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Paper: The Exact Security of PMAC

Authors:
Peter Gazi , IST Austria
Krzysztof Pietrzak , IST Austria
Michal Rybár , IST Austria
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DOI: 10.13154/tosc.v2016.i2.145-161
URL: http://tosc.iacr.org/index.php/ToSC/article/view/569
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Abstract: PMAC is a simple and parallel block-cipher mode of operation, which was introduced by Black and Rogaway at Eurocrypt 2002. If instantiated with a (pseudo)random permutation over n-bit strings, PMAC constitutes a provably secure variable input-length (pseudo)random function. For adversaries making q queries, each of length at most l (in n-bit blocks), and of total length σ ≤ ql, the original paper proves an upper bound on the distinguishing advantage of Ο(σ2/2n), while the currently best bound is Ο (qσ/2n).In this work we show that this bound is tight by giving an attack with advantage Ω (q2l/2n). In the PMAC construction one initially XORs a mask to every message block, where the mask for the ith block is computed as τi := γi·L, where L is a (secret) random value, and γi is the i-th codeword of the Gray code. Our attack applies more generally to any sequence of γi’s which contains a large coset of a subgroup of GF(2n). We then investigate if the security of PMAC can be further improved by using τi’s that are k-wise independent, for k > 1 (the original distribution is only 1-wise independent). We observe that the security of PMAC will not increase in general, even if the masks are chosen from a 2-wise independent distribution, and then prove that the security increases to O(q<2/2n), if the τi are 4-wise independent. Due to simple extension attacks, this is the best bound one can hope for, using any distribution on the masks. Whether 3-wise independence is already sufficient to get this level of security is left as an open problem.
BibTeX
@article{tosc-2016-28135,
  title={The Exact Security of PMAC},
  journal={IACR Trans. Symmetric Cryptol.},
  publisher={Ruhr-Universität Bochum},
  volume={2016, Issue 2},
  pages={145-161},
  url={http://tosc.iacr.org/index.php/ToSC/article/view/569},
  doi={10.13154/tosc.v2016.i2.145-161},
  author={Peter Gazi and Krzysztof Pietrzak and Michal Rybár},
  year=2016
}