Okay, welcome everyone to today's TSVP talk. Today we are joined by Travis Scrimshaw, who's currently visiting from Hokkaido University. Travis completed his PhD in 2015, UC Davis, and has since had positions at University of Minnesota, University of Queensland, Osaka Metropolitan University, and now is finally in Hokkaido University, where he's an assistant professor, soon to be an associate professor. And he's been working closely with a few of the members of our unit while visiting. So we're glad to be joined by him today. And the title of his talk is Throwing Balls to Make Microscopic Waves. Oh, thank you. Thank you very much for the introduction and it's allowing me to be part of this nice program, this wonderful traveling salesman's visitors problem type solution. But I guess because I'm not talking to a bunch of computer scientists, no one understands that joke and I'll move along. So during my talk today, I'm going to just give an introduction about some very classical results from probability theory. Well, not super classical, but more than 30 years old. But the model itself actually is about 80 years old now and talk about what happens in the long-term behavior and going from the microscopic level to the macroscopic level. And then I'll talk about a very similar type of model, but that's deterministic and that's motivated from a completely different area. And then I'll at the end of the day just sketch some open questions and maybe you'll have some nice applications for whatever problem that you're interested in. And the model I'm going to start with is actually a very, very widely used model in mathematical, all sorts of mathematical contexts, but also physics, chemistry, biology, traffic congestion, engineering, things like that. So what it is is known as TASAP. So this is the Totally Asymmetric Simple Exclusion Process.  And just to save time, I'm gonna draw a car as just a dot and each car on its own gets to decide, well, okay, now it's time for me to move. But of course, you never want any cars to hit each other. So you get this particle here, this car, it can't move forward because there's a car already in the next space available. And so this is our model and what we have is each particle weights, well, to start with, let's actually make this a little easier rather than it has a stopwatch and decides how it wants to wait. It has a coin with some bias. So I'm going to fix some real number alpha greater than zero and I'm gonna flip a coin with probability alpha over alpha plus one of telling me, so that will be my success telling the particle to move. But it's just a coin flip and you just wait until the first time you get a heads and then you repeat and repeat and at every time step, every car that can move, its driver flips a coin and goes, oh, it's heads. Now I move to the next space. If there's a car already in front, well, you're stuck in traffic and well, you can call it a fun game of flipping a coin but you know you're not going to do anything so you might as well watch a TikTok video. So that's the start of the model. And so the probability of the waiting time being T is equal to P to the T times one minus P. Well, actually I've done it the other way. So let me, you have one minus P to the T number of failures and then you get a probability, you get a success and therefore you move. And well, this here is basically a geometric series. You probably learned about this in high school and if I make this infinite sum, I get one because well, an event should happen for sure. That's probability one. So this is our discrete time model but when we're doing this, we actually want to do continuous time and we wanna find a way to sample this because well, if you have a continuous, you know, say you're picking a random number between zero and one, a random real number between zero and one, the chances of you getting exactly pi over three minus one, let's say is zero. There's no way you're gonna get precisely that one real number. There's just, it's too many. So we need to find a way to actually model this. And so what we do is we scale time and the probability at the same rate by say some delta T going to zero. And when we do that, this probability, it goes to an exponential waiting time. So what that means is I get some rate alpha, e to the minus alpha T becomes the probability that we have to wait. And so this idea of flipping coins at the scale of the time step and the discrete simulation that you run on a computer, we can model this exponential waiting time in this way. And so now we have a continuous time process and that more reflects what we see in the real world of actual traffic movement. But, you know, if we also want to model electrons moving down a wire or blood flow in a capillary, anything where things are constrained, it basically looks like a one-dimensional problem where things move with some randomness, but some fixed rate, this model gives you a way to do that. And we want to now understand what's its macroscopic behavior. What happens after we wait, say 20 hours, rather than just, well, in the first five seconds, things like that. What kind of behavior should we expect? And how we're going to do that is we're actually gonna change our picture altogether. So I'm gonna simplify this a little bit more and I'm going to start with all of my particles at base, all of my cars basically out of traffic light. So if you want, you can think of this as some kind of like capacitor and then it's discharged charging its electrons and moving down the wire. And of course, you know, as one particle moves, another one needs to eventually get to the wire and things like that. So this is our starting condition. And then, well, after some amount of time, then our particles, we have our special zero spot here. And our particles will move along to say some points here, here, like this. And what we do is we will draw this tilted square grid and I'm sorry for my very poor drawing skills here. And from that point, what we do is we just say, well, this first row here is how far the first particle moves. So the first particle here is moved one, two, three, four times step, four steps. The second particle has moved two time steps, this third part, sorry, two steps. The third particle has moved one step and then the rest of the particles are still stuck over here. And so I get a figure that looks like this. Maybe what I'll do is I'll color in another color here, all of the positions of my holes. So basically what it comes down to is the steps that go down correspond to the position where there's a particle and the steps that go up correspond to a hole. And this shape that I get is known as a partition. And so this is good because now we get something that actually starts looking like a function. It's piecewise linear. It goes one step down, one step up, and we just kind of chain those together. But as we start to scale out bigger and bigger and bigger, and we rescale everything so that after, say, t time steps, we just scale it so that the things that we can see actually lie between zero and one. And we can see that there's gonna be, you know, clearly less than t particles that move. Then the shape that we get, it looks a little jagged, but it starts getting closer and closer to an actual smooth curve like this. And what we wanna do is we actually wanna describe what that curve is precisely. So now we have our model, but our model is still just a little bit complicated. And when we're in continuous time as well, two particles will basically, will never move at the same time. Again, the chances of you picking a random number between zero and one, that's exactly pi over three minus one is zero. So let's actually tweak our model a little bit and say, well, only exactly one particle moves at each time step. So we're kind of forgetting about time a little bit. And we're just saying, well, we only care about the sequence that the particles move. And we therefore, again, only one particle moves every single time, we just record that. And so we record that in the boxes of the partition. So again, the partition is cut out from this square grid.  