823 research outputs found
Families of locally separated Hamilton paths
We improve by an exponential factor the lower bound of K¨orner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has maximum degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor
Skewincidence
We introduce a new class of problems lying halfway between questions about
graph capacity and intersection. We say that two binary sequences x and y of
the same length have a skewincidence if there is a coordinate i for which
x_i=y_{i+1}=1 or vice versa. We give rather sharp bounds on the maximum number
of binary sequences of length n any pair of which has a skewincidence
Path representation of maximal parabolic Kazhdan-Lusztig polynomials
We provide simple rules for the computation of Kazhdan--Lusztig polynomials
in the maximal parabolic case. They are obtained by filling regions delimited
by paths with "Dyck strips" obeying certain rules. We compare our results with
those of Lascoux and Sch\"utzenberger.Comment: v3: fixed proof of lemma
Machine-checked proofs for cryptographic standards indifferentiability of SPONGE and secure high-assurance implementations of SHA-3
We present a high-assurance and high-speed implementation of the SHA-3 hash function. Our implementation is written in the Jasmin programming language, and is formally verified for functional correctness, provable security and timing attack resistance in the EasyCrypt proof assistant. Our implementation is the first to achieve simultaneously the four desirable properties (efficiency, correctness, provable security, and side-channel protection) for a non-trivial cryptographic primitive.Concretely, our mechanized proofs show that: 1) the SHA-3 hash function is indifferentiable from a random oracle, and thus is resistant against collision, first and second preimage attacks; 2) the SHA-3 hash function is correctly implemented by a vectorized x86 implementation. Furthermore, the implementation is provably protected against timing attacks in an idealized model of timing leaks. The proofs include new EasyCrypt libraries of independent interest for programmable random oracles and modular indifferentiability proofs.This work received support from the National Institute of Standards and Technologies under agreement number 60NANB15D248.This work was partially supported by Office of Naval Research under projects N00014-12-1-0914, N00014-15-1-2750 and N00014-19-1-2292.This work was partially funded by national funds via the Portuguese Foundation for Science and Technology (FCT) in the context of project PTDC/CCI-INF/31698/2017. Manuel Barbosa was supported by grant SFRH/BSAB/143018/2018 awarded by the FCT.This work was supported in part by the National Science Foundation under grant number 1801564.This work was supported in part by the FutureTPM project of the Horizon 2020 Framework Programme of the European Union, under GA number 779391.This work was supported by the ANR Scrypt project, grant number ANR-18-CE25-0014.This work was supported by the ANR TECAP project, grant number ANR-17-CE39-0004-01
How quickly can we sample a uniform domino tiling of the 2L x 2L square via Glauber dynamics?
TThe prototypical problem we study here is the following. Given a square, there are approximately ways to tile it with
dominos, i.e. with horizontal or vertical rectangles, where
is Catalan's constant [Kasteleyn '61, Temperley-Fisher '61]. A
conceptually simple (even if computationally not the most efficient) way of
sampling uniformly one among so many tilings is to introduce a Markov Chain
algorithm (Glauber dynamics) where, with rate , two adjacent horizontal
dominos are flipped to vertical dominos, or vice-versa. The unique invariant
measure is the uniform one and a classical question [Wilson
2004,Luby-Randall-Sinclair 2001] is to estimate the time it takes to
approach equilibrium (i.e. the running time of the algorithm). In
[Luby-Randall-Sinclair 2001, Randall-Tetali 2000], fast mixin was proven:
for some finite . Here, we go much beyond and show that . Our result applies to rather general domain
shapes (not just the square), provided that the typical height
function associated to the tiling is macroscopically planar in the large
limit, under the uniform measure (this is the case for instance for the
Temperley-type boundary conditions considered in [Kenyon 2000]). Also, our
method extends to some other types of tilings of the plane, for instance the
tilings associated to dimer coverings of the hexagon or square-hexagon
lattices.Comment: to appear on PTRF; 42 pages, 9 figures; v2: typos corrected,
references adde
Homology of the mapping class group for surfaces of genus 2 with boundary
We report on the computation of the integral homology of the mapping class
group of genus g surfaces with one boundary curve and m punctures, when 2g + m
is smaller than 6. In particular, it includes the genus 2 case with no or one
puncture.Comment: This is the version published by Geometry & Topology Monographs on 29
April 200
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