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?

  • $\begingroup$ I'm reading the {256} as meaning for SHA-256 (and derivatives). $\endgroup$
    – fgrieu
    Commented Sep 22, 2019 at 8:41

1 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

  • 1
    $\begingroup$ So these indices are nothing to do with the traditional use of the sigma symbol for summation, then. $\endgroup$
    – user258279
    Commented Sep 22, 2019 at 9:55
  • $\begingroup$ That is correct. $\endgroup$
    – kelalaka
    Commented Sep 22, 2019 at 9:58

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