65,921 research outputs found
Arithmetic circuits: the chasm at depth four gets wider
In their paper on the "chasm at depth four", Agrawal and Vinay have shown
that polynomials in m variables of degree O(m) which admit arithmetic circuits
of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m).
This theorem shows that for problems such as arithmetic circuit lower bounds or
black-box derandomization of identity testing, the case of depth four circuits
is in a certain sense the general case. In this paper we show that smaller
depth four circuits can be obtained if we start from polynomial size arithmetic
circuits. For instance, we show that if the permanent of n*n matrices has
circuits of size polynomial in n, then it also has depth 4 circuits of size
n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of
polynomial size. These results have potential applications to lower bounds and
deterministic identity testing, in particular for sums of products of sparse
univariate polynomials. We also give an application to boolean circuit
complexity, and a simple (but suboptimal) reduction to polylogarithmic depth
for arithmetic circuits of polynomial size and polynomially bounded degree
Dynamic MDS Matrices for Substantial Cryptographic Strength
Ciphers get their strength from the mathematical functions of confusion and
diffusion, also known as substitution and permutation. These were the basics of
classical cryptography and they are still the basic part of modern ciphers. In
block ciphers diffusion is achieved by the use of Maximum Distance Separable
(MDS) matrices. In this paper we present some methods for constructing dynamic
(and random) MDS matrices.Comment: Short paper at WISA'10, 201
On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
We continue studies on quantum field theories on noncommutative geometric
spaces, focusing on classes of noncommutative geometries which imply
ultraviolet and infrared modifications in the form of nonzero minimal
uncertainties in positions and momenta. The case of the ultraviolet modified
uncertainty relation which has appeared from string theory and quantum gravity
is covered. The example of euclidean -theory is studied in detail and
in this example we can now show ultraviolet and infrared regularisation of all
graphs.Comment: LaTex, 32 page
A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions
State sum constructions, such as Kuperberg's algorithm, give partition
functions of physical systems, like lattice gauge theories, in various
dimensions by associating local tensors or weights, to different parts of a
closed triangulated manifold. Here we extend this construction by including
matter fields to build partition functions in both two and three space-time
dimensions. The matter fields introduces new weights to the vertices and they
correspond to Potts spin configurations described by an -module
with an inner product. Performing this construction on a triangulated manifold
with a boundary we obtain the transfer matrices which are decomposed into a
product of local operators acting on vertices, links and plaquettes. The vertex
and plaquette operators are similar to the ones appearing in the quantum double
models (QDM) of Kitaev. The link operator couples the gauge and the matter
fields, and it reduces to the usual interaction terms in known models such as
gauge theory with matter fields. The transfer matrices lead to
Hamiltonians that are frustration-free and are exactly solvable. According to
the choice of the initial input, that of the gauge group and a matter module,
we obtain interesting models which have a new kind of ground state degeneracy
that depends on the number of equivalence classes in the matter module under
gauge action. Some of the models have confined flux excitations in the bulk
which become deconfined at the surface. These edge modes are protected by an
energy gap provided by the link operator. These properties also appear in
"confined Walker-Wang" models which are 3D models having interesting surface
states. Apart from the gauge excitations there are also excitations in the
matter sector which are immobile and can be thought of as defects like in the
Ising model. We only consider bosonic matter fields in this paper.Comment: 52 pages, 58 figures. This paper is an extension of arXiv:1310.8483
[cond-mat.str-el] with the inclusion of matter fields. This version includes
substantial changes with a connection made to confined Walker-Wang models
along the lines of arXiv:1208.5128 and subsequent works. Accepted for
publication in JPhys
Chaotic Compilation for Encrypted Computing: Obfuscation but Not in Name
An `obfuscation' for encrypted computing is quantified exactly here, leading
to an argument that security against polynomial-time attacks has been achieved
for user data via the deliberately `chaotic' compilation required for security
properties in that environment. Encrypted computing is the emerging science and
technology of processors that take encrypted inputs to encrypted outputs via
encrypted intermediate values (at nearly conventional speeds). The aim is to
make user data in general-purpose computing secure against the operator and
operating system as potential adversaries. A stumbling block has always been
that memory addresses are data and good encryption means the encrypted value
varies randomly, and that makes hitting any target in memory problematic
without address decryption, yet decryption anywhere on the memory path would
open up many easily exploitable vulnerabilities. This paper `solves (chaotic)
compilation' for processors without address decryption, covering all of ANSI C
while satisfying the required security properties and opening up the field for
the standard software tool-chain and infrastructure. That produces the argument
referred to above, which may also hold without encryption.Comment: 31 pages. Version update adds "Chaotic" in title and throughout
paper, and recasts abstract and Intro and other sections of the text for
better access by cryptologists. To the same end it introduces the polynomial
time defense argument explicitly in the final section, having now set that
denouement out in the abstract and intr
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