Transparency order for Boolean functions: analysis and construction

  • Qichun WangEmail author
  • Pantelimon Stănică


The notion of transparency order, proposed by Prouff (DPA attacks and S-boxes, FSE 2005, LNCS 3557, Springer, Berlin, 2005) and then redefined by Chakraborty et al. (Des Codes Cryptogr 82:95–115, 2017), is a property that attempts to characterize the resilience of cryptographic algorithms against differential power analysis attacks. In this paper, we give a tight upper bound on the transparency order in terms of nonlinearity, inferring the worst possible transparency order of those functions with the same nonlinearity. We also give a lower bound between transparency order and nonlinearity. We study certain classes of Boolean functions for their transparency order and find that this parameter for some functions of low algebraic degree can be determined by their nonlinearity. Finally, we construct two infinite classes of balanced semibent Boolean functions with provably relatively good transparency order (this is the first time that an infinite class of highly nonlinear balanced functions with provably good transparency order is given).


Transparency order Boolean function Nonlinearity 

Mathematics Subject Classification

11T71 11L03 



The authors would like to thank the reviewers of this manuscript for extraordinarily useful criticisms and suggestions. The first author would like to thank the financial support from the National Natural Science Foundation of China (Grant No. 61572189).


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Authors and Affiliations

  1. 1.School of Computer Science and TechnologyNanjing Normal UniversityNanjingPeople’s Republic of China
  2. 2.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

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