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I have question about how a user's identity is verified by a Certification Authority.

As I understand it, the process works as follows:

  1. User generates his private key and (with the help of private key) generates his public key.
  2. Now user want to register his public key. For this, he has to send the public key to CA.
  3. User send Certificate request to the CA.

How will the CA know the user's identity for registration?

Can someone please explain it mathematically?

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There's no mathematics involved - the identity of a user is verified by real-world measures (e.g. company registration information), although many CAs aren't that discerning. As it stands, there's no cryptography involved - perhaps you could clarify what mathematics you think are involved? –  archie Nov 7 '13 at 7:32
    
As archie says there is no mathematics involved. Just to clarify, you write: "For this, he has to send the public key to CA. --> User send Certificate request to the CA". In commonly used PKIX (PKI using X.509 certificates) you do not send the public key and then a certificate request. You send a certificate request (called a certificate signing request) which includes the public key, your desired attributes and a proof that you know the corresponding private key (typically a signature on some data). Latter to be sure that you don't take a public key of someone else (to produce a fake cert). –  DrLecter Nov 7 '13 at 7:53
    
As already indicated, no mathematics. Normally for "common" internet certificates only DNS information is checked. That's why there are certificates with "high level of assurance" for banks etc., where additional checks are performed. Other means of checking are company registration records, site visits etc. Note: certificate requests are signed by the private key of the user himself, otherwise the wrong public key may be signed by the CA. –  owlstead Nov 7 '13 at 8:58
    
Okay, could someone sum these comments up into an answer? –  Paŭlo Ebermann Nov 7 '13 at 10:25
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1 Answer

First of all, as already mentioned in the comments, there is absolutely no mathematics or cryptography involved.

Independent of the model used, i.e., public key cryptography, identity based cryptography, or certificateless public key cryptography (as i summed up in my answer here), the identity verification must be established by other means.

In the wild (currently) only public key cryptography is widely used and PKIX (PKI based on X.509 certificates) is the predominant "standard". Here, the user who generates the key pair sends a certificate signing request (including the public key, identity attributes - which should be included - and a proof that the user holds the corresponding private key - which is established by signing the request) is sent to the certification authority. As already noted in the comments, the "proof of possession" of the respective private key is required to be sure that the user does not come up with the public key of some other user (for which he does not possess the private key).

Then, how the identity verification is done before issuing the certificate mainly depends on the purpose of the certificate, i.e., the corresponding private key.

For instance, many countries, as widely encountered across Europe, have deployed national electronic identity (eID) solutions, where the respective private key can be used to sign documents, and such signatures are equivalent to handwritten signatures. There, you typically have an explicit verification of identity, e.g., by showing up with your smart card (which stores the private key) at a certified registration authority or by offering an online service for activation and the activation code is then send by traditional mail (which you can only receive by showing an official credential such as a passport).

However, this practice is quite different for SSL certificates (which makes up the biggest part) or software signing certificates.

Hopefully, this answers your question and I did not miss anything that should be said.

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