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?


1 Answer 1


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.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.