We mathematically explore a model for the shortness and security for passwords that are stored in hashed form. The model is implicitly in the NIST publication [8] and is based on conditions of the Shannon, Guessing and Min Entropy. We establish various new relations between these three notions of entropy, providing strong improvements on existing bounds such as the McEliece-Yu bound from [7] and the Min entropy lowerbound on Shannon entropy [3]. As an application we present an algorithm generating near optimally short passwords given certain security restrictions. Such passwords are specifically applicable in the context of one time passwords (e.g. initial passwords, activation codes).