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The exact complexity of factoring integers (the decision problem) is a major open question in TCS (with important implications, especially in cryptography because of the RSA algorithm), and is widely conjectured to lie "between" P and NP-complete (see the AKS algorithm and Shor's algorithm for two other key aspects of its significance).

Yet, there is so much diversity among the NP-complete problems, including some relating to factoring types of objects and problems in number theory.

What are some of the "closest" or "nearest" NP-complete problems to integer factoring?

Optionally, an answer might indicate why such problems are thought not to be equivalent to integer factoring under polynomial time reductions (even circumstantial evidence would be appreciated).

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The textbook of Computational Complexity: Modern Approach, by Arora and Barak gives such example:

They define the decision problem Integer Factoring on input of three positive integers,

$\text{Integer Factoring} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$.

They state that Alon and Kilian showed that if integer $p$ is not required to be prime number then the problem becomes $NP$-complete.

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    $\begingroup$ cstheory.stackexchange.com/questions/4769/… $\endgroup$ – Huck Bennett Jun 21 '13 at 18:01
  • $\begingroup$ p.39 in other words "Given three numbers N, L, U decide if N has a prime factor p in the interval [L, U]. The certificate is the factor p." on p.58 it is stated the NP complete case is an unpublished result from personal communication. does anyone know if its ever been published? $\endgroup$ – vzn Jun 21 '13 at 18:58
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    $\begingroup$ @vzn I don't believe that the proof has been published before. Personaly, I searched a lot for it with no luck. $\endgroup$ – Mohammad Al-Turkistany Jun 21 '13 at 19:21
  • $\begingroup$ What is the current status of this? On p2.4 (42) of theory.cs.princeton.edu/complexity/book.pdf it says "The Graph Isomorphism and Factoring problems are not known to be either in P nor NP-complete." where they have defined the Factoring problem in example 2.2 as in this answer. $\endgroup$ – Anush Jun 5 at 11:01
  • $\begingroup$ @Anush I believe there are no updates $\endgroup$ – Mohammad Al-Turkistany Jun 5 at 14:39

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