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Analysis on Boolean function in a restricted (biased) domain

Abstract : Boolean functions are usually studied under the assumption that each input bit is considered independent and identically distributed. However, in the case of some stream ciphers, a keystream bit is generated by using a nonlinear Boolean function with inputs from a restricted domain. At Eurocrypt 2016, one such stream cipher (FLIP) has been proposed, where a Boolean function on n variables was exploited with inputs of weight n/2 only. Recently, Carlet et al. studied several properties of such functions and obtained certain bounds on linear approximations of direct sum in the restricted domain. In this paper, we observe that for a direct sum like f = g+h, the inputs to each sub-function g, h do not follow a uniform distribution in the restricted domain. In this regard, we study the properties of Boolean functions by considering a general probability distribution on the inputs. We further obtain several bounds related to the biases of direct sums. Finally, we obtain a lower bound on the bias of the nonlinear filter function of FLIP. Our results provide a general framework to study security parameters of ciphers over restricted domain.
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https://hal.inria.fr/hal-02374194
Contributor : Bimal Mandal <>
Submitted on : Thursday, November 21, 2019 - 1:47:43 PM
Last modification on : Friday, January 24, 2020 - 11:56:27 AM

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Subhamoy Maitra, Bimal Mandal, Thor Martinsen, Dibyendu Roy, Pantelimon Stanica. Analysis on Boolean function in a restricted (biased) domain. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2020, 66 (2), pp.1219-1231. ⟨10.1109/TIT.2019.2932739⟩. ⟨hal-02374194⟩

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