We study the impact of selected parameters on the size of the password space for "Draw-A-Secret" (DAS) graphical passwords. We examine the role of and relationships between the number of composite strokes, grid dimensions, and password length in the DAS password space. We show that a very significant proportion of the DAS password space depends on the assumption that users will choose long passwords with many composite strokes. If users choose passwords having 4 or fewer strokes, with passwords of length 12 or less on a 5 ? 5 grid, instead of up to the maximum 12 possible strokes, the size of the DAS password space is reduced from 58 to 40 bits. Additionally, we found a similar reduction when users choose no strokes of length 1. To strengthen security, we propose a technique and describe a representative system that may gain up to 16 more bits of security with an expected negligible increase in input time. Our results can be directly applied to determine secure design choices, graphical password parameter guidelines, and in deciding which parameters deserve focus in graphical password user studies.