On some subgroups of the multiplicative group of finite rings (Q558195)

Let \(S\) be a subset of the finite field \(\mathbb F_q\) of \(q\) elements and \(h\) a polynomial over \(\mathbb F_q\) of degree at least \(2\) with no roots in \(S\). The author proves several lower bounds on the size of the group \(G\) generated by the image of \(\{x-s:s \in S \}\) in the group of units of the ring \(\mathbb F_q[X]/(h)\). These bounds are needed in the analysis of the running time of the recent polynomial time primality testing algorithm of \textit{M. Agrawal, N. Kayal} and \textit{N. Saxena} [``PRIMES is in \(P\)''. Ann. Math. (2) 160, No. 2, 781--793 (2004; Zbl 1071.11070)].