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everyone's going to give a really quick explanation of pink symmetric encryption and the RSA algorithm so if you're not familiar with the concept of asymmetric encryption it's a really cool idea the idea where if you have a public key and a private key and if anyone wants to speak to you you give them your public key back to you anyone in the world who wants to talk with you on your public key and they encrypting a cinch using the public key and it sends it to you and they decrypt you decrypt it using the private key and only the public key can encrypt and only the private key can decrypt it's really cool and the analogy for that is like a padlock and you go around to anyone in the world who wants to talk to you and you give them an open padlock and they if they want to write your message they write a message put in the box and then they lock the padlock and they give to you and notice how even though they encrypted the message even if they wanted to decrypt their own message they couldn't because they don't have what you will have which is the private key or in this example it's a literal physical King because you require a physical key to unlock the padlock so this is a really nice neat out on illustration you'd give everyone in the world one of these padlocks and in RSA the equivalent of that is called the public key and I will you would keep this key which in RSA we call the private key so so it's really consistent and where you can communicate using asymmetric keys so let me look at a rule let me show you a little bit exactly no more detail what the public and private key actually look like so whatever public key look like well they're actually two sets of numbers public key that that looks like this it's two numbers its own e to encrypt and n and that might be let's just use an example a might be seven and n might be 33 you can't really just make these numbers up there's an algorithm to to derive them and I'll show you that in a second but the private key well that's key that is made up of D and the same n from above and an example of that might be 3 and 33 notice how the end is the same for both of them so if you want to understand RSA it's really important to know how to use these these numbers so - to encrypt a message let's say you have the message on I love you normally what you do is you treat this anti message as one number so you might give each other letters a particular number through the whole thing is one massive number but just to simplify things we're going to say maybe the letter U and maybe the letter U has the number to associate it with it cool so we take that and the formula for RSA for encrypting stuff is this so you take the ciphertext which is known as C you take 2 D plaintext and you raise it to the encryption key and that value there now you just mod it by N and when it comes to decrypt well it's quite simple let's let's take on if the reverse process you take the you want the plaintext take the ciphertext raises its power of the decryption key D for decryption and when you mod it by n do Krypton keys also on the deprived key read only you the owner of this Sun key has the priority and so let me do some worst examples let's say I have to that's my plate X let me encrypt it so point X is 2 I want to raise the power of E which I know from this example is 7 modded by 33 it's pretty easy I can I can do this in my head that's 128 mod 33 and what mod means is that just means I'm divided 128 by 33 and then figure out what the remainder is so if you're not too familiar you know you can just go to google and type this in here I have an example from before you just do 128 mod 33 if you don't have that on your calculator in an exam or something quite easy to do what you do is you just do 128 divided by 33 you'd see how many times at holy 15 so you can see here it's just 3 so 128 15 three times holy so I just modified this when I say 128 minus in brackets and 3 times 33 and this will give me the reminder here 29 so that's the same that's wonderful so this answer down here is 29 so it's the socket except the value that is encrypted using IRC to decrypt it I just move it over here and we look at C so what was C C was 29 but just figured that out let's raise it to the power of T and let's see we'll see is 3 so let's raise its power of 3 again we just do the same thing we mod it by 33 so it's really elegant on on this side you guys value the encryption key and secret it and not it I should say what it by it and it's a crooked us bears the private key and I'm not it by it and really elegant let's calculate this one with is echoing 29 to the power of 3 figure out what that is that phone number there OOP copy and that number again mod 33 and surprise surprise the answer is 2 which was our original plaintext here which is you so that's what the answer is this year so it's really cool that some the example of using RSA for encryption another really cool feature of RSA which is called digital signatures and the idea is it everyone has my public key and they know that that's my public key can I do something with the private key which they can then use the public key that they have to verify that it came from me and if they can use the public key to verify that something was find with the private key then they know it must be me because only I have my private key so what they do is actually reflect the maps a bit here let me move my head and delay this is what we've seen above the encryption but to sign something instead of using and the encryption key which comes in a public key to sign something let's call it the signature you take down maybe you have like a certificate to a will and you know what you'll do is you will take a hash of it and normally on what they'll do is they'll sign - so let's say we've done that and we have the message which we want to find so to sign it you start to sign with the public eight you do exactly the same process but instead of using a up here to encrypt it we just raise its power update and you mod it by n so that's right there and they get the signed document what you'll see is on you know you have the doctrine is your yeah we have document your hash it and you'll click the hash and you'll find it and you'll send it across the other person will get the document they'll hash it themselves and they'll figure out what the hashes and then to verify what they'll do is they'll take the signature they'll raise it to your public key and your public key is e stands for encryption because it's on the user encryption so raise it to e now modest by n again what they should get is they should get the original message it was