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Can large integers be factored efficiently on a classical computer?

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Why Open
When the numbers are sufficiently large, no efficient non quantum integer factorization algorithm is known, yet it has not been proven that no such algorithm exists. The presumed difficulty of this problem is what the security of RSA public key encryption and signatures rests on, so the gap between no algorithm known and no algorithm possible carries the weight of most of the world's encrypted traffic. Peter Shor showed in 1994 that a quantum computer could factor in polynomial time, which sharpens rather than settles the classical question. 1
What Would Resolve
An efficient classical factoring algorithm, which would break RSA, or a proof that none exists, which would put its security on solid ground; large scale quantum computers would change the practical stakes either way. 1
Discipline Needed
Computational number theory, Cryptography 1
Question Kind
Mechanism 1
Question Status
Open 1
Open Question

When the numbers are sufficiently large, no efficient non quantum integer factorization algorithm is known, yet it has not been proven that no such algorithm exists. The presumed difficulty of this problem is what the security of RSA public key encryption and signatures rests on, so the gap between no algorithm known and no algorithm possible carries the weight of most of the world's encrypted traffic. Peter Shor showed in 1994 that a quantum computer could factor in polynomial time, which sharpens rather than settles the classical question.

What would resolve this An efficient classical factoring algorithm, which would break RSA, or a proof that none exists, which would put its security on solid ground; large scale quantum computers would change the practical stakes either way.
OpenComputational number theory, CryptographyWikipedia: Integer Factorization
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1. Wikipedia: Integer Factorization
Wikimedia FoundationIntroduction and Time complexity sectionsView the Source
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