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?
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$\begingroup$ I'm reading the {256} as meaning for SHA-256 (and derivatives). $\endgroup$ – fgrieu Sep 22 at 8:41
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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
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1$\begingroup$ So these indices are nothing to do with the traditional use of the sigma symbol for summation, then. $\endgroup$ – user258279 Sep 22 at 9:55
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