S1 - Foundations of Cryptography

Transcript

Hi. Um, just let's wait two minutes before starting. Um, I I have too much things open. Oh gosh. Kiss, kiss, kiss. Okay okay okay okay. What are you doing? Duck duck duck dine thank you.

Um, feel free to to put where where are you coming from as usual? Are you speaking French? [laughter] Uh to we just released a big announcement.

So that's why I'm like like too too much. Um, duck duck duck. So, everyone is coming. Uh, Jim Jim in one minute. Um, we are alive. Oh, wait.

Tuck tuck tuck. Boom, bum, boom. Ah, you are. Oh, no. It's not taker. [snorts] Yeah, it's not ticker. Uh, tuck tuck. So, uh, let's wait for a minute. uh that she's coming. Feel free as I mentioned to put where are you coming from because you will have to um like maybe tag between each others.

um compared to other boot camps, this one is more like interactive u let's say because it's like more technical concepts and so actually Mita will will help a lot on that um to make sure you can understand things and nor is be be comfortable like nor um Uh wait.

Okay. Um, where is Al alati? Al ali. Oh, okay. Nice.

Well, there's a big um I think there seem like a big destation for Solana but let me check because uh Kazang is like top five country represented for Colossum. So can be interesting for you to participate into it, right? Just let me check something like this. Okay. So what's their planning? Okay, I'll sumit the Solana. You heard about it? But normally they are thinking something else.

Um let me check. Oh no, they're not testing a specific side event like event itself. Okay. Oh no. >> You can check here. Uh t. Yeah, you can join them uh if you want. Uh or ah yeah Minita you [laughter] are there.

>> Yeah yeah yeah hi hi. >> So um before starting I will just introduce is it me? >> Uh hello. >> Hello Okay sorry.

Um, okay. Um, so yeah, uh, Minissa will be your teacher. And just yes before starting I want to uh introduce briefly um so what's going on um yeah um so normally you you join uh you join these courses um so let me know if it's not the case actually uh let me put it on the chat so it's where you will find all resources is uh itself. So uh yes, let me share a bit my screen

uh tag share. Okay, here. So basically here you will find um our resources uh travel. So uh what will happen for each week? Uh recording session. So after uh each session you will find uh like the the YouTube link to uh to play the recording itself etc etc. So um and if if Melissa has additional things to to to put uh she she will put it too. So uh yeah and other things I just like release uh that we will host um our academy in more than like uh 35 countries. So which

is pretty exciting to be honest like as a community I think kind of the first time I saw like a community really doing a global aaton like this. So um yes really really cool. I will also put it on the chat. You you have the stage. Thank you so much Seline. Um can we say that uh the majority had joined or do we want to wait for a couple more minutes?

So what's your verdict? >> Yeah, we we can wait like two two minutes something like this. Um of course like I already send the news later um and also put it on the chat. Um but I can also put a message on on Google classroom. >> Yeah. Yeah. >> Okay. Yes. Uh tag because actually we are like around 60 members on the on Telegram. Um okay, let me put that.

Yeah. Yeah. Maybe people are a bit confused, you know, when it's first class people maybe a bit confused about um time or something. So >> yeah, at the beginning every time >> um >> but I I send I send like a remainder uh on by by email. So normally it's fine in case of Okay.

All right recap. Um, so, uh, Nelli wanted to come, but why? She can't. Um, >> I know there are some, um, some people cannot join join. That's completely fine. There will be a recording, but those that can, um, just to make sure that we are, um aware that they're joining Yes.

So, let me change. I think I ejected someone by by mistake. >> Yeah. I mean, this >> that's why she >> I know words. >> Yeah. Funny funny. Um so as a me feel free to put on the chats uh where do you come from u because it's it will be like an interactive um boot camp. Um so basically Melissa will will uh give you couple of exercise

or this kind of things but chill will chill no reason. Um, okay. So, I think we can start. Mhm. Mhm. As you want. Yeah. Let me let me share my screen. Uh, all right. Just one moment.

Tuck tuck tuck. We'll see you. She was already part of the previous one. [laughter] >> Okay. Okay. Super cool. Uh, so let's see. People are joining from Argentina, Nigeria, Ecuador. Yes. Uruguay, Venezuela, India as well. India. Perfect. We have we have people uh from around the world like Seline has in her background, [laughter] right?

