for six years in the field of cryptography for the Uni Karlsruhe, Oats. Okay, so thanks for the introduction and welcome to my talk. As our Herald just said, I've been working in the area of cryptography for the better part of the last six years. And I noticed that many people today have kind of a mental image of what cryptography is and what encryption is. But sometimes this mental image does not really coincide with what is actually going on. So I wanted to give an introductory talk to cryptography that is available to a broad audience, so to an audience that does not have any prior exposure to cryptography and maybe even to an audience that does not have a background in maths or computer science. So as I said before, this talk is specifically aimed at a non-technical audience, even though at 36C3 we probably have a fairly technical audience. And this is a foundation's talk. So I will not be speaking about fancy research or cutting edge results. I will be talking about the plain basics. Okay, so apart from working in cryptography, I enjoy doing applied cryptography, number theory and ponings, so exploiting all kinds of memory corruption bugs in programs. Okay, this is a picture of my parents' cats, just because every talk should have pictures of cats in them. And this is... Thanks. So this is my checklist for this talk. The first item, I think we already did that. And so for the remainder of the talk, I want to explain what encryption is, what it does and what it does not do. I want to explain authentication, which fixes a problem that encryption does not solve. I want to explain certificates because they help a lot with both encryption and authentication. And in the end, I want to explain a little how the things that I'm going to introduce work together and can be combined to build more useful things. Okay, so let's start with the first point here. I would like to explain encryption. So encryption is basically a solution to a problem, so let's talk about the problem before we get to the solution. The problem here is, or one of the classical problems is that two parties want to communicate, so we cryptographers commonly call them Alice and Bob, and Alice wants to send a message to Bob. So in this case, it's just a very stupid message, just a simple hello, but cryptography has been used in diplomacy and military for hundreds of years, so imagine that this message is something more critical. And the problem we want to solve is that there might be an eavesdropper who is wanting to listen in on the connection and read the message, read the content that is being sent from Alice to Bob. And what some people think, how cryptography works is something like the following, which is kind of close to the real thing but not really, is that Alice applies some kind of encryption procedure to her plain text message to produce some random, unintelligible gibberish, which we call the ciphertext. And then Alice sends the ciphertext to Bob, and Bob has the decryption procedure, knows how to decrypt, so to invert the encryption procedure and recover the plain text message. And now the point that some people get wrong is some people think that the knowledge how to decrypt is actually secret, but that is not true today. So in 1883, a person named August Kalkovs formulated a couple of principles that ciphers used for military applications should adhere to. And one of these requirements became well-known as Kalkov's principle, and it reads, a cipher should not require secrecy, and it should not be a problem if the cipher falls into enemy hands. So rephrasing this a little, the cipher that you are using should be secure enough that you can even tell your enemy or the attacker how the encryption process is working and how the decryption process is working without harming the security of the encryption. Or to rephrase this yet another time, if the cipher you are using is so insecure that you cannot tell anyone how it works, then maybe you shouldn't be using it in the first place. So let's get back to this image. So now if the attacker knows how to decrypt the message, then obviously this very simple scheme does not yield anything useful. And so what people did is people introduced a key to this image. So now the encryption procedure and the decryption procedure use a key, which goes into the computation that is going on. So Alice does some kind of computation based on the message and the key to produce the cipher text, and Bob, who has the same key, can invert this encryption operation and recover the plain text. However, as long as the key is only known to Alice and Bob, but not to the attacker, the attacker cannot use the decryption procedure. So one general word here. I will not go into the details of how these boxes operate. So within these boxes which represent computations, there is some math or computer science going on, and I would like to not explain how these things operate internally in order to keep this talk available to a broad audience. Okay, so a problem that we have here is that Alice and Bob have to agree on the key in advance. So Alice cannot simply send over the key to Bob because if she did, then the attacker, who is eavesdropping on the connection, learns the key as well as the message and then the attacker could just decrypt the same as Bob. Okay, so this does not work. This is a terrible attempt. So for quite some years, actually until the 70s and 80s of the last century, people were only using this scheme and this is what we call symmetric encryption because we could just flip the image around and Bob could be sending a message to Alice instead because encryption and decryption uses the same key. And if there is symmetric encryption, you can guess there is something else which is called asymmetric encryption and there is, for asymmetric encryption, there is a pair of keys. One of them is used for encryption and one of them is used for decryption. And now if we have an asymmetric encryption scheme, we can do something like the following. So Bob generates a pair of these keys, one for encryption and one for decryption, and he keeps the decryption key to himself. This is why the decryption key is called the secret key. However, Bob can publish the encryption key for everyone to see. So for example, it could be put in a kind of public registry like a phone book or whatever. And now Bob can send his public key to Alice and an eavesdropper who is listening in on the connection will learn the public key, but that's not a problem because the key is public anyway. And now after we have done this, Alice can use Bob's encryption key to encrypt her message and send that over to Bob. And now Bob can decrypt the message with his secret decryption key. However, the eavesdropper here cannot simply decrypt the message because even though the eavesdropper has the encryption key, it does not have the decryption key and thus the eavesdropper cannot decrypt. However, this solution is still kind of risky. There is still a problem with this and it is we still have to make sure the keys are distributed in advance. So if we used the simple scheme where Bob is sending his public key to Alice, then there is a problem if the attacker is not simply passively eavesdropping on the connection but is willing to actively interfere with the connection. So for example, the eavesdropper might intercept the public key that Bob is sending to Alice and replace it with his or her own public key. And then Alice would think that the key she received belongs to Bob and use this key to encrypt her message to Bob and then suddenly the attacker can read the message again. So at this point, let's summarize about encryption. So encryption conceals the content of data and this is pretty much what it does and pretty much the only thing that it does. In particular, it does not conceal the fact that there is communication going on. So an eavesdropper who is listening in on the connection obviously can see Alice sending a message to Bob and thus the eavesdropper knows there is communication going on between Alice and Bob. And this alone could be quite dangerous for Alice and Bob. So imagine if Alice was working for an intelligence agency and Bob was a journalist and the attacker sees Alice sending lots of documents to Bob then this might be a strong indication that Alice is a whistleblower and Alice could be put into jail. So something more that is not concealed by encryption is the amount of data that is being exchanged. So if Alice is sending just a very short message to Bob then the eavesdropper can guess that the message that is being transferred is not a 20 gigabytes file or something. So all this kind of metadata is something that encryption does not conceal. And there is a couple of more problems with encryption. One of them is that the attacker might change the message. Protecting from changes to the message is not the job of encryption. Another problem is that keys must be exchanged in advance which I already talked about. And there's more problems. So for example, an attacker might simply record a message when it is sent and later just replay this message to Bob. Or an attacker might go ahead and block a message altogether so intercept the message and throw it into the trash to make sure it never arrives