We estimate the Hurst parameter H∈(0, 1) of a fractional Brownian motion from discrete noisy data, observed along a high-frequency sampling scheme. When the intensity τ n of the noise is smaller in order than n− H we establish the LAN property with optimal rate n− 1/2. Otherwise, we establish that the minimax rate of convergence is (n/τ n 2)− 1/(4 H+ 2) even when τ n is of order 1. Our construction of an optimal procedure relies on a Whittle type construction possibly pre-averaged, together with techniques developed in Fukasawa et al.(2019). We establish in all cases a central limit theorem with explicit variance, extending the classical results of Gloter and Hoffmann (2007).