Let k ≥ 2 be an integer, and define S t := {(a, b) ∈ Z 2 |0 ≤ a, b ≤ 2 k − 2, a + b = t(mod 2 k − 1), w(a) + w(b) ≤ k − 1}, where t ∈ Z, 1 ≤ t ≤ 2 k − 2. This paper gives the upper bound of the cardinality of S t in the case of w(t) = 5. With this one, we conclude that a conjecture proposed by Tu and Deng in 2011 is right when w(t) = 5. INDEX TERMS Tu-Deng conjecture, algebraic immunity, Boolean function, Hamming weight.

Wu et al. proposed a generalized Tu-Deng conjecture over F 2 rm ×F 2 m , and constructed Boolean functions with good properties. However the proof of the generalized conjecture is still open. Based on Wu's work and assuming that the conjecture is true, we come up with a new class of balanced Boolean functions which has optimal algebraic degree, high nonlinearity and optimal algebraic immunity. The Boolean function also behaves well against fast algebraic attacks. Meanwhile we construct another class of Boolean functions by concatenation, which is 1-resilient and also has other good cryptographic properties.