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An adversary who has obtained the cryptographic hash of a user's password can mount an offline attack to crack the password by comparing this hash value with the cryptographic hashes of likely password guesses. This offline attacker is limited only by the resources he is willing to invest to crack the password. Key-stretching techniques like hash iteration and memory hard functions have been proposed to mitigate the threat of offline attacks by making each password guess more expensive for the adversary to verify. However, these techniques also increase costs for a legitimate authentication server. We introduce a novel Stackelberg game model which captures the essential elements of this interaction between a defender and an offline attacker. In the game the defender first commits to a key-stretching mechanism, and the offline attacker responds in a manner that optimizes his utility (expected reward minus expected guessing costs). We then introduce Cost Asymmetric Secure Hash (CASH), a randomized key-stretching mechanism that minimizes the fraction of passwords that would be cracked by a rational offline attacker without increasing amortized authentication costs for the legitimate authentication server. CASH is motivated by the observation that the legitimate authentication server will typically run the authentication procedure to verify a correct password, while an offline adversary will typically use incorrect password guesses. By using randomization we can ensure that the amortized cost of running CASH to verify a correct password guess is significantly smaller than the cost of rejecting an incorrect password. Using our Stackelberg game framework we can quantify the quality of the underlying CASH running time distribution in terms of the fraction of passwords that a rational offline adversary would crack. We provide an efficient algorithm to compute high quality CASH distributions for the defender. Finally, we analyze CASH using empirical data from two large scale password frequency datasets. Our analysis shows that CASH can significantly reduce (up to 50%) the fraction of password cracked by a rational offline adversary.

CASH: A Cost Asymmetric Secure Hash Algorithm for Optimal Password Protection