And now I'm gonna fill those boxes with integers one, two, three, up to however many boxes I have. But I can't do this in an arbitrary order. I basically need to play a big game of very simple Tetris where one block drops at a time and it slides down into a corner. And so if I want to do an example, say, here's my partition shape. Well, the only particle that can move is the very first particle at the first time step. But now we have a choice. We can move the first particle again or move the second particle. So let's say I move the first, then the second particle moves and then the second particle moves again, first particle moves, first particle moves, then the third particle, then the first particle, then the second particle, and then the fourth particle like this. And these are known as standard Young Tableau. So now our model is basically we want to actually count the number of standard Young Tableau. So I'll write that as F lambda. This is the number of standard Young Tableau. However, it turns out that if I want to properly encode the dynamics of my original particle system with that rate alpha that I originally started with, I don't quite want F lambda by itself. I actually want the probability of the particles being at a position lambda being equal to F lambda squared. And then I need to normalize that. And there's a few questions involved. First, does this make sense? Can this normalization constant be infinity and therefore everything is zero and then things go bad? Well, I want to actually make things just a little bit easier because it will go to infinity and things are bad. So let's normalize this, but let's only do it where the size of the partition, the number of boxes of the partition is equal to a fixed value n. And then we just take n off to infinity because n is our time. Maybe I should actually call it t for that matter because it's how many steps have we done and we want to take time off to infinity and then rescale. So basically I want to take a sequence of random partitions where one's contained inside of the other and that's how I'm kind of recording these tableau. And so from this, now I need to actually, I can compute zt because I know I have a finite sum. I want to compute the sum of f lambda squared. This will be my zt where I'll just throw a little bit of math notation here, but I'm just summing over all partitions of t, all ways to put these boxes, put t boxes. And so the claim is that this is equal to t factorial. In other words, this is the number of ways of shuffling t cards. And how do we see that? Well, we have a nice way to do that. And I'll just draw yet another grid. And I think this is square. One, two, three, four, five, six, seven, eight. One, two, three, four, five, six, seven, eight. And well, the number of ways of shuffling t cards here in this grid, I have eight, I have an eight by eight, I have a standard chessboard. And I want to shuffle eight cards. Well, how I can encode that is by placing eight rooks on this chessboard such that none of them can attack any of the others. So for instance, I can do something like this. And so why is this a shuffling? Well, I can just write eight, six, three, five, four, one, seven, two. Say this three is coming from, well, let's say this five here. Well, it's in the fourth position, the fourth row fifth column. That's how I can assign these. And now how do I get the standard young tableau out? Well, what I do is I can just write eight, well, what I do is I draw straight lines going up from every point and to the right of every point such that they intersect in the first possible way. That's not quite right. That goes up. I'll go like this. And now I record one, two, seven, one, four, seven, like this. And now the next thing is I look at all of these points where they've collided and I repeat. I'll go up and escape. I'll go like that and I'll go up. And so now I get three, four, eight, two. And I repeat yet again, five, three. And I'm running out of colors. And so that's six, five. And then finally six, eight, like this. And so I started with the red and I write one, two, seven, and one, four, seven. Now I take the blue, I have three, four, and two, eight. Then I next was the green. So I have five and three. And the purple. I have six and five. And finally for the pink, I have eight and six. And so what we see is we get a pair of standard young Tableau. Out from this process. And this gives us our bijection. Because well, this, why I started with is a shuffling of eight cards. And I've given you a pair of standard young Tableau. And now I need to show that I can go in the reverse direction. I can start with a pair of standard young Tableau and go backwards. Well, how I do that is basically the sequence just done in reverse. I start with line coming in at eight line coming in at six. And that creates a point there. Then I have a line coming in at six line coming in at five. And then lines coming out at this pink point. And well, that gives me the, all of the purple lines. And you can repeat that over and over again until you get, until you're done. And then the points that you're left with are the red points.  And it turns out that this construction is also related to something called polynuclear growth where you have a line and it's somewhere along that line. You, a point comes into existence.  And you may have a second fire and you record that. And that's this polynuclear growth model. And so this basic, what I'm saying with this is, well, I've transformed the picture and I now have my normalization constant. So I now have the problem, I now have an actual probability for a random partition with inboxes in it. And I can also say a little bit more from this that the shuffle here, if I look at the things. Let's see how does it go. Well, basically I look at this longest row here. And that's going to correspond to the longest increasing, yeah, longest increasing subsequence of this permutation. So the three, the five, and the seven is an increasing subsequence. And this will come up, you know, why I'm mentioning this, it'll take me another moment to actually get there, but please bear with me. So we now have a probabilistic model. And it turns out we can, I won't bore you with some of the details, but essentially using this idea that they're particles, we can use techniques from mathematical physics to actually compute these integrals, or sorry to compute the long term behavior by studying asymptotics and doing some analysis. But the, the main results is care of. Sorry, Varshak, care of, and independently, Logan, and Shep around 1986, which says that the their theorem was that the limit shape is a semicircle. So after a really, really, really long time. If I choose a random partition with this distribution with this waiting, then what I get is something that looks basically like a semicircle. So if I, you know, could draw a perfect semicircle that touched properly, and then I sampled a random partition, it would just look, if I make n sufficiently big, even something like n equals 100, I get something that looks exactly like the semicircle here. And that's, you know, the, and basically I now know the long term behavior and how the essentially the density of particles is as I go to a macroscopic level. But we, we even know things a bit in a bit finer detail because we know what the limit shape is. But we also want to look at the, a lambda one varies from the limit shape. So we want to look at the little microscopic fluctuations of the first particle compared to where it should be. So the first particle, you know, we know how that should move, you know, it's free to move. There's no constraints. It's flipping a coin. It should follow basically this geometric distribution if we do it kind of in a, you know, we've changed our model a little bit.  And it turns out that this distribution is equivalent to the distribution of the largest. Distribution value in a random unitary matrix where we choose entries by Gaussian distribution.  And so, you know, it's a fairly simple model for you choose half the entries and then the fact that it's transpose and complex conjugate has to give you the same thing you now have all of the entries of the matrix. And then you look at its largest eigenvalue, and you look how that fluctuates from another semicircle. This is not as Wigner semicircle law, as the matrix it's really big the eigenvalues approach a semicircle and distribution for value and so. And this is known as the Tracy Wittem distribution. And this is not quite a normal distribution. It looks a little skewed. There's, you know, a little bit slower up on one part and sharper down on the other. So why is this important? Well, these random matrices. The people in high energy nuclear physics. Well, if you want to model these very heavy atoms. And how the nucleus behaves, well, you from what I understand you don't want to do it directly. Instead, you take a random matrix and essentially what I've done is I've connected just particles moving down the wire with nuclear physics. Well, I, well, this is classical results, but I'm saying that it's related. And this is. Due to by death and Johansson. I believe this is the first known proof of this. And so this fairly simple model particles moving along a line. It actually has very, very rich structure to it. Not only that, but it also has other interesting macroscopic behaviors. If we add another light particle. And by this, what I mean is I have a another particle that's lighter than the original particles I started with. And if you want to get past one of those big heavy particles, well, they can just push the light particle out of the way. They treat the light particle like a whole. But the light, the light particle, if it has a hole, it can move, but it can't get past one of the heavy particles. And this particle models in the system. So if you want to think of it like water or gas, or maybe like a gas flow, you have two gases. One is lighter than the other. In a tube, you remove the things separating them and you look at how they mix and say a current. Well, they start mixing. But what will end up happening is depending upon how you set your parameters is suddenly the density changes. I mean, it starts kind of initially in the system, but you can get that same sort of short change in density as the system propagates in time and you want to measure, well, where does that shock actually move? Well, if you want to do this microscopically, this is what this light particle