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Characterizing Linear Structures of Boolean Functions from Arithmetic Walsh Transform
Qinglan ZHAO Dong ZHENG Xiangxue LI Yinghui ZHANG Xiaoli DONG
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E100A
No.9
pp.19651972 Publication Date: 2017/09/01
Online ISSN: 17451337
DOI: 10.1587/transfun.E100.A.1965
Type of Manuscript: PAPER Category: Cryptography and Information Security Keyword: Boolean functions, linear structure, arithmetic Walsh transform, WalshHadamard transform,
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Summary:
As a withcarry analog (based on modular arithmetic) of the usual WalshHadamard transform (WHT), arithmetic Walsh transform (AWT) has been used to obtain analogs of some properties of Boolean functions which are important in the design and analysis of cryptosystems. The existence of nonzero linear structure of Boolean functions is an important criterion to measure the weakness of these functions in their cryptographic applications. In this paper, we find more analogs of linear structures of Boolean functions from AWT. For some classes of nvariable Boolean functions f, we find necessary and sufficient conditions for the existence of an invariant linear structure and a complementary linear structure 1^{n} of f. We abstract out a sectionally linear relationship between AWT and WHT of nvariable balanced Boolean functions f with linear structure 1^{n}. This result show that AWT can characterize cryptographic properties of these functions as long as WHT can. In addition, for a diagonal Boolean function f, a recent result by Carlet and Klapper says that the AWT of f can be expressed in terms of the AWT of a diagonal Boolean function of algebraic degree at most 3 in a larger number of variables. We provide for the result a complete and more modular proof which works for both even and odd weights (of the parameter c in the Corollary 19 by Carlet and Klapper (DCC 73(2): 299318, 2014).

