Mathematics of De-Identification

A function f is k-private iff no finite set of argument-value pairs {(x1,f(x1)), …, (xn,f(xn))} suffices to compute f at any other point x  {x1, …, xn}.
Example: a polynomial of degree exactly k is k-private but not (k+1)-private
Wait. This depends on knowing that f is a polynomial of degree k. That is metadata about f (semantics).
What kinds of metadata can we have about functions and what can be inferred from metadata?
A function f is totally private iff it is k-private for all k>0
Example: a general power series is totally private (without other metadata about p(x))
In general, k-privacy and total privacy are undecidable

A function f is k-private iff no finite set of argument-value pairs {(x1,f(x1)), …, (xn,f(xn))} suffices to compute f at any other point x  {x1, …, xn}.