Abstract: It is a difficult challenge to find Boolean functions used in symmetric ciphers achieving many good crypto- graphic properties. Recently, two classes of Boolean functions with maximum algebraic immunity have been proposed by Tu and Deng based on correctness of the assumption of a combinatorial conjecture about binary. One class of the func- lions are bent functions with maximum algebraic immunity, and another class of the functions are balanced and have maximum algebraic immunity,optimal algebraic degree and good nonlincarity. "hu-Deng conjecture has received a lot of attennpns from cryptographers. The conjecture in the case of wt(t)=k-3 proved. As a corollary,the case of wt(t)=k-3 was also proved.

Key words: Symmetric ciphers,I3oolean functions,Tu-Deng co川ecture, Hamming weight