does. So it's just a very slight tweak of the model. But you can actually do quite a bit more with it. It also has spontaneous symmetry breaking. So again, this is kind of long-term behaviors, but if you start looking at different pram, you start adding different parameters, local inhomogeneities to the system, particles can move with different independent rates. You can see very sudden changes in your macroscopic behavior for small changes. So you get, you know, all this behavior without actually having to do too much. Your phase transitions can come boundary conditions. Sorry. Usually you see phase transitions when you start changing internal structure, changing parameters, things like that, you know, spontaneous symmetry breaking. But it turns out that you, rather than working on a line, you work in a finite dimensional, you know, finite lattice, then your boundary values can actually play a role in how the system behaves and drastically change the macroscopic behaviors of the system. So this, this model is very, very rich and basically what's currently being studied is you know, current studies are on a ring with inhomogeneities. So as I said, the particles can have a different position. And there's still a lot of open questions about this model at a mathematical point of view and being able to rigorously prove, well, if I just start tweaking my model a little bit, I allow particles to hop over another. They hop more than one space. I'm not going to talk about that. They hop over another. They hop more than one space. They hop with different rates, all of these different things. You can get very different behaviors that occur and we can actually prove them mathematically. You can run simulations, you can see lots of good behaviors, but the rigorous proofs are actually much harder than that. There's a lot of detail that's hidden in these. And was there anything else I wanted to say with this? Yeah, maybe not. Well, these are also very recently connected with how many lines pass through a generic configuration lines. Or more generally k-dimensional subspaces of n-dimensional space and how they align with each other, things like that. And so this is a very understanding how to count these things as a classical subject known as Schubert calculus. And it's really only in the past few years that we've realized, oh yeah, the TASAP is being governed somehow by this geometry. And we go from something that's very, very rigid to something probabilistic and we don't really understand how to go between these, what it means to, on the geometry the probabilistic stuff means and vice versa. So that's kind of the first half of my talk. Are there any, well, less than half, but let's say the first part. Any questions at this point? I'll save my intro at the end. All right. So now, so that's the probabilistic side, but what if I want to make this a more deterministic process? And how we're going to do this is something called the box ball system. And this was introduced by Takahashi and Tsutsuma in 1990 and what you do is you again take the model. Now I'm going to draw it slightly differently, but it's still the same model at heart. I put a bunch of particles, like so and then what I do is I scan from left to right and when I encounter a particle for the first time I throw it into the next available spot. So I take this and this particle goes over here. Now, well, I move from this point to this point, but I've already seen that particle. So I continue on I throw this particle here continue this particle gets thrown here. This gets thrown here. This gets thrown here. And so what I end up with is this local or this configuration and now if I run that process again that particle moves one step that particle moves one step and then this collection of three particles moves three steps but it still stays stuck together and what we see as we investigate this system is that collections of balls move with speed equal to its size the number of balls in particular this one moves with speed one this one moves with speed one and this moves with speed three provided they're far enough apart here they're kind of in this interacting process mainly these four here interacting with each other and so we get you know at that point we can't really see what's going on but when they're far apart they move with speed equal to their size and they keep their same size after interaction and so we call these solitons so why do we do that well if you like to study waves and thin channels you can probably come across the coat wig the fries equation for KTV if you one if that's if speaking Dutch names is way too hard for you it is for me so I usually call it the KTV equation and the solution for this are something you can actually do at home you get you know a very long thin channel say maybe half a bamboo and you make waves through it and you'll find that the waves will propagate with speed according to their size they'll come they'll interact with each other and then they will separate precisely back into the same waves that you have before and so the box ball system is just a microscopic version of the KTV equation and these standing waves or solitary waves are known as soliton solutions to the KTV equation and essentially every solution to the KTV equation separates out into solitons and so this gives us you know this really is a microscopic version of the KTV equation and so well this is good we can study this and we can drive explicit formulas for this but it turns out we can actually do something a little bit more we can introduce a carrier to describe the dynamics and what we do well rather than throwing the balls I come I walk along and every time I see a ball I pick it up and every time I see a space I drop it off and let's assume I have a fairly large basket more than the number of balls I would ever carry so I can always pick up a ball and I and progress like that and the small change in perspective it's fairly easy to see you get the same dynamics but you can model this by basically quantum symmetries so if I if I want to be very rigorous to maybe the physicists in here I'm taking represent finite dimensional representations of a quantum of a drenfeld Jimbo quantum group of affine type bunch of fancy words basically I'm doing quantum mechanics and using stuff from quantum mechanics to how these ways move and this then allows us many generalizations sorry Leeran I'm writing that with Z and I said Z so because of these quantum symmetries I can change things around and do things like introduce antiparticles and describe now how variation of these waves move and so this is all controlled by something called our matrices so if you know what an our matrix is then you pretty much understand this statement if you don't well I have vector spaces I have maps that preserve these symmetries and an our matrix is basically such a map and it defines everything here I to be really precise I'm doing this at a combinatorial level and it turns out it's just a marketing rule I pair things up and things that are unpaired move into my carrier out of my carrier depending upon how things go so I can actually if I had more time I could very very easily describe how this procedure works but it you know I've more or less already told you it's just you know picks up a ball when it can drops off when it drops off. So we have lots and lots of this actually gives us a huge amount of power. But there's also a relation with alignment of spins so if you think about basically there's two types of spins things point one way or they point the other way and if we chain these two choices together well this is known as the spin chain and then there's a bunch of physics involves but just to keep things light well I had every position along for my box ball system had two states it had a ball or it didn't have a ball and that gives you the way to realize with the spin chains and the local energy basically telling you about the interaction of two of the adjacent spins and okay I need to fuse some of the spins together but the local but essentially the local energy coming from the physics turns out to be related to the shifting of solitons from the interaction and this shift of positions is a sign of the non-linearity of the system so just to give you a brief kind of example of what I mean by this well if I have two particles and then a particle like this this jumps here this jumps here this jumps here so here and here like this and so this is my interaction and I have this size two particle here size two soliton here and the size one soliton here and the size one soliton if it was moving just by itself would be at this position and the size two soliton moving by itself would be at this position but instead they've shifted by two from where they normally would be and this is the non-linearity of the this local interaction in this spin chain system and so we get some very nice connections there and there's still a lot of open questions involved with this where probably the biggest one is well how do we include super symmetry into this there's some very recent results that have made some progress in this for one particular case but beyond that we have no idea in large part because the analog of the symmetries the quantum group and its representations we don't really understand or even know