous to the piecewise chaotic map constructed by Theorem1, another one-dimensional discrete piecewise chaotic map is constructed below.Figure 1
Rough function diagram ofg1.Theorem 2.
Letf be a continuously differentiable strictly monotonically increasing function on the closed interval 0,1 and a be a real number on the open interval (0, 1), and define a function g2 of the following form:(6)g2x=1−fx/a−f0f1−f0,0≤x≤a,1−f1−x/1−a−f0f1−f0,a<x≤1.If the functiong2 satisfies g2′x>1 in the interval 0,a∪a,1, then g2 is a chaotic map in the sense of Li–Yorke.The proof process is similar. The function image structure ofg2 is shown in Figure 2.Next, we discuss the relationship between Theorems1 and 2. Let formula (1) be Ax and formula (3) be Bx; we can know when a=0.5 has Ax=1−Bx; Ax and Bx are symmetric piecewise mappings.Based on this, we can find a homeomorphic mappingCx:Cx=1−x, having(7)C∘A=C1−Bx=1−1+Bx=Bx,B∘C=BCx=B1−x,and because we know that Bx function is obviously symmetric with respect to x=0.5, then there is Bx=B1−x on 0,1, so C∘A=B∘C holds. Then, there is a topological conjugate relationship between the mappings of Theorems 1 and 2. Furthermore, from the fact that homeomorphic mapping Cx is a linear mapping, we can see that Theorems 1 and 2 satisfy affine conjugation and therefore have the same dynamic behavior.Figure 2
Rough function diagram ofg2.Theorem 3.
Letf be a continuously differentiable strictly monotonically increasing function on the closed interval 0,1 and a be a real number on the open interval (0, 1), and define a function g3 of the following form:(8)g3x=fx/a−f0f1−f0,0≤x<a,1−f1−x/1−a−f0f1−f0,a≤x≤1.If the functiong3 satisfies g3′x>1 in the interval 0,a∪a,1, then g3 is a chaotic map in the sense of Li–Yorke.Proof of Theorem 3.
According to the conditions, the function image structure of functiong3 is shown in Figure 3.
According to the theorem,g30=0; thus, x∗=0 is the fixed point of g3x. Since the derivative of function g3 on [0, 1] is greater than 1, x∗=0 is a repulsive fixed point.(1)
Consider the auxiliary functionh1x=g3x, with x∈a,1:Choosex1∈a,1 that satisfies hx1=g3x1=x∗=0 to obtain x1=h−1x∗=g3−1x∗=a.(2)
Consider the auxiliary functionh2x=g3x−x1, x∈0,a:
Becauselimx⟶a−h2x=limx⟶a−g3x=1>a=x1, there is δ0>0, and for ∀x∈a−δ0,a, there is always h2x=g3x>x1.
Takex¯∈a−δ0,a; then, hx¯=g3x¯−x1>0, and h0=g30−x1=−x1<0.
According to the one-dimensional intermediate value theorem, there isx0∈0,x¯ such that hx0=g3x0−x1=0, that is, g3x0=x1. Then,(9)g3x0=x1⟶g3g3x0=g3x1=x∗⟶g32x0=x∗.
In conclusion, the fixed pointx∗=0 of g3x satisfies the following conditions:(1)
Taker=x1 and satisfy x0∈0,x¯∈0,x1 such that Brx∗ becomes a closed ball including x∗. Its Jacobian matrix is Dgx=g3′x>1. Thus, for any x∈Brx∗, the modulus of the eigenvalue of Dgx is greater than 1.(2)
There exists a point0<x0<1 and a natural number m=2 in Brx∗ such that g3mx0=x∗, and x0 is nondegenerate, that is, it satisfies(10)detDgmx0=detg3x1⋅detg3x0=g3′x1⋅g3′x0≠0.
Therefore, the pointx∗ is a regressive repulsor of map g3 and thus is a chaotic map in the sense of Li–Yorke.
The proof is over. Next, we verify the above three theorems by a numerical simulation.Figure 3
Rough function diagram ofg3.


