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Some exchanges mentions they use a cold wallet to keep the private key. As far as I understand cold wallet is a wallet either in an offline machine or kept in hardware.

However, it appears that transfers can still be made online on these exchanges and the underlying system will automate the signing of transaction.

How do exchanges able to maintain a cold wallet while still able to automate the process? Is it by setting up local network with limited access or by other means?

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The exchanges that use cold storage keep most of their coins offline, but do have a small percentage of funds that is necessary for normal business in online wallets.

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They are using a master public key for generating addresses and storing master private key offline.

When you make a transaction, you are not calling to get the master private key directly and signing the transaction.

Someone either processes the transactions manually or following some security checkpoints before signing the transaction.

We can't really know what is going on exactly, but it's just an example.

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