Alan L. Selman

A comparison of polynomial time reducibilities

Abstract Various forms of polynomial time reducibility are compared. Among the forms examined are many-one, bounded truth table, truth table and Turing reducibility. The effect of introducing nondeterminism into reduction procedures is also examined.

Complexity measures for public-key cryptosystems

A general theory of public-key cryptography is developed that is based on the mathematical framework of complexity theory. Two related approaches are taken to the development of this theory, and th...

The complexity of promise problems with applications to public-key cryptography

A “promise problem” is a formulation of partial decision problem. Complexity issues about promise problems arise from considerations about cracking problems for public-key cryptosystems. Using a notion of Turing reducibility between promise problems, this paper disproves a conjecture made by Even and Yacobi (1980) , that would imply nonexistence of public-key cryptosystems with NP-hard cracking problems. In its place a new conjecture is raised having the same consequence. In addition, the new conjecture implies that NP-complete sets cannot be accepted by Turing machines that have at most one accepting computation for each input word.

P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP

The notion ofp-selective sets, and tally languages, are used to study polynomial time reducibilities onNP. P-selectivity has the property that a setA belongs to the classP if and only if bothĀ ≤mpA andA isp-selective. We prove that for every tally language set inNP there exists a polynomial time equivalent set inNP that isp-selective. From this result it follows that if NEXT ≠ DEXT, then polynomial time Turing and many-one reducibilities differ onNP.

Reductions on NP and P-selective sets

Abstract P-selective sets are used to distinguish polynomial time-bounded reducibilities on NP. In particular, we consider different kinds of “positive” reductions; these preserve membership in NP and are not a priori closed under complements. We show that the class of all sets which are both P-selective and have positive reductions to their complements is P. This is used to show that if DEXT ≠ NEXT, then there exists a set in NP−P that is not positive reducible to its complement. Various similar results are obtained. We also show that P is the class of all sets which are both p-selective and positive truth-table self-reducible. From this result, it follows that various naturally defined apparently intractible problems are not p-selective unless P = NP.

Quantitative relativizations of complexity classes

Consider the following open problems: (i)

A second step toward the polynomial hierarchy

{\text{P}} = ? {\text{ NP}}

Analogues of semirecursive sets and effective reducibilities to the study of NP complexity

(ii)

A survey of one-way functions in complexity theory

{\text{NP}} = ? {\text{ co-NP}}

Relativizing complexity classes with sparse oracles

; (iii)