二元周期序列的2-adic密码学性质
发布时间：2018-10-09 17:09
【摘要】：密码设计和密码攻击是密码研究的主要内容,随着对这两者的深入研究,密码学得到不断地提升与发展。对于序列密码来说,上世纪60年代兴起一类非线性序列,即基于线性反馈移位寄存器(LFSR)生成的非线性序列,具有理想的伪随机性质,但是随着后来发起的代数攻击和相关攻击,该类序列生成器在密码应用和研究领域中已经慢慢淡出。目前序列密码的研究热点已经转移到非线性移位寄存器(NFSR).NFSR具有良好抗代数攻击性质和良好的抗相关攻击性质,但是由于研究理论的不完善,NFSR有很多性质还得不到系统的总结和分析。 FSCR是目前研究最透彻的一类非线性移位寄存序列生成器。该类寄存器的理论研究工具与LFSR的研究理论工具(有限域)不同,它利用2-adic环理论分析序列的密码学安全特性。本文利用相对成熟的FCSR理论成果和2-adic环理论对二元周期序列的2-adic密码学性质进行研究。另外,本文还对Z/(pe)环上非线性序列进行了分析,Z/(pe)上的生成序列与目前比较热门的ZUC算法有很大的联系,并且Z／(pe)环上序列也有很好的2-adic密码学性质。 本文主要取得了以下成果： 1.主要分析相关数为q=pe的FCSR生成的l-序列进行自缩得到二元序列的性质。该类自缩序列能够很好的保留l-序列的伪随机性质,比如在一个周期T内,0,1比特基本平衡,自相关期望属于{0,1/T}、方差是O(T/ln4T)。并且通过分析,我们得到该类序列的2-adic复杂度下界能够达到安全指标。 2.以2-adic整数和二元周期序列的关联为基础,利用具有相同2-adic相关数的序列,分析m-序列自缩后得到的二元序列的2-adic复杂度,描述了该类序列2-adic复杂度的一个下界。 3.由于二元周期序列和2-adic整数之间的有一一对应关系,利用指数函数和2-adic整数的关系,给出了一种讨论二元平衡序列的周期与其2-adic复杂度的方法。 4.利用Legendre变换在环上构造一类具有良好算数相关性的序列集,第一次给出了Legendre变换与算数相关性之间的关系,并且在周期,比特分布以及平移不等价性质上对该类序列进行分析。 5.算数相关性一般是作为二元序列的2-adic基本性质来进行研究,对于布尔函数却很少提及其算数相关性。文章中介绍了一类非线性布尔函数,并且分析其算数相关性。通过分析给出了构造具有良好算数相关性布尔函数的一种方法。
[Abstract]:Cryptography design and cryptography attack are the main contents of cryptography research. For sequential cryptography, a class of nonlinear sequences, which is generated based on linear feedback shift register (LFSR), has the ideal pseudorandom property, but with the subsequent algebraic attacks and related attacks, This kind of sequence generator has gradually faded out in the field of cryptographic application and research. At present, the research focus of sequence cryptography has shifted to the nonlinear shift register (NFSR). NFSR has good anti-algebraic attack property and good anti-correlation attack property. However, many properties of NFSR can not be systematically summarized and analyzed due to the imperfect theory. FSCR is a kind of nonlinear shift register sequence generator. The theoretical research tool of this kind of register is different from that of LFSR's (finite field). It uses the 2-adic ring theory to analyze the cryptographic security characteristics of sequences. In this paper, the 2-adic cryptographic properties of binary periodic sequences are studied by using relatively mature FCSR theory and 2-adic ring theory. In addition, the nonlinear sequences over Z / (pe) rings are analyzed in this paper. The generated sequences on Z / (pe) are closely related to the popular ZUC algorithms, and the sequences on Z / (pe) rings also have good 2-adic cryptographic properties. The main achievements of this paper are as follows: 1. The properties of binary sequences derived from FCSR generated by FCSR whose correlation number is q=pe are analyzed. This kind of self-shrinking sequences can preserve the pseudorandom property of l- sequences. For example, in a period T, there is a basic equilibrium between 0 bits and 1 bit. The autocorrelation expectation belongs to {0 / 1 / T}, and the variance is O (T/ln4T). Through analysis, we get the lower bound of 2-adic complexity of this class of sequences to achieve the security index. 2. Based on the correlation between 2-adic integers and binary periodic sequences, using sequences with the same 2-adic correlation number, the 2-adic complexity of binary sequences obtained by m- sequence self-shrinking is analyzed, and a lower bound of 2-adic complexity of this class of sequences is described. 3. Because of the one-to-one correspondence between binary periodic sequences and 2-adic integers, a method to discuss the periodicity and 2-adic complexity of binary equilibrium sequences is presented by using the relation between exponential function and 2-adic integers. 4. A class of sequence sets with good arithmetic correlation is constructed on the ring by Legendre transform. The relationship between Legendre transform and arithmetic correlation is given for the first time, and the sequence is analyzed in terms of periodicity, bit distribution and translation inequivalence. 5. Arithmetic correlation is generally studied as the basic property of binary sequence 2-adic, but it is seldom mentioned in Boolean function. This paper introduces a class of nonlinear Boolean functions and analyzes their arithmetic correlation. A method of constructing Boolean function with good arithmetic correlation is given.