## 2.2. Numerical Simulation of One-Dimensional Discrete Piecewise Chaotic Map

Substitutingfx=sinex−1 into Theorem 1, the image of function g1, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 4(a)–4(c). In Theorem 2, the image of function g2, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 4(d)–4(f). In Theorem 3, the image of function g3, the bifurcation graph of parameter a and the Lyapunov exponent graph are as shown in Figures 4(g)–4(i).Figure 4
(a–c) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by makingfx=sinex−1 in Theorem 1. (d–f) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=sinex−1 in Theorem 2. (g–i) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=sinex−1 in Theorem 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Substitutingfx=e0.01sinx into Theorem 1, the image of function g1, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 5(a)–5(c). In Theorem 2, the image of function g2, the bifurcation diagram of parameter a and the Lyapunov exponent diagram are as shown in Figures 5(d)–5(f). In Theorem 3, the image of function g3, the bifurcation graph of parameter a and the Lyapunov exponent graph are as shown in Figures 5(g)–5(i).Figure 5
(a–c) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by makingfx=e0.01sinx in Theorem 1. (d–f) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=e0.01sinx in Theorem 2. (g–i) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=e0.01sinx in Theorem 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Substitutingfx=0.3xcosx+0.1ex into Theorem 1, the image of function g1, the bifurcation diagram of parameter a, and the Lyapunov exponent diagram are as shown in Figures 6(a)–6(c). In Theorem 2, the image of function g2, the bifurcation diagram of parameter a, and the Lyapunov exponent diagram are as shown in Figures 6(d)–6(f). In Theorem 3, the image of function g3, the bifurcation graph of parameter a and the Lyapunov exponent graph are as shown in Figures 6(g)–6(i).Figure 6
(a–c) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by makingfx=0.3xcosx+0.1ex in Theorem 1. (d–f) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=0.3xcosx+0.1ex in Theorem 2. (g–i) The function image, bifurcation diagram, and Lyapunov exponent diagram obtained by making fx=0.3xcosx+0.1ex in Theorem 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 3. Two Types of Piecewise Chaotic Maps with Symmetric Properties

### 3.1. Construction of Symmetric Piecewise Chaotic Map

Based on the Li–Yorke chaotic discrimination theorem, a class of piecewise chaotic maps with symmetric properties are given below.Theorem 4.
Letf be a continuous differentiable strictly monotone increasing function on a closed interval 0,1, and define the following piecewise mapping h1:(11)h1x=f2x−1−f0f1−f0=f−2x+1−f0f1−f0,0≤x≤0.5,f2x−1−f0f1−f0,0.5<x≤1.If the functionh1 satisfies h1′x=2f′2x−1/f1−f0>1 on 0.5,1, then h1 is a chaotic map in the sense of Li–Yorke.Proof of Theorem 3.
From the fact thatf is continuous in the interval 0,1, it is known that h1 is a continuous function on J:0,1⟶J:0,1. Therefore, it is necessary to prove that there exist four points satisfying the conditions of the Li–Yorke theorem:(12)gy3=y4≤y1<gy1=y2<y3=gy2.
LetFx=h1x−x, then F′x=h1′x−1, and in the interval 0.5,1, F′x=h1′x−1>0 can be obtained from h1′x=2f′2x−1/f1−f0>1.
Therefore,Fx=h1x−x≤F1=0, that is, on the interval 0.5,1, h1x≤x.
Combined with the functionh1 as a unimodal function, taking y2=1/2, y3=h1y2=h11/2=0, and y4=h1y3=h10=1, obviously(13)y4>y2>y3.
Moreover,h11/2=0<y2=1/2<1=h11. Then, by the continuity of h1 and the intermediate value theorem of continuous function ∃y1∈y2,1, such that y2=gy1, one obviously has(14)y4>y1>y2.
Combining equations (13) and (14) shows that there are four points that satisfy the conditions of the Li–Yorke theorem:(15)h1y3=y4≥y1>h1y1=y2>y3=h1y2.
In conclusion, the functionh1 satisfies the Li–Yorke discriminant theorem and is a chaotic map in the sense of Li–Yorke. Equation (11) is obviously symmetric about the straight line x=0.5; thus, the function h1 has symmetry.
Similar to Theorem4, another class of one-dimensional discrete piecewise chaotic maps with symmetric properties is given below.Theorem 5.
Letf be a continuous differentiable strictly monotone increasing function on a closed interval 0,1, and define the following piecewise mapping h2:(16)h2x=1−f2x−1−f0f1−f0=1−f−2x+1−f0f1−f0,0≤x≤0.5,1−f2x−1−f0f1−f0,0.5<x≤1.If the functionh2 satisfies h2′x=2f′−2x+1/f1−f0>1 on 0,0.5, then h2 is a chaotic map in the sense of Li–Yorke.Obviously, the mappings of Theorems4 and 5 also have topological conjugation and satisfy affine conjugation. Therefore, the mappings of Theorems 4 and 5 have the same dynamic behavior.The following constructs different functionsf and inserts them into the above two theorems, yielding specific examples of the constructed functions through numerical simulation.


