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Warrant digital signature order

in this video we're going to see how digital signatures work with RSA and so what I want to do is recall the setup that we had from our original RSA video where we showed how Alice could send a message to Bob using his public and private keys so from that video bob's keys were as follows he had chosen a public key of 571 and 1073 and he had chosen a private key of 835 again 1073 and the setup we had there was that Alice wanted to send a message to Bob and the message she wanted to send was the number 331 so the way that she did it was to send this she computed a huge power 331 ^ 571 you know which turned out to be 829 mod 1073 and then what she did is she sent that number 829 and that was the encrypted message now Bob then would receive it and he would decrypt it by raising 829 ^ his private key the situation that we want to discuss now is what if Alice wanted to send this message to Bob but there was an eavesdropper that will call Eve in the middle who received the message and tried to change it so Eve intercepts this message 829 and then changes it to maybe say 851 again this is all mod 1073 then what would happen is Bob would get this message 851 be none the wiser about the fact that Eve had intercepted and changed it and then Bob would decrypt the message in the usual way by taking the number that he received raising it to the power of his private key and then that would turn out to be 185 mod 1073 and he thinks that that's the message Alice intended dissent but in reality it wasn't up here it was 331 so digital signatures in RSA are going to be a very simple way of preventing this trickery from happening so let me explain how that works one of the key ingredients in this is going to be that Alice also has a public and private key so let's for the purpose of demonstration choose some keys for Alice so Alice has keys let's say public for her is 103 2059 and private for her Alice's private key is going to be 647 again 2059 now the way that she's going to be sure that Bob receives the message she intends no matter what Eve tries to do in the middle is that with her original message so with 331 raised to the power of Bob's public key 571 which turned out to be 829 she's also going to send something so she also sends 331 the same message raised to the power of her private key 647 which turns out to be one thousand two hundred and thirty five mod the other number in her private key 2059 now why would she do that this gives Bob a way of verifying that the message actually came from Alice because what he can do upon receiving these two numbers so Bob receives two numbers now he receives one thousand two hundred and thirty five and he receives a twenty nine now he knows that this second component is the encrypted message so he decrypt as usual by taking the power of that number raised to his private key so 829 to the power of 835 and he gets 331 again mod the other number in his private key 1073 now he takes the other number and he verifies the message so he verifies the message using the other number and the way he does that is he takes one thousand two hundred and thirty five and raises it to the power of Alice's public key 103 again this is all mod 2059 and he finds 331 again so mod 2059 so why does that work this is because the number of 1235 came from alice sort of encrypting the original message using her private key that means it can be decrypted using her public key when Bob sees that that works and he gets a message here that matches the other he knows that it must have been Alice that sent this because only she could encrypt the message using her private key so that the public key would undo that encryption now to be totally clear as to what I mean let's suppose that Eve tried to do the same trick as before so suppose as before we had Alice sending a message to Bob with an eavesdropper Eve in the middle and suppose that Eve tried to change the message in the same way so Eve changes what she sees go by so one thousand two hundred and thirty five comma eight twenty nine to one thousand two hundred and thirty five comma 851 well if that were the case then Bob would compute sort of the decryption of 851 so Bob computes 851 to the power of his private key and what he would find is that that's 185 at the beginning of this example Maude 1073 and then he would do one thousand two hundred and thirty five to the power of 103 Alice's public key and he would find that that's 331 Maude 2059 and since these numbers don't match he knows that there's been some tampering now you might ask yourself why doesn't Eve just change the other number as well in order to make it so that these do match when Bob decrypts them the point is she doesn't know what to change that other number two unless she has Alice's private key which she doesn't which means that this is not going to be something that Eve can tamper with without breaking the verification method that Bob uses to ensure it comes from Alice