Live from Copenhagen, Denmark, it's theCUBE, covering KubeCon and CloudNativeCon Europe 2018, brought to you by the CloudNative Computing Foundation and its ecosystem partners. Hey, welcome back, everyone. This is theCUBE's exclusive live coverage here in Copenhagen, Denmark for KubeCon 2018. I'm John Hurd, my co-host this week, Lauren Cooney, and our next guest head, we're going to have a distinguished consulting engineer with Cisco Systems, CUBE alum, great to see you. Welcome back to theCUBE. Good to be back. So, great developer action. End of day one, we're going to be here all day tomorrow. So, one day one's kind of coming into the books, your thoughts on what's happening here, different crowd, but active? No, extremely active. Actually, one of the things I've noticed, and this is sort of a subtle point when you've been around a lot of open source projects, is you have a lot of people who are new to the Kubernetes community you're coming in. And one of the things I found extremely heartening is, they've got a really organized approach to it. When they did their developer summit, they had an entire track for bringing new contributors on. They've just revamped their documentation to help people get here, and they're finding better and better ways to articulate the things that people need to hear to help them make the leap to Cloud Native. Because one of the underappreciated things about Cloud Native is, that is different from the move to Cloud 1.0 that we made a few years ago, is that Cloud Native is not a lift and shift behavior.  And that's a global platform. So, this is like not a, you know, just a transformation project. It's like a lifetime transformation. Absolutely. Huge personnel issue. People process technology. Technology the last one. Have you accepted Cloud Native into your heart? I have come to terms with my lift and shift problem that I have, and I'm now aware, self-aware of Cloud Native. The first step is to admit that you have a problem. The making amends to your infrastructure takes longer. I mean, look at the look. Well, Ed, you would know. So, we're all working on it. So, you know, I have a question for you here. You know, as you were talking about how, you know, you're seeing a lot of new developers coming in and things along those lines. I'm also running into a lot of new developers at the hotel, at dinner, just walking around and having discussions.  I see them coming from banks, from, you know, large technology companies that are based in Europe. Where are you seeing these folks? So, that ends up matching very closely what I'm seeing as well. I mean, like, from all over the place. From people who finance large energy projects, right? From all areas of finance. Basically, you know, all the sorts of people who have big compute problems are starting to turn up in the Cloud Native world because this is literally where you solve those problems. And I think that's part of what's driving the ecosystem is the folks in Kubernetes made a number of incredibly intelligent decisions early on about how their architecture was built in terms of the modularity and expandability of it. And the result is that you get lots of people with lots of energy coming in saying, I have a problem like this. There's an obvious well-worn path to try and put together a proposed solution for solving problems like this. And they engage with the community. You know, one of the things that you're seeing just in terms of how the community grows itself is they've got special interest groups. SIGs for various areas in Kubernetes. They've now had to spawn working groups that come under them. You're just seeing things like Kubernetes proposals for how you're going to do things coming to form. So there's a lot of the maturity process that you expect to deal with the scale of people who want to solve their problems this way. So you're actually not seeing sprawl. You're seeing highly organized groups coming together in a way that can make the platform more positive. Yeah, absolutely. Not only am I seeing sprawl, but I'm starting to see highly intelligent things being said, but the people who work at what we think of is the core of Kubernetes. So I've heard a number of people make the comment that they expect the Kubernetes core to actually shrink in terms of what it offers. Because the broader ecosystem is picking up so much of the slack. So the sort of core APIs of this is what is Kubernetes without having picked out some options that meet your needs is keeping itself very tight while having architected it in a way where you can have this broad ecosystem without the kinds of problems you sometimes get with sprawl in other communities. So you want to get bigger, but you got to get smaller to be bigger. In some sense, yeah. You have to decide what's really important to get right in the core and really nail it. What are they getting right in your opinion? What's right about it that's going on? You mentioned some of the smart decisions that they're making. So a couple of the things that they've gotten really, really right are a relentless focus on developer needs. So I see this particularly in networking. And I think we've talked about this before. Developers don't want to know about subnets. They don't want to know about L2 segments. They don't even want to know about IP addresses, frankly. What they really care about is two things, reachability and isolation. Everybody can talk to everybody unless they decide you should be isolated. And service discovery and service routing. Those are the only two things they care about. And wouldn't you know it? In Kubernetes, you have network policies that control the reachability and isolation and services that do services discovery and service routing for you. So they've absolutely nailed the fundamental developer needs. Made your pain point. Yeah. So what's your take on just the ecosystem? Obviously we've commented, and this is always a dangerous game with communities, is logo farm. Everyone's here, right? Yeah, I mean, they took the CNCF logos and probably I think they broke them into three categories now. I'm not exactly sure what that would be. They have a whole new sponsorship level for... I'm not sure. But Ed, maybe you could provide some clarity here. Well, I mean, there is a certain risk in being loved to death, right? Kubernetes is full blown into what I will sometimes call crisis of success, which is you're succeeding so wildly that it's getting to be a problem. And that's good to see. But I think you're starting to see certain categories of things that are emerging. And there was a good set of readouts from the various SIGs to Kubernetes yesterday in the developer summit. So you've got a bunch of stuff around networking.  These are sort of fundamental infrastructure issues. Then you have a bunch of things literally about how do we expand the Kubernetes platform? How does that work, right? And different, you know, how do we produce the constructs we need to solve the various problems that are arising? And those things are all sort of progressively moving forward. And we're getting to sort of the interesting point where the people who did the original turn of the APIs are being really blunt and honest saying, look, these are the things we got, right? And these are the things we got wrong. And there's a lot to be said for having that level of honesty with yourself on stage in public, right? When you're the guy who wrote the code, it's unequivocally your mistake. And being able to stand up and say, look, we got this one wrong. But that's the community trust that you have and that's what makes the community. And that trust goes both ways. It's the trust of the community in that leader standing on the stage. But it's also the trust of that leader that we're going to move fast, we're going to do things right, but there's always a turn of the crank to do things better. And we've got to be straightforward about that. And their self-awareness around the iteration is key. They're putting their egos at the door, checking in at the door, focusing on the advancement. I got to get your thoughts for both of you guys. I want to ask you guys both of questions. I know that you're doing a lot of work with some startups and you're with Cisco, the big company. What's interesting about this ecosystem is the balance between the big players and the enablement for the small startups to be successful. We had a variety of startups here with news on theCUBE. So this is the give-get between sharing and projects where there's a balance and everyone can thrive and survive and grow together. Thoughts on that balance. Startups have needs, but they're not as big as the big guys. So what's your thoughts on- Why don't you start, Ed? Well, to begin with, we can't do everything much as we would like to. And back to the self-honesty, you have to be honest with yourself about that. And nobody has a monopoly on the good ideas. And so you really have to engage with the ecosystem and figure out how different aspects of the problem knit together. So I've had a lot of interesting conversations. I personally have some interest in what I sort of call unified IO. So converged networking storage. So I'm talking to a lot of folks who are doing storage stuff, a lot of little startups that are doing really cool things for storage about things we can do to help them there from the network side. And they're excited about that, right? And that's the sort of open source spirit that makes it possible to have all these startups because I'll be really frank, most of these startups, if they were having to try and build the thing themselves are simply not resource to do it. But with so much support from the community and the broad on a relatively thin startup budget, you can move mountains. Yeah, if you tap the formula properly, that's the key. Well, and the startups are getting more and more sophisticated about tapping that formula because only getting a good product is only a very small part of the equation. You also have to get the integration with the community because you have to make sure, even if you're entirely self-interested, if you build a thing, there will be a thing in the open source that does that. And it is a fundamental truth in the modern era that 80% of the value or more of all software is its connection to everything else in the ecosystem. Lauren, I want to get your thoughts on this. You're doing this now as a new startup, you're a founder of and running, but you've built programs, modern architectures that play here. You're seeing microservices growth, phenomenal, cloud native is just a whole nother ball game, going to a whole nother