Article Dans Une Revue Information Processing Letters Année : 2006

Efficient polynomial time algorithms computing industrial-strength primitive roots

Résumé

E. Bach, following an idea of T. Itoh, has shown how to build a small set of numbers modulo a prime p such that at least one element of this set is a generator of $\mathbb{Z}/p\mathbb{Z}$ [Bach (1997), Itoh (2001)]. E. Bach suggests also that at least half of his set should be generators. We show here that a slight variant of this set can indeed be made to contain a ratio of primitive roots as close to 1 as necessary. We thus derive several algorithms computing primitive roots correct with very high probability in polynomial time. In particular we present an asymptotically $O^{\sim}\left( \sqrt{\frac{1}{\epsilon}}\log^{1.5}(p) + \log^2(p)\right)$ algorithm providing primitive roots of $p$ with probability of correctness greater than $1-\epsilon$ and several $O(\log^\alpha(p))$, $\alpha \leq 5.23$ algorithms computing "Industrial-strength" primitive roots with probabilities e.g. greater than the probability of "hardware malfunctions".

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Dates et versions

hal-00002828 , version 1 (13-09-2004)
hal-00002828 , version 2 (14-09-2004)
hal-00002828 , version 3 (14-09-2004)
hal-00002828 , version 4 (14-09-2004)
hal-00002828 , version 5 (08-12-2008)

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Jacques Dubrois, Jean-Guillaume Dumas. Efficient polynomial time algorithms computing industrial-strength primitive roots. Information Processing Letters, 2006, 97 (2), pp.41-45. ⟨10.1016/j.ipl.2005.09.014⟩. ⟨hal-00002828v5⟩

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