Interpretation of dieharder results for QRNG with Toeplitz randomicity extraction and dependence on minimum entropy.

Hi all, as part of my PhD, I am currently developing a QRNG with Toeplitz hashing as the extractor. I would gladly provide all the details, but I am currently looking to get these results published and the field is quite hot at the moment. If anyone is interested in the full details, please pm me after a month or two, by then I should have it publicly available on arxiv.

Currently, the set up is pretty much finished. I am currently waiting on minimum entropy calculations from a collaborator. Meanwhile, I am checking my extractor implementation by running statistical tests. One thing I know for sure, is that my Toeplitz extractor at the moment is running with an unrealistic extraction ratio (0.7, whereas a more realistic extraction ratio is 0.4, my initial minimum entropy estimations were incorrect). By extraction ratio I mean H\_min/adc\_bit\_depth, where then the extraction ratio is used to construct

I have ran 3 dieharder tests with this command: `dieharder -k 2 -y 1 -a -g 201 -f random_file`, the first file was 8 GB and the other two were 16 GB. The 8 GB run had a single weak result, one 16 GB had three weak p values and the last 16 GB had no weak values. I have also done QQ plots for all the cases. Here is the 8 GB:

https://preview.redd.it/61njhxqbq3qf1.png?width=905&format=png&auto=webp&s=f751ca05bcb0a27cdf199b70246c19c54e5f9499

First 16 GB run (with 3 weak p-values):

https://preview.redd.it/x1dwyxnpq3qf1.png?width=905&format=png&auto=webp&s=a58362bf56ed72a24d88d60144d7b0b978529484

And last 16 GB run (no weak results):

https://preview.redd.it/zc0zawwuq3qf1.png?width=905&format=png&auto=webp&s=34ca0faac3232e5835335c8f95760d9bcfd75d7f

Between these tests, nothing was changed, only new data was gathered for each test. My question is, are these results satisfactory enough? I am aware that these results do not prove quantum randomness, my goal here is to simply confirm whether my Toeplitz extraction is working properly. I am also aware some weak p-values are expected and I also have referred to [this post](https://www.reddit.com/r/crypto/comments/2d4m1v/how_random_is_your_data_on_interpreting_diehard/) for interpreting the QQ plots. However, the swings and the slight saturation in the 8 GB and 16 GB first test are slightly worrying me. Or is such variation expected for a QRNG? I also want to ask, is there any way that the extraction ratio can impact the results from the dieharder tests? My initial answer would be no, since as far as I understand, it mostly affects the security of the QRNG.

Lastly, I would also like to run NIST tests. Does anyone have some good resources on how to run them and interpret their results?

Thank you very much for your help.