Statistical Saturation Attacks have been introduced and applied to the block cipher PRESENT at CT-RSA 2009. In this paper, we consider their natural extensions. First, we investigate the existence of better trails than the one used in the former attack. For this purpose, we provide a theoretical evaluation of the trail distributions using probability transition matrices. Since the exhaustive evaluation of all possible distributions turned out to be computationally hard, we additionally provide a heuristic branch-and-bound algorithm that allows us to generate a large number of good trails. These tools confirm that the trail of CT-RSA 2009 was among the best possible ones, but also suggest that numerous other trails have similar properties. As a consequence, we investigate the use of multiple trails and show that it allows significant improvements of the previous cryptanalysis attempts against PRESENT. Estimated complexities indicate that PRESENT-80 is safe against key recovery, by a small security margin. We also discuss the impact of multiple trails for the security of the full PRESENT-128. We finally put forward a "statistical hull" effect that makes the precise theoretical analysis of our results difficult, when the number of block cipher rounds increases.
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