基于动力学特性的混沌密码算法研究
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摘要
自1965年美国气象学家Lorenz发现混沌现象以来，混沌理论受到越来越多的关注。混沌系统因具有对初始值和系统参数的敏感依赖性、遍历性和存在周期无限长的轨道的性质，这些是理想的密码系统所追求的性质。一个简单的混沌动力系统却有着非常复杂的行为，这些复杂行为有着很好的密码学性质，这正是混沌密码学的价值所在。另一方面，建立在数论、代数及算法复杂性理论基础之上传统密码算法正遭遇各种挑战，随着密码分析方法和研究手段的不断成熟，DES，MD5，SHA1等原来被认为安全的系统最近接连被破解，量子计算的发展也对非对称密码算法造成了巨大威胁。新的密码设计理论正成为一种迫切的需求，而从全新角度进行设计的混沌密码算法正是一种良好的替代方案。
然而，现存的混沌密码算法还存在诸多问题，他们往往因为动力学特性差或者是算法结构设计存在问题而被破译，现有的混沌密码算法还没有一个得到了广泛的应用。针对上述问题，本文从以下几个方面进行了研究：
（1）研究了混沌系统动力学特性对密码算法的影响。通过对Lyapunov指数谱的计算可以明确混沌系统在哪些参数区域是非混沌的，并且可以衡量初始时刻相邻轨道随时间变化分离的快慢程度.具有恒正的Lyapunov指数谱在密码学中是比较理想的性质。混沌系统的极限点集表明了系统长时间运行后应该保持的稳定的性质，在密码学中理想的系统其极限状况应该尽量保持均匀以免泄露系统的特征。复杂度的研究可以表明系统生成的序列的随机的程度，复杂度高的系统产生的序列往往更难以预测。
（2）在分析了上述动力学性质后，提出了一类低维的具有Markov分割性质的混沌系统T(x, p, sigma)。其将区间级别的变换作为分析的基础，可利用符号动力学对其做详细的研究。可以从理论上证明它有恒正的Lyapunov指数，而且其极限分布是均匀分布。通过参数的调整，在近似熵和符号熵的评价标准下它的复杂度可以远高于Logistic映射和Tent映射，利用它产生的序列具有更好的随机性。
（3）进一步的，分析了现有混沌图像加密算法结构存在的问题。现有的混沌图像加密算法大多是基于Fridrich结构的，它有两个部分组成，一个置乱矩阵，一个扩散函数。通过对被破译的混沌算法进行分析，发现其扩散函数的设计结构过于简单易破解，而置乱矩阵是可逆矩阵，这样一来，通过选择明文攻击可以分析出算法的密钥流，进而可以破解出原始明文图像。
（4）设计了一种新的图像加密算法。利用本文所构造的新的混沌系统去设计一个新的密钥生成函数，因具备理想的动力学性质，其产生的密钥比较理想；另外，通过采用不同群中的加法混合运算改进了图像加密算法的扩散函数使其难以破解。通过对算法基本性质的测试可以看出本文设计的算法可以通过目前图像加密算法的所有测试，另外算法可以抵抗通常的攻击。
（5）在FPGA的平台上实现了本文的混沌图像加密算法。给出了算法的实现方法，利用FPGA的平台特性优化了算法的可并行执行部分，并利用流水线技术做了进一步优化，使其在资源消耗不多的情况下有不错的计算速度。
研究结果表明，本文采用的混沌系统具有比较理想的动力学特性，这些理想的特性使得此系统特别适合用于密码算法的设计。本文设计的图像加密算法安全性高，计算效率好具有不错的应用前景。
Since the American meteorologist Lorenz discovered Chaos in1965, chaostheory attracted more and more attentions. Chaotic systems have three specialproperties that are sensitive properties dependent on initial values and systemparameters, ergodicity and infinitely long period orbit, which means the chaos haveideal cipher properties. A simple chaotic dynamical system can generate very complexbehaviors with good cryptographic properties, which shows the value of the chaoticcryptography. On the other hand, the traditional cryptographic algorithms based onnumber theory, algebra and algorithm complexity theory are now experiencingchallenges from the progress of the cryptanalysis methods. Some well-knownalgorithms such as DES, MD5, SHA1et al considered secure have been crackedrecent. While, for the development of the quantum computing the asymmetriccryptographic algorithms confront a new significant threat. As the new cipher designtheory is becoming an urgent demand, the chaotic cryptograph from a new perspectiveis a good alternative.
