【作者】 周亮；

【导师】 吴建华； 马建；

【作者基本信息】 南昌大学 ， 电子与通信工程（专业学位）， 2022， 硕士

【摘要】 科学技术尤其是数字电子技术的飞速发展极大提升了传输设备的数据吞吐量、传输速率和带宽,降低了传输时延。数字图像以其直观和综合性以及生动形象等特点成为人们日常生活中一种最为常用的信息载体。由于数字图像包含了很多信息,有很多涉及个人隐私,因此也带来了很多图像信息安全方面的问题,这就使得图像加密算法的研究极具学术价值和经济价值。图像加密是将一幅带有有用信息的原始图像经过一系列的加密算法将其变成视觉上无意义的噪声图像。由于图像数据太过于庞大,现有的加密技术很难做到具有很高的安全性的同时又兼具很高的效率,而且在很多场合,例如在浮点数变换空间,为了传输加密信息无法避免由于计算机内部数值有限精度表示所带来的量化误差。针对以上的问题,我们提出了一种基于有限域的本原多项式和分圆多项式的混沌加密算法。在有限域的框架下,构造了经过参数化改造的新的本原多项式和分圆多项式,构造了有限域上的迹函数作为Logistic混沌映射的初始值参数,运用新构造的参数化的本原多项式和分圆多项式构造对称且正交的矩阵,对称且正交的矩阵的逆矩阵是它本身,在解密的时候极大地减少了运算量,从而提高了算法的效率。使用的Logistic混沌映射的伪随机序列与原始图像进行异或操作,用于对图像进行进一步的置乱和扩散,增强加密系统的安全性。本文的主要内容和贡献如下:(1)提出了基于有限域本原多项式和分圆多项式的图像加密算法。此算法的重要意义将分圆多项式和本原多项式参数化,由参数化改造后的分圆多项式和本原多项式构造对称且正交的矩阵。对称正交矩阵既具有正交矩阵的性质又具有对称矩阵的性质,首先它不改变向量的模长,在有限域上进行运算其结果还在有限域上;其次它的逆矩阵就是它本身,在解密时能极大减少解密所需要的运算量,提高算法解密效率。(2)为了解决有限域上本原多项式和分圆多项式图像算法的安全性不足的问题,下一步对图像进行了基于Logistic混沌映射的置乱和扩散。Logistic混沌映射具有初值敏感性和伪随机性质等,针对其初值敏感性的优良特性,对Logistic映射的初始值进行参数化控制。在Logistic映射后利用混沌映射得到的随机序列对图像进行异或操作再次增加的加密结果的混乱程度。大量的安全性分析和实验数据结果表明,此算法对各种常见的攻击是安全的。本文算法在进行加密的过程中没有任何的舍入误差,且加密的每一个过程都是可逆的,从而保证了该算法图像解密出的图像是严格无损的。由于本文算法使用了基于有限域的本原多项式和分圆多项式,而有限域上的本原多项式和分圆多项式具有非线性,该操作就保证了我们提出的图像加密系统是非线性的,能够抵抗已知明文和选择明文攻击。更多还原

【Abstract】 With the rapid development of science and technology,especially the rapid development of digital electronic technology,the data throughput,transmission rate and bandwidth have been greatly improved,and the transmission delay have also been reduced.This brings convenience to the transmission of information.Digital image has become one of the most commonly used information carriers in people’s daily life because of its intuitive,comprehensive and vivid characteristics.Because the digital image contains a lot of information,where there is often a lot of personal privacy,it also brings a lot of image security problems,which makes the research of image encryption algorithm of great academic and economic value.Image encryption is to change an original informative image through a series of encryption algorithms into a snowflake or noise-like image.Because the image data amount is too large,the existing encryption technology is difficult to achieve a high security and high efficiency,and avoid quantization errors due to finite precision computation within computers.To solve these problems,we propose a chaotic encryption algorithm based on primitive polynomials and cyclotomic polynomials over finite fields.In the framework of finite field,the new primitive polynomial and cyclotomic polynomial are constructed after parameterized.The trace function on the finite field is constructed and used as the initial values of parameters of the logistics chaotic map.The symmetric and orthogonal matrix is constructed by using the newly constructed parameterized primitive polynomial and cyclotomic polynomial.The inverse matrix of the symmetric and orthogonal matrix is itself.This greatly reduces the amount of computation and improves the efficiency of the algorithm.The pseudo-random sequence of logistics chaotic map is obtained to do XOR operation with the original image,further scrambling and diffusing the image,and enhancing the security of the encryption system.The main contents and contributions of this dissertation for master degree are as follows:(1)An image encryption algorithm based on a primitive polynomials and a cyclotomic polynomials is proposed.The significance of this algorithm is to parameterize the cyclotomic polynomials and primitive polynomials,and to construct a symmetric and orthogonal matrix from the modified cyclotomic polynomials and primitive polynomials.The symmetric orthogonal matrix has the properties of both orthogonality and symmetry.First of all,it does not change the module amplitude of a vector,and the result is still within the finite field.Secondly,its inverse matrix is itself,which can greatly reduce the implementation load for decryption and improve the efficiency of the algorithm.(2)In order to further strengthen the security of the image algorithms based on primitive polynomials and cyclotomic polynomials over finite fileds,we add a logistics chaotic map for diffusion.Logistic chaotic map has the characteristics of sensitivity to initial value.After logistics mapping,a simple XOR operation on the image increases the visual randomness of the encrypted results of the algorithm again.After a large number of experiments and security analysis,the results show that the algorithm is secure against all kinds of common attacks.There is no rounding errors in the process of encryption,and every process of encryption is reversible,which ensures that the decryption image is strictly lossless.Since the algorithm uses primitive polynomials and cyclotomic polynomials based on finite fields,and the primitive polynomials and cyclotomic polynomials over finite fields are nonlinear,this ensures that the image encryption system proposed is nonlinear and can resist the attacks of known plain-text and chosen plain-text.更多还原