about and so you know within this one seemingly very simple question there's a lot that needs to be done and there'll be almost certainly a lot of very good mathematics and very good physics that will come from this and kind of the other major open question is what happens when we make this system random and so now in my last few minutes here let me actually connect the two parts of my talk because right now it seems like they've been completely separate from each other but it's actually this R matrix here this little tiny thing I'm glossed over well it turns out that pace up you can describe it in a lot of very interesting cases by our matrices that satisfy stochastic condition in other words there's a certain sum of the entries that's involved and so every the way we study all of these at least that those of us in this representation theory probabilistic area of mathematics is we use our matrices to do everything on the on pace up and then we have this which has also been the box ball system but very few people have done probabilistic versions of the box ball system and usually in some recent results have shown that if I introduce an extra probabilistic parameter I get a very interesting type of object that's related with other affine the algebras if I want to throw words around I would say Hollywood polynomials and Q Wittaker polynomials but it turns out you get other extremely interesting objects and with geometric connection so yeah I think that's where all in my talk thank you very much for your attention question I had earlier you had the at the end of your taste that thing you had kind of open questions or something like other directions current studies there we go on a ring is it a set or a set for maybe one of those on a ring somehow relates to McDonald polynomials and some kind of symmetric functions is the same thing happen here on a line like should I understand or be able to understand why come here besides partitions well the so why you should expect symmetric functions is the the connection it's the our matrix the and you have Yang-Baxter equation and Yang-Baxter equation gives you the symmetric functions but you can't if you now allow particles to hop back with some additional independent parameter that's ASAP that's removing the totally parts and there's very recent results of Mandelstrom Iyer and Martin maybe Sylvia was involved I forget for that that say that the stationary distribution is a McDonald polynomial basically and there's modified McDonald's by using Tazarp where now you change the model a little bit and you allow things to stack and you break the exclusion process part but not so badly that things become completely wild are you suggesting that like using this our matrix you point on this that it's basically just looking at this as something to do with the Li-Arbor and therefore the symmetric function stuff should fall out more basically yeah anyone else have any questions thank you Travis again


Masahiro Tanaka and you Darvish had a nice back and forth this morning and the Twitter translate just turns it into wildly awesome poetry for us. So here we go. Masahiro Tanaka said, yesterday Imazumi Yu-chan gave me a lip. So I'll think I'll do it again if you have something to ask.  And then Yu says, it seems natural that a yellow cheer from a New York fan says, it's cute like a puppy.  I think he's calling him soft. Like, oh, you're saying cute things, but you're scared to sound cute. Masahiro Tanaka says, there is no such voice. I did not sound like that. Yu quotes him. I usually feel like a puppy, but it's cool to be like a lion. Masa responds, I really wonder if there is such a voice. So please give it on your YouTube channel. And Yu says, let's appear together. It seems to be a god of observing face parts by one, one by one in live. So we're all on the same page there. Masa says, I hope the number of registered users will not decrease. So we invited him on his YouTube channel. Masahiro Tanaka says, you're gonna, I hope people don't unfollow you. And Yu Darvish says, viewers are ignored. It's fun for you. And it's a God's time. Guess that's good. I'm waiting in Dallas, Texas. Masa says, I do not go. I'm not going to Dallas.  And then Yu says, then it will be a video phone appearance. When is the best date and time? And Masa says, if that's the case, it does not matter. So in conclusion, Masahiro Tanaka will not be on Yu Darvish's YouTube channel using a puppy voice.


Hi everybody, I'm Jofa Couto and so this talk is going to be about encryption switching protocols This is a joint work with Thomas Peterson David Mosheval So secure to body computation address the address is a challenge of Computing some public function F. So you have two players a listen Bob who each holds on private Input X1 and X2 and they would like to Compute F on those inputs without revealing anything more than this result in particular They don't want to reveal anything at all on their own input and there have been various methods that have been designed in the past to address this cryptographic challenge based on a large variety of tools and primitives one of the most common Primitives that have been used to build to body computation protocols is homomorphic encryption So quickly a public key homomorphic encryption scheme is a public encryption scheme So you have a public key which allows you to encrypt and the secret key which allows you to decrypt and You have in addition an algorithm eval Which takes as input an encrypted message M The public key and some functions from some class of function and outputs an encryption of this of F of M So it allows you to evaluate functions on the plain text without having to know the secret key of the scheme And it's fairly easy to see how we can build To body computation out of homomorphic encryption So if you have a homomorphic encryption scheme for some class of function think for example of linear operations Then you can let the players and create their input broadcast does homomorphic encryption homomorphically and on the locally compute Evaluate the function on those encrypted input and they end up with an encryption of the output And then there is methods to recover the output from an encrypted output in using a joint protocol And so if for example now you want to build a to body computation protocol For like linear operations, then you can use the pie encryption scheme, which is the linear homomorphic homomorphic and if you want to evaluate a function, which is like Expansion monomials or any product function, then you can lay let the players you'll use the outer market system Which is multiplicatively homomorphic If you want to do more Well, obviously it's way harder if for example, you want to be able to evaluate both additions on multiplications Then if you can do so you can evaluate any circuit in a gate by gate fashion But it's still possible and it's called fully homomorphic encryption So fully homomorphic encryption lets you evaluate any function given a ciphertext without having to decrypt the ciphertext But current implementation of fully homomorphic encryption scheme are quite slow so the natural approach For building secure to body computation out of homomorphic encryption is to say instead Well, let's forget about having an encryption scheme, which will be multiplicatively homomorphic We'll just assume that we have an additively homomorphic encryption scheme and We will evaluate all the additive gates of our protocol Homomorphic alien locally, but we will pay for each multiplicative gates using a dedicated multiplication protocol So this will cost interaction and and communication Still this way of building to body computation schemes As produced extremely efficient protocols, so it's a it's a quite efficient method But in in this work, we try to have some different approach and so our our core of the reaction was well We already have so additively homomorphic encryption scheme Which do allow us to evaluate homomorphically and locally any kind of linear operations without interactions communication, whatever And we also already have multiplicatively homomorphic encryption scheme, which let us evaluate Multiplicate multiplications Expansion and so on locally and homomorphically So what if we called some whole makes them work together and by makes them work together, I mean could we build some Encryption switching protocol Which is a protocol that will take as input a cipher text encrypting some message M and I'll put a cipher text encrypting the same message M, but with the other crypto system and The current security requirement for such a switch protocol is that during so this will be a two-party protocol And during this two-party protocol no players must learn anything about the message M The plain text M which is contained both in the input cipher text and the in the output cipher text And if we had such a switch protocol It's quite easy to see that we can easily have Secure the party computation for any function you write your circuit as layers of linear operations and multiplicative operations each time you want to evaluate linear operations you and you use Like pie encryptions of the inputs on you evaluate homomorphic. It is over this operations And each time you need to go to the next layer Then you just have to switch to multiplicatively