Yes. So everyone um hello, my name is Melissa. I'll be leading Zeke privacy uh boot camp. I'll be there for all your questions and in chats and Google Classroom and um and alter groups. So everything um you want to ask, I'll be there. Oh, hi Rosia. And then a famous mask. We love it. Rejoining from Ecuador, right? Rosio is nodding. I would take that as a yes. Uh and so I'm doing PhD in zero knowledge proofs and I'll be there uh to guide you through

um ZK privacy concepts um uh within uh this boot camp. Um what we will try to cover is foundations and to make you prepared for um Yeah. Let's go with Yeah. Uh uh to make you prepared uh for diving into ZK on your own. Um there are a lot of tools that you can use for um creating um zero knowledge proofs um a bunch of languages. Uh but for me it's essential that you know concepts uh first before you dive in um so that you're not working on more complex

topics without having um at least some um wider picture. Um so I will um I will share my presentation with you guys. uh you have um you can always interrupt me and ask some questions if I don't see the chat because I'll be focused on on um on my presentation or the examples. You can always um join in and hop on camera and uh and uh ask your questions if I don't see it and in the first two or three minutes. Um so yeah for everyone um what do you know about ZJ proofs

>> don't be shy >> feel free yeah yeah yeah there is I I think um everyone here is um up to learning so any answer is a good answer at this stage So you know just um take your mic and and do speak. Okay, we have some um interesting answers. Proving something without actually re revealing it, right? Proving something without revealing the secret. It's it's similar. Yeah. Math magic to prove who are we without tell.

Yes. [snorts] Yeah. It's a protocol. Yeah. Yeah, super cool. Anyone else? You can always you can always hop in to our call and ask. It's it's it's not intended to be a oneman show. >> All right, we have a couple more answers. Showing the >> Yeah.

Hi. >> Okay. Um, it is the process of proving things um with cryptography without actually revealing anything when it comes to transactions and things on chain. >> Exactly. Exactly. It can um occur within or onchain or even offchain as well. There are methods of u proving offchain as well of course. >> [snorts] >> Yeah. Anyone else?

Yeah. The program the verifier. Bravo cabin. [snorts] Super cool. So people are coming with some knowledge um of zero knowledge proofs to this class. Cool. Um let me try sharing my um screen to to see the presentation. All right, [snorts] let's see.

Do do you guys see my uh presentation? >> Yes, that's it. >> Okay. So for for today uh what I'm intending to cover with you guys are ciphers um number theory foundations and we'll have a couple of demos where you will be uh on the side of um of cryptographers and uh and uh creating some of um of ciphers and coding and encoding or encrypting and encrypting.

All right. So what we we say is uh cryptography, right? That would be um so why are we even learning cryptography? It's the basic basics behind uh zero knowledge proof. So it's something that is essential for you guys to understand before we're we go and dive more into uh Starks and everything we have um regarding zero knowledge proofs that has um maybe you have um heard of um on on some platforms. Um so cryptography is well from the ancient times um people were trying to hide something right you

don't want to have um everything um about you written um in front of your home and like having uh your keys your apartment or or uh for example um I don't know the password to bank account and things so we wanted to uh make uh things u more secure and we needed uh secure communication and cryptography took um well it took the spotlight especially um around military operations where uh if some message was intercepted it would definitely mean that um uh uh that the whole operation was compromised

some battle could be lost or where some um sensitive information. Um so its core goals were confidentiality and integrity and authenticity. Um uh so why is that? So you needed to have confidentiality because um no one else should have um access to your message. Integrity because you want to know that no one intercepted it. and uh um and read it and maybe even changed it. Authenticity would be if uh if we know that a certain message has the origin um on a on a good side that we

know that that person or for example that general has sent us that message to for example um uh um let's say that we know where the enemy is and uh how uh um how many people they have where the troops are etc. So um how did how did it start? Um it started with um simple sub substitutions with uh shifting methods and now we have u modern zk proofs but um lasted for more than 2,000 years and maybe even more uh for uh for those u methods to develop

and uh also like if we are looking at cryptography on on a general level we'll we will have symmetric and asymmetric um cryptography. What does that mean? Um so a basic example of symmetric um cryptography would be if um for example you have a key to your home. That's the key that both locks and unlocks the door. And with asymmetricography you will have one key that locks something and one another pair key that unlocks it. you will have um two uh um two keys and we will see it on in further classes.