level. As you're engaging out there, what are you seeing for this modern community formula, playbook, whatever you want to call it, there's a way to do things now at a whole nother level that this is going. No, I definitely agree. I think the developer experience is really key, making it simple, making it just seamless, right? So folks don't have to wait to download something or they don't have to wait for, they could just click a couple buttons through a GUI and make it really, really simple, especially those onboarding. What I see from the startup side is a lot of, this is interesting because I think it's important, a lot of startups coming from companies that wouldn't allow them to do open source inside the companies. So they're leaving these larger companies and they're doing startups and they're raising pretty good capital for seed rounds and A rounds. And I think this is something that's pretty hot right now and we want to take a look at and the VCs are definitely looking. What about the big companies? We know all of these, South Francisco, IBM, you see Amazon here, they have huge scale. Even Microsoft has had developer programs been successful over the years. We all know that. What's the modern tweak that they're making that you're seeing work? Well I think it's a small teams. Like Adrien was on here earlier talking about microservices and microteams. And I think he's absolutely right. You have to have teams that are building these services that are moving quite quickly and doing it in a way that's rapid enough to keep up or be ahead of the market. Yeah, the microteam point I think is actually really apropos because this is going to sound very engineering for probably your head, but the management overhead gets to be quite steep when you're trying to do anything with big teams. Right, so you've got to have very loose coupling to everything else in the system which is exactly what CloudAid was about. And that's what you see not only in the startups but you see these sort of hybrid approaches that emerge where you have a startup that has a small team and another startup that has a small team that's nearby and a large company like Cisco that has a small team and there's an interaction between all of these. And we're sort of operating as the growing up of this larger team completely across boundaries in order to solve actual user problems. I mean I think it's historic time. I mean I think you guys are right on. I mean it's just, this is such an exciting time for if you're an engineer, software developer or anyone in large scale systems and building applications is going to a whole new level. I mean you look at blockchain right around the corner, decentralized applications is coming soon. We won't go there in this interview because it's KubeCon but I got to take what's your view so far of what's working here, hallway conversations you're having, what are some of the things going on here that someone who's not here might want to know about? Well so I mean I tend to be very focused on networking things so the thing that I'm most excited about this happening here is the entire world seems to be getting meshy, right? So there's a huge excitement around service mesh and Istio which I think is extremely well placed. And the fundamental thing that's really happening there is they're progressively taking parts of the problem that you're not good at if you're writing a microservice and they're pulling them out into a sidecar, Envoy. So that you don't have to worry about service discovery and service routing, you don't have to worry about the policies for how you're going to figure out what things you do about getting to the next guy in the chain of the work. You don't have to worry about even things as simple as making sure that you respond to faults well, right? And there's a whole new set of ways that you think about problems in this space that's emerging there. One of the things that I'm actually really excited about that's also meshy is when you get to things like people who have less common network problems. So the operators with Envoy, people who have more sophisticated network needs, we're starting to reimagine that stuff in the language of service mesh, right? So rather than trying to force all the legacy thinking about networking into cloud native where it's not wanted, we try and recast the problems we have into cloud native ways of thinking about them. And I think that ends up being intensely powerful and it's frankly almost overwhelming because there's so much conceptually going on in this space that you want to be able to draw on for the palette for the things that you're painting. Yeah.  Yeah, just got to buy into, the cloud native is a global platform. It is a fundamental new thing. It's not just a methodology, it's a new way. I mean, it's a new way of thinking about things. The C in cloud native does not stand for container. Container is the smallest possible chunk of this.  In what areas? Like just like what? I'll give you a really simple example. Let's say that I have an application that I'm running in VMs, right? And I've got my big database VM, I've got my big web front in VM. So I pick them up, I containerize them, I drop them into Kubernetes. So I've got one replica of my database VM and one replica of my web front in VM. And that's going to break sometime in the first 24 hours. Because I need to basically pick them up and say, okay, I need a bunch of replicas that are dynamically coming up for all of these things. I need to have the services to wire mesh them together so that for whatever reason I lose some number of my replicas that everything comes back up and goes forward and we never even notice, right? In some sense the ideal situation is you have a major bug in your code, right? Let's say you have a piece of code that's leaking memory and it dies every 24 hours.  But if you just try and lift and shift, you're not going to have a happy experience because it's not going to work the way you expect it to. And the monitoring tools are getting better too. So even if you're coming in on the other side, well, Ed, thanks so much for the commentary. Great summary of the event. Any surprises here for you? Any aha moments or revelations or epiphanies or any kind of surprises, good or bad or ugly? So one of the things I was very impressed with is I'm very impressed with what you can do with NoCode. I don't know if you saw that keynote this morning. With Kelsey. But in response to Dan Cohen's point about all the sort of total attack surface area, you know, Kelsey got on stage and did the NoCode project, which has perfect security for whatever it is you deploy it for. So the fact that you can get on code, do something like that, move an entire audience of thousands of people, that's impressive. You don't see speakers who can do that very often. That was, I wouldn't say shocking, but very much a pleasant surprise. And it speaks very much to the tone of the community. The keynotes today were some of the best I've ever seen. I am not a keynote person. I seldom attend them. The keynotes today were extremely well done. They had good energy and they were relevant. The walking through of the evolution of the community in brief punctuated explanations of what's going on and why they're important. Yeah, I've never seen it done better. Yeah, they were hitting their marks well. Well, great. Thanks for coming on, Ed. Great to see you. Yep. Thank you, Ed. This is commentary from Copenhagen, Denmark. It's the CUBE coverage of the CNCF Cloud Native Compute Foundation, part of the Linux Foundation. CUBECon 2018 in Europe. I'm John Furrier, Lauren Cooney. Thanks for watching. Be right back.


So, you know how to manufacture bullets, but you want to upgrade to missiles, not a problem. Assuming you watched the last video about bullets, you should have two blueprints, one for the bullet and one for the turret. From this point, it's actually very easy to change the bullet into a missile. To start, go into the bullet blueprint, and let's start with aesthetic changes. Since we are using some older special effects, we're going to delete the Niagara particle and replace it with a Cascader particle, and then set it to the P rocket launcher trail and add a static mesh, which should be set to your missile object. And make sure that collision is set to none. Now at this point, just make sure you got everything in the right order. The box collider should be up here, the static mesh right under it, then the particle, then the radial force, then make sure the rocket trail is positioned at the back of the missile. And now that it looks like a missile, go into projectile movement and slow down the initial speed to something like 350. Make sure that is homing projectile is checked to true and decide how fast you want the missile to turn. I'm just going to set it to a thousand. Now we probably want to change the blast radius to something bigger like 20,000. And we can keep all the old blueprint stuff the same. The only difference is we should probably change the sounds of the collision. So I'm going to use this and the sound of the launch, which I am deciding is this. And we don't really need a muzzle flash anymore. So I'm just going to get rid of it. Let's change the particle effect for hitting something to something a little bigger. I'm just going to pick this one. And if you drag your missile onto the scene now and press play, it should look pretty good. Awesome. Now we need to give it a target to do that under the event begin play, drag your projectile movement out and then drag out a homing target component, promote this into a variable and name it something like missile target and make sure it's instance editable and exposed on spawn save. And now you can go into your launcher and you'll see the simple code that we use, which just says if you press the E key, spawn the bullet at our turrets location and our turrets rotation. So all we got to add to this turret is a way for it to give a target to the missile. So we're going to create a new actor variable. We're going to name it target to send, make sure it's editable and exposed on spawn, drag it in and then drag out a component by class, set this to scene component and connect the missile target. You're done. So here's how all this works. If you click on your turret, you will now see the target to send variable is available. And if you drag or use the color picker to give it an object, that object will be sent to the missile as a target. So if we pick the cube as the target and we play the game and press E, hope that helps. And as always, hope you have a fantastic day and I'll see you around.


