On Generalized Nega-Hadamard Transform and Nega-crosscorrelation

In the construction of cryptosystems, we need functions with some cryptographically significant properties. It is known that the Global Avalanche Characteristic (GAC) is one of them, which is based on the correlation of functions. In this paper, we present two indicators: the nega-sum-of-squares-of-modulus indicator (NSSMI) \(\sigma ^q_{h,k}\) ${\sigma }_{h,k}^q$ and the nega-modulus indicator (NMI) \(\Delta _{h,k}^q\) ${\mathrm{\Delta }}_{h,k}^q$ of the nega-crosscorrelation between two q-ary functions to determine the GAC of cryptographic functions. Using the properties of nega-crosscorrelation, we give a relation between the nega-Hadamard transform and the nega-autocorrelation of two q-ary functions. We deduce some bounds on NSSMI and NMI. A relationship among \(\sigma ^q_{h,k}\) ${\sigma }_{h,k}^q$ , \(\sigma ^q_{h}\) ${\sigma }_h^q$ and \(\sigma ^q_{k}\) ${\sigma }_k^q$ is established for two q-ary functions. Further, we find a relationship among the nega-crosscorrelation of four q-ary functions. Finally, we evaluate the nega-crosscorrelation for a subclass of Maiorana-McFarland type q-ary bent functions with good bounds on their NSSMI and NMI.