Notion of Algebraic Immunity and Its evaluation Related to Fast Algebraic Attacks

It has been noted recently that algebraic (annihilator) immunity alone does not provide sufficient resistance against algebraic attacks. In this regard, given a Boolean function f , just checking the minimum degree annihilators of f, 1 + f is not enough and one should check the relationships of the form fg = h, and a function f , even if it has very good algebraic immunity, is not necessarily good against fast algebraic attack, if degree of g becomes very low when degree of h is equal to or little greater than the algebraic immunity of f . In this paper we theoretically study the two currently known constructions having maximum possible algebraic immunity from this viewpoint. To the end, we also experimentally study some cryptographically significant functions having good algebraic immunity.