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.
This allows APN functions to be characterized in terms of the sum-of-square-indicator defined as
Then any function satisfies
and equality occurs if and only if is APN.
Similar techniques can be used to characterize permutations and APN functions with plateaued components.
- ↑ Thierry Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy, On Almost Perfect Nonlinear Functions Over GF(2^n), IEEE Transactions on Information Theory, 2006 Sep,52(9),4160-70