So uh a bit more about historical context uh we will see um Caesar cipher uh that is a shift cipher uh you will maybe maybe someone heard about something called skal as well that's also a shift um uh a shift cipher but um using some uh stick for example um why was Caesar famous for that uh um uh cipher it's because he used it for communicating with his generals. how uh Roman Empire had a cipher system for for that and uh with uh we'll see it in in

further examples but uh text was actually ciphered with um shifting uh shifting everything by a fixed number um for example um instead of A you will have not B but C etc but we'll show everything in um in a plain example later and we will play with it. Uh with substitution cipher you will see that you will have um unique letter mapping. So one um one uh um letter would have um it's mirroring letter that you can use

and with visionire cipher you will have a keyword shifts using uh a passcode to say so or a secret word. Um so uh one of the examples was um let's say the plain text would be hello and if we use um three shift method it would be something that we we really cannot recognize what this is in in English maybe it means something in some other language but uh with this like we wouldn't know and that would be uh a message that would be uh sent across the Roman emperor which was a vast one. So um it is um it was um very good for that

time uh so more than 2,000 years ago uh where people were first of all not all people were literate and then uh even if they were they would see some well to um non u trained eye it would be some gibberish words that wouldn't mean anything and then uh it would uh be sent like that and uh some other person would decipher it on the other end of the empire. Uh the weaknesses of course exist. It's um frequency analysis. Um why is that? Um every language um has frequency of letters that are being used

in um in language. So for example, A has the biggest frequency in most of um European languages and that can be used to um while some crypto analysts advantage if we're using some um easy brute force ciphers like uh Caesar cipher is. Why is that? Um if we are shifting um letters let's say by three uh then if someone has text um with more than thousand uh words they will see that uh that uh um the frequency is retaining but in

some other letter and uh they would say they would take for example the the biggest frequency letter and they see maybe it's a letter C or D and they would exactly know the shift from uh the most frequent letter and they would use it um to guess uh what your uh shift number is any question around this >> um looks like no if I saw something I will I will tell you >> okay okay sure uh just uh because I want to wrap uh this uh this part. So maybe someone has some a question around

Caesar cipher that um or regarding uh frequency analysis or something that we can um answer immediately and of course um we can always come back to to this uh slide. Um okay so then um what do we have? We have substitution cipher. um that means that we have mappings of letters. So for example like A can uh become Q or B can become some other other letter and uh we would need to have um some sort of matrix that we know uh which letter is is becoming uh uh what letter it's of

course it's stronger than Caesar cipher but of course um as we have mentioned uh with frequency analysis it could be uh easily breakable because it would retain ained uh frequency properties of um letters in in the language. Um every language has a different uh frequency uh of letters and um and yeah if you know um if you know which language is used then you would know the frequency of letters and like it's it's uh it can be retained within um yeah thousand words. it would be uh easy for that.

Then we have visioner cipher. Um why is uh why is this important and why is it interesting? Um it's using keywords. So uh or as we use it as we call it passwords or you can use you can use whatever word for it but the the thing is that you need to know methods right. So um how does it look like? You will have a plain text and then you will have keywords under it and um you will uh do uh you will you will uh process it with uh keywords and then you will have output of text and that's how it's being sent. Uh yeah, for uh for big enough keywords, it's uh been

um uh it's been considered unbreakable, but of course uh with advanced methods um nothing uh um is is secure and I have to tell you as someone who worked uh um in cryptography for the military that there are basically like nothing is um considered um unbreakable. It's just that we do not have some tools at the time or computing power that uh we can decipher all the things but um uh with enough uh time and resources um even if it's for um I don't know for for um some of uh

the ciphering methods that we have cryptography methods we have today sometimes it's like thousand years of processing think it's something crazy. So, it's like for now we're good, but we never know if um we'll have some uh breakthrough like what's uh uh quantum computing or something similar that um can um uh that we can use for for uh finding out the plain text. >> [snorts] >> So yeah, uh never take that for granted and uh have that in mind.

