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I have bought some bitcoins off local bitcoins and stored them in an Electra wallet in my computer. Given that I have now paid hard cash, how am I to determine that these are real?

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Enter your Bitcoin address (the one you were paid into) into a public blockchain browser such as https://blockchain.info/. That will show you the relevant entries in the global shared Bitcoin ledger, which confirms that you, as the holder of the private key for that address, are the owner of those funds.

If you lose control of your private key for any reason, then your bitcoins might not be yours for much longer.

  • Great answer, thank-you. Where can I see my private key? I had thought that public-private key pairs were newly generated for each transaction and after the transaction became obsolete. From what you say it sounds like I have one master key which is written somewhere that I need to keep safe. – Aerinmund Fagelson Aug 30 '16 at 9:14
  • @AerinmundFagelson: I'm afraid I'm not familiar with the Electra wallet software. Each Bitcoin address has its own private key, which will be stored in a wallet data file somewhere. – Greg Hewgill Aug 30 '16 at 9:18
  • I am a now confused because I am only just learning about these things, but I have been taught that private-public keys work using the RSA algorithm which means that in order to impersonate a private-key one would need to factor a large number into primes, which is only impossible computationally given a short time period. If I only ever have one private key, what stops someone determining it form my public-key given enough time? – Aerinmund Fagelson Aug 30 '16 at 9:22
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    Bitcoin does not use RSA, but ECDSA. Every address has its own associated private and public key. Computing the private key from the public key for ECDSA requires solving an EC discrete logarithm. The expected time needed with current hardware and algorithms is around 14000000000000000000000000 cpu years. – Pieter Wuille Aug 30 '16 at 10:23
  • @greg make sure to NEVER enter your private key into any website. Better to not even have it in your clipboard or in any file on your computer. Also by using an online site to check your balance you are trusting them n ot to lie ti you as well as giving away your privacy to them and possibly evesdroppers. The only real way to know the truth and be reasonably private, is to run a full node yourself (Bitcoin Core). – Jannes Aug 31 '16 at 13:23
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The best way to know your own private key is to go to a wallet generation site like bitaddress.org, and it will generate one of those key pairs for you. It will let you see both the public and the private keys.

If you use a wallet software it is difficult to see the private key; they are trying to make things less complicated for end users so they keep the key out of sight.

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If you have your bitcoins in your phone, tablet or laptop, you can buy yourself some coffee in the nearest Bitcoin-accepting coffee shop. If they accept your bitcoins, you can feel confident they are real.

Alternatively, you can make a small online donation to help Ross Ulbricht. He's a Bitcoin martyr and a political prisoner. He is serving a double life sentence for creating the Silk Road, a free market that helped popularize Bitcoin in its early days.

Finally, you can also donate to a Bitcoin-accepting foundation:

Again, if any of these entities accept your donation, you can feel confident your bitcoins are real.

You can also transfer all your bitcoins to me. I'll check them for free and if they're real I'll transfer them back to you ;)

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