Factoring with Hints

We introduce a new deterministic factoring algorithm, which could be described in the cryptographically fashionable term of “factoring with hints”: we show that, given the knowledge of the factorisations of $O(N^{1/3+\epsilon})$ terms surrounding $N=pq$ product of two large primes, we can recover deterministically $p$ and $q$ in $O(N^{1/3+\epsilon})$ bit operations. Although this is slower than the current best factoring algorithms, this method shows that the factorisations of close integers are related and that consequently one can expect more results along this line of thought.

💡 Research Summary

The paper proposes a deterministic integer‑factorisation method that the author calls “factoring with hints”. The central claim is that if one already knows the complete prime factorizations of O(N^{1/3+ε}) integers that lie in a small neighbourhood of a semiprime N = p·q, then the two unknown primes p and q can be recovered in O(N^{1/3+ε}) bit operations. The author presents this as a new paradigm: the factorisations of nearby numbers are somehow correlated, and exploiting this correlation could reduce the effort needed to factor N itself.

The paper begins with a brief survey of the state of the art. It mentions the Number Field Sieve (NFS) and its variants as the best known probabilistic algorithms, as well as deterministic methods such as Shank’s SQUFOF (running in O(N^{1/4+ε}) time) and conditional results under the Extended Riemann Hypothesis (ERH) that achieve O(N^{1/5+ε}). The author stresses that all these algorithms are probabilistic or heuristic, and that a deterministic sub‑exponential algorithm is still unknown.

The main technical development starts by introducing the multiplicative function σ_{1/2}(n) = Σ_{d|n} d^{1/2}. For a semiprime N = p·q with p < √N < q, one has the exact identity  σ_{1/2}(N) = √N + 1 + √p + √q. Thus, if σ_{1/2}(N) can be computed to within an error of O(1/N), the two unknown primes can be recovered by solving a simple quadratic equation.

To obtain σ_{1/2}(N) the author constructs a smooth test function f(t) = (1−t)^ν (supported on