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Abstract. Algebraic attack has recently become an important tool in cryptanalysing different stream and block cipher systems. A Boolean function, when used in some cryptosystem, should be designed properly to resist this kind of attack. The cryptographic property of a Boolean function, that resists algebraic attack, is known as Algebraic Immunity (AI). So far, the attempt in designing Boolean functions with required algebraic immunity was only ad-hoc, i.e., the functions were designed keeping in mind the other cryptographic criteria, and then it has been checked whether it can provide good algebraic immunity too. For the first time, in this paper, we present a construction method to generate Boolean functions on n variables with highest possible algebraic immunity n 2. Such a function can be used in conjunction with (using direct sum) functions having other cryptographic properties.In a different direction we identify that functions, having low degree subfunctions, are weak in terms of algebraic immunity and analyse some existing constructions from this viewpoint.

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Abstract. We provide two new construction methods for nonlinear resilient functions. The first method is a simple modification of an elegant construction due to Zhang and Zheng and constructs n-input, m-output resilient S-boxes with degree d > m. We prove by an application of the Griesmer bound for linear error correcting codes that the modified Zhang-Zheng construction is superior to the previous method of Cheon in Crypto 2001. Our second construction uses a sharpened version of the Maiorana-McFarland technique to construct nonlinear resilient functions. The nonlinearity obtained by our second construction is better than previously known construction methods.