13 research outputs found
Even Delta-Matroids and the Complexity of Planar Boolean CSPs
The main result of this paper is a generalization of the classical blossom
algorithm for finding perfect matchings. Our algorithm can efficiently solve
Boolean CSPs where each variable appears in exactly two constraints (we call it
edge CSP) and all constraints are even -matroid relations (represented
by lists of tuples). As a consequence of this, we settle the complexity
classification of planar Boolean CSPs started by Dvorak and Kupec.
Using a reduction to even -matroids, we then extend the tractability
result to larger classes of -matroids that we call efficiently
coverable. It properly includes classes that were known to be tractable before,
namely co-independent, compact, local, linear and binary, with the following
caveat: we represent -matroids by lists of tuples, while the last two
use a representation by matrices. Since an matrix can represent
exponentially many tuples, our tractability result is not strictly stronger
than the known algorithm for linear and binary -matroids.Comment: 33 pages, 9 figure
On the Memory-Hardness of Data-Independent Password-Hashing Functions
We show attacks on five data-independent memory-hard functions (iMHF)
that were submitted to the password hashing competition. Informally, an MHF is a
function which cannot be evaluated on dedicated hardware, like ASICs, at
significantly lower energy and/or hardware cost than evaluating a single
instance on a standard single-core architecture. Data-independent means the
memory access pattern of the function is independent of the input; this makes
iMHFs harder to construct than data-dependent ones, but the latter can be
attacked by various side-channel attacks.
Following [Alwen-Blocki\u2716], we capture the evaluation of an iMHF as a directed
acyclic graph (DAG). The cumulative parallel pebbling complexity of this
DAG is a good measure for the cost of evaluating the iMHF on an ASIC. If n
denotes the number of nodes of a DAG (or equivalently, the number of operations
--- typically hash function calls --- of the underlying iMHF), its pebbling
complexity must be close to n^2 for the iMHF to be memory-hard. We show that
the following iMHFs are far from this bound: Rig.v2, TwoCats and
Gambit can be attacked with complexity O(n^{1.75}); the
data-independent phase of Pomelo (a finalist of the password hashing
competition) and Lyra2 (also a finalist) can be attacked with complexity
O(n^{1.83}) and O(n^{1.67}), respectively.
For our attacks we use and extend the technique developed by [Alwen-Blocki\u2716],
who show that the pebbling complexity of a DAG can be upper bounded in terms of
its depth-robustness
Quantitative weak compactness
In this thesis we study quantitative weak compactness in spaces (C(K), τp) and later in Banach spaces. In the first chapter we introduce several quantities, which in different manners measure τp-noncompactness of a given uniformly bounded set H ⊂ RK . We apply the results in Banach spaces in chapter 2, where we prove (among others) a quantitative version of the Eberlein-Smulyan theorem. In the third chapter we focus on convex closures and how they affect measures of noncompactness. We prove a quantitative version of the Krein-Smulyan theorem. The first three chapters show that measuring noncompactness is intimately related to measuring distances from function spaces. We follow this idea in chapters 4 and 5, where we measure distances from Baire one functions first in RK and later also in Banach spaces.