So well um I hope you have a pen and paper [gasps] because uh we will have uh some uh exercises. Um, for me I don't think it's well like I always like to have some sort of examples uh for uh uh for what we are speaking of here because usually people are forgetting what they're hearing but when they actually execute something it's something else right so yeah um let's uh pick a simple cipher we'll uh go through both Caesar and Visionaire cipher as well um encode

and decode messages and we will use model arithmetic for for shifts. Uh let me share um yeah um so what we will do uh first of all show you with krypai okay um I don't know if you guys see my screen with krypie okay Kevin says you you can all right I'll share this with you guys in chat [snorts] message everyone one. So this is website.

Someone is commenting quantum computers are near. Cool. So guys, if we go here, uh we can um try and see like okay like in this case um if we're having shift of three, this is Caesar cipher. You should be able to uh once you go to this website, it it will offer you enigma or something else uh for for the first um um as the first option. But go and find um Caesar cipher. It's under ciphers here.

Okay. Um the alphabet that they have used back in Roman Emperor had 25 letters because we didn't have J uh at that point. Yeah. Uh J was added later uh to the letters. But now the alphabet has 26 of them. Uh but um still um in some more traditional cryptography schools J is not used and I is used instead. But um very little context is being lost due to that.

So yeah, let's uh try um using some uh text. [snorts] Let's see. Let's see. We can use crypto with shift five. Let's try. Let's let's encrypt crypto, right? and we can say it's five. So you should get something like this. So how does it work guys? A is becoming F. So you have five shifts.

A is becoming F. That means that C here is becoming H. Someone is do do we have Julius Caesar here? Nice. Yeah. White space betterment. Yeah. uh for um for whites spaces um uh once you have uh when you're creating um secret

text you are omitting white spaces because um then someone who's trying to ease drop will will see uh that you have um certain structure of a sentence it would be easier for them um to decrypt it. how white spaces are being ignored in most cases. Okay. Um so once you have so we we should um we should do something that's manual as well. So when you're writing uh uh when you're using Caesar cipher I'll show you how to do it

manually. So you will take alphabet and you'll write that down underneath your original alphabet. You will start from instead of A you will start with F. If you have five shifts [snorts] that way your alphabet will start with F G H I J K L M etc. Once you come to uh once you reach Z it will start A B CDE E. So basically Z would be mapped to E. You go around the clock, right?

>> [snorts] >> And uh then uh when you see C you will see paired underneath you will see paired H because you started with A F F G H and that will be easy for you then you that you every time you have to calculate plus 5 + 5 and all doing all the shifts maybe maybe in some cases you will um you will miss something and this is much easier way to do uh because maybe you will have a long much longer text like maybe we can copy all of this and and it would be you will have output like this one and it's um it's much more

headache to have every you have to calculate for every uh letter you will have to map it with something else um uh and maybe you can uh just read what's underneath and write it like this uh but yeah for for smaller ones like crypto it's works perfectly with um even with uh uh shifting and calculating it. Okay, so we had crypto with shift five we got the answer. Um let's try this one and decoding with decryptting with shift five and let's see your answers.

So I will copy this and Mhm. Yes. Yes. Sebastian, it's circular Z continuous with AB. Yes. Yeah. It's um it is is it like modulus? Um yeah. Yeah. Yeah. We can say here we can say it is because um let's say you have now we have 26 letters we don't have 27 letters so we need to do modules 26 and start with start with um with a yes [snorts] so we should uh here you have um encrypted [clears throat] text in your

chat and we should um using Caesar cipher and uh shift off let me check five uh we should have a plain text. Let's see. Feel free to write that down. And uh reads.

So what would m map to uh the original uh alphabet? Yeah. Now you have to now you have to think in decryting method. So instead of adding five, you'll need to subtract five uh to come to your answer since it's decryption not encryption. So yeah.

