## CryptoDB

### Melek D. Yücel

#### Publications

**Year**

**Venue**

**Title**

2007

EPRINT

Generalized Rotation Symmetric and Dihedral Symmetric Boolean Functions - 9 variable Boolean Functions with Nonlinearity 242
Abstract

Recently, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. In this paper, we present several 9-variable Boolean functions having nonlinearity of 242, which we obtain by suitably generalizing the classes of RSBFs and Dihedral Symmetric Boolean Functions (DSBFs).

2007

EPRINT

Balanced Boolean Functions with Nonlinearity > 2^{n-1} - 2^{(n-1)/2}
Abstract

Recently, balanced 15-variable Boolean functions with nonlinearity 16266 were obtained by suitably modifying unbalanced Patterson-Wiedemann (PW) functions, which possess nonlinearity 2^{n-1}-2^{(n-1)/2}+20 = 16276. In this short paper, we present an idempotent interpreted as rotation symmetric Boolean function) with nonlinearity 16268 having 15 many zeroes in the Walsh spectrum, within the neighborhood of PW functions. Clearly this function can be transformed to balanced functions keeping the nonlinearity and autocorrelation distribution unchanged. The nonlinearity value of 16268 is currently the best known for balanced 15-variable Boolean functions. Furthermore, we have attained several balanced 13-variable Boolean functions with nonlinearity 4036, which improves the recent result of 4034.

2006

EPRINT

There exist Boolean functions on $n$ (odd) variables having nonlinearity $> 2^{n-1} - 2^{\frac{n-1}{2}}$ if and only if $n > 7$
Abstract

For the first time we find Boolean functions on 9 variables having nonlinearity 241, that remained as an open question in literature for almost three decades. Such functions are discovered using a suitably modified steepest-descent based iterative heuristic search in the class of rotation symmetric Boolean functions (RSBFs). This shows that there exist Boolean functions on $n$ (odd) variables having nonlinearity $> 2^{n-1} - 2^{\frac{n-1}{2}}$ if and only if $n > 7$. Using the same search method, we also find several other important functions and we study the autocorrelation, propagation characteristics and resiliency of the RSBFs (using proper affine transformations, if required). The results show that it is possible to get balanced Boolean functions on $n=10$ variables having autocorrelation spectra with maximum absolute value $< 2^{\frac{n}{2}}$, which was not known earlier. In certain cases the functions can be affinely transformed to get first order propagation characteristics. We also obtain 10-variable functions having first order resiliency and nonlinearity 492 which was posed as an open question in Crypto 2000.

2006

EPRINT

Enumeration of 9-variable Rotation Symmetric Boolean Functions having Nonlinearity > 240
Abstract

The existence of $9$-variable Boolean functions having nonlinearity
strictly greater than $240$ has been shown very recently (May 2006)
by Kavut, Maitra and Y{\"u}cel. The functions with nonlinearity 241 have been identified by a heuristic search in the class of Rotation Symmetric Boolean Functions (RSBFs). In this paper we efficiently perform the exhaustive search to enumerate the 9-variable RSBFs having nonlinearity $> 240$ and found that there are such functions with nonlinearity 241 only and there is no RSBF having nonlinearity $> 241$. Our search enumerates $8 \times 189$ many 9-variable RSBFs having nonlinearity 241. We further show that there are only two functions which are different up to the affine equivalence. Towards the end we explain the coding theoretic significance of these functions.

#### Coauthors

- Selçuk Kavut (4)
- Subhamoy Maitra (2)
- Sumanta Sarkar (1)