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

摘要

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 × 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 √n × 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 1/2 based on √n × 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 1/2 based on only √n/ℓ × 2~(n/2) encryption queries, where ℓ is the number of blocks present in encryption queries.