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SHA-2 is a family of cryptographic hash functions designed by the NSA and published by NIST in 2001. The family includes various output lengths (224, 256, 384, and 512 bits).
7
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Why SHA-512/256 when we already have SHA-384?
Both of them are just SHA-512 with different IV's and truncation. What's the point of having 2 of them? (Actually we have 3 of them as SHA-512/224 is the exact same thing.)
12
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Why does SHA2-224 use different IV's than SHA2-256?
Given that it's otherwise just a truncation, I can guess that being able to compute the 224 value from the 256 value is an unwanted property, but that's just speculation.