**On the Depth-Robustness and Cumulative Pebbling Cost of Argon2i**

*Jeremiah Blocki and Samson Zhou*

**Abstract: **Argon2i is a data-independent memory hard function that won the password hashing competition. The password hashing algorithm has already been incorporated into several open source crypto libraries such as libsodium. In this paper we analyze the cumulative memory cost of computing Argon2i. On the positive side we provide a lower bound for Argon2i. On the negative side we exhibit an improved attack against Argon2i which demonstrates that our lower bound is nearly tight. In particular, we show that

(1) An Argon2i DAG is $\left(e,O\left(n^3/e^3\right)\right))$-reducible.

(2) The cumulative pebbling cost for Argon2i is at most $O\left(n^{1.768}\right)$. This improves upon the previous best upper bound of $O\left(n^{1.8}\right)$ [Alwen and Blocki, EURO S&P 2017].

(3) Argon2i DAG is $\left(e,\tilde{\Omega}\left(n^3/e^3\right)\right))$-depth robust. By contrast, analysis of [Alwen et al., EUROCRYPT 2017] only established that Argon2i was $\left(e,\tilde{\Omega}\left(n^2/e^2\right)\right))$-depth robust.

(4) The cumulative pebbling complexity of Argon2i is at least $\tilde{\Omega}\left( n^{1.75}\right)$. This improves on the previous best bound of $\Omega\left( n^{1.66}\right)$ [Alwen et al., EUROCRYPT 2017] and demonstrates that Argon2i has higher cumulative memory cost than competing proposals such as Catena or Balloon Hashing.

We also show that Argon2i has high {\em fractional} depth-robustness which strongly suggests that data-dependent modes of Argon2 are resistant to space-time tradeoff attacks.

**Category / Keywords: **secret-key cryptography / Argon2i, Memory Hard Functions, Depth-Robustness

**Date: **received 20 May 2017, last revised 3 Aug 2017

**Contact author: **jblocki at purdue edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20170804:042135 (All versions of this report)

**Short URL: **ia.cr/2017/442

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