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SHA1's compression function has two distinct parts: one where 16 words are expanded into 80 words, and one where the 80 words are used over 80 rounds. Are there specific names or terms for each of these parts, and if so, what are they?

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The part where 16 words are expanded into 80 words is often called message schedule (the name in FIPS 186-4) or message expansion; that's also the key expansion phase of the cipher at the core of the compression function (SHA1's message block is the key of that cipher, due to the Merkle-Damgård + Davies-Meyer structure of SHA-1).

The rest of the compression function would be the compression itself; or the encryption step of the cipher followed by the Davies-Meyer final combination (which, in SHA-1, is five additions modulo $2^{32}$, rather than XOR).

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