Hi, I'm Zor. Welcome to a new Zor education. I would like to talk about something which personally I consider quite beautiful in mathematics. It usually happens when one particular part of mathematics is somehow is related to another, and all of a sudden you see things which you didn't really see looking at each of these two parts separately. Well, just for example, if you take a concept of a function, for instance, and then through coordinate systems, you are examining the graph of this function, you understand how the function behaves much better looking at its graph than looking at its formula. So that's where we connect together algebra and geometry. And some very interesting curves, geometrical curves like ellipse, for instance, or parabola are expressed in algebraic language in certain formulas, and you see this connection. So today, I'm going to talk about connection between trigonometry and the algebra of complex numbers. Well, first of all, I do recommend you to refresh your knowledge about complex numbers. There are lectures and exercises in this course in the algebraic section dedicated to complex numbers. So I presume that we know what complex numbers are, how they look like, etc. So now let's talk about connection between trigonometry and complex numbers, which again, personally I never expected to tell you the truth. And when I saw the first time of this beautiful formula, so let me just put the formula. E is a very special number in algebra and analysis and differential in tego mathematics. Ix is exponent and we don't really know what complex exponent actually is. And this is how it's expressed in trigonometry. Well, it's unexpected. It's called the Euler's formula. The name is, I think that's how it's spelled, Euler or Euler, depending on the pronunciation. It's a Swiss mathematician who went to work to Russia during the, I think it was the 18th century and basically spent his whole life in Russia. And he is one of the people who was in the beginning of the Russian mathematical schools. And so anyway, that's the formula which he suggested. It's called by his name. And I think it looks very unexpectedly, at least for me. Now I'm not going to talk about this formula, but I will talk about something which is very, very close to this formula. So again, complex numbers and trigonometry. But first of all, you know that the general way to express the complex number is real part plus imaginary part which has a real number multiplied by number i, which is invention of mathematicians. And the only invention has the restriction is this one. To be able to actually extract the square root of negative numbers, they have invented this number i, square, which is equal to minus 1. So these are numbers which we call complex numbers. And I was talking about geometrical interpretation of the complex number. Namely, if you have the system of coordinates, then point with coordinates a, b. So this is a, and this is b. Can represent this particular complex number. I did not talk much about how certain operations are reflected in the geometrical interpretation. Today I will. Now, if you consider not only this Cartesian system of coordinates, x, y, but also the polar system of coordinates. And if you remember in the polar system of coordinates, the point is defined by also two parameters, which are the angle, the polar angle, and the distance from the pole itself. So if this polar system of coordinate is coincides, if you wish, with the Cartesian coordinate system, the polar axis is coinciding with x-axis. Then this particular point can be defined in both systems, basically. a and b would be the coordinates in the Cartesian system. And angle phi and the lengths of this distance to the pole is another coordinate. So not only a, b, but also r and phi are legitimate coordinates at this point. OK. Now I will consider, at least temporarily, only numbers, complex numbers, where this length is equal to 1, which means that I'm considering the numbers which are on the unit circle. So r is equal to 1. And using the Pythagorean theorem, obviously, it means that a squared plus b squared, or square root of them, is equal to 1. Right? a squared plus b squared is r squared, so if r is equal to 1, that's what it means. It's called modulus. So modulus of this complex number is 1. It means it lies on the unit circle. OK, fine. But if that's true, then what is abscissa and coordinate? What are a and b in this particular case? Well, by definition of sine and cosine, a, which is abscissa, is equal to cosine of phi. Right? That's the definition. And b is equal to sine phi. So all of a sudden, we came to conclusion that our number can be represented as cosine phi plus i sine phi. Well, this is the first connection with trigonometry. But this is just the connection based on definition. We're just expressing one particular way of describing our complex number with another Cartesian system and the polar system of collisions. It's just a conversion from one system to another, no big deal. But here is a big deal. Let's consider we have two different complex numbers on the unit circle. I don't need this. So this would be my first number. And you know what I'll use the letter alpha instead of phi. It would be easier for me. So this is m dot alpha. Now let's consider we have two different complex numbers. Both are on the unit circle. So this is alpha and this is b. Now, what I would like to do, I would like to multiply them. So I know that one number on the unit circle is expressed this way. Another number is expressed that way. And now I'm looking for representation of their product. So this is called z1 and this is z2. Now z1 times z2 equals. Well, let's just multiply these two things together. Now, you remember i times i, which is i squared, gives minus 1. So members which do not contain the imaginary part, members without i are cosine times cosine and sine times sine, but with a minus sign because it's i times i would be minus 1.  OK, what's the imaginary part? With i, I will have cosine alpha sine theta and sine alpha cosine theta, right? Sine alpha cosine theta plus cosine alpha sine theta. That's the result of multiplication. Well, what is this? Well, you should really recognize this formula. It's a cosine of alpha plus beta. And this formula is sine of alpha plus beta. So this is, again, one of the very, very beautiful part of this theory. When you multiply two numbers with a modulus of 1 and the angles alpha and beta, you get another number, which is also of modulus 1. And the angles are some of these angles. So multiplication by another number is basically a rotation by a corresponding angle. This is a really very unexpected and very strange, at least personally. That's how I feel about this. And that's what makes these mathematics very, very interesting and beautiful, if you wish. So what does it mean for us? Well, let's just think about the consequences of this interesting conclusion. So multiplication of the complex numbers actually results in addition of the angles which are associated, the polar angles which are associated with these numbers. Does it remind you anything? Well, let me express this rule slightly differently. And you will recognize what I'm talking about. Let's consider a function. Now, this is a function of real argument. x is a real argument. It's an angle, right? Angle measured in radians. And the value of this function is a complex number. So it's a function which has domain real numbers and codomain complex numbers. And what's an interesting property this function has? F of 0 is equal to what? This is 0, so cosine of 0 is 1. And sine is 0, right? What I have also proved right now, F of alpha times F of beta equals F of alpha plus beta, right? If I multiply these two, I get also cosine of alpha plus beta plus sine of alpha plus beta. I just proved it, right? So in this particular notation, does it remind you anything? Well, remember the function, exponential function? Well, a to the 0 is equal to 1, right? And a to alpha times a to beta is equal to a. And the exponents are heading together, right? So if you don't remember this, go ahead and re-examine the exponential functions. They do exist in the course. So you see the resemblance. The only difference is this is a function of real argument with real value. And this is a function with real argument with complex value. Well, yes, I understand there is some difference, but there is a huge similarity in the properties of this function, right? And that's what actually prompted Loyler to his formula I was talking about at the very beginning. All right, so be it as it may, I was just trying to show how interesting trigonometric functions can be related to algebra, which we have already learned, the algebra of exponents, OK? What else I can actually extract from this interesting thing that if I multiply two numbers like this, their polar angles are added together. Here is, for instance, very interesting thing. Let's again consider the unit circle. And we have agreed that if I have some kind of a number, which is complex number, which is represented by a point on this unit circle, and then I have another number, so this is angle alpha, I have another number, then their product would be something like this, which is alpha plus beta. So when I multiply, I add the angles, OK? So if I multiply by itself, so if I will do a square of a complex number, I will do cosine alpha plus I sine alpha squared. So what does that mean? Well, it means that the polar angle alpha, which was associated with this number, with this complex number, becomes two alpha, right? Alpha plus alpha for the square, OK? And alternatively, if I want to take a square root of something, instead of doubling the angle, I have to halve it, right? So if I want to do something like square root of cosine two alpha plus I sine two alpha, that would be cosine alpha plus I sine alpha. So this is two alpha, and this is one alpha, right? So that's quite interesting. What does it mean in this particular case? I would like actually to use it for one interesting purpose. Let's say I start with angle this one, angle pi. Now what's the coordinate of this point? Minus 1 comma 0, right? So we are talking about ax plus bi equals minus 1 plus 0i, which is minus 1, right? So this particular point represents this particular complex number. It has, in polar coordinates, r is equal to 1, obviously, because that's the unit circle. And pi is equal to pi. Pi is equal to pi. Now what if I want to have the square root of it? Well, as I was just explaining, I have to divide the angle in half. So half of pi is pi divided by 2, which is this point, right? Now this point, this is pi divided by 2. It has coordinates 0, 1, right? Now 0, 1 is 0 plus 1 times I, that's I. So according to my rules, square root of minus 1 is equal to I, right? Which is actually absolutely correct thing. That's how we defined I, that I square is equal to minus 1. So it completely corresponds to whatever the theory we started from, the theory of complex numbers. Now here is a slightly different last trivial. I mean this is trivial because this is basically a definition. But this is a little bit less trivial. Let's start with I. Now in one of the lectures about complex numbers, I was presenting you with a problem. Express square root of I in the regular A plus Bi form of the complex numbers. Now what did they do to find out A and B? Well, I squared both sides. I got I is equal to A plus Bi square, which is A square minus B square, right? Bi square would be B square and I square, and I square is minus 1, so it's minus plus. So this is the real part. 2AB5, this is the imaginary part. Now I wanted to be equal to I, which means this supposed to be equal to 0 and this supposed to be equal to 1, right? So I had a system of two equations with two variables. Then I solved these equations and I found that A is equal to 1 over square root of 2 or square root of 2 over 2 and B equals 1 over square root of 2. Actually, with the minus sign it's also true because I can both have minus 1 over square root of 2 and minus, and the square will be exactly the same. But let's concentrate on this solution. So it's not a very complicated solution, but it's a solution, right?  