Android's graphical authentication mechanism requires users to unlock their devices by "drawing" a pattern that connects a sequence of contact points arranged in a 3x3 grid. Prior studies demonstrated that human-generated 3x3 patterns are weak (CCS'13); large portions can be trivially guessed with sufficient training. An obvious solution would be to increase the grid size to increase the complexity of chosen patterns. In this paper we ask the question: Does increasing the grid size increase the security of human-generated patterns? We conducted two large studies to answer this question, and our analysis shows that for both 3x3 and 4x4 patterns, there is a high incidence of repeated patterns and symmetric pairs (patterns that derive from others based on a sequence of flips and rotations), and many 4x4 patterns are expanded versions of 3x3 patterns. Leveraging this information, we developed an advanced guessing algorithm and used it to quantified the strength of the patterns using the partial guessing entropy. We find that guessing the first 20% (G0.2) of patterns for both 3x3 and 4x4 can be done as efficiently as guessing a random 2-digit PIN. While guessing larger portions of 4x4 patterns (G0.5) requires 2-bits more entropy than guessing the same ratio of 3x3 patterns, it remains on the order of cracking random 3-digit PINs. Of the patterns tested, our guessing algorithm successful cracks 15% of 3x3 patterns within 20 guesses (a typical phone lockout) and 19% of 4x4 patterns within 20 guesses; however, after 50,000 guesses, we correctly guess 95.9% of 3x3 patterns but only 66.7% of 4x4 patterns. While there may be some benefit to expanding the grid size to 4x4, we argue the majority of patterns chosen by users will remain trivially guessable and insecure against broad guessing attacks. Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of the United States government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

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