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It seems like email is at least analogous to public-key cryptography, the public portion of the key being your email address, and the private portion of the key being the password to the email account. Is this merely an analogy, or is email in fact an instance of public-key cryptography?

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The components being compared are not analogous in that they do not serve the same purposes, or similar processing or similar behaviour in the two systems being compared. I think the link you are making between them is too weak to be called an analogy, but there is a degree of metaphor regarding secret versus published components, and the consequences for the secrecy failing. –  Neil Slater Jun 21 at 17:40
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As Neil stated, email is not an instance of PK crypto because anyone looking at the email in transit, or even your email provider, can see the message sent to you. Also there is no way to prove an email sent from you is indeed from you without a digital signature, but this line of reasoning quickly gets blurred by practical concerns. Until you introduce the concept of security or authentication, email is a better initial and intuitive analogy to PK crypto, in the sense that worse analogies (eg. box locked with two keys) are being used. –  rath Jun 21 at 19:16
    
The lockbox analogy is usually used for symmetric encryption or commitments, and it is a little off for public key encryption (everynoe can create the lock, but only the owner has the key to unlock it). However, I think email is an even worse example, since it provides a false sense of confidentiality or is based on wrong assumptions about email. There is no confidentiality for transmission of emails. There is no proof that an email actually came from a certain email address or that the email wasn't tempered with. Email doesn't fulfill any security definition on its own. –  tylo Jun 23 at 12:09
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This is a very abstract and wrong analogy and I assume you are talking about encryption. Often for introducing public key encryption one encounters the "box with lock" example: You publish a box and a lock (your public key) and everyone can put something in the box and close the lock and send it to you. You (the only guy that has the key to the lock - the private key) can unlock the lock and get the stuff that has been put in there, but no one else ( because you are the only guy with the key to the lock) can do so.

But your email example lacks the main desired feature of this analogy (which is called confidentiality), i.e., anyone who runs your email server or listens in the wire can read all the mails intended to you without the need to know your password. Public key encryption relates the public and private key in a mathematical way (by means of some hard to compute function) such that decryption without knowledge of the private key is computationally infeasible, which is obviously not the case in your example.

Interestingly, identity based encryption does something that comes near to what you describe although cryptographically secure. There your email address (or any arbitrary string) can be your public key. But your private key is not an arbitrary independently chosen password, but some secret that is derived from some trusted party based on your email address. Then, anyone can use your email address to encrypt messages for you and only you knowing the corresponding private key can decrypt.

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