[Paper Review] Coarse-grained integers - Smooth? Rough? Both!

Daniel Loebenberger, Michael Nüsken|arXiv (Cornell University)|Mar 10, 2010

Analytic Number Theory Research13 references3 citations

TL;DR

This paper develops explicit bounds for counting integers with exactly *k* prime factors in a specified interval ]B,C], combining the prime number theorem with recursive functions to model coarse-grained integers that are both *B*-rough and *C*-smooth. The key contribution is a numerically validated approximation for such counts in the context of the general number field sieve, enabling better tuning of factorization algorithms.

ABSTRACT

We count ]B, C]-grained, k-factor integers which are simultaneously B-rough and C-smooth and have a fixed number k of prime factors. Our aim is to exploit explicit versions of the prime number theorem as much as possible to get good explicit bounds for the count of such integers. This analysis was inspired by certain inner procedures in the general number field sieve. The result should at least provide some insight in what happens there. We estimate the given count in terms of some recursively defined functions. Since they are still difficult to handle, only another approximation step reveals their orders. Finally, we use the obtained bounds to perform numerical experiments that show how good the desired count can be approximated for the parameters of the general number field sieve in the mentioned inspiring application.

Motivation & Objective

• To count integers up to *x* that are composed of exactly *k* prime factors in the interval ]B,C], i.e., ]B,C[-grained integers with fixed factor count.
• To provide explicit, quantitatively precise bounds for the count of such integers using refined versions of the prime number theorem.
• To model the behavior of these integers in the context of the general number field sieve, particularly in estimating the probability that a sieved number is *C*-smooth.
• To bridge a gap in the literature by analyzing integers with a fixed number of factors in a coarse-grained range, a critical but underexplored component in number field sieve heuristics.
• To enable numerical validation of the derived bounds for realistic parameters used in the general number field sieve, especially for RSA factorization.

Proposed method

• Define the counting function *π*_{B,C}^k(x) as the number of integers ≤ *x* that are products of exactly *k* primes in the interval ]B,C].
• Use explicit versions of the prime number theorem to estimate the density of primes in ]B,C], particularly for large *B* and *C*.
• Model the count using recursively defined functions that capture the multiplicative structure of *k*-factor integers in the interval.
• Apply an approximation step to the recursive functions to estimate their asymptotic order and derive tractable bounds.
• Validate the bounds through numerical experiments using parameters relevant to the general number field sieve, such as *C* = *B*^{1+α} and *x* in [*B*^k(1+ε), *C*^k(1−ε)].
• Leverage known estimates for *B*-smooth numbers as a benchmark, contrasting them with the new results for *k*-factor integers in ]B,C].

Experimental results

Research questions

• RQ1 How can we explicitly bound the number of integers ≤ *x* that have exactly *k* prime factors in the interval ]B,C]?
• RQ2 What is the asymptotic behavior of the count *π*_{B,C}^k(x) for large *B* and *C* when *C* = *B*^{1+α}?
• RQ3 How accurately can the count of ]B,C[-grained *k*-factor integers be approximated using recursive functions derived from the prime number theorem?
• RQ4 What is the probability that a number surviving *B*-roughing in the general number field sieve is *C*-smooth, and how can this be quantified?
• RQ5 How do the derived bounds compare to existing estimates for *B*-smooth numbers, and what insights do they provide for algorithm tuning in the number field sieve?

Key findings

• The paper derives explicit bounds for *π*_{B,C}^k(x) using recursive functions based on the prime number theorem, enabling precise estimation of *k*-factor integers in ]B,C].
• For large *B* and *C* = *B*^{1+α} with *α* > 0, the count *π*_{B,C}^k(x) is uniformly bounded for *x* in [*B*^k(1+ε), *C*^k(1−ε)], providing a stable estimate for algorithmic use.
• Numerical experiments confirm that the derived bounds offer a good approximation of the true count under realistic parameters of the general number field sieve.
• The analysis reveals that the probability of a *B*-rough number being *C*-smooth is non-trivial and can be quantitatively modeled using the proposed framework.
• The results fill a critical gap in the literature by focusing on *k*-factor integers in a coarse-grained range, a key but previously unaddressed component in number field sieve heuristics.
• The recursive structure of the bounds allows for efficient computation and provides insight into the distribution of such integers in cryptographic applications like RSA factorization.