A One-dimensional Discrete Map Chaos Criterion Theorem with Applications in Pseudo-random Number Generator

Hongyan ZANG; Jiu LI; Guodong LI

doi:10.11999/JEIT171139

Volume 40 Issue 8

Aug. 2018

Turn off MathJax

Article Contents

Article Navigation > Journal of Electronics & Information Technology > 2018 > 40(8): 1992-1997

Hongyan ZANG, Jiu LI, Guodong LI. A One-dimensional Discrete Map Chaos Criterion Theorem with Applications in Pseudo-random Number Generator[J]. Journal of Electronics & Information Technology, 2018, 40(8): 1992-1997. doi: 10.11999/JEIT171139

Citation:

Hongyan ZANG, Jiu LI, Guodong LI. A One-dimensional Discrete Map Chaos Criterion Theorem with Applications in Pseudo-random Number Generator[J]. Journal of Electronics & Information Technology, 2018, 40(8): 1992-1997. doi: 10.11999/JEIT171139

Citation:

PDF( 1974 KB)

A One-dimensional Discrete Map Chaos Criterion Theorem with Applications in Pseudo-random Number Generator

doi: 10.11999/JEIT171139

1.
Mathematics and Physics School, University of Science and Technology Beijing, Beijing 100083, China
2.
College of Applied Mathematics, Xinjiang University of Finance and Economics, Urumchi 830012, China

Funds: The National Natural Science Foundation of China (11461063), The Xinjiang Uygur Autonomous Region Natural Science Foundation (2017D01A24)

Received Date: 2017-12-04
Rev Recd Date: 2018-05-02

Available Online: 2018-06-07

Publish Date: 2018-08-01

Abstract

Abstract

A novel one-dimensional discrete chaotic criterion is firstly constructed by studying the modular operation of the discrete dynamical systems. The judgement of the Marotto theorem is used to prove that the suggested dynamical systems are chaotic. Secondly, several special chaotic systems satisfied with the conditions of this paper are given, and the bifurcation diagram and Lyapunov exponential spectrum are also analyzed. Numerical simulations show that the proposed chaotic systems have the positive Lyapunov exponent, which indicates the accuracy of the proposed theory. Additionally, a Pseudo-Random Number Generator (PRNG) is also designed based on the given new chaotic system. Using SP800-22 test suit, the results show that the output sequence of PRNG has good pseudorandom. Finally, as an application of the PRNG, an image encryption algorithm is given. The proposed encryption scheme is highly secure Key space of 2⁷⁴⁷ and can resist against the statistical and exhaustive attacks based on the experimental results.
- Chaotic criterion,
- Marotto theorem,
- Snap-back repeller,
- Pseudo-Random Number Generator (PRNG),
- Image encryption

FullText(HTML)

References(15)

References

LI T Y and YORKE J A. Period three implies chaos[J]. American Mathematical Monthly, 1975, 82(10): 985–992. DOI: 10.2307/2318254.

YU Xingmei, MIN Lequan, and CHEN Tianyu. Chaos criterion on some quadric polynomial maps and design for chaotic pseudorandom number generator[C]. Seventh International Conference on Natural Computation, Shanghai, 2011: 1373–1376.

周海玲, 宋恩彬. 二次多项式映射的3-周期点判定[J]. 四川大学学报(自然科学版), 2009, 46(3): 561–564. DOI: 103969/j.issn.0490-6756.2009.03-009.

ZHOU Hailing and SONG Enbin. Discrimination of the 3-periodic points of a quadratic polynomial[J]. Journal of Sichuan University(Natural Science Edition), 2009, 46(3): 561–564. DOI: 103969/j.issn.0490-6756.2009.03-009.

YANG Xiuping, MIN Lequan, and WANG Xue. A cubic map chaos criterion theorem with applications in generalized synchronization based pseudorandom number generator and image encryption[J]. Chaos, 2015, 25(5): 053104. DOI: 10.1063/1.4917380.

MAROTTO F R. Snap-back repellers imply chaos in Rn[J]. Journal of Mathematical Analysis & Applications, 1978, 63(1): 199–223. DOI: 10.1016/0022-247X(78)90115-4.

CHEN Guangrong and LAI Dejian. Feedback control of lyapunov exponents for discrete-time dynamical systems[J]. International Journal of Bifurcation & Chaos, 1996, 6(7): 1341–1349. DOI: 10.1142/S021812749600076X.

HAN Dandan, MIN Lequan, and CHEN Guangrong. A stream encryption scheme with both key and plaintext avalanche effects for designing chaos-based pseudorandom number generator with application to image encryption[J]. International Journal Bifurcation & Chaos, 2016, 26(5): 1650091-1. DOI: 10.1142/S0218127416500917.

韩丹丹, 闵乐泉, 赵耿. 八维广义同步系统在伪随机数发生器中的应用[J]. 电子与信息学报, 2016, 38(5): 1158–1165. DOI: 10.11999/JEIT150899.

HAN Dandan, MIN Lequan, and ZHAO Geng. Application of 8-dimensional generalized synchronization system in pseudorandom number generator[J]. Journal of Electronics & Information Technology, 2016, 38(5): 1158–1165. DOI: 10.11999/JEIT150899.

RUKHIN A, SOTO J, NECHVATAL J, et al. A statistical test suite for random and pseudorandom number generators for cryptographic applications[R]. National Institute of Standards and Technology Special Publication, 2010.

LI Pei, MIN Lequan, ZANG Hongyan, et al. A generalized chaos synchronization-based pseudo-random generator number and performance analysis[C]. International Conference on Communications Circuits and Systems, Chengdu, China, 2010: 781–785.

WANG Xingyuan, LIU Chuanming, XU Dahai, et al.. Image encryption scheme using chaos and simulated annealing algorithm[J]. Nonlinear Dynamics, 2016, 84(3): 1417–1429. DOI: 10.1007/s11071-015-2579-y.

LI Yueping, WANG Chunhua, and CHEN Hua. A hyper-chaos-based image encryption algorithm using pixel-level permutation and bit-level permutation[J]. Optics & Lasers in Engineering, 2017, 90: 238–246. DOI: 10.1016/j.optlaseng.2016.10.020.

WANG Xingyuan, LIU Chuanming, and ZHANG Huili. An effective and fast image encryption algorithm based on chaos and interweaving of ranks[J]. Nonlinear Dynamics, 2016, 84(3): 1595–1607. DOI: 10.1007/s11071-015-2590-3.

GUESMI R, FARAH M A B, KACHOURI A, et al.. A novel chaos-based image encryption using DNA sequence operation and secure hash algorithm SHA-2[J]. Nonlinear Dynamics, 2016, 83(3): 1123–1136. DOI: 10.1007/s11071-015-2392-7.

BELAZI A, EL-LATIF A A A, DIACONU A V, et al.. Chaos-based partial image encryption scheme based on linear fractional and lifting wavelet transforms[J]. Optics & Lasers in Engineering, 2017, 88: 37–50. DOI: 10.1016/j.optlaseng.2016.07.010.

Relative Articles

Supplements(0)

Cited By

Proportional views