@inbook {IOPORT.05648891, author = {Simonetti, Ilaria}, title = {On the non-linearity of Boolean functions.}, year = {2009}, booktitle = {Gr\"obner bases, coding, and cryptography}, isbn = {978-3-540-93805-7}, pages = {409-413}, publisher = {Berlin: Springer}, doi = {10.1007/978-3-540-93806-4_30}, abstract = {Introduction: Any function from $(\Bbb F_2)^n$ to $\Bbb F_2$ is called a Boolean function (Bf). Boolean functions are important in symmetric cryptography since they are used in the confusion layer of ciphers. An affine Bf does not provide an effective confusion. To overcome this, we need functions which are as far as possible from being an affine function. The effectiveness of these functions is measured by a parameter called nonlinearity''. Usually, to compute the nonlinearity of a Bf $f$, we have to compute the discrete Fourier transform $\widehat{f}_\chi$ of the function $f_\chi(x)= (-1)^{f(x)}$. Then the nonlinearity of $f$ is $N(f)= 2^{n-1}- \frac12 \max_{a\in(\Bbb F_2)^n} |\widehat{f}_\chi(a)|$. In this paper, we compute the nonlinearity of Bf's with Gr\"obner bases.}, identifier = {05648891}, }