Abstract
In this paper, we explore quantum algorithms to check the resiliency property of a Boolean function (in particular, when it is non-resilient). First we explain that Deutsch-Jozsa algorithm can be immediately used for this purpose. We further analyse how the quadratic improvement in query complexity can be obtained using Grover’s technique. While the worst case quantum query complexity to check the resiliency order is exponential in the number of input variables of the Boolean function, in our strategy one requires polynomially many measurements only. We also describe a subset of n-variable Boolean functions for which the algorithm works in polynomially many steps, i.e., we can achieve an exponential speed-up over best known classical algorithms.
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Notes
Our analysis here does not require any specialized knowledge of the quantum paradigm. Instead the work is mostly related to combinatorial properties of Boolean functions. We show how such properties of Boolean functions can be exploited to achieve novel and improved results in the field of quantum algorithms.
For quantum algorithms, we write “query complexity” instead of “time complexity” as we need to query some oracles, e.g., \(U_{f}, \mathcal {O}_{g}\) as described in Section 2.
For more details on query complexity and measurements, refer to [11].
We go for similar abuse of notation for the phase inversion oracle later.
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Acknowledgments
The authors like to thank the anonymous reviewers for their valuable comments. The authors also acknowledge the Centre of Excellence in Cryptology, Indian Statistical Institute, for relevant support towards this work.
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This is a thoroughly revised version of the paper “Quantum algorithms to check Resiliency of a Boolean function” that has been presented in WCC 2013, April 15-19, 2013, Bergen, Norway.
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Chakraborty, K., Maitra, S. Application of Grover’s algorithm to check non-resiliency of a Boolean function. Cryptogr. Commun. 8, 401–413 (2016). https://doi.org/10.1007/s12095-015-0156-3
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DOI: https://doi.org/10.1007/s12095-015-0156-3