Communication with secrecy constraints

Let <italic><bold>x, y, z</bold></italic> be finite sets, <italic>X,Y</italic> random variables uniformly distributed over <italic><bold>x×y</bold></italic>, <italic>f</italic> a function from <italic><bold>x×y</bold></italic> to <italic>Z</italic> and 0≤ε&le1. A person P<subscrpt>X</subscrpt> knows <italic>X</italic> and a person P<subscrpt>Y</subscrpt> knows <italic>Y</italic> and they want to exchange <italic>X</italic> and <italic>Y</italic>. An eavesdropper who knows their protocol listens to their communication in order to obtain information about <italic>f(X, Y). P<subscrpt>X</subscrpt></italic> and <italic>P<subscrpt>Y</subscrpt></italic> want to ensure that for every value (<italic>x,y</italic>) of (<italic>X,Y</italic>) the eavesdropper's a priori and a posteriori probabilities of {<italic>f(X,Y)&equil;j</italic>} are ε-close for all <italic>j</italic>. Therefore, they encrypt some of the transmitted bits. The problem is to find a protocol that minimizes the number of bits encrypted in the worst case. Two kinds of protocols are considered: <italic>deterministic</italic> and <italic>randomized.</italic> For deterministic protocols it is shown that for all <italic><bold>x,y,</bold></italic> Boolean <italic>f (|Z|&equil;2) and ε>0, there exists a protocol that requires no more than 2• log(1/ε) + 16 bits.</italic> An example where log(1/ε) − 1 bits must be encrypted is given. For <italic>K</italic> valued functions (<italic>|Z|&equil;K</italic>) it is shown that at most C<subscrpt>K</subscrpt>(ε) bits must be encrypted (independent of <italic>x, y</italic> and <italic>f</italic> ). The results are extended to <italic>N</italic> persons communicating over a broadcast channel. The proofs rely on results concerning partitions of <italic>K</italic> valued matrices. For randomized Protocols it is shown that for all <italic>x,y</italic> Boolean <italic>f,</italic> and all possible joint distributions of <italic>X,Y</italic> (not only uniform), <italic>total secrecy</italic> (ε&equil;0) can be achieved using only two secret bits.