Complexity Measures for Public-Key Cryptosystems (Preliminary Report)

The first part of this paper gives results about promise problems. A "promise problem" is a formulation of a partial decision problem that is useful for describing cracking problems for public-key cryptosystems (PKCS). We prove that every NP-hard promise problem is uniformly NP-hard, and we show that a number of results and a conjecture about promise problems are equivalent to separability assertions that are the natural analogues of well-known results in classical recursion theory. The conjecture, if it is true, implies nonexistence of PKCS having NP-hard cracking problems. The second part of the paper studies more appropriate measures for PKCS. Among the results obtained are the following: One-way functions exist if an only if P /spl ne/ U and one-way functions f such that range f /spl epsiv/ P exist if and only if U /spl cap/ co-U /spl ne/ P. It will allow that there exist PKCS that cannot be cracked in polynomial time (and that satisfy other reasonable assumptions) only if P /spl ne/ U.