Let me start with I, which is this point, which has r equals to 1 and phi is equal to pi over 2. And let's extract the square root of it using the rule I was just talking about by dividing the angle in half, right? So dividing angle in half means that the new angle would be here. It has pi over 4. And what's the cohesion that's at this point? Well, cosine of pi over 4 and sine of pi over 4, right? So square root of I is equal to cosine pi over 4 plus I sine pi over 4, which is equal to square root of 2 over 2 plus I square root of 2 over 2, the same solution. So purely trigonometrically, I can come up with exactly the same solution I've got from solving the algebraic system of two equations with two variables. Isn't that wonderful? I love it. So basically, that's the general introduction to this very interesting and intricate relationship between trigonometry and the algebra of complex numbers. I did not derive completely the Euler formula because I didn't really explain what the number E is. I didn't talk about this, but I didn't really define it properly yet. But I did actually hinted that the properties of the complex numbers expressed in the trigonometric form are very much like the properties of the exponential functions. And the only difference is that exponential functions and algebra functions from real numbers to real numbers. And these are from real numbers to complex numbers. But that's a normal expansion. I mean, that's how we always expand our field of knowledge. We realize that we can do something this way, and then we invent something else. We define new numbers. We define new operations, et cetera. Now, I've not really touched what to do with numbers which are not on the unit circle. But this is really a very easy extension. Because if the number is not in the unit circle, let me just make this presentation quite quickly. I can always represent this number as scaled up, or if it's closer to 0 point, or scaled down the unit radius, the radius which has the length of 1. So I can always say that this particular AB, A plus BI, is equal to some kind of a modulus, which is this distance, which is greater than 1, or less than 1, or whatever, times this particular number, which is, again, cosine phi plus i sine phi. So when I multiply two numbers like this, my modulus plural is moduli. My moduli are multiplied by each other.   I mean, that's obvious, right? Because this is just the multiplication m1 times m2. This is a new modulus. And this part, the normalized part of the complex number, contains some of two angles, some of two polar angles. Well, that completes this particular lecture. I hope it was entertaining and fun, not only interesting from the informational standpoint. And again, the more you know, and this is not just for mathematics, it's like everywhere, the more you know, the more you realize the connections between certain things. And it really kind of opening your mind to significant growth. I mean, that's the very important part of learning experience. First, you start with learning this, this, this, and this.  On this now, I would like to end this lecture. Thank you very much, and good luck.


What is up you guys welcome back to another one if you are new to the channel I am Gold Pony and today we are in the new 2019 Genesis G80 Sport courtesy of Jack G Mbalvo Genesis in York PA and actually just got done talking with them a little bit they do have plans to build a standalone Genesis dealership now in the area that is pretty darn exciting I gotta admit so typically I start with pricing but today before we get to the pricing I did want to go over everything that you actually get with a purchase of a Genesis including free scheduled maintenance for three years a free valet service meaning the Genesis dealership actually comes to you picks up your car drops you off a loaner if you need it and then returns your car when it is done if you wanted to go that route that is also free navigation updates for three years free 24-7 roadside assistance again for three years and also free serious XM radio once again for three years quite a bit of free complimentary features included with a purchase of a Genesis so I did want to mention that to start but now let's start with pricing pricing for the Genesis G80 Sport we'll start at $55,250 for the rear wheel drive setup all wheel drive will start at $57,750 but then perhaps the best part about the Genesis G80 Sport is the engine setup and this is where the sport of course differentiates itself from the standard G80 powering this beast is a 3.3 liter twin turbocharged v6 putting out 365 horsepower at 6000 rpm 376 pound-feet of torque available at 1300 rpm power again sent to rear wheels or all wheels through Genesis's H-track all wheel drive system that powers sent to the ground through an eight-speed automatic with paddle shifters which you guys already know I am definitely gonna be testing those out in a little bit see how quickly their reaction times are and zero to sixty time is gonna commit at approximately 5.1 seconds with the top speed of 155 miles per hour and MPG numbers coming in at 17 in the city 24 on the highway taking premium unleaded fuel but so that before we get into the paddle shifter test or acceleration test there is a drive mode button located directly behind the shifter there but so when you press that you will find driving modes like normal eco and sport I did just put it in sport driving mode it did immediately downshift for me holding the RPMs at a much higher level giving you more power on-demand it's also gonna adjust the rotary spawn steering sensitivity and the suspension via some damping control adjustments the G80 sport does come with an adaptive damping system monitoring each shock absorber individually adjusting the ride based on your driving style and the roads in perfection so to speak not only giving you a smoother ride but also tightening up that suspension around heavy cornering really essentially giving you the best of both worlds but so now that we have mentioned all of that I think you guys know we have to do next let's go ahead and hit one of these paddle shifters that is gonna put me in paddle shift mode I am in fourth gear I'm gonna have to downshift a couple times and let's test out the paddle shifters here it's slight delay to the paddle shifters they're not that bad they're still there for a little bit of fun there but yeah there is gonna be an ever so slight delay there but now let's get the control back to the car all you need to do to go ahead and do that is just slide the shifter all the way to the back once again that gives the control back to the car and let's do a quick little acceleration in the new G80 sport see how quickly we can get this one here up to speed nice engine sound to this thing but yeah acceleration definitely not an issue absolutely no issues emerging onto the highway this thing is definitely quite quick but so then to go along with that acceleration as always braking is equally important and so of course brakes have been updated for the G80 sport comparatively speaking to the standard G80 at least for example with the standard G80 you have 13.6 inches up front 12.4 in the back with the G80 sport those rotor sizes are going to be bumped up to 14.2 inches up front even 13 inches in the back and as far as the braking feel goes definitely no issues there Hyundai and Genesis always kill it with the braking feel so definitely a big fan there but so then touching on handling and suspension a little bit up front you will find an independent multi-link front suspension with coil springs and shock absorbers in the back and independent multi-link rear suspension with coil springs and shock absorbers also front and rear stabilizer bars and like I was previously mentioning the adaptive damping suspension which again gives you the best of both worlds that is probably going to make the most difference that is why the G80 sport has such a nice ride quality to it and at the same time handles well when you're going through those back roads too but overall my short driving stint today again ride quality has been excellent comes to the steering feel with that sport driving mode the steering feel definitely is nice not the heaviest weighted steering wheel in the world which probably is a good thing but definitely a nice weight to it the only thing when it comes to steering feel if I had any constructive criticism is the steering wheel itself the grips are kind of small for being a little more geared towards a performance car so wouldn't of mind a little bit thicker grips on the G80 sport at least and touching on cabin noise you guys could probably tell there isn't a whole lot of exterior noises whatsoever coming into the cabin on this one so definitely impressed there as well and touching on visibility I can see perfectly fine out the back but perhaps my favorite part about visibility is the head-up display that I got in front of me here showing me things like safety features how fast I'm going of course but also the speed limit of any given road so definitely appreciative of that but what do you guys say now let's take a look at the exterior of the G80 sport because there are a ton of differences that really differentiate this one comparatively speaking to the G80 and so starting up front the most obvious way you can really pinpoint a G80 sport comparatively speaking to a G80 is the copper accents and they are going to be located throughout the car but specifically in the perimeter of that thick black mesh front grille that is where I see most of the copper accents and speaking of that front grille if you were curious what that rectangular box is that is kind of painted like the front grille up there that is going to be the sensor for the adaptive cruise control system so I do kind of like how they incorporated that into the front grille definitely looks nice it doesn't look obvious like it doesn't belong or anything like that so well done Genesis there to the sides you will find full LED headlights with dynamic bending meaning when you're going around a bend at night those headlights will swivel based on your steering angle better illuminating what is around the corners so you don't go hitting any deer or any pedestrians or anything like that it's always a good thing and of course with those headlights they do come with the automatic feature so you don't have to worry about turning them on when it does start to get dark out as well as LED daytime running lights and once again copper accents found within the front headlights definitely a nice little accent