【学位授予单位】：北京邮电大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TN918.1
本文编号：2260112
[Abstract]:Cryptography design and cryptography attack are the main contents of cryptography research. For sequential cryptography, a class of nonlinear sequences, which is generated based on linear feedback shift register (LFSR), has the ideal pseudorandom property, but with the subsequent algebraic attacks and related attacks, This kind of sequence generator has gradually faded out in the field of cryptographic application and research. At present, the research focus of sequence cryptography has shifted to the nonlinear shift register (NFSR). NFSR has good anti-algebraic attack property and good anti-correlation attack property. However, many properties of NFSR can not be systematically summarized and analyzed due to the imperfect theory. FSCR is a kind of nonlinear shift register sequence generator. The theoretical research tool of this kind of register is different from that of LFSR's (finite field). It uses the 2-adic ring theory to analyze the cryptographic security characteristics of sequences. In this paper, the 2-adic cryptographic properties of binary periodic sequences are studied by using relatively mature FCSR theory and 2-adic ring theory. In addition, the nonlinear sequences over Z / (pe) rings are analyzed in this paper. The generated sequences on Z / (pe) are closely related to the popular ZUC algorithms, and the sequences on Z / (pe) rings also have good 2-adic cryptographic properties. The main achievements of this paper are as follows: 1. The properties of binary sequences derived from FCSR generated by FCSR whose correlation number is q=pe are analyzed. This kind of self-shrinking sequences can preserve the pseudorandom property of l- sequences. For example, in a period T, there is a basic equilibrium between 0 bits and 1 bit. The autocorrelation expectation belongs to {0 / 1 / T}, and the variance is O (T/ln4T). Through analysis, we get the lower bound of 2-adic complexity of this class of sequences to achieve the security index. 2. Based on the correlation between 2-adic integers and binary periodic sequences, using sequences with the same 2-adic correlation number, the 2-adic complexity of binary sequences obtained by m- sequence self-shrinking is analyzed, and a lower bound of 2-adic complexity of this class of sequences is described. 3. Because of the one-to-one correspondence between binary periodic sequences and 2-adic integers, a method to discuss the periodicity and 2-adic complexity of binary equilibrium sequences is presented by using the relation between exponential function and 2-adic integers. 4. A class of sequence sets with good arithmetic correlation is constructed on the ring by Legendre transform. The relationship between Legendre transform and arithmetic correlation is given for the first time, and the sequence is analyzed in terms of periodicity, bit distribution and translation inequivalence. 5. Arithmetic correlation is generally studied as the basic property of binary sequence 2-adic, but it is seldom mentioned in Boolean function. This paper introduces a class of nonlinear Boolean functions and analyzes their arithmetic correlation. A method of constructing Boolean function with good arithmetic correlation is given.
【学位授予单位】：北京邮电大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TN918.1
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