Okay, let's uh give everyone the chance to focus and decrypt it. Let me see. Uh, we got a question. So,

to encode you add and to decode you subtract, right? Exactly. Great question Evans. Can we decrypt a text without knowing how many rounds of encryption it has gone through? Um yeah how many how many rounds of encryption you mean how many shifts is that right PI okay um so yeah then you will have to do brute force for um all of them um and uh try mapping from A to Z um all of them yeah but you can do

All right. Um let's say that uh now we are uh quite positive that everyone went through um decryting and we got hello. Let's see. So then this would be our text and we we would get hello cryp. Cool. Um let's see another one. Um let's see. Visioner cipher. Uh we'll have uh keywords and we will have um um a plain text to encrypt and um let's try it.

What did we say? Uh it's proof and math. So here in um encryptile you will see instead of Caesar cipher you can choose visionire cipher. Okay. And then we will use proof here. Um we will use instead of cryp we'll use math and this is what we get. So if you try to analyze this you will see that we'll have double O here and it corresponds to two different letters and you will see in um in encrypted text

that you have um R on in second and fifth place and that's not the same case here. Uh why is that? because we will we have in the second place we have um a and a if we u take a look at it it's um it's seen as zero and it adds nothing. So uh what you have is um you write down math underneath proof and then you repeat m and for every kind of text you will just have it under your text and you will uh do sub uh you will do addition and that's how you will get this uh text uh here the the encrypted

text um we can try for example with a aaa aa And it will mean uh the same thing because nothing else changed. But if we try with um let's say um a a b you will see that the third letter became something else. It went uh uh it went to one up to say so. So so instead of o that we have here it became p. it shifted.

But if we uh continue with B here, you will you see that uh uh this one shifted as well. And if we do another B, F shifted to G. If we say it's B BB B, uh we'll have all of them shifted uh for one place. And you can play with this key. you can write uh something else but let's say it's uh for example it's um it's math uh but yeah like uh you can uh play with it in uh uh with um even bigger keys that you have. It's not um um it's not usually practical, but um um

yeah, anything that's bigger than the plain text is um it's not that um useful because it won't be used the rest of the key won't be used for it. But um yeah, of course, you can try it out. Um, anything any questions around this? Mhm. [clears throat] description of engineer.

Someone said, "I'm starting to see how it's hard to crack." Yeah, you need to uh you need to you need to know the methods what can be uh uh what can be done. So practically how visioner works is that you have a plain text. So something you want to encrypt underneath you would have your keyword that repeats for every letter that you have in your plain text. What happens then is that you get um all of your uh uh all of your letters. You can turn it from uh 0 to 25.

um as an alphabet. So A signifies zero, B signifies two, zed uh signifies 25, right? And then you can um you can do addition and you will get uh uh you can also uh uh first of all also need to uh turn your keyword into um an array of numbers as well uh just for practicality. Uh for example, if your uh um if your uh uh keyword is math, uh it would have um it would have uh its own mirror in um in in in alphabet, right? And then you do uh

addition and you will get uh what's uh what would that uh uh new words or new encrypted tax be. [snorts] So uh let me show you once again uh see okay sharing my screen. So uh for example we have let's try hello and we have key math. Um so now you have now you have the tool right but if you want to write that down let me see let me see I have some papers here.

Okay. I'll um I'll write that down. Okay. So, this is yellow. And then we will have Matt. It looks like this. Let's see. Okay. It doesn't work. Let's see. Uh maybe because you have like a blue blue. >> Yeah. Yeah. Yeah. Yeah. Yeah. Just let's see. I will see.

Yeah. Will just give me a second to get back to um to Zoom workplace. Set up video without blurring. Now it should work. So, do you see it? I think it's inverted right? Maybe I can write it inverted. So this is hello and underneath you will say you will have math and then you will repeat the the first character M and you will have it um uh at the end.

So all of these letters which is plain text you will turn all of them into a number in uh in alphabet. So uh what number would be h if a is zero and zed is 25. So that means that if this is if this one is zero, a will not move.

