ON NONLINEARITY AND GLOBAL AVALANCHE CHARACTERISTICS OF VECTOR BOOLEAN FUNCTIONS

It is well known that the nonlinearity of vector Boolean functions F on n-dimensional vector space $GF(2)^n$ to $GF(2)^m$ is bounded above by $2^{n-1} - 2 ^{\frac{n}{2}-1}$. In this paper we derive upper bounds and a lower bound on the nonlinearity of vector Boolean functions in terms of auto-correlations. Strengths and weaknesses of each bounds are examined. Also, we modify the notions of the sum-of-square indicator and absolute indicator for Boolean functions to the case of vector Boolean functions to measure global avalanche characteristics of vector Boolean functions. Using those indicators we compare the global avalanche characteristics of DES (Data Encryption System) and Rijndael.