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I don't distrust https://www.password-hashing.net/, but I have a hard time gauging its credibility. Why should I trust the results of its password hashing competition?

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You shouldn't. You should trust the subsequent research (or lack thereof of results) and compare its record to other similar algorithms.

Why should you trust AES? an algorithm chosen by a competition?

Tbf, NIST has a much bigger reputation, and the aes competition was a much larger affair attracting a large swath of experts.

PHC had some of the biggest names in the crypto field, had NIST representation and multi-stage evaluations with top notch testing.

PHC was far more mature in many ways over AES.

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