On fractional correlation immunity of majority functions

The correlation immunity is known as an important cryptographic measure of a Boolean function with respect to its resist against the correlation attack. This paper generalizes the concept of correlation immunity to be of a fractional value, called fractional correlation immunity, which is a fraction between 0 and 1, and correlation immune function is the extreme case when the fractional correlation immunity is 1. However when a function is not correlation immune in the traditional sense, it may also has a nonzero fractional correlation immunity, which also indicates the resistance of the function against correlation attack. This paper first shows how this generalized concept of fractional correlation immunity is a reasonable measure on the resistance against the correlation attack, then studies the fractional correlation immunity of a special class of Boolean functions, i.e. majority functions, of which the subset of symmetric ones have been proved to have highest algebraic immunity. This paper shows that all the majority functions, including the symmetric ones and the non-symmetric ones, are not correlation immune. However their fractional correlation immunity approaches to 1 when the number of variable grows. This means that this class of functions also have good resistance against correlation attack, although they are not correlation immune in the traditional sense.