Yeah, that's true. [snorts] And then you are adding h to m with their own numbers in alphabet and you will get something else. That something else you will get uh would look like this. this text that you have, right? Um, do you see it here in cryp? So, this one did not change places, but this one did. [snorts] Let's see

the text. Yes, exactly. So, with the keyword BB BB, it will all add one and you will shift all of the letters to the next uh letter in the alphabet that's being followed. For example, uh P would be followed by Q. R would be followed with S and um um both O's would be followed by P. The thing is that the last letter needs to be covered uh with the repeating keyword. So F would be uh uh followed with um uh with G.

Let's try it and let's show it. Um let's use it here. Proof the key would be even if you have three of them. Uh it's it's still repeating. So you will need to uh uh you need to um uh to um to to put it to write it under proof all the beads and and yeah and then you will have shift by one in this case.

Cool. Um, do you guys want to challenge to try and decrypt now that we have uh learned the encryption of visioner, people are putting thumbs up. So you will have um text to decrypt.

Okay. Keyword I will even give you keyword. How much time we have for decryption? Um, take your time. No pressure. And we'll have methods. Bisher. Here it goes.

So we have put um our um text um keyword and methods of encryption in uh the chat. The do we see it? text to decrypt keyword and method. Don't cheat. Don't use cryp.

Try try writing it down. Uh keyword is not that bad. Ah, someone's commenting that the first

few letters gave it away. Yes. Yes. You see um for um uh decryption it's important that you do not see uh um the spaces. Yes. The key math is repetitive. Yes. Math. Math. Math. Math. It's difficult to do brute forcing for for visioner [snorts] especially if uh the key is as long as

text. Good luck. All right, let's see. Uh, let's go further. Is there uncomfortable now with uh visioner and with encoding and decoding? I mean

encryption and decryption. When I decrypt manually, I get it wrong. Uh Alexia, do you want to share your method with us? Happy to help here. Bravo, Sebastian. You're you're on a on a on a good spot. Yeah.

Okay, let's um let's go further with um with our class. So, I'll be sharing uh more of modern cryptography. Um do you guys see my screen? I hope so. Put some thumbs up so that I can see as well. Cool. Thank you, Kevin. Um, so um you have seen the traditional uh ciphers uh but there is of course more to it. Um when it comes to modern cryptography

um everything we have seen is more or less deprecated or it has been used um in as as a basis um uh for for some of uh but it's important for you to know that um it exist and how it works because we cannot um actually um play like we we did now with um [snorts] some more uh modern cryptographic systems um even like with Enigma that has been a key uh to um encrypting [clears throat] messages and world the second um now that is uh being uh debunked uh uh for

for a long time um I mean since u the this the war the second um so what we can see here is um is uh something that's been used before but of course uh we'll devote more time to modern cryptography and um and uh uh the the basis uh of uh of zero knowledge proofs as well. So uh what we can do is uh shifting from um puzzles that we got and um uh diving more into maths and I'll show you what um tools we have now. Um most of it is built on number theory

which is using prime numbers, mod modular arithmetic and discrete um logarithms etc. Uh we'll see it um as we go. Uh but yeah, let's um yeah, another another good thing for um cryptography is uh that sometimes you're even hiding what are you hiding? And uh that method is called where sometimes you have um a plain text that doesn't say much. It's some simple message but if you know where to look or you have um some uh padded paper you

would be able to read the message and um um in the past steganography has been used to hide messages in uh pictures in books and like um in some let's say plain sight but Um but uh uh unknown to an unt on trade eye. Um so yeah let's um let's uh go further with this and let's start with um number theory. Um so why is number theory um important? Um unfortunately um we cannot uh cover the the whole class of it but

uh we'll go through basics of it and of course if you are um more interested you can uh always ask for more materials I'll be happy to provide. Um so number theory is the foundation of RSA defy helman uh elliptic curves and it deals with primes and modular arithmetic and um we'll show some of the examples and we'll see that we also use modular arithmetic in everyday uh world as well.

Um so um what do we know about prime numbers? Um they are uh by definition that not nothing divides prime numbers and um you why are they used uh because if you have big enough prime number you will need to check if it's a prime number by trying to divide it many many times uh until you reach um that last one. Um here are the examples. I'm uh sure we all had uh prime numbers in in school but um uh prime numbers are something that is foundational block of

uh cryptography especially the modern one and um we will see um some examples as we go uh through our lessons. Um also a greatest common divisor um that is uh dividing uh both of um uh uh numbers and uh we will see it and um later uh with key generation and and checks and I want to uh give you the examples of it and um well the concept so that's uh when once we see it it's not something that is uh new for you.

