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On some subgroups of the multiplicative group of finite rings. (English) Zbl 1078.11069
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 M. Agrawal, N. Kayal and N. Saxena [“PRIMES is in \(P\)”. Ann. Math. (2) 160, No. 2, 781–793 (2004; Zbl 1071.11070)].

11T55 Arithmetic theory of polynomial rings over finite fields
11Y11 Primality
Full Text: DOI Numdam EuDML
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