at Bob's site. And the first problem here, an attacker might change the message will actually lead me to the second part of my talk which is authentication. So on my talk checklist let's mark encryption as done. So now what is authentication? Authentication enables the detection of changes to data. It does not prevent changes from happening. It only enables the recipient to detect the changes after they have happened. Okay. So for example, one example where something like authentication was needed is when Bob was sending his public key to Alice but this is by far not the only scenario where authentication is needed. So imagine if Alice is running a charitable organization so for example she's saving refugees from drowning in the Mediterranean Sea and Bob wants to donate to Alice to help her do that then Alice has to send her bank account number to Bob that Bob can make the donation. And notice that in this scenario the message that Alice is sending to Bob her bank account number is nothing that is secret it does not have to be encrypted because this information is public knowledge. However, we do want to make sure that the message that arrives at Bob is indeed the correct bank account number. So to prevent something like this from happening where a criminal might intercept the message and replace the bank account number so Bob would send his money to the criminal's bank account instead of Alice's. And one way to realize authentication is again by having a pair of keys one of them is used for authentication and one of them is used for verification so checking if a message has been changed or not and the authentication key must be kept secret thus it is called the secret key whereas the verification key can be made public and it is called the public key. And now if you have a setup like this then Alice can go ahead and take the message that she wants to send to Bob and apply some computation to it together with the secret key, the authentication key to produce something that we call a signature or a digital signature. And then Alice sends this signature over to Bob along with her bank account number and Bob will take the signature that he receives and the bank account number he receives and apply some kind of computation to them and this computation will determine if the bank account number has been changed or is in fact original. So if the attacker changes the bank account number then Bob will be able to detect this change by checking the signature and this holds even if the attacker does not only change the bank account number but also the signature. So these things are designed in a way which hopefully makes it impossible for any attacker to come up with a valid signature for anything else than the original message. So the only thing that Bob will the only way that Bob will accept the signature is if the attacker does not in fact change the bank account number and in this case it is safe for Bob to transfer the money. So here is a different solution to this problem and it's actually pretty much the same except that now we have just a single key which is used for both authentication and verification and in this case things simply have a different name they work in exactly the same way except that the signature is called a message authentication code or MAC for short. But in both of these scenarios whether we have two distinct keys or just one key we still have the problem of key distribution. Imagine if in the scenario with two keys Alice was sending her public key to Bob then we would have the same attack as before namely the attacker could just go ahead and change the key that Alice is sending to Bob and exchange it for his own key. So if the attacker is sending his own public key his own verification key to Bob then obviously the attacker can create a valid signature for his forged bank account number and Bob would accept this. So again we have this problem of key distribution which is that the verification key must be known to Bob in advance. And this leads me to the next section of my talk so let's mark authentication as done and go on with certificates. So a certificate is a document that confirms that a specific public key belongs to a specific entity for example a person or an organization. And if we want to use certificates let's just go back to the scenario we had before so Alice wants to send her bank account number her public key and a signature for her bank account number to Bob and an attacker might change the public key and the bank account number and the signature and now if we add certificates into this we need to add something that we call a certificate authority and this is a trusted third party which will create certificates which confirm the association between a person and a public key. So before Alice is sending a message to Bob she will walk up to the certificate authority and say hey certificate authority this is my public key I'm Alice please give me a certificate and then the certification authority will check that Alice is indeed Alice and that Alice indeed owns this public key and if Alice passes these checks then the certification authority will create a certificate and hand that to Alice and this certificate is just a document which says that the certification authority has verified that the silvery key here on the slides belongs to Alice. And now once Alice has the certificate she can just send her public key to Bob together with the certificate and then Bob if he knows the certificate authority's public key can check that the certificate is indeed correct so it was indeed created by the certificate authority and if he trusts the certificate authority he will know that the silvery key is in fact Alice's and then afterwards Bob will be convinced that the silvery key is Alice's and he can check the message that Alice is sending to Bob and make sure it has not been changed. Okay so we're not completely free from the key distribution problem yet however because still Bob has to know the public key of the certification authority in advance so Bob does not need to know Alice's public key in advance but he needs to know the public key of the certification authority in advance and in practice there's not just a single certification authority but there's a whole bunch of them and certification authorities can even create certificates for other certification authorities and so on. So now Bob does not have to know all the public keys of everyone he's communicating with but he only has to know the public keys of a couple of certification authorities. Okay so let's summarize about certificates so as I said before certificates confirm that a specific public key belongs to a specific entity like a person or an organization but we're still not completely free from the key distribution problem because people have to know the certificate authorities public keys and another problem here is that this scheme gives an enormous amount of power to a certification authority so if an attacker can compromise a certification authority then he could force the certification authority to create fake certificates connecting fake keys to real identities so he could create a fake certificate which says that the certification authority has checked that the attacker's public key belongs to Alice and fixing this problem about the certification authorities power is something that cryptographers are still working on so that's still a problem today and in fact this problem is not just theoretical there's a number of incidents that have happened with certification authorities so one famous example is the Digi-NOTAR case where in fact a certification authority named Digi-NOTAR was hacked and the attacker has created a fake certificate for a Google.com domain or one of the other Google domains I don't exactly remember and then these certificates showed up being used in Iran so this is not just a theoretical problem this has in fact happened before so this concludes what I wanted to say about certificates so let's move on and see how these things can be put together to build more complex but also more useful tools so one of the tools I want to introduce is called authenticated encryption and it's basically a combination of encryption and authentication so for some reason people use this phrase mostly in the symmetric case where there is one key for encryption and decryption and one key for authentication and for verification but you could pretty much recreate the same scheme in an asymmetric fashion and that is also being done in practice in this case people just don't call it authenticated encryption so one way to build authenticated encryption is so if Alice wants to send a message to Bob then she will encrypt the message using the encryption key and send the ciphertext over to Bob and then she will use a copy of the ciphertext and compute a message authentication code from it using the second key that she has and then Alice is going to send over that message authentication code to Bob too and now Bob can decrypt the message using the key he has and additionally Bob can check if this message has been changed or whether it is original by using the verification procedure so again this kind of authentication does not prevent changes from happening but Bob can check whether