**Bernstein Bound on WCS is Tight - Repairing Luykx-Preneel Optimal Forgeries**

*Mridul Nandi*

**Abstract: **In Eurocrypt 2018, Luykx and Preneel described hash-key-recovery and forgery attacks against polynomial hash based Wegman-Carter-Shoup (WCS) authenticators. Their attacks require $2^{n/2}$ message-tag pairs and recover hash-key with probability about $1.34 \times 2^{-n}$ where $n$ is the bit-size of the hash-key. Bernstein in Eurocrypt 2005 had provided an upper bound (known as Bernstein bound) of the maximum forgery advantages.
The bound says that all adversaries making $O(2^{n/2})$ queries of WCS can have maximum forgery advantage
$O(2^{-n})$. So, Luykx and Preneel essentially analyze WCS in a range of query complexities where WCS is known to be perfectly secure. Here we revisit the bound and found that WCS remains secure against all adversaries making $q \ll \sqrt{n} \times 2^{n/2}$ queries. So it would be meaningful to analyze adversaries with beyond birthday bound complexities.

In this paper, we show that the Bernstein bound is tight by describing two attacks (one in the ``chosen-plaintext model" and other in the ``known-plaintext model") which recover the hash-key (hence forges) with probability at least $\frac{1}{2}$ based on $\sqrt{n} \times 2^{n/2}$ message-tag pairs. We also extend the forgery adversary to the Galois Counter Mode (or GCM). More precisely, we recover the hash-key of GCM with probability at least $\frac{1}{2}$ based on only $\sqrt{\frac{n}{\ell}} \times 2^{n/2}$ encryption queries, where $\ell$ is the number of blocks present in encryption queries.

**Category / Keywords: **Authenticator, WCS, GCM, Polynomial hash, AXU, key recovery, forgery.

**Original Publication**** (with minor differences): **IACR-CRYPTO-2018

**Date: **received 26 May 2018, last revised 5 Jun 2018

**Contact author: **mridul nandi at gmail com

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

**Note: **..

**Version: **20180606:055847 (All versions of this report)

**Short URL: **ia.cr/2018/520

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