https://hal.inria.fr/hal-02374194Maitra, SubhamoySubhamoyMaitraIndian Statistical Institute [Kolkata]Mandal, BimalBimalMandalCARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms - Inria Nancy - Grand Est - Inria - Institut National de Recherche en Informatique et en Automatique - LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry - LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications - Inria - Institut National de Recherche en Informatique et en Automatique - UL - Université de Lorraine - CNRS - Centre National de la Recherche ScientifiqueMartinsen, ThorThorMartinsenNPS - Naval Postgraduate School Roy, DibyenduDibyenduRoySTQC - Standardisation Testing and Quality CertificationStanica, PantelimonPantelimonStanicaNPS - Naval Postgraduate School Analysis on Boolean function in a restricted (biased) domainHAL CCSD2020[MATH] Mathematics [math][INFO.INFO-CR] Computer Science [cs]/Cryptography and Security [cs.CR]Mandal, Bimal2019-11-21 13:47:432023-03-24 14:53:132019-11-21 13:47:43enJournal articles10.1109/TIT.2019.29327391Boolean 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.