How to store private user data in google chrome extension?

Question

Google Chrome

There is some data, such as the user's private key, that needs to be used in the extension. We take it, encrypt it with the user's password and store it in LocalStorage. But at some point we need to ask the user for a password, recover the private key and use it.
How to prevent the ability to intercept the password / key by a script from an open tab?