Balanced Boolean functions with optimum algebraic degree, optimum algebraic immunity and very high nonlinearity

It is a difficult challenge to construct Boolean functions with good cryptographic properties. In this paper, we construct an infinite class of even-variable balanced functions with optimum algebraic degree, optimum algebraic immunity and very high nonlinearity (higher than all other known balanced functions with optimum algebraic immunity). For any balanced Boolean function with optimum algebraic immunity, it is still unknown what is the highest nonlinearity possible. We achieve a higher nonlinearity than previous methods which gives a new lower bound on the maximum possible nonlinearity of balanced Boolean functions with optimum algebraic immunity.