### 3.2. Numerical Simulation of Symmetric Piecewise Chaotic Map

Insertingfx=x1/2 into Theorem 4, the function image and the time-domain waveform of h1 are as shown in Figures 7(a) and 7(b); if instead substituted into Theorem 5, the function image and the time-domain waveform of h2 are as shown in Figures 7(c) and 7(d).Figure 7
Insertingfx=x1/2 into Theorems 4 and 5, and the function image (a, b) and the time-domain waveform diagram (c, d) are obtained.
(a)(b)(c)(d)Insertingfx=x5+x into Theorem 4, the function image and the time-domain waveform of h1 are as shown in Figures 8(a) and 8(b); inserting it into Theorem 5, the function image and the time-domain waveform of h2 are as shown in Figures 8(c) and 8(d).Figure 8
Insertingfx=x5+x into Theorems 4 and 5, the function image (a, b) and the time-domain waveform diagram (c, d) are obtained.
(a)(b)(c)(d)Insertingfx=x⋅10sinx+ex into Theorem 4, the function image and time-domain waveform of h1 are as shown in Figures 9(a) and 9(b); if instead substituted into Theorem 5, the function image and time-domain waveform of h2 are as shown in Figures 9(c) and 9(d).Figure 9
Insertingfx=x⋅10sinx+ex into Theorems 4 and 5, the function image (a, b) and the time-domain waveform diagram (c, d) are obtained.
(a)(b)(c)(d)


## 3.1. Construction of Symmetric Piecewise Chaotic Map

Based on the Li–Yorke chaotic discrimination theorem, a class of piecewise chaotic maps with symmetric properties are given below.Theorem 4.
Letf be a continuous differentiable strictly monotone increasing function on a closed interval 0,1, and define the following piecewise mapping h1:(11)h1x=f2x−1−f0f1−f0=f−2x+1−f0f1−f0,0≤x≤0.5,f2x−1−f0f1−f0,0.5<x≤1.If the functionh1 satisfies h1′x=2f′2x−1/f1−f0>1 on 0.5,1, then h1 is a chaotic map in the sense of Li–Yorke.Proof of Theorem 3.
From the fact thatf is continuous in the interval 0,1, it is known that h1 is a continuous function on J:0,1⟶J:0,1. Therefore, it is necessary to prove that there exist four points satisfying the conditions of the Li–Yorke theorem:(12)gy3=y4≤y1<gy1=y2<y3=gy2.
LetFx=h1x−x, then F′x=h1′x−1, and in the interval 0.5,1, F′x=h1′x−1>0 can be obtained from h1′x=2f′2x−1/f1−f0>1.
Therefore,Fx=h1x−x≤F1=0, that is, on the interval 0.5,1, h1x≤x.
Combined with the functionh1 as a unimodal function, taking y2=1/2, y3=h1y2=h11/2=0, and y4=h1y3=h10=1, obviously(13)y4>y2>y3.
Moreover,h11/2=0<y2=1/2<1=h11. Then, by the continuity of h1 and the intermediate value theorem of continuous function ∃y1∈y2,1, such that y2=gy1, one obviously has(14)y4>y1>y2.
Combining equations (13) and (14) shows that there are four points that satisfy the conditions of the Li–Yorke theorem:(15)h1y3=y4≥y1>h1y1=y2>y3=h1y2.
In conclusion, the functionh1 satisfies the Li–Yorke discriminant theorem and is a chaotic map in the sense of Li–Yorke. Equation (11) is obviously symmetric about the straight line x=0.5; thus, the function h1 has symmetry.
Similar to Theorem4, another class of one-dimensional discrete piecewise chaotic maps with symmetric properties is given below.Theorem 5.
Letf be a continuous differentiable strictly monotone increasing function on a closed interval 0,1, and define the following piecewise mapping h2:(16)h2x=1−f2x−1−f0f1−f0=1−f−2x+1−f0f1−f0,0≤x≤0.5,1−f2x−1−f0f1−f0,0.5<x≤1.If the functionh2 satisfies h2′x=2f′−2x+1/f1−f0>1 on 0,0.5, then h2 is a chaotic map in the sense of Li–Yorke.Obviously, the mappings of Theorems4 and 5 also have topological conjugation and satisfy affine conjugation. Therefore, the mappings of Theorems 4 and 5 have the same dynamic behavior.The following constructs different functionsf and inserts them into the above two theorems, yielding specific examples of the constructed functions through numerical simulation.


