Distributed point function

In cryptography, a distributed point function is a cryptographic primitive that allows two distributed processes to share a piece of information, and compute functions of their shared information, without revealing the information itself to either process. It is a form of secret sharing.^[1]

Given any two values ${\displaystyle a}$ and ${\displaystyle b}$ one can define a point function ${\displaystyle P_{a,b}(x)}$ (a variant of the Kronecker delta function) by

${\displaystyle P_{a,b}(x)={\begin{cases}b\qquad {\text{for }}x=a\\0\qquad {\text{for }}x\neq a\end{cases}}}$

That is, it is zero everywhere except at ${\displaystyle a}$, where its value is ${\displaystyle b}$.^[1]

A distributed point function consists of a family of functions ${\displaystyle f_{k}}$, parameterized by keys ${\displaystyle k}$, and a method for deriving two keys ${\displaystyle q}$ and ${\displaystyle r}$ from any two input values ${\displaystyle a}$ and ${\displaystyle b}$, such that for all ${\displaystyle x}$,

${\displaystyle P_{a,b}(x)=f_{q}(x)\oplus f_{r}(x),}$

where ${\displaystyle \oplus }$ denotes the bitwise exclusive or of the two function values. However, given only one of these two keys, the values of ${\displaystyle f}$ for that key should be indistinguishable from random.^[1]

It is known how to construct an efficient distributed point function from another cryptographic primitive, a one-way function.^[1]

In the other direction, if a distributed point function is known, then it is possible to perform private information retrieval. As a simplified example of this, it is possible to test whether a key ${\displaystyle a}$ belongs to replicated distributed database without revealing to the database servers (unless they collude with each other) which key was sought. To find the key ${\displaystyle a}$ in the database, create a distributed point function for ${\displaystyle P_{a,1}(x)}$ and send the resulting two keys ${\displaystyle q}$ and ${\displaystyle r}$ to two different servers holding copies of the database. Each copy applies its function ${\displaystyle f_{q}}$ or ${\displaystyle f_{r}}$ to all the keys in its copy of the database, and returns the exclusive or of the results. The two returned values will differ if ${\displaystyle a}$ belongs to the database, and will be equal otherwise.^[1]