used an original signature now this message matches the hash of the document then they know what to do this is actually the legitimate original document that was originally signed so that's a really cool application of RSA it's digital signatures is used absolutely everywhere including in your browser and to make sure that you're communicating with the right person because at every certificate in your browser is signed using this process exactly here so I mentioned before that these are the public and private keys but we don't just make these numbers up there is actually an algorithm to figure them out so these stuffs we went through here are all very important very critical to know and I'm going to go through that more detail here about how to calculate those numbers it can be quite scary but how bear with me will will survive I believe in you you can do it I always recommend on following along with a little tutorial so there's one I will recommend I will just this one here and you can find it by googling an RSA crypto worked example and the CS Virginia website will come up it's a really good one and they step through all the maps here which is really good follow along Wikipedia is fine as well but let's step through it a problem the first way to figure out your on public key which is your a and your n and your private which is your D in your n is you start by picking on two numbers so the two numbers are called P and Q now you really have to mind your P's and Q's because these should be on prime numbers typically they should be really large prime numbers but because I don't hate myself I'm just going to pick really small numbers on so one prime number will I'll pick three and I might pick 11 so they're both prime numbers and what I do is I simple I figure out one of my stores numbers and we just figure that out by multiplying T times Q so it can do that quite easily 3 times 11 equals 33 excellent I've done my first part of RSA figure out my first number next thing I need is a really fun one it's good oil is poisoned which is just lovely thing here it's a great fun it can be quite scary but it's really well this portion is take P that you calculated before or subtract 1 and then Q they picked before up here subtract on as well and multiply them together so let's do that really quickly 3 minus 1 and 11 minus 1 equals that's 2 times 10 or this portion is 20 and that's the number that we use it immediately to figure out some other values so we remember that or this question equals 20 all right now we've got M which is equal to 33 but always push it now what we do is we have to UM just randomly pick our encryption value just randomly pick a number so pick P now there are a few restrictions on e it needs to be between 1 and what is cushioned so basically X between with th350 1 and 20 and it can't have any [Music] shared factors of 20 with that make sense so for a number between 1 and 20 we'll all the options are as 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 what's that relatively curb prime means is that a II can't have any shared factors of 20 so what are the factors of 20 will 20 you can get 20 by multiplying 1 and 20 you can get by multiplying 1 and 10 we get a pulse coupling for and we're going to have 405 Sigma factors of 20 so that means we can't use those so exactly 20 which is 1 not just to current choose 10 can choose full can't choose 5 so let's take the value 7 so we're going to do running out of space lightly and so raise this or faculty will say let e equal seven now what we need to do is mean that's all so we've got everything we need for our public key right but II just seven we just decided that NN which is 33 over here cool we've got our public key now we need to figure out our private key so private key and xD to find D you need to solve this question here which is T times e when divided by 33 the remainder should be 1 so the way we write that is 1 mod why do I say 33 see with me and it's actually said that it should be known as as well as well this matches because equal so let me say that again we need to solve this puzzle here this one this mathematical expression which says what number I hid when x by AE has a remainder 1 when it's divided by or Distortion very complicated I'll just fill that so we're saying D times a which we have seven has remainder 1 when divided by or the store chain which remember we said up ta lotion is 20 what about 20 so I hope that make sense on let me meant let me just show that let's for example look at the case of 1 so 1 times 7 well that's 7 and I'll just do this half of it 7 divided by 20 is 0 with some remainder seven so that doesn't equal one so once an option gibble is group let me try 2 so 2 times 7 equals 14 14 divided by 20 equals 0 with the remainder of 14 so the remainder here doesn't equal 1 so that's not a solution either oh the joy is enjoys oh I love that so let's try 3 D equal 3 if T equals 3 then that comes 3 times 7 which equals 21 now 21 divided by 20 where the solution equals 1 with a remainder of 1 and that's that remainder there that we're looking for here so whenever we find a solution which has remainder 1 perfect we've found the correct version of DDS suitable number D so there you go we have all the values we need now so we have our public key that's made up of the e from Krypton key and the n also known in this example as on e which we just picked which was 7 and 33 private key is on its D and n which is I just calculated should remember this do you and 33 there you go it can be quite daunting we got all these numbers in Greek letters and look where this math on the page it's quite simple T come to prime numbers multiply them together and you put in then subtract 1 from each of those that add the peas and Q values multiply them together that gives you this intermediary value called Euler's portion take a random a that's less than euler's totient and greater than 1 and just remember it has that side condition that ii can't have any of the same factors as just 20 value here or this portioned and then just solve this formula here d times a so the public in the private key encryption the decryption T when they multiply together when you divide that number by euler's totient that intermediate value of the here should only have a remainder one and that's essentially an RSA algorithm if it doesn't make sense and it's important to know this how to calculate that I definitely recommend reading a worked example such as this and getting a head wrapped around it there are some shortcuts to solve this column here but we won't go into them for this video so I hope that makes sense and it's a little clear and good luck with your RSA questions