homomorphic cipher text and you can keep on evaluating everything homomorphically and so on until you get an encryption of the output and so Okay, but what will be the benefit of doing so Because we already have as I said extremely efficient multi-party computation protocols based on Paying for each multiplicative gates. So here the advantage is essentially that You you won't have to pay for each multiplicative gates. So think think for example of your circuit Where at some point many multiplicative gates are grouped together You can't think for example of a circuit which at some point computes an expensation this involves many multiplications and Instead of paying in terms of communication for each of those multiplicative gates. He you will just have to switch to To a multiplicatively homomorphic encryption scheme evaluate everything locally and then switch back so essentially doing two-party computation out of Two complementary homomorphic encryption scheme together with an encryption switching protocols allows to have Two-party computation protocols with which might be sub linear in the size of the circuits If the circuit is well structured like multiplication gates are kind of grouped together and so on Okay, so I Said that we won't we would like to build such a non encryption switching protocol This is our goal in this work for the Pi encryption scheme and the L gamma encryption scheme. So let me present them in slightly more details so the pi acrypto system is a Semantically secure crypto system whose semantic security relies on the decisional composite residuosity assumption Which states that if you take an RSE modulus and the product of to save prime It's computationally infeasible to distinguish and spore from random elements over Zin and squared On the other hand, we are the air gamma crypto system whose semantic security rely on the decisional DfL man assumption Which states that you cannot distinguish Turbles of the form g g to the a g to the b g to the a times b from uniformly random to pose and Which which is multiplicatively homomorphic and the important thing is The biacrypto system allows you to encrypt any plain text over Zn while the Algama crypto system allows you to encrypt Well any plain text which belongs to a group over reach the decisional DfL man assumption is believed to hold And you might already see the problem here when I ask can we build encryption switching protocol The question that should come first is does that even make sense? Because my encryption switching protocol ideally will take an encryption of some message and outputs another encryption of the same message So it must at least be meaningful to talk about an encryption with the other scheme of the same message So we must at least have encryption scheme where the two plain text spaces Have some intersection and moreover we really need that the two plain text spaces are essentially the same because if it's not the case and Our encryption switching protocol could be used to distinguish during a two-part computation protocol between inputs on which We can switch we which does belong to the intersection of the plain text spaces from inputs on which we cannot switch Which would cause a failure and so this will leak information on the inputs of the players in our two-party computation protocol So what we would need would be to our first two crypto system an additively homomorphic one on the multiplicatively homomorphic one that operate on the same plain text space and Unfortunately while pay it does allow you to encrypt over ZN The decision of the film and assumption doesn't hold over ZN nor does it hold over ZN star the asset of invertible elements over ZN and so Our first task if we want even our goal to make sense Will be to design a variant of the L gamma crypto system which remains multiplicatively homomorphic But which will allow us to encrypt message over ZN so as to complement the bio-crypto system That makes sense Okay So we will start with a slightly Simpler goal which would be to design a male gamma like multiplicatively homomorphic crypto system over ZN star and So let's dig a bit into the structure of ZN star So you can see ZN star has been divided into four equal size parts So the two columns here correspond to element having Jacobi symbol one or minus one Why the two lines here corresponds to element having what I call sine one or minus one and The group of elements with Jacobi symbol minus one here is Jn. This is a larger subgroup over ZN star and So as our goal is to build an L gamma like encryption scheme, we need to have some hardness assumption to rely on So let's see what is hard about that over ZN star So not DDH as I we said before DH doesn't hold over ZN star But the decision only feel man assumption is believed to hold over the subgroup of squares Even if the players know the factorization Because it reduces to the decision only feel man assumption over the subgroup of squares of both that P and that Q Where P and Q are the factors of our models Moreover, it is believed to be computationally infeasible to distinguish between squares and minus squares over Jn This is the quadratic residuality assumption and so but this holds When players do not know the factorization it it's at least Easier than factorization Computing the Jacobi symbol is always easy Even if players do not know the factorization So the hard task essentially will be to hide this Jacobi symbol when encrypting the plain text from ZN star Because you can easily see that if we can encrypt with L gamma over the squares If it's impossible to distinguish squares from minus squares then essentially it means that if no players know the factorization We can use an L gamma like crypto system over Jn Everything works fine. The hard part is Encrypting and hiding the Jacobi symbol of the plain text So let's see how we will do it So To encrypt a message M we first add to the public key some uniformly random element with Jacobi symbol minus one so from the second column here and a generator G of Jn and With a decomposition of M will be a topo a M one So that a is a uniformly random value which will be even if M as Jacobi symbol one and Odd if M as Jacobi symbol minus one so that key to the A as the same Jacobi symbol than M And M one is the value that satisfies M equals key to the A times M one and you can easily see that M one always has Jacobi symbol minus one Because the product of two elements with Jacobi symbol. Let's say alpha beta is alpha beta. So So this M one part does belong to Jn So we can simply encrypt it over Jn using a standard algorithmal crypto system The hard part is hiding this key to the A and This is done as follow While it is easy to distinguish between key to the 2x and key to the 2x plus one This is just computing the Jacobi symbol It's believed to be infeasible to distinguish between our two lines here So we will encode this key to the A as G to the A So that key to the A as Jacobi symbol one is equivalent to G to the A being a square So from G to the A you cannot recover the Jacobi symbol unless you break the quadratic residue of this assumption and It's quite easy to see that this encryption Remains multiplicatively homomorphic because that one is multiplicatively homomorphic and that one we just change the basis So what remains to see is whether we can decrypt that because it looks like we might have to compute some discrete log here to decrypt So the intuitive way of avoiding to have to compute some discrete log would be to add Let's say the discrete log of key in base G to our secret key So as to read to construct key to the A from G to the A using this discrete log But this doesn't make sense as a key and G doesn't belong to the same group at all G is the generator of Jn key as Jacobi symbol minus one But what we do is essentially that mod p and mod q so We generate key by picking two uniformly random value t1 and t2 with opposite parities and We set key to be G to the t1 mod p and G to the t2 mod q And we reconstruct key mod n using the Chinese remainder theorem and Ask you want it to have opposite parities. This is guaranteed to be an element with Jacobi symbol minus one But now from G to the A using t1 and t2 we can construct key to the a mod p and mod q and again reconstruct key to the A Using the Chinese remainder theorem Why they think on part is quite easy to decrypt so we just We just reconstruct this key to the A decrypt the second part and recover the message M Okay, so this works. It's fine, but I promised that then For the moment we only have an encryption scheme over Z and star So a first step a very small step toward getting Z then would be to add a zero So that's what we will do first How to have and how to build a multiplicative a homomorphic encryption scheme over Z and star union zero and The trick here is quite simple Any element over then a star union zero will be encoded as a pair of element both over Z and star So that we don't lose the multiplicative the home the multiplicative properties and so to encode a message M and uncoding would be either M one if M is An M zero and on zero and if M is zero then our encoding will be random and random Why because then if you multiply 10 by 10 many such encodings Either the first part is what you're looking for the product of all the messages or at some