Weightwise perfectly balanced functions with high weightwise nonlinearity profile

  • Jian Liu
  • Sihem MesnagerEmail author


Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the recent FLIP stream cipher (Méaux et al.: in Lecture Notes in Computer Science, vol. 9665, pp. 311–343, Springer, Berlin, 2016). In this paper, we propose a large class of weightwise perfectly balanced (WPB) functions, which is 2-rotation symmetric. This new class of WPB functions is not extended affinely equivalent to the known constructions. We also discuss the weightwise nonlinearity profile of these functions, and present general lower bounds on k-weightwise nonlinearity, where k is a power of 2. Moreover, we exhibit a subclass of the family. By a recursive lower bound, we show that these subclass of WPB functions have very high weightwise nonlinearity profile.


FLIP cipher Boolean function Weightwise perfectly balance Weightwise nonlinearity 

Mathematics Subject Classification

11T71 94A60 06E30 



The authors thank the anonymous reviewers for their valuable comments which have highly improved the manuscript.


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Authors and Affiliations

  1. 1.School of Computer SoftwareTianjin UniversityTianjinPeople’s Republic of China
  2. 2.CNRS, UMR 7539 LAGAParisFrance
  3. 3.Department of MathematicsUniversity of Paris VIII, University of Paris XIII, CNRS, UMR 7539 LAGAParisFrance
  4. 4.Telecom ParisTechParisFrance

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