二元周期序列的2-adic密碼學性質
發(fā)布時間：2018-10-09 17:09
【摘要】：密碼設計和密碼攻擊是密碼研究的主要內容,隨著對這兩者的深入研究,密碼學得到不斷地提升與發(fā)展。對于序列密碼來說,上世紀60年代興起一類非線性序列,即基于線性反饋移位寄存器(LFSR)生成的非線性序列,具有理想的偽隨機性質,但是隨著后來發(fā)起的代數攻擊和相關攻擊,該類序列生成器在密碼應用和研究領域中已經慢慢淡出。目前序列密碼的研究熱點已經轉移到非線性移位寄存器(NFSR).NFSR具有良好抗代數攻擊性質和良好的抗相關攻擊性質,但是由于研究理論的不完善,NFSR有很多性質還得不到系統(tǒng)的總結和分析。 FSCR是目前研究最透徹的一類非線性移位寄存序列生成器。該類寄存器的理論研究工具與LFSR的研究理論工具(有限域)不同,它利用2-adic環(huán)理論分析序列的密碼學安全特性。本文利用相對成熟的FCSR理論成果和2-adic環(huán)理論對二元周期序列的2-adic密碼學性質進行研究。另外,本文還對Z/(pe)環(huán)上非線性序列進行了分析,Z/(pe)上的生成序列與目前比較熱門的ZUC算法有很大的聯(lián)系,并且Z／(pe)環(huán)上序列也有很好的2-adic密碼學性質。 本文主要取得了以下成果： 1.主要分析相關數為q=pe的FCSR生成的l-序列進行自縮得到二元序列的性質。該類自縮序列能夠很好的保留l-序列的偽隨機性質,比如在一個周期T內,0,1比特基本平衡,自相關期望屬于{0,1/T}、方差是O(T/ln4T)。并且通過分析,我們得到該類序列的2-adic復雜度下界能夠達到安全指標。 2.以2-adic整數和二元周期序列的關聯(lián)為基礎,利用具有相同2-adic相關數的序列,分析m-序列自縮后得到的二元序列的2-adic復雜度,描述了該類序列2-adic復雜度的一個下界。 3.由于二元周期序列和2-adic整數之間的有一一對應關系,利用指數函數和2-adic整數的關系,給出了一種討論二元平衡序列的周期與其2-adic復雜度的方法。 4.利用Legendre變換在環(huán)上構造一類具有良好算數相關性的序列集,第一次給出了Legendre變換與算數相關性之間的關系,并且在周期,比特分布以及平移不等價性質上對該類序列進行分析。 5.算數相關性一般是作為二元序列的2-adic基本性質來進行研究,對于布爾函數卻很少提及其算數相關性。文章中介紹了一類非線性布爾函數,并且分析其算數相關性。通過分析給出了構造具有良好算數相關性布爾函數的一種方法。
[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.
【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2014
【分類號】：TN918.1
本文編號：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.
【學位授予單位】：北京郵電大學
【學位級別】：博士
【學位授予年份】：2014
【分類號】：TN918.1
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，本文編號：2260112
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