Yes, you don't want to to DAC from the Arduino, just use the pseudorandom bit stream directly. One output pin with the random bit on it.
The sequence will repeat periodically, period depending on the length of your shift register and where you tap it.
I have a wave thing on a clock and it repeats every 12-15 seconds, no good!
A simple transistor circuit will give you real random noise that will not repeat. Besides costing a great deal less.
This
https://modwiggler.com/forum/viewtopic.php?t=209024
shows the circuit I had in mind - google
white noise circuit
and look at images, you will see several many two-transistor white noise generators, small variations in component values &c., but all exploiting the fact that transistors be noisy.
If the first transistor you try isn't (noisy) or you don't like the noise it makes try a few different ones for the noise generating transistor, the one w/ only the base and emitter connected.
I did a white noise blanket the room thing, well two sources, each had its own noise circuit, very much the one linked above +/-.
Sleep well. 
a7