**To Hash or Not to Hash Again? (In)differentiability Results for H^2 and HMAC**

*Yevgeniy Dodis and Thomas Ristenpart and John Steinberger and Stefano Tessaro*

**Abstract: **We show that the second iterate H^2(M) = H(H(M)) of a random oracle H cannot achieve strong security in the sense of indifferentiability from a random oracle. We do so by proving that indifferentiability for H 2 holds only with poor concrete security by providing a lower bound (via an attack) and a matching upper bound (via a proof requiring new techniques) on the complexity of any successful simulator. We then investigate HMAC when it is used as a general-purpose hash function with arbitrary keys (and not as a MAC or PRF with uniform, secret keys). We uncover that HMAC’s handling of keys gives rise to two types of weak key pairs. The first allows trivial attacks against its indifferentiability; the second gives rise to structural issues similar to that which ruled out strong indifferentiability bounds in the case of H^2 . However, such weak key pairs do not arise, as far as we know, in any deployed applications of HMAC. For example, using keys of any fixed length shorter than d − 1, where d is the block length in bits of the underlying hash function, completely avoids weak key pairs. We therefore conclude with a positive result: a proof that HMAC
is indifferentiable from a RO (with standard, good bounds) when applications use keys of a fixed length less than d − 1.

**Category / Keywords: **Indifferentiability, hash functions, HMAC

**Publication Info: **Advances in Cryptology - Crypto 2012

**Date: **received 12 Jun 2013, last revised 8 Aug 2013

**Contact author: **rist at cs wisc edu

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

**Note: **Full version of Crypto 2012 paper.

**Version: **20130808:232612 (All versions of this report)

**Short URL: **ia.cr/2013/382

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