Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cusick, T.W., Stănică, P.: Cryptographic Boolean functions and applications. Elsevier–Academic Press (2009)
Erdős, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)
Friedgut, E., Kalai, G.: Every monotone graph property has a sharp threshold. Proc. AMS. 124(10), 2293–3002 (1996)
Keller, N., Mossel, E., Schlank, T.: A note on the entropy/influence conjecture. Discrete Math. 312(22), 3364–3372 (2012)
Lu, Y., Desmedt, Y.: Bias analysis of a certain problem with applications to E0 and Shannon ciper. ICISC LNCS 6829(2011), 16–28 (2010)
Montanaro, A., Osborne, T.J.: Quantum Boolean functions, Chicago J. Theor. Comput. Sci. Article 1, pages 1–45,. http://cjtcs.cs.uchicago.edu/ , arXiv:0810.2435v5 (2010)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)
O’Donnell, R.: Analysis of Boolean functions. Cambridge University Press (2014)
O’Donnell, R., Tan, L-Y.: A composition theorem for the Fourier entropy-influence conjecture, ICALP (1) 2013 780–791. arXiv:1304.1347v1 (2013)
Parker, M.G.: Generalised S-box nonlinearity, NESSIE Public Document, 11.02.03: NES/DOC/UIB/WP5/020/A
Parker, M.G., Pott, A.: On Boolean functions which are bent and negabent. In: Proc. Int. Conf. Sequences, Subsequences, Consequences, vol. 4893, pp 9–23 (2007)
Riera, C.: Spectral Properties of Boolean functions, Graphs and Graph States. PhD. thesis University of Bergen (2005)
Riera, C., Parker, M.G.: Generalized bent criteria for Boolean functions. IEEE Trans. Inf. Theory 52(9), 4142–4159 (2006)
Rothaus, O.S.: On bent functions. J. Combin. Theory, Ser. A 20, 300–305 (1976)
Schmidt, K.U., Parker, M.G., Pott, A.: Negabent functions in the Maiorana–McFarland class. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) . SETA 2008, LNCS 5203, pp 390–402. Springer, Heidelberg (2008)
Stănică, P., Gangopadhyay, S., Chaturvedi, A., Gangopadhyay, A.K., Maitra, S.: Investigations on bent and negabent functions via the nega–Hadamard transform. IEEE Trans. Inf. Theory 58(6), 4064–4072 (2012)
Acknowledgments
The authors thank Dr. Aalok Misra of the Department of Physics, Indian Institute of Technology Roorkee for extremely helpful discussions on quantum mechanics.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gangopadhyay, S., Gangopadhyay, A.K., Pollatos, S. et al. Cryptographic Boolean functions with biased inputs. Cryptogr. Commun. 9, 301–314 (2017). https://doi.org/10.1007/s12095-015-0174-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-015-0174-1