a change has happened and in fact this kind of authenticated encryption can actually boost the security of the encryption scheme so another thing I wanted to talk about is called hybrid encryption and this is a combination of symmetric encryption and asymmetric encryption and the reason why this is interesting is that asymmetric encryption is usually quite slow compared to symmetric encryption so if you wanted to send a very long message to Bob and you only had a public key encryption scheme so an asymmetric encryption scheme then it would take a very long time to encrypt the message and to decrypt the message so however you can combine asymmetric encryption and symmetric encryption in a way that makes the encryption process faster and the way you do this is so if Alice wants to send a message to Bob Alice first generates a new key for the symmetric encryption scheme and Alice will encrypt her message with this key and send the ciphertext over to Bob and afterwards Alice will take the symmetric key that she has just generated and encrypt this key with Bob's public key and then that is sent over to Bob as well and now Bob can decrypt the symmetric key using his secret decryption key the kind of golden one here on the slides to recover the symmetric key and afterwards Bob can use the freshly recovered symmetric key to decrypt the actual message however a neistropper listening in on the connection cannot decrypt the message because it does not have the symmetric key and it cannot decrypt the symmetric key because it does not have Bob's secret decryption key you can continue to build on these kind of things and what you end up with is something called transport layer security or TLS for short and transport layer security is a network protocol that combines much of the things that I've introduced so far so it combines encryption either symmetric or hybrid and it combines it with authentication so max and signatures and certificates and all the other things and it adds in a couple of more things to detect replays of messages so if an attacker was to simply replay a message recorded earlier this is something that TLS can detect and TLS can also detect if a message has been suppressed by the attacker so within a connection so what TLS does is it kind of establishes a secure connection between two entities so say Alice and Bob over an insecure network which is controlled by the attacker and one application where TLS is commonly used is for sending emails so for example when you're sending an email from say Alice wants to send an email to Bob then the email is usually not sent directly but Alice sends the message to her own email server and then Alice's email server will forward this email to Bob's email server and when Bob goes online and checks his emails the email will be downloaded to his device like his phone or desktop computer or whatever device Bob is using and while Alice can make sure that when she's uploading her message, her email to her own server that this connection is secure just by encrypting the message so essentially using TLS and all the things that involves like encrypting the message and authenticating the message and so on however Alice cannot check if her own email server also uses a secure connection to forward this email to Bob's server okay so let us take a more detailed look here so each of these green locks signifies a secure connection this means that when a message is sent then or each time a message is sent over a secure connection there is some encryption and authentication going on on the sender side so if Alice wants to send an email to Bob then Alice will build up a secure connection and send the email over it and this will involve encrypting the email and authenticating the email and Alice's server will decrypt the email and verify that it has not been changed and then Alice's server will forward the email to Bob's server which involves again encrypting it and authenticating it and Bob's server will decrypt and verify it and then again the same process repeats when the email is sent or is downloaded by Bob from his server however in this case so even though the message is encrypted every time it has sent over network it is known in plain text by Alice's server and by Alice's server and by Alice's server and by Alice's server and by Alice's server and by Alice's server and by Alice's server and by Alice's server and by Alice's server and Bob's server because Alice is sending the message so she's encrypting it and Alice's server is decrypting it so Alice's server can read the message in plain text and the same goes for Bob's server and this is what we call transport encryption because the email is encrypted every time it is being sent over network and a concept opposed to this is what we call end-to-end encryption where Alice before sending the message Alice encrypts it but not with a key that is known to her server but directly with Bob's public key and she might even sign it with her own secret authentication key and then Alice sends this already encrypted message over a secure channel to her own server which involves encrypting the message again and authenticating it again and then Alice's server will decrypt the message and verify that it has not been changed however the server cannot remove the second layer of encryption so the email is encrypted two times one time with Bob's key and the second time so that the server can decrypt it and now the server can remove the second encryption but the first one is still there so Alice's server cannot read the email and then the process repeats the already encrypted message is encrypted a second time and decrypt it again at Bob's server and then it is when it is downloaded by Bob it is encrypted again and decrypt it again and then Bob who has the secret the secret decryption key can remove the inner layer of encryption and so Bob can read the message however the servers in between cannot read the message just because it is still encrypted with Bob's public key okay so with that I would like to wrap up sorry so I would like to wrap up so here are a couple of take home messages so the first one is encryption conceals the content of data and that's pretty much all it does it does not conceal the metadata and it does not prevent the message that is being sent from being changed that is the job of authentication authentication enables the detection of changes to data and both for encryption and for authentication you need to have pre-shared keys or maybe not really pre-shared keys but key distribution has to happen beforehand and one way to make this problem simple of key distribution simpler is with certificates so certificates confirm that the specific public key is owned by a specific entity and if you have all these things encryption and authentication and certificates you can build a network protocol which takes care of securely transmitting a message from one place to another and you can apply that to get transport encryption but transport encryption is inferior to end-to-end encryption and with transport encryption all the intermediaries can still read the email or the message being sent however with end-to-end encryption they cannot and with that I'd like to close and I will be happy to answer your questions if there are any questions which you cannot ask today you can send me an email at this email address on the slides and I will keep that email address open for one or two years for your talk and now we would come to the question parts if you have any question you can come up to microphones in the middle of the rows are there any questions from the internet we have plenty of time if anyone comes up with a question you're invited ah we have a question on microphone too please thanks for your good talk I would like to know how can you change a message that was properly decrypted without the other receiving part noticing that the decryption doesn't work anymore that depends on the encryption scheme that you're using but for quite a number of encryption schemes changing the message is actually quite simple so there is a really large encryption scheme which just work by changing a couple of bits so the message is made up of bits and your encryption scheme gives you a way to determine which of the bits to change and which not to change so when you're encrypting you use the encryption scheme to figure out which bits must be flipped from 1 to 0 and you just apply this bit change to the message that is being sent and then the receiver can just undo this this change to recover the original message whereas an attacker who does not know which bits have been flipped cannot however so still the attacker can just flip a couple of the bits and in this case so say the bit has been flipped by Alice and it is being flipped another time by the attacker so the bit is at its original value again and then Bob who knows how to decrypt will flip another time so it's changed again and thus the message has been changed and there's a couple of things you can do with this kind of with this kind of changes to the messages so the decryption simply does not fail it just maybe outputs the wrong message ok next question from microphone 6 please you stated that encryption does not cover metadata are there any thoughts