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According to wikipedia:

https://en.wikipedia.org/wiki/Rijndael_S-box

AES is doing interesting thing (where $<<<$ is circular shift):

$s = b oplus (b <<< 1) oplus (b <<< 2) oplus (b <<< 3) oplus (b <<< 4)$

and this is equal to ($times$ is multiplication in $GF(2^8)$):

$s = b times 31 mod 257$

This provides great bit mixing to my eye. Let’s say I have 128 bit $x$ and $y$ and I want to compute something similar:

$x = y oplus (y <<< 1) oplus (y <<< 2) oplus (y <<< 3) oplus … oplus (y <<< 64)$

Can I do it faster using multiplication in $GF(2^{128}) mod 2^{128}+1$? I don’t know theory behind this, so I have two types of multipliers for this:

$2^{125}-1$

and

$2^{65}-1$

I think this second one may work in the same way in $GF(2^{128})$, this is the rule. So is there similar number which I can use? What is that number?