基于半张量积的几类密码算法的研究

发布时间：2018-05-07 21:30

本文选题：序列密码 + 半张量积　；参考：《北京交通大学》2014年博士论文

【摘要】：密码技术已经被广泛应用于政治、军事、外交、商务等各个领域,成为确保信息安全的重要手段。近年来随着新技术与新的数学工具的出现,密码学研究出现了许多新的热点。作为一门以数学为基础学科,数学领域新的进展经常能够促进密码学研究的发展。本论文将半张量积这种新的数学工具引入到密码学的研究当中,主要做了以下两方面的工作：一方面,基于半张量积的序列密码的周期问题的研究；另一方面,基于半张量积运算构建的灵活格基加密算法的研究。本文的主要研究工作如下： 1、研究了序列密码设计中的常用器件：NFSR的状态周期问题。给出了求解NFSR状态空间的一般性方法。以往关于NFSR圈结构的研究一直没有很好的方法,本文给出了一种可以精确描述NFSR圈结构、圈长度和分析NFSR非奇异性的方法,然后通过仿真实验证明了所提方法的正确性。 2、研究了以NFSR器件为基础的Grain型序列密码算法的周期问题。分析了有外界输入情况下,NFSR状态的变化情况,给出有效输入序列的判别条件和状态可达性的判断条件,并给出了构建输入序列的方法。提出有外界周期性输入的情况下,NFSR状态空间的描述的方法,并进行了实验仿真。 3、研究另一类以NFSR器件为基础的序列密码算法—Trivium型算法的周期问题。建立了级联NFSR的圈长度的求解方法。研究了有大量外界输入的NFSR的状态演变问题,给出缩小状态矩阵规模的方法,并提出一种利用可达状态矩阵对NFSR器件的状态平衡性进行评测的方法。 4、研究了利用半张量积运算构建灵活的格基加密算法。利用半张量积可以进行不同维度矩阵相乘的性质,以格基密码中的重要困难问题：LWE问题为基础,提出了新的困难问题STP-LWE问题,并证明了新提出的困难问题的复杂性。在此基础上,提出基于STP-LWE问题的公钥加密方案,并证明了方案的安全性和正确性。此方案实现了在同一套安全参数下使用不同规模的密钥进行加密。
[Abstract]:Cryptography technology has been widely used in political, military, diplomatic, commercial and other fields, and has become an important means to ensure information security. In recent years, with the emergence of new technologies and new mathematical tools, cryptography research has emerged many new hot spots. As a mathematics-based subject, new advances in mathematics can often promote the development of cryptography. In this paper, a new mathematical tool, semi-tensor product, is introduced into the research of cryptography, which mainly includes the following two aspects: on the one hand, the period problem of sequential cryptography based on semi-tensor product; on the other hand, The research of flexible lattice base encryption algorithm based on semi-tensor product operation. The main work of this paper is as follows: 1. The state period of the common device in the design of sequence cryptography:: NFSR is studied. A general method for solving NFSR state space is given. There has been no good method for the study of NFSR loop structure in the past. This paper presents a method that can accurately describe the NFSR cycle structure, cycle length and analysis of NFSR nonsingularity, and then proves the correctness of the proposed method by simulation experiments. 2. The periodic problem of Grain type sequence cipher algorithm based on NFSR device is studied. In this paper, the variation of NFSR state in the presence of external input is analyzed, and the conditions for judging the effective input sequence and the state reachability are given, and the method of constructing the input sequence is given. A description method of NFSR state space with periodic external input is proposed, and the experimental simulation is carried out. 3. The periodicity of another kind of sequential cipher algorithm-Trivium type algorithm based on NFSR devices is studied. A method to solve the cycle length of cascaded NFSR is established. The state evolution of NFSR with a large number of external inputs is studied. The method of reducing the size of state matrix is given. A method of evaluating the state balance of NFSR devices by using reachable state matrix is proposed. 4. Using semi-tensor product operation to construct flexible lattice-based encryption algorithm. By using the property of semi-tensor product to multiply different dimensional matrices, a new difficult problem STP-LWE problem is proposed based on the important difficult problem in lattice-base cryptosystem: LWE, and the complexity of the new difficult problem is proved. On this basis, a public key encryption scheme based on STP-LWE problem is proposed, and the security and correctness of the scheme are proved. This scheme implements encryption with different size keys under the same set of security parameters.
【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TN918.1

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