However, the existing chaotic cryptographic algorithms still have many problemssuch as poor dynamic properties and structure weakness. None of the existing chaoticcryptographic algorithms have been widely used yet. In response to these problems,this paper was studied from the following aspects:
Firstly, the effects on the chaotic cipher from the chaotic dynamic properties arestudied. By the calculation of Lyapunov exponents of the chaotic systems can make itclear that the system in which area is non-chaotic, which can also measure the speedover time of the separation of the adjacent orbit from initial time. The constantpositive Lyapunov exponent spectrum in cryptography is an ideal property. Thelimit-point-set of chaotic system shows a stable long-running nature of the system. In cryptography, the ideal situation of the chaotic system should keep evenly to preventleakage of the characteristics of the system. Research shows that the degree ofcomplexity can be a measure of the degree of the randomness of the random sequencegenerated by the system, which means the higher complexity of the system is thesystem is more difficult to predict.
Secondly, based on the analyzing of the dynamic properties of the chaos, a newclass of low-dimensional chaotic system with Markov property of T (x, p, sigma)was propose. It used the interval level transformation as the basis for analysis of thesystem which can make detailed studies by symbolic dynamics. It can be proved thatthe theoretically Lyapunov exponent is positive and the limiting distribution of thechaos is uniform. By adjusting the parameters, the chaos is more complex than theLogistic map and Tent map measuring by approximate entropy, which means thesequence it produced has better randomness.
Thirdly, the weaknesses of some existing chaotic image encryption algorithmsare proposed. Mostly, the existing chaotic image encryption algorithms are based onthe Fridrich‘s structure which has two components, a permutation matrix and adiffusion function. When researched in the broken algorithms, it is found that thedesigns of the diffusion functions‘structure are too simple to break. Also, thepermutation matrixes are reversible so that the key stream can be got by chosenplaintext attack, and then the original image can be recovered.
Then, a new image encryption algorithm is designed. A new key generationfunction is designed by use of the new chaotic system with ideal dynamical propertyconstructed in this paper, which can generate ideal key stream. What is more, thediffusion function is improved through the mixed operations of the additions indifferent groups, which is hard to break. It can be seen that the algorithm designed inthis article can pass all the tests use in the image encryption algorithms. While, thealgorithm can resist the common attacks either.
Finally, the chaotic image encryption algorithm of this paper is realized on theFPGA platform. The realization method of the algorithm is given. The performance ofthe algorithm is optimized in parallel part. The pipeline technology has also beenfurther optimized so that it can have better computing speed while consumed lessresources.
The results show that the chaotic system proposed in this article has idealdynamical properties which make the system particularly suitable for the design of cryptographic algorithms. This image encryption algorithm designed in the article issafe and efficiency which has great potential usage.
然而，现存的混沌密码算法还存在诸多问题，他们往往因为动力学特性差或者是算法结构设计存在问题而被破译，现有的混沌密码算法还没有一个得到了广泛的应用。针对上述问题，本文从以下几个方面进行了研究：
（1）研究了混沌系统动力学特性对密码算法的影响。通过对Lyapunov指数谱的计算可以明确混沌系统在哪些参数区域是非混沌的，并且可以衡量初始时刻相邻轨道随时间变化分离的快慢程度.具有恒正的Lyapunov指数谱在密码学中是比较理想的性质。混沌系统的极限点集表明了系统长时间运行后应该保持的稳定的性质，在密码学中理想的系统其极限状况应该尽量保持均匀以免泄露系统的特征。复杂度的研究可以表明系统生成的序列的随机的程度，复杂度高的系统产生的序列往往更难以预测。
（2）在分析了上述动力学性质后，提出了一类低维的具有Markov分割性质的混沌系统T(x, p, sigma)。其将区间级别的变换作为分析的基础，可利用符号动力学对其做详细的研究。可以从理论上证明它有恒正的Lyapunov指数，而且其极限分布是均匀分布。通过参数的调整，在近似熵和符号熵的评价标准下它的复杂度可以远高于Logistic映射和Tent映射，利用它产生的序列具有更好的随机性。
（3）进一步的，分析了现有混沌图像加密算法结构存在的问题。现有的混沌图像加密算法大多是基于Fridrich结构的，它有两个部分组成，一个置乱矩阵，一个扩散函数。通过对被破译的混沌算法进行分析，发现其扩散函数的设计结构过于简单易破解，而置乱矩阵是可逆矩阵，这样一来，通过选择明文攻击可以分析出算法的密钥流，进而可以破解出原始明文图像。
（4）设计了一种新的图像加密算法。利用本文所构造的新的混沌系统去设计一个新的密钥生成函数，因具备理想的动力学性质，其产生的密钥比较理想；另外，通过采用不同群中的加法混合运算改进了图像加密算法的扩散函数使其难以破解。通过对算法基本性质的测试可以看出本文设计的算法可以通过目前图像加密算法的所有测试，另外算法可以抵抗通常的攻击。
（5）在FPGA的平台上实现了本文的混沌图像加密算法。给出了算法的实现方法，利用FPGA的平台特性优化了算法的可并行执行部分，并利用流水线技术做了进一步优化，使其在资源消耗不多的情况下有不错的计算速度。
研究结果表明，本文采用的混沌系统具有比较理想的动力学特性，这些理想的特性使得此系统特别适合用于密码算法的设计。本文设计的图像加密算法安全性高，计算效率好具有不错的应用前景。
