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The algebraic immunity of cryptographic Boolean functions is studied in this paper. Proper modifications of functions achieving maximum algebraic immunity are proved, in order to yield new functions of also maximum algebraic immunity. It is shown that the derived results apply to known classes of functions. Moreover, two new efficient algorithms to produce functions of guaranteed maximum algebraic immunity are developed, which further extend and generalize known constructions of functions with maximum algebraic immunity. Boolean functions constitute important building blocks for cryptographic systems, either as S-boxes in block ciphers or as filter/combiner functions in stream ciphers. The security of these systems is mainly attributed to the properties of the underlying functions.
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