Thomas Thierauf

Threshold Computation and Cryptographic Security

Threshold machines are Turing machines whose acceptance is determined by what portion of the machines computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machines accepting computation paths. In 1975, Simon proved that for unbounded-error polynomial-time machines these two notions yield the same class, PP\@. Perhaps because Simons result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored. In this paper, we compare the bounded-error probabilistic class BPP with the analogous threshold class,

On the correlation of symmetric functions

\bpppath

NONDETERMINISTICALLY SELECTIVE SETS

, and, more generally, we study the structural properties of

The Boolean isomorphism problem

\bpppath

The Formula Isomorphism Problem

. We prove that

On functions computable with nonadaptive queries to NP

\rm BPP_{path}

Bipartite perfect matching is in quasi-NC

contains both

Reachability In K 3,3 -free graphs and K 5 -free graphs is in unambiguous log-space

\np^{\bpp}

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

and

The polynomially bounded perfect matching problem is in NC 2

\p^{\rm NP[\log]}