2,603 research outputs found
A recursive construction of t-wise uniform permutations
We present a recursive construction of a (2t + 1)-wise uniform set of
permutations on 2n objects using a (2t + 1) - (2n, n, \cdot) combinatorial
design, a t-wise uniform set of permutations on n objects and a (2t+1)-wise
uniform set of permutations on n objects. Using the complete design in this
procedure gives a t-wise uniform set of permutations on n objects whose size is
at most t^2n, the first non-trivial construction of an infinite family of
t-wise uniform sets for t \geq 4. If a non-trivial design with suitable
parameters is found, it will imply a corresponding improvement in the
construction
Probabilistic existence of regular combinatorial structures
We show the existence of regular combinatorial objects which previously were
not known to exist. Specifically, for a wide range of the underlying
parameters, we show the existence of non-trivial orthogonal arrays, t-designs,
and t-wise permutations. In all cases, the sizes of the objects are optimal up
to polynomial overhead. The proof of existence is probabilistic. We show that a
randomly chosen structure has the required properties with positive yet tiny
probability. Our method allows also to give rather precise estimates on the
number of objects of a given size and this is applied to count the number of
orthogonal arrays, t-designs and regular hypergraphs. The main technical
ingredient is a special local central limit theorem for suitable lattice random
walks with finitely many steps.Comment: An extended abstract of this work [arXiv:1111.0492] appeared in STOC
2012. This version expands the literature discussio
Non-Local Probes Do Not Help with Graph Problems
This work bridges the gap between distributed and centralised models of
computing in the context of sublinear-time graph algorithms. A priori, typical
centralised models of computing (e.g., parallel decision trees or centralised
local algorithms) seem to be much more powerful than distributed
message-passing algorithms: centralised algorithms can directly probe any part
of the input, while in distributed algorithms nodes can only communicate with
their immediate neighbours. We show that for a large class of graph problems,
this extra freedom does not help centralised algorithms at all: for example,
efficient stateless deterministic centralised local algorithms can be simulated
with efficient distributed message-passing algorithms. In particular, this
enables us to transfer existing lower bound results from distributed algorithms
to centralised local algorithms
Higher Order Correlations in Quantum Chaotic Spectra
The statistical properties of the quantum chaotic spectra have been studied,
so far, only up to the second order correlation effects. The numerical as well
as the analytical evidence that random matrix theory can successfully model the
spectral fluctuatations of these systems is available only up to this order.
For a complete understanding of spectral properties it is highly desirable to
study the higher order spectral correlations. This will also inform us about
the limitations of random matrix theory in modelling the properties of quantum
chaotic systems. Our main purpose in this paper is to carry out this study by a
semiclassical calculation for the quantum maps; however results are also valid
for time-independent systems.Comment: Revtex, Four figures (Postscript files), Phys. Rev E (in press
Explicit near-Ramanujan graphs of every degree
For every constant and , we give a deterministic
-time algorithm that outputs a -regular graph on
vertices that is -near-Ramanujan; i.e., its eigenvalues
are bounded in magnitude by (excluding the single
trivial eigenvalue of~).Comment: 26 page
The t-wise Independence of Substitution-Permutation Networks
Block ciphers such as the Advanced Encryption Standard (Rijndael) are used extensively in practice, yet our understanding of their security continues to be highly incomplete. This paper promotes and continues a research program aimed at *proving* the security of block ciphers against important and well-studied classes of attacks. In particular, we initiate the study of (almost) -wise independence of concrete block-cipher construction paradigms such as substitution-permutation networks and key-alternating ciphers. Sufficiently strong (almost) pairwise independence already suffices to resist (truncated) differential attacks and linear cryptanalysis, and hence this is a relevant and meaningful target. Our results are two-fold.
Our first result concerns substitution-permutation networks (SPNs) that model ciphers such as AES. We prove the almost pairwise-independence of an SPN instantiated with concrete S-boxes together with an appropriate linear mixing layer, given sufficiently many rounds and independent sub-keys. Our proof relies on a *characterization* of S-box computation on input differences in terms of sampling output differences from certain subspaces, and a new randomness extraction lemma (which we prove with Fourier-analytic techniques) that establishes when such sampling yields uniformity. We use our techniques in particular to prove almost pairwise-independence for sufficiently many rounds of both the AES block cipher (which uses a variant of the patched inverse function as the -box) and the MiMC block cipher (which uses the cubing function as the -box), assuming independent sub-keys.
Secondly, we show that instantiating a key-alternating cipher (which can be thought of as a degenerate case of SPNs) with most permutations gives us (almost) -wise independence in rounds. In order to do this, we use the probabilistic method to develop two new lemmas, an *independence-amplification lemma* and a *distance amplification lemma*, that allow us to reason about the evolution of key-alternating ciphers
Low-Memory Algorithms for Online and W-Streaming Edge Coloring
For edge coloring, the online and the W-streaming models seem somewhat
orthogonal: the former needs edges to be assigned colors immediately after
insertion, typically without any space restrictions, while the latter limits
memory to sublinear in the input size but allows an edge's color to be
announced any time after its insertion. We aim for the best of both worlds by
designing small-space online algorithms for edge-coloring. We study the problem
under both (adversarial) edge arrivals and vertex arrivals. Our results
significantly improve upon the memory used by prior online algorithms while
achieving an -competitive ratio. In particular, for -node graphs with
maximum vertex-degree under edge arrivals, we obtain an online
-coloring in space. This is also the
first W-streaming edge-coloring algorithm for -coloring in sublinear
memory. All prior works either used linear memory or colors.
We also achieve a smooth color-space tradeoff: for any , we get an
-coloring in space,
improving upon the state of the art that used space for
the same number of colors. The improvements stem from extensive use of random
permutations that enable us to avoid previously used colors. Most of our
algorithms can be derandomized and extended to multigraphs, where edge coloring
is known to be considerably harder than for simple graphs.Comment: 32 pages, 1 figur
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