The Sponge function is known to achieve 2~(c/2) security, where c is its capacity. This bound was carried over to keyed variants of the function, such as SpongeWrap, to achieve a min{2~(c/2), 2~κ} security bound, with κ the key length. Similarly, many CAESAR competition submissions are designed to comply with the classical 2~(c/2) security bound. We show that Sponge-based constructions for authenticated encryption can achieve the significantly higher bound of min {2~(b/2),2~c,2~κ} asymptotically, with b > c the permutation size, by proving that the CAESAR submission NORX achieves this bound. Furthermore, we show how to apply the proof to five other Sponge-based CAESAR submissions: Ascon, CBEAM/STRIBOB, ICEPOLE, Keyak, and two out of the three PRIMATEs. A direct application of the result shows that the parameter choices of these submissions are overly conservative. Simple tweaks render the schemes considerably more efficient without sacrificing security. For instance, NORX64 can increase its rate and decrease its capacity by 128 bits and Ascon-128 can encrypt three times as fast, both without affecting the security level of their underlying modes in the ideal permutation model.
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