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The core of their premise appears to hinge upon the assumption that people implementing Sphinx (and the original paper) suggest using Anderson/Biham's experimental setup from the original BEAR/LION paper, which is flat out wrong. They're also seem to have missed the point of the BEAR/LION paper, in that BEAR/LION/LIONESS are generic constructs, and the experimental setup in the 90s with SHA-1/SEAL was part of an experimental performance evaluation setup.
Other notes:
The LRW construct they propose is utterly unusable for a lot of use cases due to the rather short limit on AD/plaintext size (1023/1023 - alpha respectively).
SHA-3 (or more specifically Keccak-f 1600) is ridiculously slow.
AEZ has a per-key usage cap of 2^48 bytes (2^44 blocks). I'm not sure why they're freaking out over birthday bound issues.
The one useful thing they're doing is "Have the per hop mac that authenticates the header, also cover the payload". But at that point, I question the need for a fragile/wide-block construct for payload encryption in general.
http://www.cs.ru.nl/~bmennink/pubs/16cans.pdf
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