Background and definition
Almost perfect nonlinear (APN) functions are the class of Vectorial Boolean Functions that provide optimum resistance to against differential attack. Intuitively, the differential attack against a given cipher incorporating a vectorial Boolean function is efficient when fixing some difference and computing the output of for all pairs of inputs whose difference is produces output pairs with a difference distribution that is far away from uniform.
The formal definition of an APN function is usually given through the values
which, for , express the number of input pairs with difference that map to a given . The existence of a pair with a high value of makes the function vulnerable to differential cryptanalysis. This motivates the definition of differential uniformity as
which clearly satisfies for any function . The functions meeting this lower bound are called almost perfect nonlinear (APN).
Autocorrelation functions of the directional derivatives
Given a Boolean function , the autocorrelation function of is defined as
Any -function satisfies
for any . Equality occurs if and only if is APN.