Ronald V. Book

The verifiability of two-party protocols

Public key encryption as used in network communication has been investigated extensively. The main advantage of the techniques developed in this a rea is the potential for secure communication. However, while public key systems are often effective in preventing a passive saboteur from deciphering an intercepted message, protocols must be designed to be secure when dealing with saboteurs who can impersonate users or send copies of intercepted messages on the public channel. Dolev and Yao [3] have shown how informal arguments about protocols can lead to erroneous conclusions, and they have developed formal models of two-party protocols, both cascade protocols and name-stamp protocols. Recall that a protocol is a set of rules that specify what operators a pair of users, the sender and the receiver, need to apply in an exchange of messages for the purpose of transmitting a given plaintext message from the sender to the receiver. In terms of their models, Dolev and Yao developed an elegant characterization of cascade protocols that are secure, a characterization with conditions that can be checked by inspection.

On the Unification Hierarchy

We are interested in first order unification problems and more specifically in the hierarchy of equational theories based on the cardinality of the set of most general unifiers.

On complexity classes and algorithmically random languages

Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of Martin-Lof. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classes P, BPP, AM, and PH in terms of reducibility to such oracles. These characterizations lead to the following result: (i) P = NP if and only if there exists an algorithmically random set that is ≤ btt P -hard for NP. (ii) P = PSPACE if and only if there exists an algorithmically random set that is ≤ btt P -hard for PSPACE. (iii) The polynomial-time hierarchy collapses if and only if there exists k>0 such that some algorithmically random set is σ k P -hard for PH. (iv) PH = PSPACE if and only if there exists a algorithmically random set that is PH-hard for PSPACE.

On Sets with Small Information Content

The purpose of this paper is to review and summarize a number of results relating to sets with small information content such as sets with small generalized Kolmogorov complexity. The emphasis is on the role of such sets as oracle sets in the context of structural complexity theory.

The Global Power of Additional Queries to Random Oracles

It is shown that, for every k≥0 and every fixed algorithmically random language B, there is a language that is polynomialtime, truth-table reducible in k+1 queries to B but not truth-table reducible in k queries in any amount of time to any algorithmically random language C. In particular, this yields the separation Pk-tt(RAND) ⫋ P(k+1)-tt(RAND), where RAND is the set of all algorithmically random languages.

Additional Queries to Random and Pseudorandom Oracles

We know that a pseudorandom generator (PRG) takes a random string as input, and produces a longer string of output that is indistinguishable from random in probabilistic polynomial time (PPT). We have proved that if G is a PRG, then G ◦G is also a PRG, however for each extra use of G we can only get a fixed amount of additional string. Definition: An oracle algorithm, indicated by A?, may make queries to its oracle function and receive an answer in one step. For example Af(·) can query “x” and receive y = f(x) in one step, regardless of how f(·) works and without any knowledge of how f(·) works.

Monadic String-Rewriting Systems

Certain restrictions on the form of the rules of string-rewriting systems have yielded some extremely interesting results. Any finitely presented group can be presented by a finite special string-rewriting system (with no restriction to the notion of reduction). But special systems have a few undesirable limitations. Thus we turn to the study of “monadic” string-rewriting systems which extend the power of special string-rewriting systems in useful ways.

Relativizations of the P =?NP and other Problems: Some Developments in Structural Complexity Theory

The P =?NP problem has provided much of the primary motivation for developments in structural complexity theory. Recent results show that even after twenty years, contributions to the P=?NP problem, as well as other problems, still inspire new efforts. The purpose of this talk is to explain some of these results to theoreticians who do not work in structural complexity theory.

On Random Hard Sets for NP

The problem of whether NP has a random hard set (i.e., a set in RAND) is investigated. We show that for all recursive oracle A such that PA ≠ NPA, NPA has no hard set in RAND. On the other hand, we also show that for almost all oracle A, PA ≠ NPA and NPA has a hard set in RAND.

Relativizing Complexity Classes With Random Oracles

It is known that for almost every oracle set A, P(A) ≠ NP(A) and PH(A) ≠ PSPACE(A); but there are no known results that relate these facts to the P =?NP or PH =?PSPACE problems. Here the following result is shown (Theorem 6):