and again it ties in all the other copper accents I'm going to be mentioning as well then make your way to the side of the G80 sport you will find sport trim specific side skirts along with gloss black side mirrors with LED integrated turn signals zooming out a little bit you're going to find chrome window surrounds definitely adds to that high-end look on this one taking a look down at the wheel setup 19 inch alloy wheels will come standard and believe it or not they are a staggered fitment meaning the front wheels are going to be slightly smaller as far as with goes comparatively speaking to the rear wheels even with the all-wheel drive setup that's not just for the rear wheel drive so overall wheel size comes in at 19 by 8.5 inches up front 19 by 9 inches in the back more than likely helping with a little better acceleration there but again copper accents found on the center wheel caps and I got to love kind of the carbon fiber look with the genesis logo in the middle there as well so wheels definitely look good make your way to the back led taillights will come standard along with led license plate lighting always have to mention the small stuff like that gloss black rear diffuser just below and to the sides there dual exhaust outlets with chrome quad integrated exhaust outlets so they're integrated into the rear bumper always a fan of that so you guys know what we have to do next as always here is that exhaust clip it's open now since we are round back to open that rear trunk simply just press the button on the key five that's what I did at least but it is a power rear trunk there but once opened up cargo capacity is going to come in at 15.3 cubic feet and I was curious so I did see there was a little handle I lifted up underneath that and I did find a spare tire anyways make your way up to the rear leg room that comes in in even 35 inches also for those rear passengers they will find rear ventilation along with the rear center armrest with cupholders and when there's rear passengers put that rear center armrest down they will also find they have the power to turn on their heated rear seats along with a power rear sunshade as well and to continue that sunshade trend rear side window sunshades will come with this one as well so if you are a vampire and you are considering being in the back seat of one of these you are going to do just fine but say then make your way up to the front seats the copper stitching continues throughout not only on the seating both front and rear seats but on the side doors as well definitely a nice look there and those front seats are power adjustable with power lumbar that will come with memory settings for up to two different drivers they are also heated and ventilated and of course come standard with a leather finish taking a look at the steering wheel it is tilt and telescoping wrapped in a perforated leather meaning that steering wheel will come heated as well it's definitely nice and when it comes to the startup let me first start by showing you guys the key here you do have your genesis logo on the one side and when you flip it over lock unlock and again that button to pop the rear hatch but it is all actually keyless entry so all you need to do is simply just put your foot on the brake and press that engine start button located just by the driver's right knee there and what's started up tachometers on your left speedometers on your right and there is a fairly large digital display front and center to control what is on that digital display simply use the steering wheel mount and controls on the right side there and that's going to give you things like your outside temperature how many miles you have left until you hit empty there's your digital speedometer up there if you wanted it as well as when you need your next oil change and there's a ton of other stuff honestly you could check out there making our way to overall interior quality though now first thing i noticed at least was the analog clock again with the copper accents and the genesis logo front and center definitely appreciated that also an alcantara headliner will come standard on the g80 sport again absolutely in love with that you usually see alcantara headliners on very high-end cars like Porsche Lamborghini McLaren things like that so that was definitely nice aluminum foot pedals come standard along with the panoramic sunroofs to both the front seats and the rear seats have access to the sky there homeland controls will come standard up to three different garage doors and there is carbon fiber trimmed throughout this one which of course goes along well with the twin turbo v6 of course but now let's make our way to the tech display you will find a 9.2 inch high resolution color touchscreen display this will come standard with bluetooth and audio streaming as well as android auto and apple carplay factory navigation system also standard and again part of the genesis brand is you get three years of free updates there you will also get blue link on that tech display which is kind of genesis this version of chevy's on star and by the way to control all of this it is a color touchscreen display but if you didn't want to smudge up the screen there's actually a circular dial and buttons just behind the shifter there so feel free to use those as well but you can of course also check out your radio settings up there and by the way when it comes to the sound system you do get a lexicon 17 speaker surround sound system and i gotta say the speakers on this lexicon sound system they're very high end looking most of them i'm looking at have aluminum covers to them so a very high end look there but enough with the speakers let's just test out the sound system here see what we got playing on our three years of serious xm radio here and let's test out the clarity of this one bass is nice i think i'm more impressed with the clarity though 17 speakers i guess it is as expected that it would be a much clearer sound system than i'm used to hearing but very nice clarity on this lexicon sound system in the bass of course is nice as well but so then last thing on the tech display i wanted to mention ills when you do put the genesis g80 sport in reverse you will find a rear view camera with a 360 degree view as well letting you know who or what is behind you which is always is going to leave me into safety and so when it comes to safety i first wanted to start with the 2019 genesis g80 sport was awarded a 2019 ihs top safety pick plus recognition and that is the very highest safety recognition by that group so that is definitely nice but front side inside curtain airbags will come standard along with a driver's knee airbag as well but it continues in the back rear seat side impact airbags also standard you usually almost never see that right now and since we're in the back now latch a.k.a lower anchors and tether surcharger for the rear car seats also there automatic emergency braking with pedestrian detection comes standard along with adaptive cruise control with stop and go and essentially what that is is after coming to a complete stop the speed and the distance control are then reengaged when the car in front of you starts moving again so it's a very intelligent system so that's pretty cool too but lane departure warning also standard along with lane keep assist driver in attention monitor so it's going to detect when you maybe start to get drowsy start swerving a little bit and it's going to yell at you essentially and last standard safety feature i wanted to mention is the blind spot monitoring system with rear cross traffic alert and so but that is about it for this one you guys thank you so much for watching be sure to like the video and subscribe for free to follow me on social media at the bottom of the screen there and i will see you guys in the next video stay gold


You said something that you see that man right there to park you see him.   Why are you trying to figure it out? I know not my dog. I talked to my time He just he he got it.  I told him.  Yes Yeah, I know I know I know Ryan I know Ryan but I'm still had a producer cow I Said I said listen to it before you drop your album. May my album out.  It's not it and I and I know you understand, but he if he drop an album right now, right? Then on the flip side he can make it say hey It's for work. I got 20,000 for the hottest artist in the city To come sign to sign to me, right? He'll get more. He'll get more love on this post than this post Wow Yeah, you don't say think about how many people tagging the hottest artists. You know, yeah, yeah Do you you say some of the stuff don't hear the district? Well, I've heard you to take them to serve in some very disturbing thing You said something that you see that man right there to park you see him. Yeah, young boy You know, I heard the statement I just I just cut the damn video young boy got more plaster I said man, it just man crazy young boy as more you understand what this man was doing in the tonality and And and how he wrapped in the way he touched people lives and the people the kids We're not talking we're not talking impact. We're not some bigger than music You just we're not we're not talking about the style of who went harder. We're talking about the numbers So you saying he got he got more plaques.  Let me tell you like this Drake on the way to pass Michael Jackson Yeah, you can't say nothing about this game. I don't know about that You'll never be able to touch Michael Jackson on number wise No performance wise for them. We're talking about number wise I'm talking about Show up Michael Jackson show up. Yeah, yeah, Michael Jackson. We're just gonna be like to pop show up and then what show I don't know Come out he really late. He don't play no game When he come out don't kids go be yeah, but that's what you just said kids.   Man NBA young boy. He's a good rapper and he's a good guy And I respect what he's done in his era, but he's no to pop at all He couldn't touch him with a tent.  No, you didn't see all of well was recorded Yeah, but the impact you came count numbers when you start talking about impact and you and I both know that the numbers This is I know more people don't want to jail for listening to Young boy He have influenced the kids to kill each other He have influence. I don't mean to people died on young boy He got the kids rocker at her killing each other And To pop did to to pop had people wanted to be one of the biggest us He the one you and I had all these shirts off and all that's to pop shirts off come to do pop Maybe it was not in Texas. It was hot It was to pop man. It was hot so to pop the two pocket floor It's so so so without to pop begins from having sure though. No, he wouldn't have a shirt Rick Ross either I gotta ask you something. So yeah, I know but Where did you know how rappers love to wear like a lot of jewelry and the way how they dress and stuff like that? Where did that come from? Where do you think that came from? No, I'm talking rappers period when the rap game started And they used to wear all of this stuff. Where do you think all of that came from didn't know mr. T It wasn't it wasn't around with pimping was pimping pimps. Who pimps? We've been interviewing these pimps lately. That's where it come from They I've looked at this Roses the fans and it's them niggas got it bro He said um mr. T got it from He used to be the driver from for Don one the bling bling We brought it to the table, you know Bishop You see mr. T all that jewelry mr. T. Yeah, yeah for sure. It came from Bishop Hmm You know mr. T was a good brother, but he came out of the neighborhood one was working at Dean man's at the time You know, he was Bishop by the God at first His security My thank you all tomorrow. So let me ask you a question. So y'all up here interviewing pimps Why pimps getting eager as pregnant That's the part of the game Mm-hmm. What's the baby?  