# Tag Info

16

The number field sieve has never been analyzed rigorously. The complexity that you quote is merely heuristic. The only subexponential algorithm which has been analyzed rigorously is Dixon's factorization algorithm, which is very similar to the quadratic sieve. According to Wikipedia, Dixon's algorithm runs in time $e^{O(2\sqrt{2}\sqrt{\log n\log\log n})}$. ...

8

In the past few months, a version of the number field sieve has been analyzed rigorously: http://www.fields.utoronto.ca/talks/rigorous-analysis-randomized-number-field-sieve-factoring Basically the worst-case running time is $L_n(1/3, 2.77)$ unconditionally and $L_n(1/3, (64/9)^{1/3})$ under GRH. This is not for the "classic" number field sieve, but a ...

6

You can look at the survey paper by Bogdanov and Trevisan, and this survey talk by Luca. The main open question is whether $\mathsf{P} \neq \mathsf{NP}$ implies that there exist hard on average problems in $\mathsf{NP}$. There are also more concrete conjectures about specific problems, two of which were mentioned in the comments: The planted clique problem: ...

4

The assumptions often used in crypto are a form of average-case complexity assumptions, such as hardness of factoring or discrete log in various groups. Many lattice problems used for strong hardness guarantees for lattice based cryptography are qualitatively somewhat different than the factoring/discrete log based cryptographic assumptions. The ...

3

The bounded post correspondence problem is complete for RNP (Randomized NP). Therefore, it is believed to be hard on average over uniformly random distributions of inputs. Y. Gurevich, Average case completeness, J. Comp. Sys. Sci., 42 (3): 346–398, 1991

2

Let $n$ be the total number of elements in all sets in $F$, basically your input size. Maintain a priority queue of the remaining sets, prioritized by cost / number of uncovered elements. Every time you cover an element, update the cost of all sets that cover it. Then there are n total updates, so the total time for the algorithm using a priority queue in ...

2

The first thing that comes to mind is "Smoothed Analysis" of Spielman and Teng: arxiv.org/pdf/cs/0111050.pdf. Their main result is Theorem 5.0.1, which bounds the expected (over "typical instances") runtime of a version of the Simplex algorithm by a polynomial, though the degree of the polynomial is not stated there.

2

If Graph Isomorphism is randomly self-reducible in the sense of the question (clarified in the comments), then it could be solved in poly time. The reason is that there is in fact an average-case linear time algorithm for GI (even a canonical form) [BK]. For Group Isomorphism, this is not known. However, it's also somewhat of a funny question, because of ...

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