Hi Julian Wilson, thanks for the feedback.

What I meant by “checkpoints is a centralized solution” is that they might be directly centralized, or implemented in a decentralized way, but will cause problems down the line that will still require centralized solutions. So in effect, checkpointing is centralizing.

There’s actually a difference between ‘old private keys attack’ & ‘current private keys attack’. You can check out my Part 2 which discusses this more in-depth.

https://medium.com/@hugonguyen/proof-of-stake-private-keys-attacks-and-unforgeable-costliness-the-unsung-hero-5caca70b01cb

Cheers!

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