Almost Perfect Nonlinear (APN) Functions

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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).

Characterizations

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.