No, not it the thing is that they want to get pregnant They want to get pregnant by the pin because they come just like what you talk about like in the music game When you have all these other guys getting some of these females head they try to run stuff if they don't ones in their heads They control everything and also They are only used to dealing with those type of women. They don't know how to deal with it Simple girls or green girls. Yeah, that's how I was going. I was I'll not deal with no grown lady.  No, man. They have In on Harry Hines here. That's their girlfriends. That ain't they take all of my girlfriends It's a lot more time with them than everybody. Yeah, I said, I thought I was pimping but I really didn't Remember heard about you. Yeah, man. Make my day. Make my day.  Oh, yeah, I was I was the king of back I didn't bring you new money. Yeah, man. It's one girl listen in 2008, right? She said she said in my car and I met a sir when she said dad I want you now.  I was in my under now And I was I was homeless about what I'm gonna talk about go back to my mama I'm gonna take her to see the hill more Let me crave to get a girl sit here. They tattoo smoke cigarettes and she was skyline too So he was like I go over here Over here was over here Hans. So she said it was at the 7-Eleven.  She came back about five hundred dollars. Damn.  Where you get that from I Thought she stole it up some and then she put it in my hand And I ain't nothing and it wasn't that nothing was the same and you kept a bite side the whole time. I loved Yeah, we on boss talk one-on-one. Yeah, we gonna talk


We are at the Crawley Leisure Centre on the track here, preparing for the Crawley Run Crew Fun Run League Five-Miler. So this is an event that starts on the track and heads out into Tillgate Park here in Crawley. We do five miles and we finish on the track here. There's a kids race first and then our race. Stay in there Alice, well done. Just once around the track Alice, go, go, go. Alice was just pipped on the line. Fourth place for Alice. Good effort. Fourth place though. Fourth. There you go. That's good isn't it? Oh I'm fed. Race briefing done and we are off. And away we go. One kilometre in, 403 for that first kilometre. 2K markers early, it's not 2K yet. 434 for 2 kilometres. Big hill in that one though. And we've got a big hill on the way back by the looks of it. 407 for kilometre 3 down by the lake now in Tillgate Park. Very beautiful. 406 for kilometre 4. 5km in, didn't see the time for that one. 426, 4km 6, 2 to go uphill now. That 7km mark is a bit early, not quite there yet. That 7km is 434 for that kilometre. It's not all on road though. I love it on muddy path as well. Coming back down towards the track now. Back on the track. 300 metres to go. Up to the finish line. 35, 32. 8.3K. So just over 5 miles. First lady Amelia. Of course. But not by much. It was a really good race at the end. Well done Rachel. If you want to watch any of the other runs in the Fun Run League series then click that link right there. That will take you to the playlist for those. But a great time here at the Crawley Run Crew 5miler through Tillgate Park. Very enjoyable. So we'll see you for the next one. Next time. Take care. Bye bye.


I am here to discuss with you on topic transportation sector, part of infrastructure. Learning outcomes of today's session, at the end of the session students will be able to explain status of development of transportation sector in India. Introduction Dear students, you know transportation sector is important sub sector of infrastructure and again there are sub sectors of transportation like roads and ground transportation is there, airports are there, ports are there, railways are there. Let us see these sub sectors one by one. Let us go to roads or highways. Now what is the status? Central road sector under this India has got 4 million kilometers of roads nowadays. Nation or India has got 4 million kilometer of road and still the number is increasing. Roads carry 61 percent of rate and 85 percent of passenger traffic remember. This is the huge load on roads and we are spending 18,000 crores annually on roads. This number or this amount is increasing day by day, which are our flagship projects golden quadrilateral GQ national highways. So this is a project of national highways, this is our flagship project going on. North main ramp, NHDP phase 1, then north, south, east, west corridor, NHDP phase 2. So these projects are complete, almost complete and the works for new highways is going on. Then what are the key players in national highways, which are the key players NHAI National Highway Authority of India, a key player which is backed by government of India remember this organization NHAI this looks after development of national highways in India. Then Morse Ministry of Surface Road Transportation and Highways. This is also a government ministry which looks after development of roads. Then key programs NHDP national highways development program is conducted in seven stages right and central road fund comes for this NHDP is the development program which is undertaken by government of India. Now what is proposed in national highways, see lot of work is going on and for instance four laning of 7900 kilometer NSCW corridor, then two laning of 20,000 kilometer NHDP phase 4 of highways with only one lane, then six laning of 6500 kilometers of four lane GQ. All these are the developments taking place, lot of money is going to be spent on roads remember NHDP 2,20,000 crore to be invested by 2012, they have invested that means 10th plan under 10th plan 60,000 crore were invested under 11th plan 1,80,000 crores were invested. See such a huge investment is there in highways and roads. Now rate of development of roads in India it was in 2010 only 12 kilometers per day. Now it has accelerated remember it is in 2018,19 it was 30 kilometers per day, our target is 40 kilometers highways to be constructed per day, nowadays the rate is 38 kilometers remember, nowadays we have reached 38 kilometer construction of roads per day such a huge rate is there. Then NH AI given more independence to select and implement projects in order to aid speedy development of infrastructure. See now some reforms are there which government time to time gives so that this sector can develop itself. So NH AI has given some independence to the authorities so as to develop infrastructure. 100% FDI is permitted, remember for this sector. Then 100% income tax exemption for a period of 10 years such a concession is given. Tolling proposed to reduce operational costs on toll roads. Then planning for expressways is undertaken. So many reforms time to time are taken by government so that this particular sector flourishes. Now let us see again some provisions which are given rural roads which is large part of India about 40% are not connected by roads remember rural roads about 40% are not yet connected by roads they are to be connected. Now several plans to do so in 11th fire plan 1.7 to 2 lakh unconnected habitations are connected in the 11th plan Pradhan Mantri,Gram Sadak Yojana and the lot of work is going on in this sector. Rural infrastructural development fund is set up to provide funds for rural road developments remember. Then what are the issues challenges in front of this particular development of roads? First is acquisition of land remember acquisition of land is the challenge or issue which this sector is facing then this involves the political will to acquire land then time consuming process is there this is time consuming process which leads to delays acquisition of land etc. Then environmental and societal concerns regarding displacement of people deforestation these also are hurdles in this particular sector. NGOs and special interest groups often raise these issues and this leads to delays. Then exact tariff that should be charged according to varying categories of consumers is a difficult decision that must be made this is also one more hurdle and in addition there are always risks that by adding a toll charge the ridership on the road might decrease leading to an infeasible project. So these are the hurdles or issues which are faced by this particular sector.  Now increase in container terminals to the tune of Indian rupees 10,000 crores is planned remember JNPT that is Jawarlal Nehru port test is planning to spend 3,000 crore in expanding the capacity since the plan is to attract a large volume of maritime traffic the objectives in the port sector are to increase capacity of the existing ports and to add new ports in order to decrease turn around time of ships and to increase productivity. So this is the status of port sector again port sector is focusing on private sector participation remember model concession agreement is being prepared for this fees will be collected on a licensing and revenue sharing. Model TAMP that is tariff authority for major ports is the regulator in the sector and does a good job for specifying the tariffs. Plans are up put to improve road and rail connectivity to the ports thereby reducing transportation costs on goods and making the ports sector more attractive remember. Plans are being made to corporatize ports to increase operational efficiency. This will lead to independence from a central authority like port trust of India. So these are some of the measures being taken by government of India. Now let us go to railways sector which are the some of the key interventions of railways session increasing the utilization of existing capacity boogies by cutting cost fares etc tying up with private players to run trains offering volume based discount to boost sales developing owned land and generating profits computerizing operations lower passenger prices right these are the key interventions in this particular sector. Here we can see some of the highlights see Cochrane railway project is the major railway project after independence okay independence of our country. Now rail reforms on the annual a new investment about 60,000 crore in current plan for a dedicated Mumbai Delhi freight corridor then other dedicated corridors may come up soon. Private participation is being sought in track laying, freight maintenance etc. Plans are being formulated to bring in world class train and stations and that are to be built in our country right which will be at par with the standards of the foreign countries. Now let us see or compare our infrastructure with that of China. See everywhere China is ahead of us but we are developing nation we are trying to make up with the gap and these are the comparative values. Now let us stop a while here dear friends here are two review questions answer them neatly. Answer of the first question is D right all of these answer of the second question is C 38 kilometer road of rate of road construction in India these are the references for today's session. Thank you.


