ECDSA Private Key from SSS
In ECDSA one randomly selects the private key $d_A$ from interval $[1,n-1]$, where the $n$ is the order of the Elliptic curve with $n-1$ non-trivial points, and with the point at infinity, $\mathcal O$ as the trivial point.

$n = \texttt{FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141}$ that is 64 bytes, or as an integer

$n = 115792089237316195423570985008687907852837564279074904382605163141518161494337$

one should prefer $S = \operatorname{Hash}(f(0)) \pmod n$ so that all the information mixed before the modulus. Actually, one can still use the original number, however, if the number is bigger than the order, it generates unnecessary bigger calculations, when this step is performed

$$s=k^{-1}(z+rd_{A})\,{\bmod {\,}}n$$

Note that the $p$ is used to represent the elements of the curve and an easy observation is that it limits the number of points of any curve $\leq p^2$. The Hasse theorem gives a tighter relation;

$$|n-(p+1)|\leq 2{\sqrt  {p}}$$