point Some random value happen in the computation it and it's a uniformly random value that loses all information on the message And the second part of the encoding allows you to check whether this is a zero in which case you would have a Random value or it Would the All the messages that were multiplied were non-zero in which case the second part will be a one And so the what we do is simply we encode its message From that inside union zero as a pair of message over Z and star and we encrypt Both element a boss element of our pair with the scheme that we just designed over Z and star This is still not Z then but almost and by almost I mean that if no players knows a factorization Then it's computationally infeasible To find an element which will be in Z then but not in Z then star union zero Because any such element is a multiple of either P or Q and so finding such an element is perfectly equivalent to the findings of actualization of M and so It is possible to assume that no players will know the factorization By relying by using a threshold encryption scheme instead of just a standard encryption encryption scheme So in our construction Rather than using standard homomorphic encryption scheme We will use threshold schemes in which the secret key of the scheme is secretly shared between the two players so that each player individually has no knowledge on the full secret key and Then decryption of some message is performed by using a joint decryption procedure Which is a an interactive protocol which outputs the plain text containing the ciphertext and such Distributed description procedure are known for most homomorphic public encryption schemes and so That's all what we can do is simply with a threshold scheme instead of a standard scheme and we can just assume that everything will be exactly as if Our El Gamal scheme over Z then star union zero was in fact a scheme over Z then and we know that we have a very mean probability This won't cause any error in the protocol and this can be formally proven Okay, so now we we kind of have an El Gamal a multiplicative homomorphic variant of the El Gamal encryption scheme and We can assume that it works over Z then somehow So our goal was to be an encryption switching protocol out of that and so rather than presenting the full construction I will give a toy scheme Which will give you the intuition on how we can do that and in particular our tool scheme won't even handle the case where M is 0 But the core ID and this encryption switching protocol is If we have an additive scheme on a on a multiplicative scheme, they are likely to have very different algebraic properties However, they must share at least one common algebraic property Which is that in both schemes? You can have external multiplication either on the additive scheme by using a square on multiple algorithm or on in the multiplicative scheme by just like Encrypting your external value before and then performing the multiplication Okay, so so we have two players and they start they start with an additive encryption of some message M and To make the presentation simpler. We assume here that M is non-zero on the players knows that The first player picks a uniformly random value R and using the fact that it can do external multiplication on the additive scheme It multiplies are inside this encryption of M, which gives an encryption of R times M And he sends a multiplicative lyomorphic encryption of R minus 1 Then both players perform a joint encryption protocol using their shares of the secret key on The additive scheme think first of all of the paya scheme so that only the second players learn the result and As we assume here that M was non-zero this R times M leaks no information on M. It's a uniformly random value over the So with this R times M We can again use this external Multiplicate this external multiplication, but on the other scheme So the second player simply multiplies this R times M In the encryption of R minus 1 and sends back the result which is an encryption of M And that's it So that's a toy scheme you can see many problems here related to the fact that we would have to randomize thing, but okay And so what do we have to do next to solve all the other problems related to encryption switching protocol? Well, first we have to deal with the other direction which essentially requires to build a joint encryption procedure to the To the multiplicative scheme that we designed We need to extend the construction so that it under the case where the plain text is zero So that add some more technicalities in the paper Proformally that if we have all of that complimentary homomorphic encryption scheme and Encryption switching protocol we can evaluate any function That's quite long on technical, but there is nothing particularly exceptional with this with this and add security against malicious adversaries This is quite interesting because at some point it requires to be able to prove statements of the form two different Encryption scheme do encrypt the same message and no Efficient proof for such a statement do appear in the literature So we have to design one and it appears that this gives a new zero knowledge proof for various more classical statements, so this is a side contribution of the paper and That's all. Thank you for your attention


Yes. So before the break, we were looking at what the cross has done for us, the new legal status which is ours, because of what Jesus Christ has done on the cross. So the finished work of the cross has given us a new legal status and we need to be very, very aware that we are no longer under the law. We are under a new master and our new master by his grace, he enables us to live in a new and holy way. We saw that the power of Satan over us is broken. He no longer has any legal power and authority over us. We also saw that we have been redeemed and brought out from that kingdom of darkness into the kingdom of light and so the old laws of the previous kingdom no longer have any control over us. We have a right to assert ourselves in Jesus Christ and declare that now we are citizens of a new kingdom and we can command Satan and his minions to leave in Jesus' name. They have no choice but to vacate our areas, those areas of our lives that they are trying to dominate. So this is what we have in Christ because of the finished work of the cross. We will also very briefly look at the things that are said in scripture about this new identity that we now have because we are in Christ. Because of our union with Christ, what is our new identity? Our new identity is that we are a blessed people. We have been blessed with spiritual blessings and those spiritual blessings will help us to live a victorious life. So we have a scripture which talks about that. If someone could read out for us, Ephesians chapter 1, 3 and 4. Ephesians 1 verses 3 and 4, Blessed be the God and Father of our Lord Jesus Christ who has blessed us with every spiritual blessing in the heavenly places in Christ just as he chose us in him before the foundation of the world that we should be holy and without blame before him in love. So over here, we have been blessed it says with every spiritual blessing and the NIV rather than using the word just, it uses the word for. So why have we been given these spiritual blessings? Because he has chosen us in him to be holy and blameless. Now all these spiritual blessings have been given to us not just to make us rich people in material ways, not just to see that we always get healed from sickness. Those are also vital important parts of our new Christian walk but these things, these spiritual blessings have been given us so that we can be holy and blameless so that we can live an overcoming life. What are some of the spiritual blessings that we have? The spiritual blessing of being able to stand over there in complete freedom and make a choice to be able to choose. Shall I sin or shall I not sin? Earlier when we were slaves of sin, we had no choice. We had no say in the matter. It was a struggle but now we are free. We can actually stand over there in complete freedom and choose whether we wish to sin or not. It is a spiritual blessing, this freedom that we have in Christ. We have the spiritual blessing of being able to hear from Jesus Christ so clearly because he says, my sheep will be able to hear my voice. So when the time of temptation comes, he will guide us. He will tell us how we can escape from that temptation because we have the scripture which says that when temptation comes, he will also show us the way of escape. So we have the spiritual blessing of being able to hear from him. So in our time of crisis, he will tell us what to do, what to say and he will show us the way of escape so that we can overcome that particular temptation. We have the spiritual blessing of using his word, the word of God as a sword and that can be a very powerful weapon. So we have all of these spiritual blessings that have been given to us. Why were they given to us? Because God has chosen us in Jesus Christ to be holy and blameless. So now we are blessed.  In the same way, we also have been enriched. Let us look at a passage which talks about that. If someone could read out for us, 1 Corinthians 1, 4 to 9. 