The topic of this talk is traceable PRFs, full collision resistance, and active security. This is joint work with David Wu. The primary goal of traceable cryptography is to prevent unauthorized distribution of software. And this is done by preventing unauthorized redistribution of cryptographic functionalities. Typically, in such a scheme, we want to embed a mark within cryptographic objects. And this can come in various forms, for example, decryption keys in an encryption scheme, signing keys in a signature scheme, and so on and so forth. There is a pirate device which is attempting replication of the underlying functionality. And the security requirement is the following. If such a successful replication is achieved, then we should be also able to successfully trace back to the set of corrupt identities that were used to build this device. The notion of traceable pseudo random functions was introduced by Goyal et al. in a very recent work in 2021. Informally, it says that if there is a program C that can distinguish from a pseudo random function from a random function on randomly sampled inputs, then marked identities used to build the device should be traceable, used to build the pirate device as we just saw. So more formally or more generally, this is the diagram. So suppose an adversary is given the marked version of a PRF, then it outputs a program C, which we call the distinguisher. And it can see arbitrarily many input-output pairs. We say that the distinguisher C is good if given input-output behavior of a pseudo random function and a random function, C should be able to tell these two distributions apart. So in essence, it should be able to break the security of the pseudo random function. However, Goyal et al. discussed in their work that this notion is not in general sufficient. For example, if the distinguisher C has hardwired in it, the tuple x star comma PRF k comma x star, then this notion does not make sense because it can always break the security of the pseudo random function game. So they relax this notion to say that C should be able to differentiate between input-output pairs of those evaluations from a pseudo random function and that from a random function on randomly sampled inputs of the domain, which they called weak pseudo randomness. So the tracing security informally means that the distinguisher C, if can break weak pseudo randomness of the underlying traceable PRF, then the marked identities used to construct the PRF should be traceable. The syntax of a traceable PRF consists of the following four algorithms. First is the setup algorithm, which samples a master secret key and a tracing key. The tracing key can be public or secret. Then comes the key generation algorithm, which embeds an identity into the key. So it takes as input an identity ID and uses the master secret key to come up with this marked key SKID. The eval algorithm plays the evaluation of the PRF, the pseudo random function. And finally, we have the trace algorithm, which is given oracle access to a weak PRF distinguisher. With this access, the trace algorithm should be able to output an identity or a set of identities, T. The requirements for a traceable PRF as introduced in that work also involve the following correctness, which basically catches the essence of functionality preserving. It says that the marked and unmarked keys agree almost everywhere. So the eval MSKX and eval SKID, comma X should indeed match everywhere, almost everywhere. And of course, we need the notion of pseudo randomness, which means that the eval algorithm is actually a pseudo random function. The tracing security, as I already mentioned previously, means that if the adversary is able to produce a useful distinguisher D, which can differentiate between the input-output behavior of the pseudo random function from the truly random function on randomly sampled inputs, that means it can break the security of the weak pseudo randomness. It can break weak pseudo randomness. Then the identities that were used to construct D should be traceable, at least one of them. So here is the game. The adversary queries to the challenger an identity ID to which the challenger responds a marked key. And finally, the adversary returns a distinguisher D, outputs a distinguisher D. And the security property precisely states that if D breaks weak pseudo randomness of the eval algorithm with non-negligible advantage epsilon, then the tracing algorithm should output identity with probability, which is as close as possible to epsilon. This is the single key setup, which means that the adversary is querying to the challenger on only one identity. So natural's next case is what happens if the adversary queries multiple times. That brings us to the idea of collusion resistance. We say that a traceable PRF, or in general any such scheme, is a fully collusion resistant if security holds against adversaries that can obtain any unbounded polynomial number of keys. So the diagram is still the same. The major change is here, the adversary is querying multiple times, and these queries can be adaptive as well. And for every such query, the challenger responds with the corresponding marked key. Finally, the adversary outputs a distinguisher D. If D breaks weak pseudo randomness of eval with advantage non-negligible epsilon, then the tracing algorithm should output at least one of the IDJs. So while it might seem more natural to require that the tracing algorithm should output all the identities which were queried by the adversary, this requirement is in general not possible, because the adversary may choose to construct D depending only on the response to the first query and ignores all the rest of it. So as long as we are able to trace at least one of the identities, the tracing security should be good. This is a reasonable expectation to have. So Coel et al in their work proved the following. Assuming LWE, they proved that there exists a single key traceable PRF. Further, assuming indistinguishability of first-cation and injective one-way function, they showed the existence of a fully collusion resistant traceable PRF. I should mention here that the construction they have from standard lattice assumptions, the single key traceable PRF construction is completely insecure if the adversary obtains even just two marked keys. In fact, what happens is that the adversary can recover the PRF secret key from any two marked keys. Thus, the natural question which motivates this work is, can we obtain a fully collusion resistant traceable PRF without going through obfuscation? Such a construction would have the advantage of being plausibly post-quantum secure and also will provide a more direct instantiation than going through the full power of indistinguishability obfuscation. This led us to our work, as I said, and this is the topic of this talk. We proved this theorem that from LWE, we construct a fully collusion resistant traceable PRF. So we provide an algorithm to how to generically augment a single key traceable PRF to a fully collusion resistant traceable PRF and our identity space is polynomial. As I said, collusion resistance is actually meaningful and non-trivial to achieve even when the identity space is polynomial. Because as I mentioned, the existing construction of a single key traceable PRF is completely insecure even in the presence of two identities queried by the adversary. So polynomial identity space is a reasonable start. And here is a very quick overview. I will elaborate on this very soon. So what we do is we combine a single key traceable PRF with a collusion resistant fingerprinting code. Essentially, single key security of the traceable PRF binds the adversary to strategies that conform to the restrictions of the fingerprinting code model. This, in turn, yields a fully collusion resistant traceable PRF. So before we move on, we need to review what a collusion resistant fingerprinting code is. And that brings us to this slide. So fingerprinting codes are information theoretic objects. The syntax consists of two algorithms. First, the gen algorithm, which samples a codebook gamma and a tracing key tk. The code words in the codebook gamma are l tuples. They belong to 0 comma 1 power l. And this l is called the length of the fingerprinting code or the code length. There is also the trace algorithm, which takes as input the tracing key and the word W star and outputs an S, which should be a subset of the identity space. For the purposes of this work, we can consider this to be our identity space. We also require the notion of feasibility to better understand the security of the fingerprinting code. So it goes as follows. Suppose we are given a subset W of gamma. The gamma is the codebook, remember. So a word W star is feasible for the set W. It is often denoted by the notation F of W, capital F of W. If every bit in W star agrees with the corresponding bit in one of the code words in W, so there should exist at least one code word in W. So that if you look at W star one, one of the code words has the same bit in the first position. So for example, if W is this set, which consists of