Paper 2018/520

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.

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