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what algorithm does ethereum use to generate public and private key pair. If i have both the keys how can i verify the match of the key in solidity, and how much gas does it cost? is it an expensive operation?

Nameless
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    to know more about how ethereum generate keys. see https://ethereum.stackexchange.com/questions/3542/how-are-ethereum-addresses-generated – bordalix May 22 '18 at 11:54
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    I think this answer will help you https://ethereum.stackexchange.com/questions/15149/get-public-key-of-msg-sender-in-a-smart-contract/15190#15190 – Dmytro Zarezenko May 22 '18 at 14:06

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