Toward a theory of unknown functions (Corresp.)

A function
f(x)
is chosen from a finite set of functions
\cal F
. An outsider observer knows
\cal F
but not the actual choice
f(x)
. He is, however, able to make a limited number of observations
(x,y)
satisfying the unknown function
y = f(x)
. The uncertainty of the outside observer with respect to the unknown function is measured as the entropy of the output variable
y
when the function
f(x)
is regarded as a random choice in
\cal F
. With this measure, an upper bound on the uncertainty is derived. The existence of unknown functions satisfying this bound is investigated, and necessary and sufficient conditions are derived. The problem is shown to be closely related to the problem of finding algebraic codes with high minimum Hamming distance. The theory can be applied to cryptography, identifying mechanisms, access control in computers, and possibly also to reliability analysis.