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On the Hardness of the NTRU Problem
2021The 25 year-old NTRU problem is an important computational assumption in public-key cryptography. However, from a reduction perspective, its relative hardness compared to other problems on Euclidean lattices is not well-understood. Its decision version reduces to the search Ring-LWE problem, but this only provides a hardness upper bound.We provide two ...
Pellet-Mary, Alice, Stehlé, Damien
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The Hard Problem is Mainly Hard Work
Cognitive Neuroscience, 2020Doerig et al. point out that there is now a great deal of evidence bearing directly on our understanding of consciousness. However, they argue that the multiplicity of theories suggest that we have a 'lack of stringent criteria specifying how empirical data constrains ToCs.'
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Hard Instances of Hard Problems
2000This paper investigates the instance complexities of problems that are hard or weakly hard for exponential time under polynomial time, many-one reductions. It is shown that almost every instance of almost every problem in exponential time has essentially maximal instance complexity.
Jack H. Lutz +2 more
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SIAM Journal on Computing, 1995
Summary: A weak completeness phenomenon is investigated in the complexity class \(E= \text{DTIME}(2^{\text{linear}})\). According to standard terminology, a language \(H\) is \(\leq^P_m\)-hard for \(E\) if the set \(P_m(H)\), consisting of all languages \(A\leq^P_m H\), contains the entire class \(E\). A language \(C\) is \(\leq^P_m\)-complete for \(E\)
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Summary: A weak completeness phenomenon is investigated in the complexity class \(E= \text{DTIME}(2^{\text{linear}})\). According to standard terminology, a language \(H\) is \(\leq^P_m\)-hard for \(E\) if the set \(P_m(H)\), consisting of all languages \(A\leq^P_m H\), contains the entire class \(E\). A language \(C\) is \(\leq^P_m\)-complete for \(E\)
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Operations Research, 1980
We consider a class of algorithms which use the combined powers of branch-and-bound, dynamic programming and rudimentary divisibility arguments for solving the zero-one knapsack problem. Our main result identifies a class of instances of the problem which are difficult to solve by such algorithms.
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We consider a class of algorithms which use the combined powers of branch-and-bound, dynamic programming and rudimentary divisibility arguments for solving the zero-one knapsack problem. Our main result identifies a class of instances of the problem which are difficult to solve by such algorithms.
openaire +1 more source