A note on the fast algebraic immunity and its consequences on modified majority functions

Boolean functions used as nonlinear filters and/or combiners in LFSR-based stream ciphers should satisfy several desired cryptographic properties simultaneously, to withstand all known cryptographic attacks. In the past decade, the algebraic and fast algebraic immunities are the most infusive criteria on the design of cryptographic Boolean functions, due to the high efficiency of the algebraic and fast algebraic attacks on stream ciphers. Up to now, Boolean functions with optimal algebraic immunity have been built in several ways, but there are not many known results on their fast algebraic immunities. In this paper, we first derive a relation on the fast algebraic immunity between a Boolean function f and it’s modifications f + s, which shows that if f has low fast algebraic immunity and s has low algebraic immunity then f + s may also have low fast algebraic immunity in general. Thanks to this relation, we obtain some upper bounds on the fast algebraic immunity of several known classes of modified majority functions.