724 research outputs found
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
We present two algorithms to compute the endomorphism ring of an ordinary
elliptic curve E defined over a finite field F_q. Under suitable heuristic
assumptions, both have subexponential complexity. We bound the complexity of
the first algorithm in terms of log q, while our bound for the second algorithm
depends primarily on log |D_E|, where D_E is the discriminant of the order
isomorphic to End(E). As a byproduct, our method yields a short certificate
that may be used to verify that the endomorphism ring is as claimed.Comment: 16 pages (minor edits
Classification of 3-dimensional integrable scalar discrete equations
We classify all integrable 3-dimensional scalar discrete quasilinear
equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An
equation Q=0 is called integrable if it may be consistently imposed on all
3-dimensional elementary faces of the 4-dimensional lattice.
Under the natural requirement of invariance of the equation under the action
of the complete group of symmetries of the cube we prove that the only
nontrivial (non-linearizable) integrable equation from this class is the
well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2
has been made.) (Version 3: A few small corrections: one more reference added,
the main statement stated more explicitly.)Comment: 20 p. LaTeX + 1 EPS figur
On Zone-Based Analysis of Duration Probabilistic Automata
We propose an extension of the zone-based algorithmics for analyzing timed
automata to handle systems where timing uncertainty is considered as
probabilistic rather than set-theoretic. We study duration probabilistic
automata (DPA), expressing multiple parallel processes admitting memoryfull
continuously-distributed durations. For this model we develop an extension of
the zone-based forward reachability algorithm whose successor operator is a
density transformer, thus providing a solution to verification and performance
evaluation problems concerning acyclic DPA (or the bounded-horizon behavior of
cyclic DPA).Comment: In Proceedings INFINITY 2010, arXiv:1010.611
Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
We present an efficient proof system for Multipoint Arithmetic Circuit
Evaluation: for every arithmetic circuit of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
Interactive Oracle Proofs with Constant Rate and Query Complexity
We study interactive oracle proofs (IOPs) [BCS16,RRR16], which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs or IPs alone. Our main results are:
1. Circuit satisfiability has 3-round IOPs with linear proof length (counted in bits) and constant query complexity.
2. Reed-Solomon codes have 2-round IOPs of proximity with linear proof length and constant query complexity.
3. Tensor product codes have 1-round IOPs of proximity with sublinear proof length and constant query complexity.
For all the above, known PCP constructions give quasilinear proof length and constant query complexity [BS08,Din07]. Also, for circuit satisfiability, [BKKMS13] obtain PCPs with linear proof length but sublinear (and super-constant) query complexity. As in [BKKMS13], we rely on algebraic-geometry codes to obtain our first result; but, unlike that work, our use of such codes is much "lighter" because we do not rely on any automorphisms of the code.
We obtain our results by proving and combining "IOP-analogues" of tools underlying numerous IPs and PCPs:
* Interactive proof composition. Proof composition [AS98] is used to reduce the query complexity of PCP verifiers, at the cost of increasing proof length by an additive factor that is exponential in the verifier\u27s randomness complexity. We prove a composition theorem for IOPs where this additive factor is linear.
* Sublinear sumcheck. The sumcheck protocol [LFKN92] is an IP that enables the verifier to check the sum of values of a low-degree multi-variate polynomial on an exponentially-large hypercube, but the verifier\u27s running time depends linearly on the bound on individual degrees. We prove a sumcheck protocol for IOPs where this dependence is sublinear (e.g., polylogarithmic).
Our work demonstrates that even constant-round IOPs are more efficient than known PCPs and IPs
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
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