Relativizations of Unambiguous and Random Polynomial Time Classes

Abstract

We study relationships between ${\bf P}$, $NP$ , and the unambiguous and random time classes, ${\bf U}$, and ${\bf R}$. Questions concerning these relationships are motivated by complexity issues in public-key cryptosystems. We prove that there exists a recursive oracle A such that ${\bf P}^A \ne {\bf U}^A \ne {\bf NP}^A $, and such that the first inequality is “strong,” i.e., there exists a ${\bf P}^A $-immune set in ${\bf U}^A $. Further, we construct a recursive oracle B such that ${\bf U}^B $ contains an ${\bf R}^B $-immune set. As a corollary, we obtain ${\bf P}^B \ne {\bf R}^B \ne {\bf NP}^B $ and both inequalities are strong. By use of the techniques employed in the proof that ${\bf P}^A \ne {\bf U}^A \ne {\bf NP}^A $, we are also able to solve an open problem raised by Book, Long and Selman [Quantitative relativizations of complexity classes, SIAM J. Comput. 13 (1984), pp 461–487].