一個(gè)一維離散混沌判定定理及其在偽隨機(jī)數(shù)發(fā)生器中的應(yīng)用
doi: 10.11999/JEIT171139
-
1.
北京科技大學(xué)數(shù)理學(xué)院 ??北京 ??100083
-
2.
新疆財(cái)經(jīng)大學(xué)應(yīng)用數(shù)學(xué)學(xué)院 ??烏魯木齊 ??830012
A One-dimensional Discrete Map Chaos Criterion Theorem with Applications in Pseudo-random Number Generator
-
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
-
摘要: 該文研究了一類取模運(yùn)算的1維離散動(dòng)力系統(tǒng),提出了一個(gè)這類離散映射的混沌判據(jù),利用Marotto定理證明了其混沌的存在性。給出了幾個(gè)滿足該判據(jù)的特殊形式的系統(tǒng),分析了其分岔圖、Lyapunov指數(shù)譜等基本動(dòng)力學(xué)性質(zhì),通過模擬結(jié)果驗(yàn)證了理論的正確性?;谛孪到y(tǒng)設(shè)計(jì)了一個(gè)偽隨機(jī)數(shù)發(fā)生器(PRNG), SP800-22隨機(jī)性檢測(cè)結(jié)果表明了該序列具有良好的偽隨機(jī)性。進(jìn)一步給出了一個(gè)圖像加密方案,其密鑰空間可以達(dá)到2747。該文提出的新系統(tǒng)的系統(tǒng)參數(shù)可以無窮多,所以理論上該加密方案的密鑰空間可以無窮大。
-
關(guān)鍵詞:
- 混沌判據(jù) /
- Marotto定理 /
- 返回?cái)U(kuò)張不動(dòng)點(diǎn) /
- 偽隨機(jī)數(shù)發(fā)生器 /
- 圖像加密
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. -
表 1 SP800-22隨機(jī)性檢驗(yàn)結(jié)果
檢測(cè)項(xiàng)目 給定初始值 參數(shù)擾動(dòng)100次 P-值 檢測(cè)結(jié)果 擬合優(yōu)度P-值 通過率 頻率測(cè)試 0.1493 通過 0.8978 0.97 塊內(nèi)頻率測(cè)試 0.6436 通過 0.3041 0.99 向前累積和測(cè)試 0.2777 通過 0.3191 0.97 向后累積和測(cè)試 0.1971 通過 0.8832 0.98 游程測(cè)試 0.3875 通過 0.1453 0.97 塊內(nèi)最長(zhǎng)連續(xù)1測(cè)試 0.5564 通過 0.4559 1.00 二元矩陣秩測(cè)試 0.0523 通過 0.9463 1.00 離散傅里葉變換測(cè)試 0.1445 通過 0.4373 0.99 非重疊模板匹配測(cè)試 0.5761 通過 0.7598 0.99 重疊模板匹配測(cè)試 0.2961 通過 0.1719 0.98 全局通用統(tǒng)計(jì)測(cè)試 0.2253 通過 0.9963 0.99 近似熵檢測(cè) 0.5118 通過 0.6579 1.00 隨機(jī)偏移測(cè)試 0.0246 通過 0.7598 1.00 隨機(jī)偏移變量測(cè)試 0.5542 通過 0.0072 0.98 線性復(fù)雜度測(cè)試 0.4339 通過 0.2023 0.99 串行測(cè)試 0.9780 通過 0.4190 0.98 下載: 導(dǎo)出CSV
-
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. 二次多項(xiàng)式映射的3-周期點(diǎn)判定[J]. 四川大學(xué)學(xué)報(bào)(自然科學(xué)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. 韓丹丹, 閔樂泉, 趙耿. 八維廣義同步系統(tǒng)在偽隨機(jī)數(shù)發(fā)生器中的應(yīng)用[J]. 電子與信息學(xué)報(bào), 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. -