1 Corinthians 1, 4 to 9. I thank my God always concerning you for the grace of God which was given to you by Jesus Christ that you were enriched in everything by him in all utterance and all knowledge. Even as the testimony of Christ was confirmed in you so that you come short in no gift eagerly waiting for the revelation of our Lord Jesus Christ who will also confirm you to the end that you may be blameless in the day of our Lord Jesus Christ. God is faithful by whom you were called into the fellowship of his son Jesus Christ our Lord. So if we look in verse 5, 1 Corinthians 1 verse 5, the actual words used over there, the Greek words used over there are words that we are actually familiar with foreign him. You have been enriched in every way. How have we been enriched with all kinds of logos and with all gnosis that is G-N-O-S-I-S. So these are two things, these are two words that we are kind of familiar with already. It says we have been enriched with all kinds of logos and with all gnosis. Now the word ogos we know it is applied to words, it is applied to speech, it also talks about the word of God. So we have been given all kinds of logos, all kinds of words, all kinds of speech and in the translation that Rosalind read out it says utterances. So here it is talking about the spiritual gifts of word of prophecy, word of knowledge and word of wisdom. It is talking about that. It is also talking about the written word of God which is also another form of speech, another form of utterance. So we have been given all kinds of utterances, all kinds of words or speech. We have been given all kinds of logos and something else. We have also been given gnosis, g-n-o-s-i-s that is basically your knowledge. So he also gives us the ability to know how to handle each situation. So we have the spiritual gifts, we have the written word of God, we have the divine knowledge which is given to us on how to tackle each situation that we face. So it goes on to say in verse 7, therefore you do not lack any spiritual gift. So everything that you require to live a victorious life, it has been given to you. So we do not lack in any way and that is why it says in verse 8, he will also keep you firm to the end so that you will be blameless on the day of our Lord Jesus Christ. So God will use these, if we choose to use these giftings that we have, if we choose to use these logos that we have, if we choose to use these things, then through them God will keep us firm to the end. If we choose that we do not want to use them, then of course we will not be able to walk in victory. But if we are making use of all the giftings and logos which is given to us, if we make use of the knowledge which God imparts to us when we cry out to him and say, Lord help me, show me how to overcome this stronghold, how to overcome this area of weakness in my life. When we reach out to him and ask him actively for these things, it says he will keep us firm to the end. Why? Because it says in verse 9, God is faithful who has called you into fellowship with his Son. So the Lord will be faithful. He will enable us to live victoriously. This is something that we can have in him because he is very, very faithful to his word. He will enable us and that is why in 2 Corinthians 3-5, it says something very nice over there. If someone could read out, 2 Corinthians 3 verse 5. 2 Corinthians 3 verse 5. Not that we are sufficient of ourselves to think of anything as being from ourselves, but our sufficiency is from God. Alright, so look at the immediate context of this particular verse.  It's talking about how Paul and his leaders have been enabled by God to be good leaders. So they're saying, you know, we're not saying that we're good leaders on our own. God has enabled us. So that is why we have what we have. So that is, of course, the immediate context of the verse. But then if you were to take that verse more broadly, all the sufficiency, all the competence, that's the word that NIV uses. The competence that we have to be able to handle things is because God has given that to us. So, you know, in that sense, NIV says not that we are competent in ourselves to claim anything for ourselves, but our competence comes from God. He has given us all competency. So this is something that we need to remember. We have been made competent to overcome every single kind of temptation. So we don't have to say, I lack competency in this particular area. No, no, all competency comes from God. And when we reach out to him and we ask him, he is faithful. He will grant us what we require to overcome, you know, even those areas of sin and temptation. Another thing that we need to remember regarding our identity in Christ is that we are victorious. Second Corinthians chapter 2 verse 14. Second Corinthians 2, 14, if someone could read out. Second Corinthians chapter 2 verse 14. Now, thanks be to God who always leads us in triumph in Christ and through us diffuses the fragrance of his knowledge in every place.  So it kind of tries to make the meaning of that sentence clear by putting it in this way. This is what NIV says. But thanks be to God who always leads us as captives in Christ's triumphal procession and uses us to spread the aroma of the knowledge of him everywhere. Because you see in those times, in those ancient times, when you would have a procession being held by a king, that's actually a procession of victory. Where the king is, you know, walking ahead is come back from a battle and now is, you know, walking in a royal procession towards his palace. Okay, so he walks through the city to show off what he has achieved in that battle. So behind him are all these captives whom he has captured.  And all his nobles and commanders and officers, they're all, you know, now in chains. And this royal king was walking in the front, he's saying, see, I have won the victory. These guys, they're my captives. This is how powerful I am. So it's that kind of a procession. And so it says, thanks be to God who always leads us as captives in Christ's triumphal possession and uses us to spread the aroma of the knowledge of him everywhere.  But over here, you know, because generally speaking, in a triumphal procession, the people who are getting led, I have nothing to be happy about. They are now captives. They are now slaves, not a very nice place to be.  I mean, the condition that he is in, they actually put a hook, a metal hook in his nose.  So in 2 Chronicles, chapter 33 was 11, we see that he is a hook is put through his nose and he is bound in shackles of bronze. And in that condition, he is not taken away. So basically, in the natural sense, being a captive is something very horrible, very terrible. But here, in Christ's triumphal procession, we are captives who were in a very, very pathetic condition. We were like this Manasseh, but now we are captives who have been set free by our new master, Jesus Christ. So yes, we are still now slaves, we are slaves of our new master. But our new master, why did he set us free? Not to throw us in some dungeon and make our lives miserable, but so that he could restore our lives, so that he could rebuild us. So it's a whole different kind of a procession. It's like a king who has gone into battle and conquered all these people and they all were enslaved by slavery and misery. Now he's bringing them out, he has set them free. So yes, now he is their new master, but he's not brought them out to make them miserable, he's brought them out to build their lives. And so that is why in this procession, when these captives are being led in victory, they're not following Christ as defeated people who are now helpless. What to do now? They're under this new king. No, they're in fact walking with great joy, like royalty.  But now Christ has led us out and we are walking in this triumphal procession as conquerors, as people who have been set free. So that is why when this procession is going along, you don't smell defeat. You don't smell shame and humiliation. Rather, it says over here, the aroma of the knowledge of him is spread everywhere.  We go over there as Christ's captive, but we are not shamed and defeated or humiliated. We say, once upon a time we were slaves of sin, but now we are slaves of our new master Christ. And what is he doing? He's causing us to walk in victory. He's causing us to walk in spiritual blessings. We have been enriched with all kinds of utterances and knowledge. And so we, in fact, use our giftings among the people to help them, to be a blessing to them.  Being a slave of Christ has a very pleasant aroma. It smells very good. It's not the smell of defeat.  