about that any thoughts any solution for maybe encrypting metadata don't know so much of this is pretty hard to come by so I mean for emails there is the idea of also encrypting the subject which is usually not encrypted however so if you want to hide the length of the message what you can do is simply pad the message so just random garbage at the end of it to kind of hide how long it is exactly so the attacker will still have an upper bound on your message length so it knows that the message you're sending is at most as long as the refer text you're sending but it does know maybe it's shorter so if you want to hide your identity while communicating you should be going for something like Tor where you're not connecting directly to the person you want to communicate with but via a couple of intermediaries in a way that none of the intermediaries know both you and the final recipient Thank you. Okay, then I believe we had a question from the internet. Yes, the internet is asking can you say anything about the additional power consumption of the encryption layers on a global scale? Sadly I think I cannot so I do not exactly know how much power consumption is caused by the encryption however so in terms of computation at least symmetric encryption is quite cheap in the sense that it takes a couple of processor circuits to decrypt, I don't know 16 blocks or something so you can usually decrypt hundreds of megabytes per second with a processor at least with a modern one so I don't know any numbers but I mean you can guess that if everyone in the first world countries is using encryption then in the sum there is a pretty large amount of energy going into it Next question, microphone 2 please You mentioned a couple of times that an attacker might be able to replay a message I haven't really understood if I'm an attacker how does this benefit me that I am able to do that So imagine if Alice is sending an order to her bank to transfer some money and every time the bank is receiving such an order to initiate the bank transfer then as an attacker that would be pretty cool to exploit because you once eavesdrop on Alice sending such an order to the bank and then later on you can just repeat the same order to the bank and more money will be sent so if you were the recipient that would be pretty cool and you could just deplete Alice's bank account Then from microphone 3 I was in the talk about elliptic curves and cryptography and I'm wondering where this would be applied now in your example or in your process you showed us Let me maybe just go to another slide So typically encryption is applied or elliptic curves are applied within these encryption and decryption boxes so there is a lot of mathematics going on in these computations which I did not explain because I wanted to keep this talk accessible to a broad audience but one way to realize such an encryption procedure is by using elliptic curves within these boxes Microphone 1 please Another limitation I could think of or how to overcome this is devices like IoT devices that have low power and limited processing capability so how do you adapt this complex encryption-decryption computation for these devices There is some research going on on encryption schemes that are particularly lightweight particularly suited for resource constrained devices but as far as I know pretty much all of them have some weaknesses that came out of them so security-wise they do not offer the same guarantees as the ones that you use if you have the resources for it On microphone 2 please You mentioned the enormous power that certificate authorities have in the picture of certificate and authentication I was wondering if what are the possible solutions or the proposed solutions at the moment what is the state of the art on solving that problem So one solution that is currently being pushed is called certificate transparency and that works by basically creating or lots of public log files where each certificate that is ever created must be added to the log file and so if you're google.com or if you're google and you see someone entering google certificate into the log file and you know that you didn't ask for the certificate then you know that there is a fake certificate out there and so whenever you get a certificate you're expected to check if the certificate is actually contained in one of the public log files Does that answer the question? Yes but how does appending a certificate work So for example making sure that a certificate is recognized as legitimate Okay so the idea is whenever you get a certificate it will be put in a log file and everyone who gets the certificate is expected to check that the log file is actually there So is the certificate also pushes the certificate to the log? That's how it's expected to work Okay thank you You're welcome Then we have one more question from the internet The internet wants to know can we or where can we get an authentication for a PGP key and how to apply it on a key afterwards Is there a possibility somehow? I guess that depends so with PGP the common model is that there is not there's not a central certification authority or a bunch of them but so you have kind of a social graph of people who know each other and exchange emails and each of these emails each of these users should authenticate the public keys of their peers Okay so when you want to communicate with someone who you do not already know but who's maybe a friend of yours then hopefully your friend will have authenticated the public key of his friend and if you trust your friend you can then check that your friend in fact has created a kind of certificate for his friend Are there more questions from the internet? One more? Yes please I don't know if it's a question regarding your talk really but someone wants to know would you recommend start the TLS or SSL TLS in email? So as far as I'm concerned I would always opt for using encryption on the outermost layer so first building a secure connection to your email server and then doing SMTP or whatever over that connection so I think directly establishing the connection in a secure manner is a better way to do it than with start TLS I believe that was it for questions please have a last round of applause for Oats


seek to have innovative solutions that they're going to have a fair playing field and I think this addresses that. First, my first question is for Ms. Sleiman and I want to know how you would apply traditional antitrust law what exists today, the goal of which is to protect consumers from high prices to the subject of today's hearing which is the power of gatekeepers online. Ms. Sleiman, do you think that there's room in existing law for courts to use existing antitrust law to protect consumers or do we need changes to the law? Thank you so much, Congressman. We absolutely do need changes to the law. I have very much appreciated the antitrust cases that have been brought already by the Department of Justice, the Federal Trade Commission and state enforcers against Google and Facebook and I think that those cases are doing a great job in the direction of trying to address this problem but really the problem is much broader than any particular antitrust violations that may have occurred so even if those cases are very successful and we get really strong remedies out of them I think we would still be left with this fundamental gatekeeper problem and the structure of the market that tends towards monopoly so that's why it's so important to add on top these regulatory tools like interoperability and non-discrimination. And even if current antitrust laws could address the monopolistic conduct of gatekeepers and it sounds like your opinion is that it cannot, are they sufficiently fast enough and clear enough to deter that conduct in any meaningful way? The length of time that it takes to do antitrust enforcement is also I think a real limitation and that's something that other witnesses have talked about as well. We can work on trying to speed that up but I do think that having X anti-rules set in advance can save so much time and provide more certainty for businesses as they are deciding how to operate in the market for them to know interoperability is going to be required or discrimination is not going to be allowed that can really save a lot of litigation time and help those businesses. Ms. Lehmann I listened to your opening statement where you argued that we need new sector specific laws. Since the beginning of antitrust policy though we've had broadly applicable laws which apply to every industry equally so why does technology require its own laws and if we do it that way wouldn't every industry come up with its own complaints and want its own laws? Well many industries are already subject to pro-competition regulation. A lot of these ideas are inspired by the communications industry where we have pro-competition regulation. I'm also thinking of a black where we have pro-competition regulation so although this is new we want to learn the lessons from those other industries but many industries are regulated for competition and I think that is fully justified to do that here. Great thank you and I yield back. The gentleman who moves back I now recognize the gentleman from Florida.


