85 research outputs found
Secret Key Agreement from Correlated Data, with No Prior Information
A fundamental question that has been studied in cryptography and in
information theory is whether two parties can communicate confidentially using
exclusively an open channel. We consider the model in which the two parties
hold inputs that are correlated in a certain sense. This model has been studied
extensively in information theory, and communication protocols have been
designed which exploit the correlation to extract from the inputs a shared
secret key. However, all the existing protocols are not universal in the sense
that they require that the two parties also know some attributes of the
correlation. In other words, they require that each party knows something about
the other party's input. We present a protocol that does not require any prior
additional information. It uses space-bounded Kolmogorov complexity to measure
correlation and it allows the two legal parties to obtain a common key that
looks random to an eavesdropper that observes the communication and is
restricted to use a bounded amount of space for the attack. Thus the protocol
achieves complexity-theoretical security, but it does not use any unproven
result from computational complexity. On the negative side, the protocol is not
efficient in the sense that the computation of the two legal parties uses more
space than the space allowed to the adversary.Comment: Several small errors have been fixed and the presentation has been
improved, following the reviewers' observation
Extracting the Kolmogorov Complexity of Strings and Sequences from Sources with Limited Independence
An infinite binary sequence has randomness rate at least if, for
almost every , the Kolmogorov complexity of its prefix of length is at
least . It is known that for every rational , on
one hand, there exists sequences with randomness rate that can not be
effectively transformed into a sequence with randomness rate higher than
and, on the other hand, any two independent sequences with randomness
rate can be transformed into a sequence with randomness rate higher
than . We show that the latter result holds even if the two input
sequences have linear dependency (which, informally speaking, means that all
prefixes of length of the two sequences have in common a constant fraction
of their information). The similar problem is studied for finite strings. It is
shown that from any two strings with sufficiently large Kolmogorov complexity
and sufficiently small dependence, one can effectively construct a string that
is random even conditioned by any one of the input strings
Linear list-approximation for short programs (or the power of a few random bits)
A -short program for a string is a description of of length at
most , where is the Kolmogorov complexity of . We show that
there exists a randomized algorithm that constructs a list of elements that
contains a -short program for . We also show a polynomial-time
randomized construction that achieves the same list size for -short programs. These results beat the lower bounds shown by Bauwens et al.
\cite{bmvz:c:shortlist} for deterministic constructions of such lists. We also
prove tight lower bounds for the main parameters of our result. The
constructions use only ( for the polynomial-time
result) random bits . Thus using only few random bits it is possible to do
tasks that cannot be done by any deterministic algorithm regardless of its
running time
On approximate decidability of minimal programs
An index in a numbering of partial-recursive functions is called minimal
if every lesser index computes a different function from . Since the 1960's
it has been known that, in any reasonable programming language, no effective
procedure determines whether or not a given index is minimal. We investigate
whether the task of determining minimal indices can be solved in an approximate
sense. Our first question, regarding the set of minimal indices, is whether
there exists an algorithm which can correctly label 1 out of indices as
either minimal or non-minimal. Our second question, regarding the function
which computes minimal indices, is whether one can compute a short list of
candidate indices which includes a minimal index for a given program. We give
some negative results and leave the possibility of positive results as open
questions
List Approximation for Increasing Kolmogorov Complexity
It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. But is it possible to construct a few strings, not longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that on input a string x of length n returns a list with O(n^2) many strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than n. We obtain similar results for other parameters, including a polynomial-time construction
- …