Stuart A. Kurtz

A discrete logarithm implementation of perfect zero-knowledge blobs

Brassard and Crépeau [BCr] introduced a simple technique for producing zero-knowledge proof systems based on blobs. As is to be expected, their implementation rests on an unproven cryptographic assumption, specifically, that it is easy to generate numbers that are difficult to factor. In this paper we present an implementation of blobs based on a different cryptographic assumption, specifically, that it is easy to generate primes p over which it is difficult to compute discrete logarithms. If, in addition, we can produce a generator for Zp*, this implementation then has the advantage that it leads to proof systems which are perfect zeroknowledge, rather than only almost perfect zero-knowledge.The relationship between factoring and finding discrete logarithms is not well understood, although Bach [Bac1] is an important contribution. Given our current state of number theoretic knowlege, there is no reason to prefer the cryptographic assumption required by one of these implementations over that required by the other. In any event, we introduce the notion of a product blob, whose favorable properties depend only on at least one of these assumptions holding.

The isomorphism conjecture fails relative to a random oracle

Berman and Hartmanis [1977] conjectured that there is a polynomial-time computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [1985] gave a structural definition of a class of NP-complete sets-the k-creative sets-and defined a class of sets (the K f k s) that are necessarily k-creative. They went on to conjecture that certain of these K f t s are not isomorphic to the standard NP-complete sets. Clearly, the Berman-Hartmanis and Joseph-Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K f k is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions-the annihilating functions-exist relative to a random oracle. Random oracles are the first examples of oracles relative to which the isomorphism conjecture fails with respect to higher classes such as EXP and NEXP.

Extremes in the degrees of inferability

Most theories of learning consider inferring a function f from either (1) observations about f or, (2) questions about f. We consider a scenario whereby the learner observes f and asks queries to some set A. If I is a notion of learning then I[A] is the set of concept classes I-learnable by an inductive inference machine with oracle A. A and B are I-equivalent if I[A] = I[B]. The equivalence classes induced are the degrees of inferability. We prove several results about when these degrees are trivial, and when the degrees are omniscient (i.e., the set of recursive function is learnable).

The ismorphism conjecture fails relative to a random oracle

Berman and Hartmanis [BH77] conjectured that there is a polynomial-time computable isomorphism between any two languages m-complete (“Karp” complete) for NP. Joseph and Young [JY85] discovered a structurally defined class of NP-complete sets and conjectured that certain of these sets (the <italic>K<supscrpt>k</supscrpt><subscrpt>ƒ</subscrpt></italic>s) are not isomorphic to the standard NP-complete sets for some one-way functions ƒ. These two conjectures cannot both be correct. We introduce a new family of strong one-way functions, the <italic>scrambling</italic> functions. If ƒ is a scrambling function, then <italic>K<supscrpt>k</supscrpt><subscrpt>fnof;</subscrpt></italic> is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions--the <italic>annihilating</italic> functions--exist relative to a random oracle.

The Structure of Complete Degrees

The notion of NP-completeness has cut across many fields and has provided a means of identifying deep and unexpected commonalities. Problems from areas as diverse as combinatorics, logic, and operations research turn out to be NP-complete and thus computationally equivalent in the sense discussed in the next paragraph. PSPACE-completeness, NEXP-completeness, and completeness for other complexity classes have likewise been used to show commonalities in a variety of other problems. This paper surveys investigations into how strong these commonalities are.

Collapsing degrees

An m-degree is a collection of sets equivalent under polynomial-time many-one (Karp) reductions; for example, the complete sets for NP or PSPACE are m-degrees. An m-degree is collapsing iff its members are p-isomorphic, i.e., equivalent under polynomial time, 1-1, onto, polynomial time invertible reductions. L. Berman and J. Hartmanis showed that all the then known natural NP-complete sets are isomorphic, and conjectured that the m-degree of the NP-complete sets collapses, in essence claiming that there is only one NP-complete set. However, until now no nontrivial collapsing m-degree was known to exist. In this paper we provide the first examples of such degrees, In particular, we show that there is a collapsing degree which is btt-complete for EXP (the exponential time decidable sets) and that, for every set A, there is a collapsing degree which is hard for A. We also obtain analogous results for noncollapsing degrees.

On 1-truth-table-hard languages

Abstract The polynomial-time 1-tt-hard sets for EXP and RE are polynomial-time many—one-hard

The isomorphism conjecture holds relative to an oracle

The authors introduce symmetric perfect generic sets. these sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. They then show that the Berman-Hartmanis (1977) isomorphism conjecture holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set A, all NP/sup A/-complete sets are polynomial-time isomorphic relative to A. As part of the proof that the isomorphism conjecture holds relative to symmetric perfect generic sets they also show that P/sup A/=FewP/sup A/ for any symmetric perfect generic/sup /A.<<ETX>>

The Isomorphism Conjecture Holds Relative toan Oracle

The authors introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the Berman--Hartmanis isomorphism conjecture holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set

Recursion theory and ordered groups

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