Welcome to the session on partition join operation in parallel databases. It is one of the inter-operation technique. Talking about the learning outcome of this session, at the end of this session you will be able to apply this partition join operation on a given relation, inter-operation parallelism. What is this inter-operation parallelism? In this one, usually for the parallel databases, every individual operation in a query is parallelized. Whatever the operations are there, say for example the operation may be sorting, the operation may be searching, join, whatever a single operation is now taken for parallelism. So that operation is parallelized, that is called as inter-operation parallelism. So in this one, basically for the operations like parallel sorting technique is there, parallel join techniques are there. So for parallel sort, we have already videos are there, you can refer those videos. Today we will be studying this partition join operation which is one of the parallel join technique. Let us go ahead for this partition parallel join. Now this parallel join, basically what the join operation is doing? Now for the join operation, basically if more than one relation are there, even self-join is also there, the pairs of tuples are tested, one-to-one tuples are tested and if a particular condition they satisfy, then that pair is added to the output. So let us go ahead for the parallel join operation, that is a partition parallel join operation. Basically it has the three steps you can say, first one is partitioning where we are splitting the pairs whatever we want to do the operation, those pairs we are splitting for multiple processors. So that is called as a partitioning, first we are doing the partitioning, second we are going for the actual parallel process that is now we are doing the operation a join operation. So every processor parallely computes the join locally because already the data they have according to the partitions and finally once the parallel join operation in all the processors parallely has done, then the result is collected at each processor at one of the place. So that is the third step. Now this partition join operation is basically for equi-joins and natural joins, equi-joins it is possible that the partitions have two input relations across the processors and compute the join locally at each processor, that is every processor has partitions say consider that more than one relations are there, so if two relations are there then those are partitioned into processors and locally every processor is doing the join operation. Let us say consider that there are two input relations say here the relation R and the relation S and those are we are joining on the attributes A of relation R and B of relation S. So and now the relation R and S each are now partitioned into say some n partitions are there we have n disks, so we have n partitions, so the partitions are denoted by 0 to n minus 1. So you may say that the relation R0, R1, Rn minus 1 are partitioned on across n processors. Similarly the relation S is partitioned as S0, S1, Sn minus 1 across the n processors. So now every processor has one partition of R and another partition of S which is locally it is going to take the join operation. So the same thing here, now what we can do for the partitioning is we can either use range partitioning or we can use hash partitioning. These are the partitioning techniques. Basically there are three partitioning techniques, we have seen that round robin partitioning, range partitioning, hash partitioning so either we can use range partitioning or we can use hash partitioning here for taking the partitions, so the join attribute now how to take the partition the partition should be specifically on the join attributes. So what are the join attributes we have seen here that the join attribute for relation R is attribute A and the joint attribute for relation S is B. So, based on R should be partitioned by A attribute and S should be partitioned by B attribute because based on this only we are taking the joint operation. So, what is the scenario now that partition R i that is relation R is partition say R i is one of the partition of relation R and S i is one of the partition of relation S then this R i and S i we are sending to processor P i. Now, processor P i locally is computing the joint operation based on this. So, partition R i joins with partition S i on the attribute of A of R i with B of S i ok. Now, any of the standard joint operation we can apply here. So, this is the scenario you can see that diagrammatically I have represented here. So, this is our joint operation which parallely is going across these many processors. So, processor P is 0, processor 0, processor 1, processor n minus 1. So, the partitions are what 0 partition, 1 partition n minus 1 partition these are the disks D 0 to do n d n minus 1 you can say. Now, relation R and S are there. So, relation R is partitioned as R 0, R 1, R n minus 1 across these n disks. Similarly, relation S is partitioned as S 0, S 1 across these n processors till S n minus 1. Now, what every partition will do now? Every processor say processor P 0 it is locally doing the joint operation of R 0 joins with S 0.  So, parallely every processor is doing the joint operation that is what the parallel partitioning partition joint operation is ok. This is another diagram depiction you can see here that relation R is there and these are partitioned as R 0, R 1 till R n minus 1. Similarly, relation S is here that is partitioned as S 0, S 1, S 2 till S n minus 1 you can say. So, now what processor P 0 is doing it is doing the joint operation of R 0, S 0. Similarly, P 1, R 1, S 1 like this. So, the parallely these are executing. Consider the example here. This is our example like we have two relations student and courses you can see the attributes are there. We want to take the joint operation between these one ok. So, the scenario is like this we have two relations student and courses the joining attribute we are taking here is roll number. So, based on roll number we want to do the joint operation. Number of disk we are considering for example as 2 and therefore, the number of partitions are also 2. So, therefore, the partitions the processors are also 2 you can say. Now, the partitioning technique we are applying here is hash partitioning technique. Partitioning attribute is roll number. So, as it is hash partitioning technique we are taking the hash function like this H of roll number. So, that is mod n and what is our n? n is our 2. So, roll number mod 2 whatever the result will come that hash value disk that is containing that tuple ok. So, we are applying the hash partitioning on student as well as on courses both the tables. Let us see now. So, this is the partitioning what we have done we may say that this is the partition 0 for the process of P 0 ok. So, partition that is the student relation is partitioned as student 0 and courses relation is partition as courses 0 in. So, this all thing will go to disk 0 and the processor P 0 is working on this one ok. So, the number of partitions and the partitioning technique we have applied is hash partitioning here. So, you can see this is partition 0. Now, what I would say is pause this video and think of the second partition. Can you show the partition 1 of student and courses relations? We have this table and we want to do the partition 1 on this one based on the hash value hash function mod 2. Take a minute and do this one. Let us see. So, this is partition 1 ok. So, here this will go to disk 1 which will be parallelly operated on processor P 1. So, what the processor P 1 will do? It will do the join operation of partition 1 of student 1 with courses 1 ok. So, that is what the parallel partitioning is. So, you can see here that processor P 0 is doing this is partition 0 of student and this is partition 0 of courses. Similarly, this is partition 1 of student and this is partition 1 of courses. So, parallelly both the processors are doing their join operation and finally, you can see this is the result of the join operation whatever we have seen earlier ok. According to the example this is our result. You can just go through the result, but now what we have done here that parallelly we have done the join operation on processor P 0 and processor P 1. These are my references. Thank you.


