Efficient polynomial time algorithms computing industrial-strength primitive roots

Jacques Dubrois (1) , Jean-Guillaume Dumas (2)

1 Axalto - Axalto
2 LMC - IMAG - Laboratoire de Modélisation et Calcul

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".

Mots clés: Cryptography; Randomized algorithms; Computational complexity

Domaines: Calcul formel [cs.SC]; Théorie des nombres [math.NT]

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⟩