## 3.2. Numerical Simulation of Symmetric Piecewise Chaotic Map

Insertingfx=x1/2 into Theorem 4, the function image and the time-domain waveform of h1 are as shown in Figures 7(a) and 7(b); if instead substituted into Theorem 5, the function image and the time-domain waveform of h2 are as shown in Figures 7(c) and 7(d).Figure 7
Insertingfx=x1/2 into Theorems 4 and 5, and the function image (a, b) and the time-domain waveform diagram (c, d) are obtained.
(a)(b)(c)(d)Insertingfx=x5+x into Theorem 4, the function image and the time-domain waveform of h1 are as shown in Figures 8(a) and 8(b); inserting it into Theorem 5, the function image and the time-domain waveform of h2 are as shown in Figures 8(c) and 8(d).Figure 8
Insertingfx=x5+x into Theorems 4 and 5, the function image (a, b) and the time-domain waveform diagram (c, d) are obtained.
(a)(b)(c)(d)Insertingfx=x⋅10sinx+ex into Theorem 4, the function image and time-domain waveform of h1 are as shown in Figures 9(a) and 9(b); if instead substituted into Theorem 5, the function image and time-domain waveform of h2 are as shown in Figures 9(c) and 9(d).Figure 9
Insertingfx=x⋅10sinx+ex into Theorems 4 and 5, the function image (a, b) and the time-domain waveform diagram (c, d) are obtained.
(a)(b)(c)(d)


## 4. Design of PRNG

In order to design the pseudorandom number generator using the above construction system, we first give a chaotic example satisfying the chaotic condition.Corollary 1.
Takefx=sinx+p⋅x, a=0.5, then when p≥sin2, the chaos condition can be satisfied in Theorems 1–5, so that equations (1)–(8) are chaotic maps.Therefore, ifp=0.04>sin2 and a=0.5 are introduced into formula (1), then(17)g1x=2sinx+0.08xsin1+0.08,0≤x≤0.5,2sin1−x−0.08x+0.08sin1+0.08,0.5<x≤1.It is a chaotic system.Based on system (17), a transformation of binary pseudorandom sequence is proposed,(18)TranXk=modroundXk,256,sk=binaryTranXk,where L=2552×108, function roundX means to round x to get an integer, modx,n means to modulo n operation on x, and function binaryx means to convert integer x into binary number.


### 4.1. Randomness Test of PRNG

PRNG plays an important role in most chaotic cryptosystems. Because PRNG with good performance is unpredictable and similar to pseudorandom sequence, it has good statistical performance. The following mainly used NIST SP800-22 detection standard [25] proposed by NIST to test the random property of binary pseudorandom sequences generated by chaotic system.According to system (17), we select 1000 groups of different parameters and initial values, generate 1000 groups of different binary pseudorandom sequences by PRNG, and test the randomness of NIST SP800-22. The results are shown in Table 1.Table 1
The test results of NIST SP800-22.
No.Test indexPRNG based on system (17)ProportionP valueResult1Frequency0.99300.544254Success2Block frequency0.99200.869278Success3Cumulative sums10.99200.361938Success4Runs0.99100.966626Success5Longest run0.99200.757790Success6Rank0.99200.916599Success7FFT0.9880.973718Success8Nonoverlapping template10.98100.678686Success9Overlapping template0.99000.516113Success10Universal0.99000.721777Success11Approximate entropy0.98600.295391Success12Random excursions10.99830.186164Success13Random excursion variant10.98110.202783Success14Serial10.99200.984881Success15Linear complexity0.99100.735908Success1The test item contains several submodules, of which the worst results are listed here.In Table1, we give the pass rate of the detection sequence and the P value of the uniformity test (denoted as the UP value). If the pass rate is in the interval 1−α−3α/n,1−α+3α/n and all UP values are greater than α, the PRNG is considered to have passed the detection, where α is the significant level, α=0.01 is taken here.It can be seen from Table1 that the passing rate of each data of PRNG based on system (17) is within the acceptable range, and it has passed the NIST SP800-22 randomness test, so it is suitable for the design of PRNG and has good randomness.