This video is going to talk about composition of functions. The composition of functions can be written that looks like a degree sign between the f and g of x. Or it can be written as f in place of its x we put the function g of x. What that really means is that we're going to take f and g function. And we take the g function and we're going to plug it into here. So I take an input and then I get an output and that goes in here. So the output of g will become the input of f. So the domain of the composition is all the x's in the domain of g which g of x is in the domain of f. What that really means is that here's my domain of f. And I have my g but it's going to have some things overlap. That's the range, the output is going to be part of the input. And it's these values in here. So the output of g has to be part of the input of f. That's what that means. This is the range of g. Alright so let's just put it into practical terms. Here we have an f of x is equal to x squared minus 2x. g of x is x plus 3. And we know that to begin with both functions have all real as their domains. So it wants us to find h of x which is this composition and then determine the domain of the result. It might be easier for you to think about it this way. g of x goes inside f of x. So wherever I see x it's x squared minus 2 times x. But my x becomes this g function. So it's x plus 3 that's being squared and minus 2 times x plus 3. g went into the red outside f. Let's work it out. This is going to give us x squared. This is 3x times 2. It's a shortcut twice the product. So plus 6x. 3 times 3 would be 9. Minus and then I've got to distribute the negative 2. So it's minus 2x and minus 6. And combining my like terms I have x squared. And 6 minus 2 is going to be plus 4x. And 9 minus 6 is going to be plus 3. And we see this result as a domain of x being an element of negative infinity to infinity. This can have all reels because all of these had all reels. So given those same two functions, find h of x which is g composed with f. So this time if you want to rewrite it it would be g. And I'm going to write it this way. Composed with f of x. So the g function has x and then plus 3. And the f function in here is x squared minus 2x. So I've plugged in the f function into the g. And we find out that we get x squared minus 2x plus 3. So the result is that the domain has an x that is an element of all reels. So since this has a domain of all reels and g has a domain of all reels, when you put one inside the other one you can still have all reels. And q of x is x plus 2 and we want to find the h of x is q of p of x. They want us to find this p of q of negative 2. But if you notice these are the two ways you can write that. And I want to do it with one way using the h of x and one way just using the function. So since this one is really h of x, let's go and figure out what that is. So that's p which is x squared minus 8. And q goes inside the p function. The second one always goes inside. So that's going to be x plus 2. Now though I have this negative 2 so I can just plug it right in. Negative 2 plus 2 quantity squared minus 8 is going to be 0 minus 8 or just negative 8. Okay, let's do this one. p of q of negative 2. Technically what this one is saying is I need to say that q of negative 2 is equal to negative 2 plus 2 and that's going to be equal to 0. And then I want to take and say p of this now is equivalent to q of negative 2. So I want to know p of 0. x squared or 0 squared minus 8 will give us the same answer of negative 8. So we want to find f of g of x and then determine the domain. So f of g of x is going to be f of g of x. So inside the f function I'm going to put the g function. Well the f function is the square root of x plus 5. But what is my x? It's the g of x function. So inside here I have to put 4x minus 1. So really I just have the square root of 4x plus 4. And I would be happy with that. This is okay for me. We'll talk about the domain in a minute. But if you're going to look in the back of the book they would do this. They would factor it into the common factor of 4 then x plus 1. And then each one of these gets their own. The square root of 4 since they're being multiplied. The square root of x plus 1. So they're going to say that it's 2 square root x plus 1. Either one works. I'm okay with this one too. Because we're not really simplified in this class. So let's talk about the domain. The g function, its domain is all reals. The f function, its domain is going to be x plus 5 has to be greater than or equal to 0. So x has to be greater than or equal to negative 5. Let's think about this. We are really using a square root function which is f. So it has to be greater than negative 5. So I have to take the g function and say that's what I was inputting. That's the x of that function. So it has to be greater than or equal to negative 5. Let me see if I can help you think through that. I really have the square root of g of x plus 5. And this is my x instead of x plus 5. So we say g of x plus 5 has to be greater than or equal to 0. So g of x has to be greater than or equal to negative 5. That's another way to think about it. So if I do that continuing on, then g of x is 4x minus 1. And that has to be greater than or equal to negative 5. So if I add 1 to both sides, I'm going to have 4x is greater than or equal to 4. It's greater than or equal to 1, which means that the domain is going to be from 1 and including 1 to infinity. So the purple and the red are my two answers. Now we have to go think back and we want to do g of f of x. So we have the g function is 4x minus 1. So 4 times my x, whatever it is, minus 1. My f function is the square root of x plus 5. And when I simplify that, I get 4 times the square root of x plus 5 minus 1. I can't really do anything more with that. So now let's go think about these functions again. Again the g function is all reals. The f function we know we already figured out that that has to be x is greater than or equal to negative 5. If we think back to what we did before, we have 4 times f of x minus 1. So I'm inputting that f of x and that's what my x is. So my x says that it has to be greater than or equal to negative 5 and then I have this function has to be greater than or equal to 5 or the domain would be negative 5 to infinity. So here's my domain and here is my function. This is probably a good idea to try putting your function in function notation. So what is your x? That's the square root. Well what's the square root's restriction? It's negative 5. If we look back to the one before, g of x is my x underneath the radical. Well what do we have to do there? Well then we have to say that g of x plus 5 is greater than or equal to 0.


Welcome back to my channel. I have a guest come on in. I guess that you guys Hello, this is my neighbor the neighbor that I always refer to as my neighbor neighbor Stefan Yeah, that's me and in this video I'm gonna give you my first impressions on dusk diver which I'm playing on the PlayStation 5 and you will give us a three Underrated hidden gems that you can find and play on the PlayStation PlayStation 4 for your case.  No, do you want one? Well, actually I do because it's so fast. I know it's so fast.  Okay, so dusk diver 2 it came out like almost a year ago But I haven't gotten around to play it before now on the PlayStation 5 I got a review called from idea factory. I played the first game and now they have given out I mean a year ago. I'm so late to the party. So the story is a continuation from what you did in dusk diver one where you Play as this girl who works at this door and in night times She dives into the dusk through these dimensional rifts fishers And she brawls away to help the world be a world basically That's a terrible explanation of the story. Yeah, you're just gonna have to show me that Yeah, so the city that you play in it is based on Taipei in Taiwan and walking around the streets there Very fun. So the gameplay it isn't brawler action RPG where you have a lot of skills and you can level up your characters Which you can also change between it's a brawler game and you know what the brawler is To me it sounds like wrestling but super smash bros brawl Well, it's basically an hack and slash with your fists. Well, it sounds like fun You just have to throw me and a couple of characters that you get later on in the game They are long-range characters So you also have a bit of that now the graphics the actual little portraits when you're talking to characters super pretty That is just so up my alley. If you guys know me. I love my anime style games You always comment on my videos and you say that looks like a very Cheek-on kind of game. Yeah, every single video. Well, what I mean by cheek-on is so yeah, it's it's like very childish and very Well cheek-on you like more mature looking games Yeah, it has RPG elements where you can level up into individual all those regulars. That's that we know Games like fun fancy. Yeah, you can level up within stats that you wanna do Excellent around the town there are a lot of food stalls and stuff like that Now this game is hard.  