In many schemes for digital watermarking, an attacker in possession of a small fixed number of differently marked copies of a single document can easily produce an unmarked copy. In contrast, we show in this paper that if the watermarks have a component-wise normal distribution, then $Omega ig( {sqrt{n/ln m}},/ig)$ differently marked copies are required to successfully erase any mark with non-negligible probability. Here n is the dimensionality of the watermarks, and m is the number of differently marked copies distributed. This result holds in a specific but natural statistical model of watermarking.