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To browse Academia. This talk will be devoted to Boolean functions for cryptography, and more precisely to those which can be used for filtering linear feedback shift registers, bringing nonlinearity in the pseudo random generators of stream ciphers using them. This way of using Boolean functions is currently the only one for which are known infinite classes of Boolean functions allowing to resist all the known efficient attacks the so-called combiner model is another possible way but requires an extra criterion, high order resiliency, and no construction of functions is known which ensures all criteria for this model.
After the improvement by Courtois and Meier in of the algebraic attacks on stream ciphers and the introduction in of the related notion of algebraic immunity, a first series of constructions of infinite classes of balanced Boolean functions with optimum algebraic immunity had been proposed.
All of them gave functions whose nonlinearities are insufficient for allowing resistance to fast correlation attacks and whose behavior with respect to fast algebraic attacks a new version of algebraic attacks, introduced by Courtois in as well is not good either. An infinite class of balanced functions achieving optimum algebraic immunity, optimum algebraic degree and a much better nonlinearity than all the previously obtained infinite classes of functions was then proposed by the author and K.
The nonlinearity which has been computed for small numbers of variables small, but quite sufficient for cryptographic applications gives good results but still poses an open problem since all the lower bounds which could be proved mathematically, and are then valid for any numbers of variables, are significantly lower than the computed values.
Two researchers, Tu and Deng, proposed later a modification of these functions which allowed to reach a still much better nonlinearity and to keep optimum algebraic immunity. Unfortunately, it was proved by Johansson and Wang at INSCRYPT that these functions and all functions based on the same principle of construction had weak behavior against fast algebraic attacks.