573 research outputs found
Chunky and Equal-Spaced Polynomial Multiplication
Finding the product of two polynomials is an essential and basic problem in
computer algebra. While most previous results have focused on the worst-case
complexity, we instead employ the technique of adaptive analysis to give an
improvement in many "easy" cases. We present two adaptive measures and methods
for polynomial multiplication, and also show how to effectively combine them to
gain both advantages. One useful feature of these algorithms is that they
essentially provide a gradient between existing "sparse" and "dense" methods.
We prove that these approaches provide significant improvements in many cases
but in the worst case are still comparable to the fastest existing algorithms.Comment: 23 Pages, pdflatex, accepted to Journal of Symbolic Computation (JSC
Multivariate sparse interpolation using randomized Kronecker substitutions
We present new techniques for reducing a multivariate sparse polynomial to a
univariate polynomial. The reduction works similarly to the classical and
widely-used Kronecker substitution, except that we choose the degrees randomly
based on the number of nonzero terms in the multivariate polynomial, that is,
its sparsity. The resulting univariate polynomial often has a significantly
lower degree than the Kronecker substitution polynomial, at the expense of a
small number of term collisions. As an application, we give a new algorithm for
multivariate interpolation which uses these new techniques along with any
existing univariate interpolation algorithm.Comment: 21 pages, 2 tables, 1 procedure. Accepted to ISSAC 201
Parallel sparse interpolation using small primes
To interpolate a supersparse polynomial with integer coefficients, two
alternative approaches are the Prony-based "big prime" technique, which acts
over a single large finite field, or the more recently-proposed "small primes"
technique, which reduces the unknown sparse polynomial to many low-degree dense
polynomials. While the latter technique has not yet reached the same
theoretical efficiency as Prony-based methods, it has an obvious potential for
parallelization. We present a heuristic "small primes" interpolation algorithm
and report on a low-level C implementation using FLINT and MPI.Comment: Accepted to PASCO 201
Computing sparse multiples of polynomials
We consider the problem of finding a sparse multiple of a polynomial. Given f
in F[x] of degree d over a field F, and a desired sparsity t, our goal is to
determine if there exists a multiple h in F[x] of f such that h has at most t
non-zero terms, and if so, to find such an h. When F=Q and t is constant, we
give a polynomial-time algorithm in d and the size of coefficients in h. When F
is a finite field, we show that the problem is at least as hard as determining
the multiplicative order of elements in an extension field of F (a problem
thought to have complexity similar to that of factoring integers), and this
lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650
POPE: Partial Order Preserving Encoding
Recently there has been much interest in performing search queries over
encrypted data to enable functionality while protecting sensitive data. One
particularly efficient mechanism for executing such queries is order-preserving
encryption/encoding (OPE) which results in ciphertexts that preserve the
relative order of the underlying plaintexts thus allowing range and comparison
queries to be performed directly on ciphertexts. In this paper, we propose an
alternative approach to range queries over encrypted data that is optimized to
support insert-heavy workloads as are common in "big data" applications while
still maintaining search functionality and achieving stronger security.
Specifically, we propose a new primitive called partial order preserving
encoding (POPE) that achieves ideal OPE security with frequency hiding and also
leaves a sizable fraction of the data pairwise incomparable. Using only O(1)
persistent and non-persistent client storage for
, our POPE scheme provides extremely fast batch insertion
consisting of a single round, and efficient search with O(1) amortized cost for
up to search queries. This improved security and
performance makes our scheme better suited for today's insert-heavy databases.Comment: Appears in ACM CCS 2016 Proceeding
ObliviSync: Practical Oblivious File Backup and Synchronization
Oblivious RAM (ORAM) protocols are powerful techniques that hide a client's
data as well as access patterns from untrusted service providers. We present an
oblivious cloud storage system, ObliviSync, that specifically targets one of
the most widely-used personal cloud storage paradigms: synchronization and
backup services, popular examples of which are Dropbox, iCloud Drive, and
Google Drive. This setting provides a unique opportunity because the above
privacy properties can be achieved with a simpler form of ORAM called
write-only ORAM, which allows for dramatically increased efficiency compared to
related work. Our solution is asymptotically optimal and practically efficient,
with a small constant overhead of approximately 4x compared with non-private
file storage, depending only on the total data size and parameters chosen
according to the usage rate, and not on the number or size of individual files.
Our construction also offers protection against timing-channel attacks, which
has not been previously considered in ORAM protocols. We built and evaluated a
full implementation of ObliviSync that supports multiple simultaneous read-only
clients and a single concurrent read/write client whose edits automatically and
seamlessly propagate to the readers. We show that our system functions under
high work loads, with realistic file size distributions, and with small
additional latency (as compared to a baseline encrypted file system) when
paired with Dropbox as the synchronization service.Comment: 15 pages. Accepted to NDSS 201
Detecting lacunary perfect powers and computing their roots
We consider solutions to the equation f = h^r for polynomials f and h and
integer r > 1. Given a polynomial f in the lacunary (also called sparse or
super-sparse) representation, we first show how to determine if f can be
written as h^r and, if so, to find such an r. This is a Monte Carlo randomized
algorithm whose cost is polynomial in the number of non-zero terms of f and in
log(deg f), i.e., polynomial in the size of the lacunary representation, and it
works over GF(q)[x] (for large characteristic) as well as Q[x]. We also give
two deterministic algorithms to compute the perfect root h given f and r. The
first is output-sensitive (based on the sparsity of h) and works only over
Q[x]. A sparsity-sensitive Newton iteration forms the basis for the second
approach to computing h, which is extremely efficient and works over both
GF(q)[x] (for large characteristic) and Q[x], but depends on a number-theoretic
conjecture. Work of Erdos, Schinzel, Zannier, and others suggests that both of
these algorithms are unconditionally polynomial-time in the lacunary size of
the input polynomial f. Finally, we demonstrate the efficiency of the
randomized detection algorithm and the latter perfect root computation
algorithm with an implementation in the C++ library NTL.Comment: to appear in Journal of Symbolic Computation (JSC), 201
- …