Correlation Immunity and the Summation Generator

It is known that for a memoryless mapping from GF(2)N into GF(2) the nonlinear order of the mapping and its correlation-immunity form a linear tradeoff. In this paper it is shown that the same tradeoff does no longer hold when the function is allowed to have memory. Moreover, it is shown that integer addition, when viewed over GF(2), defines an inherently nonlinear function with memory whose correlation-immunity is maximum. The summation generator which sums N binary sequences over the integers is shown as an application of integer addition in random sequence generation.