44 research outputs found
АСИМПТОТИЧЕСКОЕ ПОВЕДЕНИЕ РЕЗИСТОРНЫХ РАССТОЯНИЙ В ГРАФАХ КЭЛИ
In the present paper, we prove asymptotically exact bounds for resistance distances in families of Cayley graphs that either have a girth of more than 4 or are free of subgraphs K2,t, assuming that the growth function is at least subexponential, and either the diameter or the inverse value of the spectral gap are polynomial with respect to degrees of a graph.В настоящей работе доказаны асимптотически точные оценки для резисторных расстояний в некоторых семействах графов Кэли при условии, что функция роста является как минимум субэкспоненциальной, а диаметр либо обратная величина к спектральному пробелу полиномиальны по степени графа.
Large components in random induced subgraphs of n-cubes
In this paper we study random induced subgraphs of the binary -cube,
. This random graph is obtained by selecting each -vertex with
independent probability . Using a novel construction of
subcomponents we study the largest component for
, where , . We prove that there exists a.s. a unique largest
component . We furthermore show that , and for , holds.
This improves the result of \cite{Bollobas:91} where constant is
considered. In particular, in case of , our
analysis implies that a.s. a unique giant component exists.Comment: 18 Page
General Impossibility of Group Homomorphic Encryption in the Quantum World
Group homomorphic encryption represents one of the most important building
blocks in modern cryptography. It forms the basis of widely-used, more
sophisticated primitives, such as CCA2-secure encryption or secure multiparty
computation. Unfortunately, recent advances in quantum computation show that
many of the existing schemes completely break down once quantum computers reach
maturity (mainly due to Shor's algorithm). This leads to the challenge of
constructing quantum-resistant group homomorphic cryptosystems.
In this work, we prove the general impossibility of (abelian) group
homomorphic encryption in the presence of quantum adversaries, when assuming
the IND-CPA security notion as the minimal security requirement. To this end,
we prove a new result on the probability of sampling generating sets of finite
(sub-)groups if sampling is done with respect to an arbitrary, unknown
distribution. Finally, we provide a sufficient condition on homomorphic
encryption schemes for our quantum attack to work and discuss its
satisfiability in non-group homomorphic cases. The impact of our results on
recent fully homomorphic encryption schemes poses itself as an open question.Comment: 20 pages, 2 figures, conferenc
Homomorphic encryption and some black box attacks
This paper is a compressed summary of some principal definitions and concepts
in the approach to the black box algebra being developed by the authors. We
suggest that black box algebra could be useful in cryptanalysis of homomorphic
encryption schemes, and that homomorphic encryption is an area of research
where cryptography and black box algebra may benefit from exchange of ideas
Random induced subgraphs of Cayley graphs induced by transpositions
In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, . Our main
result is that for any minimal generating set of transpositions, for
probabilities where , a random induced subgraph has a.s. a unique
largest component of size , where
is the survival probability of a specific branching process.Comment: 18 pages, 1 figur