What is the bit complexity of finding all the irreducible factors $f_1, ..., f_r$ of a degree-$d$ polynomial $f(x) = \sum_{i=0}^d a_i\cdot x^i \in \mathbb{Q}[x]$ whose all coefficients are $B$-bit values (that is, $-2^B < a_i < 2^B$)?

If I don't mistake:

  • Berlekamp-Zassenhaus Algorithm solves this in time $O(2^{r-1} \cdot poly(d, B))$, but I don't know the polynomial.
  • An algorithm by Arnold Schönhage finds these factors in $O(d^8 + d^5 \cdot B^3)$.

Furthermore, I have found some modern papers about this problem, but the bit complexity is not explicit given, instead, they provide the complexity in terms of "LLL switches"...

So, do you know the bit complexity of the best algorithm nowadays?


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