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 leastbased 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.