### 4.2. Key Sensitivity Analysis

Key sensitivity means that the small change of key will also lead to the substantial change of output. A well-designed PRNG should have good key sensitivity; even if a bit changes, it will output completely different sequences. Therefore, it is necessary to detect and analyze the key sensitivity.After fixingn=1050, we disturb the initial value of the key. For the three generated sequences, the first 20 arrays are extracted and plotted in Figure 10. It is obvious that the sensitivity of the key to the initial value is above 10−16, and slight changes in the initial value will lead to great differences in the sequence. Therefore, the above test results and analysis show that our generator has strong key sensitivity.Figure 10
The sequences under different initial secret key.


## 4.1. Randomness Test of PRNG

PRNG plays an important role in most chaotic cryptosystems. Because PRNG with good performance is unpredictable and similar to pseudorandom sequence, it has good statistical performance. The following mainly used NIST SP800-22 detection standard [25] proposed by NIST to test the random property of binary pseudorandom sequences generated by chaotic system.According to system (17), we select 1000 groups of different parameters and initial values, generate 1000 groups of different binary pseudorandom sequences by PRNG, and test the randomness of NIST SP800-22. The results are shown in Table 1.Table 1
The test results of NIST SP800-22.
No.Test indexPRNG based on system (17)ProportionP valueResult1Frequency0.99300.544254Success2Block frequency0.99200.869278Success3Cumulative sums10.99200.361938Success4Runs0.99100.966626Success5Longest run0.99200.757790Success6Rank0.99200.916599Success7FFT0.9880.973718Success8Nonoverlapping template10.98100.678686Success9Overlapping template0.99000.516113Success10Universal0.99000.721777Success11Approximate entropy0.98600.295391Success12Random excursions10.99830.186164Success13Random excursion variant10.98110.202783Success14Serial10.99200.984881Success15Linear complexity0.99100.735908Success1The test item contains several submodules, of which the worst results are listed here.In Table1, we give the pass rate of the detection sequence and the P value of the uniformity test (denoted as the UP value). If the pass rate is in the interval 1−α−3α/n,1−α+3α/n and all UP values are greater than α, the PRNG is considered to have passed the detection, where α is the significant level, α=0.01 is taken here.It can be seen from Table1 that the passing rate of each data of PRNG based on system (17) is within the acceptable range, and it has passed the NIST SP800-22 randomness test, so it is suitable for the design of PRNG and has good randomness.


## 4.2. Key Sensitivity Analysis

Key sensitivity means that the small change of key will also lead to the substantial change of output. A well-designed PRNG should have good key sensitivity; even if a bit changes, it will output completely different sequences. Therefore, it is necessary to detect and analyze the key sensitivity.After fixingn=1050, we disturb the initial value of the key. For the three generated sequences, the first 20 arrays are extracted and plotted in Figure 10. It is obvious that the sensitivity of the key to the initial value is above 10−16, and slight changes in the initial value will lead to great differences in the sequence. Therefore, the above test results and analysis show that our generator has strong key sensitivity.Figure 10
The sequences under different initial secret key.


## 5. Conclusions

In this work, we construct three types of one-dimensional discrete piecewise maps, and based on the Li–Yorke discriminant theorem and Marotto’s theorem, we provide sufficient conditions for these three types of maps to become chaotic maps. Then, we designf and further construct several specific examples, after which numerical simulations are carried out. The bifurcation diagram and Lyapunov exponent diagram of the function with the change in parameters are given.In view of the construction method of the one-dimensional discrete piecewise chaotic maps proposed in this paper, considering the piecewise chaotic map with symmetric properties, we construct the functions on this basis. Sufficient conditions for them to become chaotic maps are given, several examples satisfying the theorem conditions are given, the theory is verified by numerical simulation, and the design idea is proved to be correct. This method can provide a theoretical basis for further constructing a new one-dimensional discrete chaotic system.Finally, from the perspective of cryptographic application, the PRNG algorithm proposed in this paper is tested. The test results show that the PRNG designed in this paper has passed the sp800-22 randomness test, and the test index value is equivalent to that of the literature [18, 19, 22, 23]. The random performance and key sensitivity are analyzed. The results show that the proposed pseudorandom sequence generator can meet the performance requirements of good PRNG and has strong key sensitivity. Therefore, the PRNG proposed in this paper is practical and reliable and can further design a high security encryption scheme, which has high application potential.



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*Source: 1375346-2021-08-20.xml*