VSH and multiplicative modular relations between small primes with polynomial exponents

Abstract

We study the frequency of the positive integers $n$ that are products of two primes of the same order of magnitude and such that the congruence

$$\begin{aligned} \prod _{i=1}^k p_i^{f_i (n)} \equiv 1 \pmod {n} \end{aligned}$$

holds with some fixed nonzero polynomials $f_1(X), \ldots , f_k(X) \in {\mathbb {Z}}[X]$, where $p_i$ denotes the $i$th prime. The question is motivated by collision finding in the so-called Very Smooth Hash function, introduced by Contini et al. (Lecture notes in computer science, vol. 4004. Springer, Berlin, pp 165–182, 2006).