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

臧鴻雁; 李玖; 李國東

doi:10.11999/JEIT171139

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

基金項(xiàng)目: 國家自然科學(xué)基金(11461063)，新疆維吾爾自治區(qū)自然科學(xué)基金(2017D01A24)

詳細(xì)信息

作者簡(jiǎn)介:
臧鴻雁：女，1973年生，副教授，研究方向?yàn)榛煦缦到y(tǒng)理論及混沌密碼學(xué)

李玖：男，1995年生，碩士生，研究方向?yàn)榛煦缦到y(tǒng)理論及圖像加密

李國東：男，1972年生，教授，研究方向?yàn)榧?xì)胞神經(jīng)網(wǎng)絡(luò)和混沌密碼學(xué)

通訊作者:
李國東? lgdzhy@126.com

中圖分類號(hào): O415.5; TP309.7
計(jì)量
- 文章訪問數(shù): 2187
- HTML全文瀏覽量: 672
- PDF下載量: 75
- 被引次數(shù): 0
出版歷程
- 收稿日期: 2017-12-04
- 修回日期: 2018-05-02
- 網(wǎng)絡(luò)出版日期: 2018-06-07
- 刊出日期: 2018-08-01

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

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

摘要

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。
- 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.
- Chaotic criterion /
- Marotto theorem /
- Snap-back repeller /
- Pseudo-Random Number Generator (PRNG) /
- Image encryption

HTML全文

圖 1 系統(tǒng)式(15)的分岔圖

下載: 全尺寸圖片幻燈片

圖 2 系統(tǒng)式(15)的Lyapunov指數(shù)譜

下載: 全尺寸圖片幻燈片

圖 3 系統(tǒng)式(16)的分岔圖

下載: 全尺寸圖片幻燈片

圖 4 系統(tǒng)式(16)的Lyapunov指數(shù)譜

下載: 全尺寸圖片幻燈片

圖 5 圖像的加解密效果圖

下載: 全尺寸圖片幻燈片

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
檢測(cè)項(xiàng)目	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

表 2 密鑰空間對(duì)比表

	本文	文獻(xiàn)[11]	文獻(xiàn)[12]	文獻(xiàn)[13]	文獻(xiàn)[14]	文獻(xiàn)[15]
密鑰空間	${2^{747}}$	${2^{{\rm{640}}}}$	${2^{{\rm{273}}}}$	${2^{{\rm{270}}}}$	${2^{{\rm{267}}}}$	${2^{{\rm{208}}}}$

下載: 導(dǎo)出CSV

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