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Is it true that the cryptography which is used in bitcoin allows for leakage of information which could be used to recalculate the private key - IF you are continuously reusing the same public address for transactions?

I think there were faulty client implementations in the past which have mishandled the protocol and leaked the keys when reusing same address.

If that is correct, is it possible to make any kind of mathematical assumption, how many times of reusing the same address could practically enable the reengineering of the private key?

After how many times of reusage would you strictly advice a user to migrate to another wallet with a new private key? Two times reusage is enough to break the key?

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Yes, this happens when the same key is used perform multiple signatures with the same R value - simply using the same key for multiple, sanely produced signatures with a different R value will not result in leakage, no matter how many such signatures you make.

Modern wallets use a deterministic nonce via RFC 6979.

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  • Note that no competent software would ever re-use a nonce. – Anonymous Aug 1 at 16:46
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    @Raghav Sood: note that OP was asking about reusing keys, not reusing nonces. Reusing keys will not leak your private keys (as long as the nonces are securely generated) - but may reduce your privacy. – Pieter Wuille Aug 3 at 0:17
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    @PieterWuille True, but I suspect the OP was referring to the nonce issues, since they specifically called out faulty clients, which is how several keys were leaked in the past – Raghav Sood Aug 3 at 1:55

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