Since the American meteorologist Lorenz discovered Chaos in1965, chaostheory attracted more and more attentions. Chaotic systems have three specialproperties that are sensitive properties dependent on initial values and systemparameters, ergodicity and infinitely long period orbit, which means the chaos haveideal cipher properties. A simple chaotic dynamical system can generate very complexbehaviors with good cryptographic properties, which shows the value of the chaoticcryptography. On the other hand, the traditional cryptographic algorithms based onnumber theory, algebra and algorithm complexity theory are now experiencingchallenges from the progress of the cryptanalysis methods. Some well-knownalgorithms such as DES, MD5, SHA1et al considered secure have been crackedrecent. While, for the development of the quantum computing the asymmetriccryptographic algorithms confront a new significant threat. As the new cipher designtheory is becoming an urgent demand, the chaotic cryptograph from a new perspectiveis a good alternative.
However, the existing chaotic cryptographic algorithms still have many problemssuch as poor dynamic properties and structure weakness. None of the existing chaoticcryptographic algorithms have been widely used yet. In response to these problems,this paper was studied from the following aspects:
Firstly, the effects on the chaotic cipher from the chaotic dynamic properties arestudied. By the calculation of Lyapunov exponents of the chaotic systems can make itclear that the system in which area is non-chaotic, which can also measure the speedover time of the separation of the adjacent orbit from initial time. The constantpositive Lyapunov exponent spectrum in cryptography is an ideal property. Thelimit-point-set of chaotic system shows a stable long-running nature of the system. In cryptography, the ideal situation of the chaotic system should keep evenly to preventleakage of the characteristics of the system. Research shows that the degree ofcomplexity can be a measure of the degree of the randomness of the random sequencegenerated by the system, which means the higher complexity of the system is thesystem is more difficult to predict.
Secondly, based on the analyzing of the dynamic properties of the chaos, a newclass of low-dimensional chaotic system with Markov property of T (x, p, sigma)was propose. It used the interval level transformation as the basis for analysis of thesystem which can make detailed studies by symbolic dynamics. It can be proved thatthe theoretically Lyapunov exponent is positive and the limiting distribution of thechaos is uniform. By adjusting the parameters, the chaos is more complex than theLogistic map and Tent map measuring by approximate entropy, which means thesequence it produced has better randomness.
Thirdly, the weaknesses of some existing chaotic image encryption algorithmsare proposed. Mostly, the existing chaotic image encryption algorithms are based onthe Fridrich‘s structure which has two components, a permutation matrix and adiffusion function. When researched in the broken algorithms, it is found that thedesigns of the diffusion functions‘structure are too simple to break. Also, thepermutation matrixes are reversible so that the key stream can be got by chosenplaintext attack, and then the original image can be recovered.
Then, a new image encryption algorithm is designed. A new key generationfunction is designed by use of the new chaotic system with ideal dynamical propertyconstructed in this paper, which can generate ideal key stream. What is more, thediffusion function is improved through the mixed operations of the additions indifferent groups, which is hard to break. It can be seen that the algorithm designed inthis article can pass all the tests use in the image encryption algorithms. While, thealgorithm can resist the common attacks either.
Finally, the chaotic image encryption algorithm of this paper is realized on theFPGA platform. The realization method of the algorithm is given. The performance ofthe algorithm is optimized in parallel part. The pipeline technology has also beenfurther optimized so that it can have better computing speed while consumed lessresources.
The results show that the chaotic system proposed in this article has idealdynamical properties which make the system particularly suitable for the design of cryptographic algorithms. This image encryption algorithm designed in the article issafe and efficiency which has great potential usage.
引文
[1] http://observe.chinaiiss.com/html/201311/30/a651c5.html
[2] http://www.xinhuanet.com/world/ljm2013/
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