Rationale of the SHA-256 sigma function definitions

Question

When the sigma functions are defined in equations (4.4) and (4.5) in FIPS PUB 180-4, the defined functions show the traditional sigma summation symbol, with limits 0 to {256} and 1 to {256}. The definitions, however, are in terms of circular right shift and XOR operations. Why is a summation symbol used here?

Answer 1

It is the naming of functions and the functions are already defined - rotations (ROTR) and x-ors ($\oplus$). There are two $\Sigma$ and $\sigma$ functions for SHA-256 and SHA-512 series. And, in Greek, the $\Sigma$ is capital of $\sigma$.

The upper index represents the SHA family as 256 and 512. The sub-index, selects the function. $\Sigma_{0}^{\{256\}}$ means the first $\Sigma$ function of SHA-256 series. Note the curly braces.

Note: the functions are started to appear with SHA-2 series. It is not listed with SHA-1, the document can be seen from archive.org