Tools in Analyzing Linear Approximation for Boolean Functions Related to FLIP

  • Subhamoy Maitra
  • Bimal Mandal
  • Thor Martinsen
  • Dibyendu RoyEmail author
  • Pantelimon Stănică
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11356)


For cryptographic purposes, we generally study the characteristics of a Boolean function in n-variables with the inherent assumption that each of the n-bit inputs take the value 0 or 1, independently and randomly with probability 1 / 2. However, in the context of the FLIP stream cipher proposed by Méaux et al. (Eurocrypt 2016), this type of analysis warrants a different approach. To this end, Carlet et al. (IACR Trans. Symm. Crypto. 2018) recently presented a detailed analysis of Boolean functions with restricted inputs (mostly considering inputs with weight \(\frac{n}{2}\)) and provided certain bounds on linear approximation, which are related to restricted nonlinearity. The Boolean function used in the FLIP cipher reveals that it is actually a direct sum of several Boolean functions on a small number of inputs. Thus, with a different approach, we start a study in order to understand how the inputs to the composite function are distributed on the smaller functions. In this direction, we obtain several results that summarize the exact biases related to such Boolean functions. Finally, for the nonlinear filter function of FLIP, we obtain the lower bound on the restricted Walsh–Hadamard transform (i.e., upper bound on restricted nonlinearity). Our techniques provide a general theoretical framework to study such functions and better than previously published estimations of the biases, which is directly linked to the security parameters of the stream cipher.


Bias Boolean function FLIP Homomorphic encryption Restricted domain Stream cipher 



We would like to thank the anonymous reviewers of Indocrypt 2018 for their valuable suggestions and comments, which considerably improved the quality of our paper. The work of T.M. and P.S. started during an enjoyable visit to ISI-Kolkata in March 2018. They would like to thank the hosts and the institution for the excellent working conditions. T.M. also acknowledges support from the Omar Nelson Bradley foundation officer research fellowship in mathematics.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Subhamoy Maitra
    • 1
  • Bimal Mandal
    • 1
  • Thor Martinsen
    • 2
  • Dibyendu Roy
    • 3
    Email author
  • Pantelimon Stănică
    • 2
  1. 1.Indian Statistical InstituteKolkataIndia
  2. 2.Naval Postgraduate SchoolMontereyUSA
  3. 3.National Institute of Science Education and Research (HBNI)BhubaneswarIndia

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