Quote from: katatunix on April 26, 2015, 05:55:32 PM

Why don't we sign receivers' addresses also?

We do. Why do you think that we don't?

Quote from: katatunix on April 26, 2015, 05:55:32 PM

Let's consider following scenario:

Alice (a honest girl) is buying a product from the merchant Apple.
Alice create a new transaction TxA to send her bitcoins to Apple.
Bob (a bad guy) receives TxA and modifies the receiver's address to create TxB.

Not possible. When Alice created the transaction, the signature covered her inputs AND all the outputs. If Bob changes the outputs then the transaction is no longer valid because the signature no longer matches the signed content.

Quote from: katatunix on April 26, 2015, 05:55:32 PM

As far as I know, to create TxA, Alice must sign her bitcoins which is identified by the output index in the former transaction.

This is not correct. In each input, Alice supplies a signature of all the relevant parts of the transaction (inputs and outputs). The signature must be created with the private key that is associated with the address that the output being spent was sent to.

Quote from: katatunix on April 26, 2015, 05:55:32 PM

So why doesn't Alice sign the receiver's address (Apple) also when creating TxA?

She does.

Quote from: katatunix on April 26, 2015, 05:55:32 PM

This will make Bob cannot modifies the receiver's address and thus can prevent the scenario above.

Correct.

Exactly how did you think Bitcoin lasted 7 years and grew to its significant amount of usage without this basic protection?

Perhaps you should try reading the original bitcoin whitepaper?

https://bitcoin.org/bitcoin.pdf