user21351: One question I'm not able to properly answer when asked is

How exactly can  the blockchain verify the identity of the user without a centralized trusted party such as VeriSign?

I know this is done in a decentralized way and the ECDSA algorithm should do the trick, but other applications still need a centralized trusted party in order to verify if the user is really who he says he is even when they're using ECDSA as well.
But how exactly can I guarantee that nobody can pretend to be someone else and spend their bitcoin instead?
Yes, I know the basic of how public-key cryptography works, but Information Security is not strong with me.

Edit: let me explore the question further.
Suppose I own a coffee house with intense flow of customers. How can I be sure about costumers that had already paid and others how didn't pay?
Let's say that Bob and Anna are my customers. Bob orders a Macchiato, Anna orders a Capuccino. Both coffees have the same price. How can I know which payment came from each one of them?

user32091: simple answer: give each customer your different address
never reuse addresses

user23123: The only practical way to verify the identity of a user is to make the user sign something with their private key. The only identity information that you get at that point is that the user has access to the private key. On the bitcoin blockchain, the owner of the private key is the only one that can move the bitcoins locked to that key because a signature is needed to do so. Essentially, if you try to pretend to be someone else, you won't succeed since you won't be able to produce a signature.

user5464: Different orders have different Bitcoin payment addresses.
The order is paid when the address in question receives the requested amount of Bitcoins.

user5406: The payer's identity is often irrelevant in Bitcoin payments. The important check is that the payment was authorized by the owner. In Bitcoin, whoever controls the private key owns the funds, and therefore it is sufficient that the transaction was signed with the correct private key (which is a given for a valid transaction).
What's important to the shop owner is to know that they got paid. As Bitcoin payments are irreversible, it's not important whether Bob gives his name, just like the barista wouldn't care to get Bob's name when Bob walks into the café and pays for a Macchiato with cash.
To distinguish payments, you give out a new receiving address for each invoice. That way, when an address receives money, you know which invoice was settled. Luckily, the Bitcoin address space is sufficiently huge that everyone can make as many addresses as they need.

user208: Do you really care which customer paid, or do you care which bill was paid?
Say you have customers Anna and Bob, who both order something. Bob notices he's out of money, so Anna says she'll pay for both. If you really want the sender to reveal their identity, your system would already fail, as Bob never pays anything.
In almost all cases, all you care about is which bill is paid, and you have no need for the payer's identities (which could be a privacy violation in some cases).
The normal way to do this in practice is by creating a new payment address for each payment you want to receive. You track which bill was paid by tracking at which addresses you received money. Creating new addresses is very cheap, and they are sufficiently long that it is exceedingly unlikely that the same one will ever be produced twice.

21351: <p>One question I'm not able to properly answer when asked is</p>
<blockquote>
<p>How exactly can  the blockchain verify the identity of the user without a centralized trusted party such as VeriSign?</p>
</blockquote>
<p>I know this is done in a decentralized way and the ECDSA algorithm should do the trick, but other applications still need a centralized trusted party in order to verify if the user is really who he says he is even when they're using ECDSA as well.</p>
<p>But how exactly can I guarantee that nobody can pretend to be someone else and spend their bitcoin instead?</p>
<p>Yes, I know the basic of how public-key cryptography works, but Information Security is not strong with me.</p>
<hr />
<h2>Edit: let me explore the question further.</h2>
<p>Suppose I own a coffee house with intense flow of customers. How can I be sure about costumers that had already paid and others how didn't pay?</p>
<p>Let's say that Bob and Anna are my customers. Bob orders a Macchiato, Anna orders a Capuccino. Both coffees have the same price. How can I know which payment came from each one of them?</p>
 32091: <p>simple answer: give each customer your different address</p>

<p>never reuse addresses</p>
 23123: <p>The only practical way to verify the identity of a user is to make the user sign something with their private key. The only identity information that you get at that point is that the user has access to the private key. On the bitcoin blockchain, the owner of the private key is the only one that can move the bitcoins locked to that key because a signature is needed to do so. Essentially, if you try to pretend to be someone else, you won't succeed since you won't be able to produce a signature.</p>
 5464: <p>Different orders have different Bitcoin payment addresses.</p>

<p>The order is paid when the address in question receives the requested amount of Bitcoins.</p>
 5406: <p>The payer's identity is often irrelevant in Bitcoin payments. The important check is that the payment was authorized by the owner. In Bitcoin, whoever controls the private key owns the funds, and therefore it is sufficient that the transaction was signed with the correct private key (which is a given for a valid transaction).</p>
<p>What's important to the shop owner is to know that they got paid. As Bitcoin payments are irreversible, it's not important whether Bob gives his name, just like the barista wouldn't care to get Bob's name when Bob walks into the café and pays for a Macchiato with cash.</p>
<p>To distinguish payments, you <strong>give out a new receiving address for each invoice</strong>. That way, when an address receives money, you know which invoice was settled. Luckily, the Bitcoin address space is sufficiently huge that <a href="https://bitcoin.stackexchange.com/q/8804/5406">everyone can make as many addresses as they need</a>.</p>
 208: <p>Do you really care which customer paid, or do you care which bill was paid?</p>

<p>Say you have customers Anna and Bob, who both order something. Bob notices he's out of money, so Anna says she'll pay for both. If you really want the sender to reveal their identity, your system would already fail, as Bob never pays anything.</p>

<p>In almost all cases, all you care about is which bill is paid, and you have no need for the payer's identities (which could be a privacy violation in some cases).</p>

<p>The normal way to do this in practice is by creating a new payment address for each payment you want to receive. You track which bill was paid by tracking at which addresses you received money. Creating new addresses is very cheap, and they are sufficiently long that it is exceedingly unlikely that the same one will ever be produced twice.</p>
