504 research outputs found
On the Complexity of Computing Two Nonlinearity Measures
We study the computational complexity of two Boolean nonlinearity measures:
the nonlinearity and the multiplicative complexity. We show that if one-way
functions exist, no algorithm can compute the multiplicative complexity in time
given the truth table of length , in fact under the same
assumption it is impossible to approximate the multiplicative complexity within
a factor of . When given a circuit, the problem of
determining the multiplicative complexity is in the second level of the
polynomial hierarchy. For nonlinearity, we show that it is #P hard to compute
given a function represented by a circuit
The Advice Complexity of a Class of Hard Online Problems
The advice complexity of an online problem is a measure of how much knowledge
of the future an online algorithm needs in order to achieve a certain
competitive ratio. Using advice complexity, we define the first online
complexity class, AOC. The class includes independent set, vertex cover,
dominating set, and several others as complete problems. AOC-complete problems
are hard, since a single wrong answer by the online algorithm can have
devastating consequences. For each of these problems, we show that
bits of advice are
necessary and sufficient (up to an additive term of ) to achieve a
competitive ratio of .
The results are obtained by introducing a new string guessing problem related
to those of Emek et al. (TCS 2011) and B\"ockenhauer et al. (TCS 2014). It
turns out that this gives a powerful but easy-to-use method for providing both
upper and lower bounds on the advice complexity of an entire class of online
problems, the AOC-complete problems.
Previous results of Halld\'orsson et al. (TCS 2002) on online independent
set, in a related model, imply that the advice complexity of the problem is
. Our results improve on this by providing an exact formula for
the higher-order term. For online disjoint path allocation, B\"ockenhauer et
al. (ISAAC 2009) gave a lower bound of and an upper bound of
on the advice complexity. We improve on the upper bound by a
factor of . For the remaining problems, no bounds on their advice
complexity were previously known.Comment: Full paper to appear in Theory of Computing Systems. A preliminary
version appeared in STACS 201
Online Multi-Coloring with Advice
We consider the problem of online graph multi-coloring with advice.
Multi-coloring is often used to model frequency allocation in cellular
networks. We give several nearly tight upper and lower bounds for the most
standard topologies of cellular networks, paths and hexagonal graphs. For the
path, negative results trivially carry over to bipartite graphs, and our
positive results are also valid for bipartite graphs. The advice given
represents information that is likely to be available, studying for instance
the data from earlier similar periods of time.Comment: IMADA-preprint-c
Online Dominating Set
This paper is devoted to the online dominating set problem and its variants on trees, bipartite, bounded-degree, planar, and general graphs, distinguishing between connected and not necessarily connected graphs. We believe this paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. In most cases, we obtain tight bounds on the competitive ratios. Our results show that requiring the graphs to be presented in a connected fashion allows the online algorithms to obtain provably better solutions. Furthermore, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm\u27s disadvantage
Implementing Grover Oracles for Quantum Key Search on AES and LowMC
Grover's search algorithm gives a quantum attack against block ciphers by
searching for a key that matches a small number of plaintext-ciphertext pairs.
This attack uses calls to the cipher to search a key space of
size . Previous work in the specific case of AES derived the full gate cost
by analyzing quantum circuits for the cipher, but focused on minimizing the
number of qubits. In contrast, we study the cost of quantum key search attacks
under a depth restriction and introduce techniques that reduce the oracle
depth, even if it requires more qubits. As cases in point, we design quantum
circuits for the block ciphers AES and LowMC. Our circuits give a lower overall
attack cost in both the gate count and depth-times-width cost models. In NIST's
post-quantum cryptography standardization process, security categories are
defined based on the concrete cost of quantum key search against AES. We
present new, lower cost estimates for each category, so our work has immediate
implications for the security assessment of post-quantum cryptography. As part
of this work, we release Q# implementations of the full Grover oracle for
AES-128, -192, -256 and for the three LowMC instantiations used in Picnic,
including unit tests and code to reproduce our quantum resource estimates. To
the best of our knowledge, these are the first two such full implementations
and automatic resource estimations.Comment: 36 pages, 8 figures, 14 table
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