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Suppose someone has single private key, and from that private key he made a public key for Bitcoin, Ethereum, Dogecoin, etc. - is it possible to figure out the private key having those multiple public keys?

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It depends on how many times that private key has been used and whether the attacker has access to a side channel. Practically speaking if the attacker does not have access to a side channel and does not know a vulnerability in the secp256k1 curve (there are no known ones), then no, it's not practically possible to figure out the private key.

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