So yeah, when it comes to modular arithmetic um we use it actually every day, right? And uh the good uh uh the good thing is we're using it uh with clocks. So, um here I am in um Europe and um for me it's 700 p.m. And uh if I look if I take a look at my my uh my phone it says 1900 and that is like 7 + 12. every time uh if we say 700 p.m. we did a modulus of uh of 12 uh uh for for 19. Um so it's

sometimes it's 1 p.m. but we see it's 1300 uh on on our clock. So that is a great uh reminder that we all use arithmetic um around uh the globe. So um let's see like it's used for many many operations um due to uh while the I mean the the uh our computing systems have their own limitations. So we need to stay within certain area uh that's agreed for for using it for example elliptic curves. So we need to do additions and the

multiplications and um everything else within that um uh agreed upon space. So we need to use modular arithmetic. Every time we are trying to um let's say multiply something and then we're going over that space we need to do modulus and come back within that space. So the same goes for uh for let's say for for a clock right and another thing is like um let's say like um it's um it's easier for us to do um to use modulus every day. So because like it's much easier for us to say uh how many hours are

there in the day and what hour is it now instead of calculating all the minutes uh in the day or maybe sometimes like uh from the beginning of time or something else or like how uh how old are we in in in years and not in months or in days. But that's also um something that's legit for us to use but uh we are not uh using it because it's not practical for human comprehension. [snorts] So we need to do something for computers as well and their limitations.

Um yes it's keeping um numbers within finite range. why it matters and we will see how it's working within uh uh RSA and other methods but um as someone mentioned in the class and asked like are we using modulus uh for uh shifting methods yes we do uh because um if we do not have that uh we would not be able to uh to come back at the beginning of alphabet and um just continue after After zed we would have a b c or how however uh um how many shifts we have we will have as much as letters um at the end of

the alphabet. So let's try a quick demo and compute 7 * 8 modulus 10 and 3 to the^ of 4 modulus 5 It would be the best if um you guys are

trying to use pen and paper just to practice a bit. [snorts] Who would like to jump in and explain their answer that they're writing here in the chat?

It's 7* 8 and then the result is you we're using modulus 10. Yes. [snorts] So Sebastian like we have uh brackets around seven and eight not after mode. It's what we Feel free to join guys. Um this is um meant

to be interactive and for you guys to uh take the stage as well. So we're quite open around it. Selena and I so for the first one 7 * 8 is 56 and for more of uh 10 it means you can take uh 10 figure so I did 56 divided by 10 that gives you five remainder six that's stuff like that.

>> Exactly. Abs, you're you're right around this that we'll have six as um the remainder and we'll take it for the uh result of uh 7 * 8 modulus 10. [snorts] Anyone else for the second example? Well, Fred, do you want to take the

stage? The mic is yours if you want it. [snorts] >> Can you hear me? Yes. Yes, we can. Go ahead. >> All right. For the for the 7 * 8, it's just like I said, just like the clock exactly seven in normal arithmetic, 7 * 8 gives 56. But since this mod 10, we are like looking for the maximum number of rounds of 10 that can go into the number in question and then what remain after that?

So five rounds of 10 can go into 56. Six cannot go because no six. So five rounds can go and the remainder is six. Just like with your 24-hour clock, you have 14 as 2 p.m. something like that. So the same thing applies to the other one. 3^ 4 is 81 normally. And five can go into 81 16 times. and it's going to leave you with one as remainder.

>> So although there are other methods to do it but that's like the easiest. Yes, that is correct and that's why we have uh for the first one the answer is six. Um and um as Abins explained and for the second one as Wilfred explained it's one. So let's take that um we have resolved it and if we do have any questions we're open for it. [snorts] Okay. Um so let's dive a bit into a group discussion. Um again feel free to

uh take the stage. The mic is yours, the floor is yours. um which uh cipher seemed stronger for you uh of uh what uh of all of those that we have seen and um yeah maybe um more about discussions uh more about the uh discussing a bit where do you see weaknesses in these methods Anyone?

