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SHA-3 has been formally defined in FIPS 202 and in its reference manual. We define the sponge function denoted by $KECCAK[r,c]$ by applying the sponge construction as specified in Algorithm 1 with $KECCAK-f[r+c]$, multi-rate padding and the bitrate $r$. $$KECCAK[r,c] = SPONGE[KECCAK-f[r + c], pad10∗1, r]$$ The value $b = r + c$ is called the width of ...


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Let's start from the top and work down, using FIPS 202 as reference. Note that the KECCAK paper uses a somewhat different notation (!). $$\operatorname{SHA3-224}(M) = \operatorname{KECCAK}[448] (M | 01, 224)$$ $$\operatorname{SHA3-256}(M) = \operatorname{KECCAK}[512] (M | 01, 256)$$ $$\operatorname{SHA3-384}(M) = \operatorname{KECCAK}[768] (M | 01, 384)$$ $$...


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There is work underway to specify KMAC. It's basically just SHA-3, but with a length specification for the key and a special value to indicate that this is KMAC instead of hashing. These constructions are required to make sure that there are no unfortunate collisions with previously hashed data or - more importantly - key / message pairs where $H(K_1,M_1) = ...



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