7,665 research outputs found
A hierarchy for BPP//log* based on counting calls to an oracle
Algorithms whose computations involve making physical measurements can be modelled by Turing machines with oracles that are physical systems and oracle queries that obtain data from observation and measurement. The computational power of many of these physical oracles has been established using non-uniform complexity classes; in particular, for large classes of deterministic physical oracles, with fixed error margins constraining the exchange of data between algorithm and oracle, the computational power has been shown to be the non-uniform class BPP//logâ . In this paper, we consider non-deterministic oracles that can be modelled by random walks on the line. We show how to classify computations within BPP//logâ by making an infinite non-collapsing hierarchy between BPP//logâ and BPP . The hierarchy rests on the theorem that the number of calls to the physical oracle correlates with the size of the responses to queries.info:eu-repo/semantics/publishedVersio
On the Impossibility of Probabilistic Proofs in Relativized Worlds
We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or nondeterministic computations that make queries to the oracle.
This question is intimately related to a recent line of work that seeks to improve the efficiency of probabilistic proofs for computations that use functionalities such as cryptographic hash functions and digital signatures, by instantiating them via constructions that are "friendly" to known constructions of probabilistic proofs. Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it.
We prove several impossibility results for probabilistic proofs relative to natural oracles. Our results provide strong evidence that tailoring certain natural functionalities to known probabilistic proofs is inherent
The chaining lemma and its application
We present a new information-theoretic result which we call the Chaining Lemma. It considers a so-called âchainâ of random variables, defined by a source distribution X(0)with high min-entropy and a number (say, t in total) of arbitrary functions (T1,âŠ, Tt) which are applied in succession to that source to generate the chain (Formula presented). Intuitively, the Chaining Lemma guarantees that, if the chain is not too long, then either (i) the entire chain is âhighly randomâ, in that every variable has high min-entropy; or (ii) it is possible to find a point j (1 †j †t) in the chain such that, conditioned on the end of the chain i.e. (Formula presented), the preceding part (Formula presented) remains highly random. We think this is an interesting information-theoretic result which is intuitive but nevertheless requires rigorous case-analysis to prove. We believe that the above lemma will find applications in cryptography. We give an example of this, namely we show an application of the lemma to protect essentially any cryptographic scheme against memory tampering attacks. We allow several tampering requests, the tampering functions can be arbitrary, however, they must be chosen from a bounded size set of functions that is fixed a prior
Strengths and Weaknesses of Quantum Computing
Recently a great deal of attention has focused on quantum computation
following a sequence of results suggesting that quantum computers are more
powerful than classical probabilistic computers. Following Shor's result that
factoring and the extraction of discrete logarithms are both solvable in
quantum polynomial time, it is natural to ask whether all of NP can be
efficiently solved in quantum polynomial time. In this paper, we address this
question by proving that relative to an oracle chosen uniformly at random, with
probability 1, the class NP cannot be solved on a quantum Turing machine in
time . We also show that relative to a permutation oracle chosen
uniformly at random, with probability 1, the class cannot be
solved on a quantum Turing machine in time . The former bound is
tight since recent work of Grover shows how to accept the class NP relative to
any oracle on a quantum computer in time .Comment: 18 pages, latex, no figures, to appear in SIAM Journal on Computing
(special issue on quantum computing
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