Okay. >> Okay. >> With the ciphers. Uh, okay. I don't know there we the weakness maybe because the example you gave us was actually easy but the weakness is if you can just notice certain parts of the word like you don't you actually don't have to crack every single part if you can notice some significant parts you can you I think you can intuitively guess the rest of the word and sometimes even the sentence

>> especially the scissor the scissor the scissor cipher is very easy like you can when you see what you can already guess that >> vowels like e and a will probably appear and you can check the most frequently occurring alphabets there and maybe the first time you assume it to be e if that does not work you can assume it to be a just be checking according to how frequent each letter occurs in English language if that's the language that the original text is in but so yeah those are like the I love the three pack you gave I actually didn't have to check the whole

thing once I got the U was equal to I and the second letter of math was a and so it was Then I saw three in in between the like last in between in I saw three within the last word and I saw it there. So I just somehow knew that I didn't I didn't I didn't I actually didn't have to saw anything. I just knew instantly that it has to be along. So most times you actually don't have to do the whole thing. But the the the Vegnar cipher is stronger. It's much stronger than the scissor cipher

because it's not one constant. What was it called? Shift. Yes. Yeah, that's right. [snorts] Thank you for for sharing your opinion, Wilfred. Um, anyone else taking the stage? >> Yes, Kevin.

Oh well uh regardless of which Cifer seems stronger as you say there is not something like such a things as unbreakable because if you have enough computer power you could basically break any safer you have but of the three we see I think this the vignette is like one of the strongest for uh until now and how did number three help in in calling and decoding well I I thing is that it sets you like a range where you can use. So basically it's replicable all over the world. So you have this range that you can share with the people you have you want to

decode the message and it's more easy for them and also for the computing. And the weaknesses well I think we as human are really good like seeing patterns but of course computer are like more stronger with that. I see when you find the pattern to encode and decode like the cifer or animation animation you want you basically can replicate that in any other instances so I think the major weakness is like that pattern that you can identify that's a great answer thank you

Anyone else? [laughter] Yeah, we have some very motivated people here in the chat. So cool. All right. If we do not um have um anyone else joining, we will see um us in in two days and um until then um I'll be there to

answer all the questions uh for you guys. Um feel free to ping me in our telegram chat. um or in our Google classroom as well. I'm there for for uh uh all uh all the answers. So see you soon and thank you for your attention. Thanks man. Uh so yeah like the chat was pretty proactive. So, as I told you at the beginning, uh the thing is that um it's going to be interactive classes because it's always more fun and you are learning uh faster. Um so yeah, uh thanks Minita for for everything. Um so

yeah, let's say I see you on Thursday, right? >> Yeah, I see. >> Yes. Yes. Yes. Yes. Rosio asked for group picture if it >> of course we can do that. Family Pink. We call that a family p. >> Yes. >> So maybe remove the u screen and so like this. >> Okay. Sure. I'll stop sharing. >> Oh, [snorts] because of me is on her mobile. >> Yeah. Yeah. Yeah. Yeah. I understand.

Yeah. Oops. [laughter] Family pig. >> Yay. >> [laughter] >> Nice. Okay, let's wait uh a bit before taking the pick. Um >> yeah, >> tuck tuck. >> Three or four more. >> Yes. [laughter] >> Wait in a bit. [laughter] like there's two two two slots left. So >> yes, people please turn on the camera so

that we can have a sweet family pick. [laughter] >> Yeah, like in we have done a lot of family V and you can do that like for three death you can do that. [laughter] Yeah. All right. So let's take a peek. Uh wait. Um so one ah no I put some emoji too like you can react um on your keyboard. Yeah. All right. So let's go. One two three energy.

Uh one more one more with with more more reaction. [laughter] One two three perfect two. We are good. >> Okay. So, see you on Thursday with the same energy. Um, and yeah, of course, I will put the recording on YouTube and also on Google Classroom, so you have everything else like everything you need. So, fine. Thanks again, Melissa. And see you on Thursday then. >> Thank you, Seline, and everyone in the class joining from all around the globe.

>> Yeah, as usual. Just bye.