New results about Tu-Deng's conjecture

To design robust symmetric encryption schemes, we need to use Boolean functions with suitable properties. Among the security criteria these functions need to fulfill, we can mention algebraic immunity. A lot of papers study how to construct suitable functions, but some of them assume the validity of Tu-Deng's combinatorial conjecture [2] to estimate the algebraic immunity of the Boolean functions they design. In this paper we prove two new results about this conjecture and point out a new family of integers that satisfy it.