two applications of an incomplete additive character sum to estimating nonlinearity of boolean functions

In recent years, several classes of Boolean functions with good cryptographic properties have been constructed by using univariate (or bivariate) polynomial representation of Boolean functions over finite fields. The estimation of an incomplete additive character sum plays an important role in analyzing the nonlinearity of these functions. In this paper, we consider replacing this character sum with another incomplete additive character sum, whose estimation was firstly given by A.Winterhof in 1999. Based on Winterhof's estimation, we try to modify two of these functions and obtain better nonlinearity bound of them. © 2011 Springer-Verlag.National Natural Science Foundation of China (NNSFC); The Microsoft Corporation; Beijing Tip Technology Corporation; Trusted Computing Group (TCG)In recent years, several classes of Boolean functions with good cryptographic properties have been constructed by using univariate (or bivariate) polynomial representation of Boolean functions over finite fields. The estimation of an incomplete additive character sum plays an important role in analyzing the nonlinearity of these functions. In this paper, we consider replacing this character sum with another incomplete additive character sum, whose estimation was firstly given by A.Winterhof in 1999. Based on Winterhof's estimation, we try to modify two of these functions and obtain better nonlinearity bound of them. © 2011 Springer-Verlag