Cryptography and Communications

, Volume 9, Issue 2, pp 301–314 | Cite as

Cryptographic Boolean functions with biased inputs

  • Sugata GangopadhyayEmail author
  • Aditi Kar Gangopadhyay
  • Spyridon Pollatos
  • Pantelimon Stănică


While performing cryptanalysis, it is of interest to approximate a Boolean function in n variables \(f: {\mathbb {F}_{2}^{n}} \rightarrow \mathbb {F}_{2}\) by affine functions. Usually, it is assumed that all the input vectors to a Boolean function are equiprobable while mounting affine approximation attack or fast correlation attacks. In this paper we consider a more general case when each component of the input vector to f is independent and identically distributed Bernoulli variates with the parameter p. Since our scope is within the area of cryptography, we initiate an analysis of cryptographic Boolean functions under the previous considerations and derive expression of the analogue of Walsh–Hadamard transform and nonlinearity in the case under consideration. We observe that if we allow p to take up complex values then a framework involving quantum Boolean functions can be introduced, which provides a connection between Walsh-Hadamard transform, nega-Hadamard transform and Boolean functions with biased inputs.


Boolean functions Quantum Boolean functions Bias Walsh–Hadamard transform Nega-Hadamard transform 

Mathematics Subject Classification (2010)

94A60 94C10 81P94 06E30 



The authors thank Dr. Aalok Misra of the Department of Physics, Indian Institute of Technology Roorkee for extremely helpful discussions on quantum mechanics.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sugata Gangopadhyay
    • 1
    Email author
  • Aditi Kar Gangopadhyay
    • 2
  • Spyridon Pollatos
    • 3
  • Pantelimon Stănică
    • 3
  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia
  3. 3.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

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