two elements 0, 0, 0 and 1, 1, 1, then the feasible set consists of 0, 1 power cube. Because if you, if you choose any element from this set, 0, 1 cube, each of the bits can be matched up with one of the bits in these words. Next, we talk about collision resistant codes and the idea is exactly similar to what we have done so far, seen so far, we say a fingerprinting code is fully collision resistant. If tracing security holds against adversaries that can obtain any unbounded polynomial number of code words and the adversary can make adaptive queries. So it's the same diagram as before. The adversary makes queries, identity queries, which can be adaptive to the challenger and the challenger keeps responding with the corresponding code words. This constitutes the set W of code words obtained by the adversary. Finally, the adversary outputs a word W star. And as one can guess, the security requirement should be as follows. If W star is feasible for the set W, which is acquired by the adversary through its queries, then the tracing algorithm should at least output one of the identities, which was squared by the adversary. So this brings us to the construction of the collision resistant traceable PRF. So as I've already mentioned, the primary ingredients are number one, a single key traceable PRF, which is guaranteed by Goyal et al's work. So there were the domain range and identity space are suitably chosen so that we can, the entire construction is compatible with each other. Each of its functioning parts. And the second part is a collision resistant fingerprinting code with N code words, W I and code length L that W I belongs to zero one power L. So the primary idea that we use in this construction is the following. Take L independent copies of the traceable PRF and just X or them. And what do we use? Why do we need these L independent copies? We use each of these copies to embed a single bit of the code word. So if we take the identity map it to its code word, then look at the bits. Then embed that bit using the I it copy of the PRF. And that is how the construction goes and finally X or all of these together. So the evil algorithm goes as follows. I'm sorry, I should not have said X or all of these together. We output all those marquis as a total as we see here. So the construction goes as follows. The evil algorithm is on input X, we just X or PRF Ki comma X. So X source of PRF are secured. It is well known that this is exhaust secure as long as one of the underlying PRF is secure. That is the main idea behind this construction. The marking scheme goes as follows. The first step is we take an identity, you map it to the corresponding code word. Then we look at the bits. Then we use the corresponding PRFs to embed those bits and obtain these marquis. And finally we output SK1 to SKL as a tuple. The tracing algorithm, which one can probably guess by now is that the distinguisher D. If you're given for a distinct for the entire PRF, we construct the utility using that we construct it. Distinguish a DI for each of the components PRF Ki. Then we run the trace algorithm for each of these to obtain these sets Ti. Finally, using this whatever output we have here, we construct the word W star. And feed it to the tracing algorithm of the underlying fingerprinting code. So a limitation of our construction is that the mark keys Ki IDs are long. As you can see the PRF keys. K is a tuple of L independent keys K1 to KL and also the mark keys are SK1 to SKL. So the mark keys are quite long. So this is one of the limitations. So let me briefly talk about why this is secure. Notice that if this W star, which we construct here W bar star, if this is feasible, then we can resort to the security of the underlying fingerprinting code now. So all we need to guarantee is that the underlying word should be feasible. But now note that feasibility here means that if all of the code words corresponding to identities which are requested by the adversary have the same bit in a particular position. Say at I not the position I not all of the bits have the same bit. Then the corresponding bit in this constructed word W bar star must also match. But notice that in this case, because the bits are the same, the adversary actually receives only one marked key. Marked version. So now we can resort to the security of the underlying single key security of the underlying traceable PRF. That is, if there is a key associated with a single value, then single key security guarantees that the adversary cannot change that bit. Thus, we are guaranteed to make that the word W star is feasible. And once we have that collision resistance of the fingerprinting code ensure security for us. So recapping the construction again, the evil algorithm goes as follows, we take L independent copies of the traceable single key traceable PRF, and we XOR them. And the marking algorithm is we take an identity, map it to the code word, look at the corresponding bits, then use each of these PRFs to embed those bits and output the tuple as a whole for the mark key. The tracing algorithm goes as follows, we have a distinguisher D we construct using that we construct a distinguisher DI for each of these components. We run the trace algorithm, get a set DI suitably construct W star and feed it to the tracing algorithm of the underlying fingerprinting code. We should also mention that in the paper we actually provide a statistical test to check whether this is how this distinguisher is the efficient or not we actually provide a statistical test to check the efficacy that is needed in the security to prove also. So we also do this additional work in this paper, we considered the notion of active security. We additionally allow the adversity to make queries to the tracing record. This is an intermediate security notion, I use the word intermediate because it's intermediate between secret tracing and public tracing. And also an intermediate notion was previously also considered in the setting of water markable PRFs. And we were able to show using the same idea that using LWP, there exists a fully collision resistant traceable PRF with active security against adversities making up to queue tracing queries for any a priori bounded polynomial q, which is q of lambda lambda is the parameter here. As I mentioned, the same construction can be used as before the X or in idea the major differences we just have to use a fingerprinting code, which is fully collision resistant against adversities making queue tracing queries. Similarly for us, such codes were shown to exist very recently by Yang et al in their work in 2020, and we leveraged that in this construction to get the notion of active security with fully collision resistance of traceable PRFs. It is well known that PRFs can be used to construct authenticated encryption schemes, example via the encrypt and MAC paradigm. Surprisingly, instantiating the encryption scheme with a traceable PRF and composing with an arbitrary MAC directly is a trader tracing scheme where the underlying encryption scheme is an authenticated encryption, and hence trivially satisfy CCA security. If the underlying traceable PRF is secure against q bounded active adversities, then the resulting trader tracing scheme is also secure against q bounded active adversities. As I said, we just have to instantiate the PRF with a traceable PRF with active security. So this is one of the very important properties of traceable PRF which was also pointed out in the works of Goyal et al. That whenever we have an application which is built from a PRF, we can get a corresponding traceable version of that application by replacing the PRF with a traceable PRF and secret key trader tracing is simply an application of that. So let me summarize whatever we have discussed in this talk so far. We provide the first fully collision resistant traceable PRF from standard lattice assumptions, and we do this through the usage of fully collision resistant fingerprinting codes. We generically, generically augment of such a fully collision resistant traceable PRF using two primary ingredients. One is a single key traceable PRF. And number two is the fully collision resistant fingerprinting codes. We also showed, as we discussed, that tracing security holds even against active adversities that have oracle access to the tracing algorithm. So I would like to end with two questions, one of which I have already mentioned that one of the drawbacks of our construction is that the sizes of the marked keys is long. It's SK1 to SKL where L is the codelin. Can they be decreased efficiently? And number two is the active security result that we had, which dependent on Q. It had a polynomial bound Q on the number of tracing queries. This bound is appearing only because of the corresponding property of the fingerprinting underlying fingerprinting code that is a fingerprinting code has the same bound on the tracing number of tracing queries. If that restriction can be lifted from the fingerprinting code, the same can be done for the traceable PRF as well. So the natural question is, can we construct such codes? Thank you very much.