We analyze the fingerprinting method in Kirovski et al. (2002) and suggest attacks that are better than the ones presented therein. To do this, we observe and use some extra information available in the system. We derive the optimal subtractive estimation attack structure in the sense of minimizing the expected value of the watermark detector statistics. Selection of different attack channel distortion constraints are also discussed. See Zhao et al. (2003) for various recently proposed attacks on spread spectrum fingerprinting systems for private watermarking.