So they are very Lift or repeat lift I repeat. They're like damage to sponsors and some of them I have heard that this game performs terribly on the switch So actually from what I heard from switch watch, it is terrible on the switch Don't get it on the switch I'm playing on the PlayStation 5 and I think it is a decent game But I would still say this is a wait for sale because I don't view it as a full price type of game It's a game. It's fun. It has its moments, but it also has its flaws So wait for a sale But I want to say as a closure to that that I find it to be a big Improvement over dust diver one that I played. Oh, it's a sequel. Yeah, it's a sequel. They improved on almost everything So that's my first little PlayStation hidden gem, whatever underrated game. It's not a very commercially big game, whatever. So what are your games? Well, can I start with the my complaint? Oh, you want to complain? I want to complain And I want to complain about the trophy achievements on PS4 and PS5 because I really hate when you have like online Trophies. Oh, yeah, we are trophy hunting on the PlayStation Yeah, definitely.  Yes, that's that's what I'm thinking because you have some trophies Well, but you can never get because the servers are down. Yeah It's like I don't remember but one of these games I have like 95 percent and the last five percent are multiplayer which the servers are down The servers are down so I will never get a trophy That's terrible. Well, and also I have to complain about Drift. Oh god damn. I hate Drift in my PS4 controller I have four PS4 controllers And two of them have Drift and all of my joy cons for switch have Drift I only had Drift once in a joy con. That was terrible for the first edition joy con like the 2017 launch edition God damn.  I hate Drift, but I have never had that in a PlayStation controller I think they're really good like overall really good. Yeah, except on that.  Oh, yeah, check out my video on it It's always good to play as Kratos and I try you so I gave that to you actually. Yeah, you did. Thank you Yeah, I'm a generous neighbor. Yeah, I'm also playing Savage Moon. That's a tower defense game from 2008 it's on the PS4 classic Savage Moon nice on PlayStation 3 on PS3, but on PS4 and 5 on the classic So what are you doing the game? Well, it's a tower defense game So you defend your base against incoming alien attacks, basically That's the best way I can describe the game. What makes it fun. It's like the thing that makes it really fun Is that you can make your own strategy So it's like you are placing towers here and here and here and you see the route where the aliens are walking and you Just place towers on the fence. Wow. So you feel like that is a hidden gem Absolutely, and I'm also playing extreme works to love that game It's a lot more easy than than the first action words came out in 2021 It's a 2d indie developed a metroidvania created by one man. I guess I think Yeah, and I saw it at your place and I gotta say the music is out of this world Yeah, the music is absolutely fantastic. It's everything I want to say in a game Yeah, and you are addicted to it. It's everything that you play. Yeah, you're like I'm playing Well, it's not well, it's axiom works too mostly and it's smurfs mission wayleaf Oh, that also came out in 2021. So I recommended it to everyone. It's so much fun I think this is like the fifth video. Well, the trophies are not fun in that game. It's just a chore Okay, yeah, your trophy hunting arm. Yeah, so the only thing I've left is to complete the game on Hard Now drink this yeah fairies, and I have the best cup assassins create origins. No, I have the best cup because it's told but It's not really told it's my mushroom.  I love snow. Yeah, I mean, it's a story is my baby brother It's not a so big. He's so big.  I remember I could have an independent hand and now it's like Huge I know I know What are you feeding that testosterone steroids? Definitely. He's hunting birds already Okay, so that were some hidden gems that we consider everyone to check out and your obsession with axiom words She's so crazy right now. I have to say on axiom words, too I have completed the map hundred percent, but I don't have hundred percent items. So you're such a sweaty gamer Yeah, I'm a sweaty gamer leave down below your suggestions for PlayStation games that are sort of under the radar sort of hidden gems sort of underrated We are loving those types of games. We need to know about that I only play underrated games Yeah, like hidden games that no one is seemingly is talking about something me too.  We want to say hi to you guys Hey, hey, we got a talk soon on PlayStation party. Yeah, thank you so much for watching everyone Thank you neighbor Stefan for being here. Yeah, and we will see you later Wave tell him to like and subscribe like and like and subscribe like and subscribe Thank God for editing


International Women's Day is a day set aside annually to celebrate the cultural, political and socio-economic achievements of women. Plus, TV News went round town to get reactions of women to an endurance in general, to the bias surrounding women. But that's on Street Views. What a man can do, a man can do better. And in this society that we are now, everybody has their own rights to do what they want. It's good for everybody to have their own rights. If you feel like going to school to have your own certificates, it's okay by you. It's not by force that I say, I'm going to tell you, you don't have any rights to go to school. All these days, International Women's Day, International Fathers Day, International Whatever Day, how does it reflect into our life in Nigeria? There can never be equality anywhere, not even in Africa, not even in the western world. But however, haven't said that. We appreciate our mothers, our sisters, gender equality in political representative in Nigeria. Forget it. Don't forget it. I feel like they're supposed to be the same as men because nowadays because of the way our great grandfather did and said that they're supposed to be in the kitchen. And because of that, some people, they didn't even send their women to school. Even me, because I'm here, because of what my dad did, it's women's rights. They have rights to say they want to go to school. Because I didn't go to school because my dad said I'm a lady that I can't do anything. The only thing that they have to do is for me to get married and go to my husband house because they think that it's only the men that are useful. So I don't feel that way. I feel we are equal. As a woman, I think a woman should go as fast as she can, especially in a country where it doesn't give her the space. It's a saying that says that if you train a girl child, you train a nation. So that shows that a lot can be done when we train a girl child. A lot can be achieved when women go further in life. And you can see it in the case of Ngazi Okunjo Iwela. We can see it in the case of Chimamanda Adichie. We can see it in the case of our sister Toshola. So we can see that there are great potentials in women and if we can help them maximize it, we just need a country that can encourage them.


My daughter Colette is an athlete and she got injured playing basketball. I first knew something was wrong with Colette because she started getting headaches. When I first found out that she had a concussion, we were shocked. As a parent, finding the right physician was so important to us. The priorities that I have in treating concussion are to manage symptoms and to get children back to their normal lives as quickly as possible. Today, Colette is doing fantastic.


