For p ≤ n, let b1(n),...,bp(n) be independent random vectors in ${ℝ}^n$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If ${\widehat{b}}_1^{\left(n\right)},...,{\widehat{b}}_p^{\left(n\right)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios...