ECM and the Elliott-Halberstam conjecture for quadratic fields (Q6581190)

An integer is \(y\)-friable (or \(y\)-smooth) if all its prime factors are less than \(y\). Let \(E/\mathbb{Q}\) be an elliptic curve that has good reduction precisely at every prime number \(p\) with \(p\nmid \Delta_E\). In the paper under review, the authors provide an estimate of the form\N\[\N\psi_E(x,y)\sim \rho(u)\frac{x}{\log x},\qquad (x\to\infty),\N\]\Nfor the prime counting function \(\psi_E(x,y)=|\Psi_E(x,y)|\) defined on the set\N\[\N\Psi_E(x,y)=\{p\leq x\colon p\text{ prime, }p\nmid \Delta_E, |E(\mathbb{F}_p)|\text{ is \(y\)-friable} \},\N\]\Nwhere \(E(\mathbb{F}_p)\) denotes the set of \(\mathbb{F}_p\)-points on the reduction of \(E\) modulo \(p\), and \(\rho\) is the unique continuous function on \(\mathbb{R}_{\geq 0}\) that is differentiable on \((1,\infty)\) and satisfies \(\rho(u)\equiv 1\) on \([0,1]\) and \(u\rho'(u)=-\rho(u-1)\) on \((1,\infty)\). The main result asserts that the probability that the number of \(\mathbb{F}_p\)-points on a given elliptic curve is friable approaches asymptotically the probability for any integer to be friable. Having cryptographic motivation for their work, the authors give an example of algorithmic application for the splitting step for discrete logarithm.