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i'm still learning about bitcoin, and i learned about the "proof-of reserve"-concept, that online wallets need to accomplish to prove they are having enough bitcoins for all their liabilities.

Now, the only procedure that i know of to do that is the one described in this document: https://www.bitstamp.net/s/documents/Bitstamp_proof_of_reserves_statement.pdf

This document describes how bitstamp is giving a proof of their reserves, by sending their whole cold storage amount to an adress of themself. So far, that makes sense.

What i don't understand is: What purpose has the signing of a selected message by the third party, described at the end of the first page?

Thanks for any help.

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What i don't understand is: What purpose has the signing of a selected message by the third party, described at the end of the first page?

Without this step, it is possible that the transfer you're observing is entirely unrelated to Bitstamp. They could have watched the chain for a while, waiting for a transfer of sufficient amount, and then claim that it was theirs.

Assuming you trust the third party is in fact a third party, and is not colluding with Bitstamp, the signature proves that BitStamp in fact has control over the private key the funds were sent to, meaning they also have control over the funds.

Note that there is research around more advanced cryptographic techniques that do not rely on a third party, and don't require moving all funds around. For example, see this paper: http://www.jbonneau.com/doc/DBBCB15-CCS-provisions.pdf.

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