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SHA-384 can actually be faster than SHA-256 is some cases. The reason is that SHA-256 used 64 rounds with 32 bit words, while SHA-384 uses 80 rounds with 64 bit words. So, each block processed by SHA-384 is twice as big but uses less than twice as many rounds.

Oh yes, definitely. But SHA-384's output is 128 bits wider, which uses more bandwidth. Although for the AEAD ciphersuites, it doesn't make much of a difference, admittedly.

There is the not so widely used SHA-512/256 for that case - the speed of the 64 bit variants with a 256 bit output.

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