I'm Staff Sergeant Brian Smith. I am from Davenport, Florida. I'm Sergeant Mitchell Orr, and I'm from Fort Myers, Florida. I'm Specialist Nadia Zaz-Bazan, and I'm from Fort Lauderdale, Florida. Sergeant Stefan Sierra from Fort Myers, Florida. On the civilian side, I work for my local Chamber of Commerce, helping local businesses and tourism to our area. My primary education is in the healthcare administration. So on the civilian side, I'm trained to prepare the paperwork for the operation. I have never got a skill set or experience how to do it hands-on. The Army gave me a skill set and experience. So on the civilian side, I'm actually a Marine technician with Fort Myers Marine Max. I work on boats. I actually just recently was working at Universal Studios for their Mardi Gras event. So outside of this, I actually go into acting.  Just today, we were working with the civilian sector with paramedics and firefighters. They were processing paperwork while we were collecting samples outside of this testing facility. There's police officers controlling the traffic. Just exactly all the sectors of my battalions here are not necessary medics. Our supply sergeants, our officers are here all the time. It's an operation that's been ongoing 24-7. We're all putting 200% of an effort to keep people safe. It's a fight that every other country is fighting right now. I'm really honestly honored to be a part of this. I have a great team that I'm working with. The ability to bring peace of mind to the public in this time, in this day and age, with what everything's going on. Yes, I absolutely feel fantastic about what we're doing out here. When we put on this uniform, we're Florida soldiers and we're here for the citizens of this state. It's a brotherhood of sorts, so it's a great opportunity to show the civilian population why we're here. We're your soldiers, so we're here for you guys. A message that I would say to anybody watching is to just keep your family in the forefront of your mind. Everybody is going through different challenges right now and we just need to hold each other up, keep each other's arms held high, just support everybody and everything that they're doing. And if we come together and band together, we can get through this better as one team than individually. You don't know when, where, anything like that, but honestly that's what we signed up for. So as the unit morale as a whole has been nothing but positive.  They want to know, they have families at home, right? So we came out here with the understanding of what we would be doing without the end aspect of when it will be over. How is it myself as a father?  It's never really easy telling them, Daddy's got to go, right? They understand to the point where they can. I still talk to them every day. I encourage all my soldiers to talk to their families, wives, husband, children, parents, everybody. Just let them know that we're out here and what we're doing. I want the families to be just as proud of what they're doing as the pride that they feel in themselves for being out here. My family is actually in three different continents right now. I keep in touch with them every day, sometimes twice, three times a day. I just want to make sure we're safe, we keep ourselves safe. My family's good, my family's kind of used to the swing of things. My wife, Megan, she takes care of the house while I'm gone. My mom and my grandparents, they know what this job requires of me and I'm lucky enough to have their support. My family is very proud of us being here. They couldn't support us anymore. They're just proud that we're able to bring a sense of peace to the area that we're working in. They obviously miss us but they're fully supporting us and we have their full support to be here and helping the areas that are impacted the heaviest. What I would like everyone at home to understand about what we're doing out here, we're out here for you. We are the men and women that signed that contract that put ourselves into this situation because we want nothing more than to help. We're not out here for any other reason. We understand the position we've been put in. We understand the importance of this and for us to have that honor, that title, that position of being able to go home with that sense of pride, that we did something, that we helped something out here. I want everyone to understand at home that that's why we're here. That's what we want.  My personal take on being activated with the National Guard for this relief effort for COVID-19 is of great honor. Just being able to support the communities in this area and to provide some sense of security has been just a great honor for everything that we're doing. Every day is a proud moment for me. Every day I am running the lanes, responding to my superiors, being here and being involved in what's important, making history every day is a moment of pride for me. To my wife, Megan, I love you. Mom, grandparents, I love you guys so much. Thank you for keeping in contact. I can never thank you enough. Please take care of yourselves. Mom, brother, so yeah. I miss you all. As soon as we resolve this problem, I'll be right back home with you. I am Staff Sergeant Brian Smith. I am Sergeant Mitchell Orr. Specialist Nadia Saez-Basan. I'm Sergeant Stefan Sierra. And I'm ready. And I'm ready. And I'm ready for this mission.


Not many people know about the Brainerd Adapted Floor Hockey team, but it's about time that people start to take notice. The Warriors have won back-to-back state titles and have finished this year's regular season at 10-0. Sports reporter Anthony Scott has more on the Warriors journey to a three-peat. Adapted Floor Hockey for those that are not familiar with the sport is a varsity activity for kids with physical ailments, and it follows many of the same rules as ice hockey. The Brainerd Warriors are becoming a dynasty at the sport as they haven't lost a game since the 2016 state tournament. Brainerd hasn't done this. Their adapted team has never been this good. I think that kind of shows that it's teamwork. And teamwork has been more important than ever for this season as three of Brainerd's top players from last year were all seniors. It's a much different team than we were a year ago. We don't necessarily have all the depth that we had. The players that were coming off the bench a year ago now are starting for us. The kids have really responded well and we're just pleased with how everything is gone. Pleased may be an understatement as the Warriors have outscored their opponents 114 to 26 and root to their 10-0 record. The Warriors attribute most of that success to the way they practice. This gym is pretty much our toughest competition. We had the girls varsity hockey team come in on Monday and scrimmaged us. We've got a Baja ice hockey team here today to scrimmage so we really try to push the kids day in and day out. Brainerd is excited to get down to the state tournament next weekend and compete against the state's best. I like close games because that's when it gets interesting and it's competitive. I would like to see an overtime game maybe. Reporting from Brainerd, Anthony Scott, Lakeland News. And they'll try and get that three-peat next weekend. If you enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland PBS.


Hi, I'm Zor. Welcome to Inezor Education. I would like to start a completely new course, if you wish, or chapter of this website. It's called MES Plus and Problems. Well, let me first explain what I mean by MES Plus. We have already covered, basically, all the material, which is supposed to be in high school and maybe a little bit above, as far as the MES is concerned, in the previous course called MES 14s. So, what is MES Plus? It's not anything new, theoretically. However, the purpose of the whole course, which I'm starting today, is presenting the problem. So, it's the problems, which is the purpose. And plus means that the problems will be a little bit unusual, not typical problems, which are supposed just to check your theoretical knowledge.  Some of them might be kind of difficult. Some of them are problems which are presented on certain mathematical Olympics, so to speak, Olympiads. Well, there is an international mathematical Olympiads organization, which is organizing these Olympiads every year. So, some of them are taken from old textbooks, which I know about as really challenging ones. So, that's basically the purpose of this course. It's not typical, unusual, tricky, maybe, sometimes. They are really challenging. They're supposed to be really challenging your mind. So, that's the most important purpose of presenting all these problems. This, the whole course, will be just the problem-solving, basically. So, I will present the problem and basically allow you to stop the video and basically do it yourself, try to solve it yourself. Or you might take the textual part for each lecture, which I have here. And basically, the conditions of the problems are presented there. So, again, it's all supposed to challenge you, which means you have to really spend some time solving the problems yourself, before listening to whatever I'm saying about the solution. Also, I put in reference of the website. There is basically a menu option, references. I put the names of the books which I took it from. I think I covered all of them. I hope so. And finally, let me explain why I called this first introductory to this course. I called it Who Needs Problems. Well, basically, nobody needs problems. People need solutions. And what's the most important part is that every time you really face a real-life problem, which you need to solve somehow, it's usually not exactly the same as it was before. Or in many cases, it's not the same as before. And solutions which were offered by some other people before you, sometimes not really fit it. And you have to come up with a solution to something, at least completely or partially, maybe a new problem which you are facing. So, to have this solution, to come up with a solution, your mind must be, well, trained, must get used to facing the problems, which are not really explained to you in the theoretical material. It's really something which you have to really come up with yourself. That's your creativity. That's your logic. That's your analytical thinking. And that's the purpose, basically, of these problems which I'm going to present in this course. So, we need solutions. And to get to solutions, we have to train our mind to be able to face all these unusual circumstances. Well, that's it for an introductory. And the next thing is problems and their solutions, obviously. So, thanks very much and good luck.