1,780 research outputs found
Some results on the existence of -all-or-nothing transforms over arbitrary alphabets
A -all-or-nothing transform is a bijective mapping defined on -tuples over an alphabet of size , which satisfies the condition that the values of any input co-ordinates are completely undetermined, given only the values of any output co-ordinates. The main question we address in this paper is: for which choices of parameters does a -all-or-nothing transform (AONT) exist? More specifically, if we fix and , we want to determine the maximum integer such that a -AONT exists. We mainly concentrate on the case for arbitrary values of , where we obtain various necessary as well as sufficient conditions for existence of these objects. We consider both linear and general (linear or nonlinear) AONT. We also show some connections between AONT, orthogonal arrays and resilient functions
Polar Codes for the m-User MAC
In this paper, polar codes for the -user multiple access channel (MAC)
with binary inputs are constructed. It is shown that Ar{\i}kan's polarization
technique applied individually to each user transforms independent uses of a
-user binary input MAC into successive uses of extremal MACs. This
transformation has a number of desirable properties: (i) the `uniform sum rate'
of the original MAC is preserved, (ii) the extremal MACs have uniform rate
regions that are not only polymatroids but matroids and thus (iii) their
uniform sum rate can be reached by each user transmitting either uncoded or
fixed bits; in this sense they are easy to communicate over. A polar code can
then be constructed with an encoding and decoding complexity of
(where is the block length), a block error probability of o(\exp(- n^{1/2
- \e})), and capable of achieving the uniform sum rate of any binary input MAC
with arbitrary many users. An application of this polar code construction to
communicating on the AWGN channel is also discussed
Orthogonal Codes for Robust Low-Cost Communication
Orthogonal coding schemes, known to asymptotically achieve the capacity per
unit cost (CPUC) for single-user ergodic memoryless channels with a zero-cost
input symbol, are investigated for single-user compound memoryless channels,
which exhibit uncertainties in their input-output statistical relationships. A
minimax formulation is adopted to attain robustness. First, a class of
achievable rates per unit cost (ARPUC) is derived, and its utility is
demonstrated through several representative case studies. Second, when the
uncertainty set of channel transition statistics satisfies a convexity
property, optimization is performed over the class of ARPUC through utilizing
results of minimax robustness. The resulting CPUC lower bound indicates the
ultimate performance of the orthogonal coding scheme, and coincides with the
CPUC under certain restrictive conditions. Finally, still under the convexity
property, it is shown that the CPUC can generally be achieved, through
utilizing a so-called mixed strategy in which an orthogonal code contains an
appropriate composition of different nonzero-cost input symbols.Comment: 2nd revision, accepted for publicatio
Finding All Solutions of Equations in Free Groups and Monoids with Involution
The aim of this paper is to present a PSPACE algorithm which yields a finite
graph of exponential size and which describes the set of all solutions of
equations in free groups as well as the set of all solutions of equations in
free monoids with involution in the presence of rational constraints. This
became possible due to the recently invented emph{recompression} technique of
the second author.
He successfully applied the recompression technique for pure word equations
without involution or rational constraints. In particular, his method could not
be used as a black box for free groups (even without rational constraints).
Actually, the presence of an involution (inverse elements) and rational
constraints complicates the situation and some additional analysis is
necessary. Still, the recompression technique is general enough to accommodate
both extensions. In the end, it simplifies proofs that solving word equations
is in PSPACE (Plandowski 1999) and the corresponding result for equations in
free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As
a byproduct we obtain a direct proof that it is decidable in PSPACE whether or
not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk
at CSR 2014 in Moscow, June 7 - 11, 201
On Security Properties of All-or-nothing Transforms
All-or-nothing transforms have been defined as bijective mappings on all
s-tuples over a specified finite alphabet. These mappings are required to
satisfy certain "perfect security" conditions specified using entropies of the
probability distribution defined on the input s-tuples. Alternatively, purely
combinatorial definitions of AONTs have been given, which involve certain kinds
of "unbiased arrays". However, the combinatorial definition makes no reference
to probability definitions.
In this paper, we examine the security provided by AONTs that satisfy the
combinatorial definition. The security of the AONT can depend on the underlying
probability distribution of the s-tuples. We show that perfect security is
obtained from an AONT if and only if the input s-tuples are equiprobable.
However, in the case where the input s-tuples are not equiprobable, we still
achieve a weaker security guarantee. We also consider the use of randomized
AONTs to provide perfect security for a smaller number of inputs, even when
those inputs are not equiprobable
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