Walsh transforms and cryptographic applications in bias computing

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	Walsh transforms and cryptographic applications in bias computing
	Lu, Y ; Desmedt, Y
刊名	CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
	2016
卷号	8 期号:3 页码:435-453
关键词	Walsh transform Linear cryptanalysis Bias analysis Maximum entropy principle Piling-up lemma
ISSN号	1936-2447
中文摘要	Walsh transform is used in a wide variety of scientific and engineering applications, including bent functions and cryptanalytic optimization techniques in cryptography. In linear cryptanalysis, it is a key question to find a good linear approximation, which holds with probability (1+d)/2 and the bias d is large in absolute value. Lu and Desmedt (2011) take a step toward answering this key question in a more generalized setting and initiate the work on the generalized bias problem with linearly-dependent inputs. In this paper, we give fully extended results. Deep insights on assumptions behind the problem are given. We take an information-theoretic approach to show that our bias problem assumes the setting of the maximum input entropy subject to the input constraint. By means of Walsh transform, the bias can be expressed in a simple form. It incorporates Piling-up lemma as a special case. Secondly, as application, we answer a long-standing open problem in correlation attacks on combiners with memory. We give a closed-form exact solution for the correlation involving the multiple polynomial of any weight for the first time. We also give Walsh analysis for numerical approximation. An interesting bias phenomenon is uncovered, i.e., for even and odd weight of the polynomial, the correlation behaves differently. Thirdly, we introduce the notion of weakly biased distribution, and study bias approximation for a more general case by Walsh analysis. We show that for weakly biased distribution, Piling-up lemma is still valid. Our work shows that Walsh analysis is useful and effective to a broad class of cryptanalysis problems.
英文摘要	Walsh transform is used in a wide variety of scientific and engineering applications, including bent functions and cryptanalytic optimization techniques in cryptography. In linear cryptanalysis, it is a key question to find a good linear approximation, which holds with probability (1+d)/2 and the bias d is large in absolute value. Lu and Desmedt (2011) take a step toward answering this key question in a more generalized setting and initiate the work on the generalized bias problem with linearly-dependent inputs. In this paper, we give fully extended results. Deep insights on assumptions behind the problem are given. We take an information-theoretic approach to show that our bias problem assumes the setting of the maximum input entropy subject to the input constraint. By means of Walsh transform, the bias can be expressed in a simple form. It incorporates Piling-up lemma as a special case. Secondly, as application, we answer a long-standing open problem in correlation attacks on combiners with memory. We give a closed-form exact solution for the correlation involving the multiple polynomial of any weight for the first time. We also give Walsh analysis for numerical approximation. An interesting bias phenomenon is uncovered, i.e., for even and odd weight of the polynomial, the correlation behaves differently. Thirdly, we introduce the notion of weakly biased distribution, and study bias approximation for a more general case by Walsh analysis. We show that for weakly biased distribution, Piling-up lemma is still valid. Our work shows that Walsh analysis is useful and effective to a broad class of cryptanalysis problems.
收录类别	SCI
语种	英语
WOS记录号	WOS:000384758600007
公开日期	2016-12-09
内容类型	期刊论文
源URL	[http://ir.iscas.ac.cn/handle/311060/17316]
专题	软件研究所_软件所图书馆_期刊论文
推荐引用方式 GB/T 7714	Lu, Y,Desmedt, Y. Walsh transforms and cryptographic applications in bias computing[J]. CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES,2016,8(3):435-453.
APA	Lu, Y,&Desmedt, Y.(2016).Walsh transforms and cryptographic applications in bias computing.CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES,8(3),435-453.
MLA	Lu, Y,et al."Walsh transforms and cryptographic applications in bias computing".CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES 8.3(2016):435-453.