On the uniformity of distribution of the RSA pairs (Q2701565)

Let \(p\) and \(l\) be two distinct primes and put \(m=pl\). The author shows that for almost all \(e\) with \(\gcd(e,\varphi(m))=1\) the RSA pairs \((x,x^e)\) are uniformly distributed modulo \(m\) where \(x\) runs through \(\mathbb{Z}_m^*\). The proof is based on a new bound on the exponential sums NEWLINE\[NEWLINEW(r,s)= \sum_{\substack{ 1 \leq e \leq \varphi(m)\\ \gcd(e, \varphi(m))=1}} \biggl|\sum_{x \in \mathbb{Z}^*_m} \exp(2 \pi i (r x + sx^e)) \biggr|,NEWLINE\]NEWLINE namely NEWLINE\[NEWLINE\max_{\gcd(r,s,m)=1} W(r,s)=O(m^{23/12}).NEWLINE\]NEWLINE The author also proves an analogue result on \((x, x^e)\) where \(x\) runs through all ``\(k\)-products'' \(x=\prod_{j=1}^{k} a_{i_j}\) with \(1 \leq i_1 < \ldots < i_k \leq n\) for some fixed \(a_1, \ldots, a_n \in \mathbb{Z}_m^*\).