126 research outputs found
Arithmetic complexity via effective names for random sequences
We investigate enumerability properties for classes of sets which permit
recursive, lexicographically increasing approximations, or left-r.e. sets. In
addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably,
Schnorr, and Kurtz random sets, weakly 1-generics and their complementary
classes, we find that there exist characterizations of the third and fourth
levels of the arithmetic hierarchy purely in terms of these notions.
More generally, there exists an equivalence between arithmetic complexity and
existence of numberings for classes of left-r.e. sets with shift-persistent
elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz
non-randoms) have left-r.e. numberings, there is no canonical, or acceptable,
left-r.e. numbering for any class of left-r.e. randoms.
Finally, we note some fundamental differences between left-r.e. numberings
for sets and reals
Entanglement, quantum randomness, and complexity beyond scrambling
Scrambling is a process by which the state of a quantum system is effectively
randomized due to the global entanglement that "hides" initially localized
quantum information. In this work, we lay the mathematical foundations of
studying randomness complexities beyond scrambling by entanglement properties.
We do so by analyzing the generalized (in particular R\'enyi) entanglement
entropies of designs, i.e. ensembles of unitary channels or pure states that
mimic the uniformly random distribution (given by the Haar measure) up to
certain moments. A main collective conclusion is that the R\'enyi entanglement
entropies averaged over designs of the same order are almost maximal. This
links the orders of entropy and design, and therefore suggests R\'enyi
entanglement entropies as diagnostics of the randomness complexity of
corresponding designs. Such complexities form a hierarchy between information
scrambling and Haar randomness. As a strong separation result, we prove the
existence of (state) 2-designs such that the R\'enyi entanglement entropies of
higher orders can be bounded away from the maximum. However, we also show that
the min entanglement entropy is maximized by designs of order only logarithmic
in the dimension of the system. In other words, logarithmic-designs already
achieve the complexity of Haar in terms of entanglement, which we also call
max-scrambling. This result leads to a generalization of the fast scrambling
conjecture, that max-scrambling can be achieved by physical dynamics in time
roughly linear in the number of degrees of freedom.Comment: 72 pages, 4 figures. Rewritten version with new title. v3: published
versio
A Study of Separations in Cryptography: New Results and New Models
For more than 20 years, black-box impossibility results have been used to argue the infeasibility of constructing certain cryptographic primitives (e.g., key agreement) from others (e.g., one-way functions). In this dissertation we further extend the frontier of this field by demonstrating several new impossibility results as well as a new framework for studying a more general class of constructions.
Our first two results demonstrate impossibility of black-box constructions of two commonly used cryptographic primitives. In our first result we study the feasibility of black-box constructions of predicate encryption schemes from standard assumptions and demonstrate strong limitations on the types of schemes that can be constructed. In our second result we study black-box constructions of constant-round zero-knowledge proofs from one-way permutations and show that, under commonly believed complexity assumptions, no such constructions exist.
A widely recognized limitation of black-box impossibility results, however, is that they say nothing about the usefulness of (known) non-black-box techniques. This state of affairs is unsatisfying as we would at least like to rule out constructions using the set of techniques we have at our disposal. With this motivation in mind, in the final result of this dissertation we propose a new framework for black-box constructions with a non-black-box flavor, specifically, those that rely on zero-knowledge proofs relative to some oracle. Our framework is powerful enough to capture a large class of known constructions, however we show that the original black-box separation of key agreement from one-way functions still holds even in this non-black-box setting that allows for zero-knowledge proofs
Algorithmic randomness for Doob's martingale convergence theorem in continuous time
We study Doob's martingale convergence theorem for computable continuous time
martingales on Brownian motion, in the context of algorithmic randomness. A
characterization of the class of sample points for which the theorem holds is
given. Such points are given the name of Doob random points. It is shown that a
point is Doob random if its tail is computably random in a certain sense.
Moreover, Doob randomness is strictly weaker than computable randomness and is
incomparable with Schnorr randomness
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