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Are there any practical algorithms for integer factorization that work by solving the decision version of the problem $\log N$ times to isolate a factor? Or is the decision version of only theoretical interest?

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All of the following algorithms either find a complete factor or a complete non-trivial factorization:

  1. Trial division.

  2. Pollard rho.

  3. Pollard $p-1$ and its variants, including the elliptic curve method (ECM).

  4. Shanks' SQUFOF.

  5. Quadratic sieve (QS).

  6. Number field sieve (NFS).

As far as I understand this list includes all algorithms actually used to factor integers in real life.

Finding just one bit of a factor sounds very unnatural to me, and I can't imagine an algorithm which can find one bit without finding all bits. (Even the SAT algorithm for factorization will find all bits at once.)

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