It's a very different aroma. So we need to remember that we are not following Christ as defeated, helpless people. But rather we are following Him as victors, as victorious people who are now, who now can live in a new and different way. And as we pass by, everyone smells this new life that we have and they think, wow. So this is what it is like to be living under Jesus Christ. This is the kind of life that He gives His slaves. So it's what they learn about us. So we have to understand that our identity is one of victory, of our heads being lifted up rather than being defeated and being shamed. If we can also look at Romans chapter 5 verse 17, if someone could read out for us, Romans 5 verse 17. For if by the one man's offence, death reigned through the one, much more those who receive abundance of grace and of the gift of righteousness will reign in life through the one Jesus Christ. Here it says that now that we are under Jesus Christ, you know, earlier of course we were under death and we were in a terrible condition. But now we are under Jesus Christ. And this is what we have received. It says who receive God's abundant provision of grace. Okay. So an abundant provision of grace has been given to us. Why? To reign in life. We are meant to be rulers, ruling in life. We are no longer defeated. We are no longer helpless. So our identity is that in our hands we are holding an abundance of provision of grace for every situation, for every circumstance, for every area of weakness. An abundant provision of grace has been given so that that area of weakness no longer needs to be an area of weakness. It can become an area of strength. Why? Because we are holding in our hands an abundant provision of grace. Okay. So these are all things, the realities that we need to understand and we need to start teaching our minds these things. Because our mind is not aware of these things automatically. It's something that we would have to teach our mind. And we renew our mind by telling it these things.  Your perspective starts changing. You start seeing yourself the way Christ sees you. And that makes a huge difference. So I have an example in mind but then I think maybe we could talk about that in another class. Just moving on to Hebrews 4, 15. Yeah, if someone could read out for us Hebrews 4, 15. Hebrews chapter 4 verse 15. For we do not have a high priest who cannot sympathize with our weaknesses but was in all points tempted as we are yet without sin. Yeah. So our new representative Jesus Christ who is very different from Adam our old representative. So this new representative of our Jesus Christ, it says over here who was tempted in every way. So all the temptations that we face, he too has faced all of those temptations but of course he was without sin. So he knows the strategy. He knows how every kind of temptation can be overcome. This is something that he knows and so he can reveal that to us when we go to him in the next verse. In fact, that's what it says. It says because he's that kind of a high priest. Go to his throne with confidence to receive grace and mercy in your time of need is what it says in the next verse. So this is a high priest who has gained victory over every kind of temptation and he knows the strategy how you can overcome all types of temptation. So when we go to him and we learn to live under him, abide in him to hear from him every day because as we spend time with him we begin to hear his voice more clearly. We become more comfortable in hearing from him. We won't be as spiritually deaf as we used to be in our early days as a believer. Now we can hear him better because he gives instructions. He gives important instructions on a moment by moment basis. So he is able to equip us. He is able to guide us and show us how to live in victory because he has already been through the process. And so he knows what methods we can use in our own situations to live in victory. And that is why in Psalm 1834, you know, this is what it says if someone could read out Psalm 1834. Psalm 1834, he teaches my hands to make war so that my arms can bend a bow of bronze. A bow made of bronze is not an easy thing to bend. See if your bow is made of wood and you pull back on the string and this bronze bow which is there, the frame of that has to get stretched that way and then this string gets stretched this way. So you would need to have a certain amount of strength to be able to handle a wooden bow. But when it comes to a bow of bronze, how much greater would your strength need to be? So it says, he trains my hands for battle. My arms can bend a bow of bronze. So you gain victory over your enemy. You take their bow and you're able to bend it. You're able to maybe even break it. So it's talking about victory over here. So sometimes when I'm facing situations which I'm finding very difficult to handle, I say, Lord, this is what you promised in your word. You said you would train my hands for battle. You said that I would be strong enough to take Satan's bow and break it in two. This is what you have promised me. So Lord, you help me now in my situation. So these are promises that we can claim for ourselves. Something very similar even in Psalm 144 verse 1. If someone could read out, Psalm 144 verse 1. Psalm 144 verse 1. Praise be to the Lord my rock who trains my hand for war, my fingers for battle. Okay, so almost the same thing is mentioned in Psalm 18 and Psalm 144. God is the one who trains our hands for war. He is the one who trains our fingers for battle. Of course, it's using Old Testament imagery over here of the bow and arrow and all of that. But today, when we are facing our trials and temptations, when we need to overcome and live victoriously, he is the one who trains us. So we can go to him and say, Lord, this is what you promised. You said that you will train my fingers for battle.  How Lord can I be victorious and the Lord, you know, the Holy Spirit, he will teach us. He will train us. He will show us how to live in victory.  You know, for instance, you know, the preacher would say, you know, you must learn to read the word of God daily and meditate upon it.  So some general principles are given to us from the pulpit on how to live an overcoming life.   How to be victorious in individual situations. So this is not something that you can do if you are, if your connection with the wine is weak. So it is very important for us to build our connection with him, to be strong in him. You know, the stronger the connection, the more we are able to hear, the more our hands get trained for war and the more victories we will enjoy. And now we will be able to break what both made of bronze, we'll be able to break them into two, you know, so we will have that kind of power and authority. Which is why it says in Ephesians. Yeah, right. Yeah. Okay. If you can read out Ephesians chapter two verse six, someone could read out Ephesians two, six. Ephesians chapter two verse six. And, and he raised us up together and made us sit together in the heavenly places in Christ Jesus. Okay, so we are familiar with this, right? So we're in the spiritual realm. We are seated with Christ in the heavenly realm, you know, which is very, very nice. But what should you be doing because you're seated over there in the heavenly realm? What should be your attitude if that gets explained to us in Colossians three, one to three. Colossians three, one to three. Colossians chapter three verse one to three. Since then, you have been raised with Christ, set your hearts on things about where Christ is seated at the right hand of God. Set your minds on things about not on earthly things for you died and your life is now hidden with Christ in God. So it's all very well to say, yes, I'm now seated in the heavenly realm with Christ. But then in your thinking, you still have the unrenewed thinking, you're still chasing after the things of the world. You're still, you know, longing and desiring for sinful things. Then it's pointless. It really doesn't matter where you're sitting because in your mind you are still very much here on this earth. So it's not enough that you are that your spirit is seated with Christ in the heavenly realm. Maybe you should also take your mind over there and ask it to join the spirit. So in our thinking, we need to start desiring the things of God. We need to set our affections on divine eternal things rather than this temporary things, you know, after which the world runs. So rather than focusing on just temporal pleasures, we choose to set our desires and our hopes and our ambitions on eternal things. So if this basic fundamental change doesn't happen in our thinking, it's going to be very, very difficult to live a victorious life because your unrenewed mind and your flesh will continue to crave and long for the things which the world, you know, longs for. And so even though your spirit is seated in the heavenly realm with Christ, it will be of no use. You got to start taking your mind also to that level and say, you know, tell your mind, you know, you are now a new creation. And this is what the Bible says about you begin to start understanding this and begin to start living it. So you who are seated in your spirit in the spiritual realm with Jesus Christ need to drag your mind off from this earthly level and tell it kindly start aligning yourself with scripture, with what scripture says about you.  And that is why look at the kind of wording that is used in Romans 16, 19 and 20. Romans 16, 19 and 20, please. Romans chapter 16 verse 19 and 20. Everyone has heard about your obedience. So I'm full of joy over you, but I want you to be wise about what is good and innocent about what is evil. The God of peace will soon crush Saturn under your feet. The grace of our Lord Jesus be with you. All right. So here there's something very practical that, you know, Paul is saying to the Roman believers is saying, be wise about what is good and innocent about what is evil. So, you know, you're now sitting in this, seated in the heavenly realm. Start focusing on the things of the heavenly realm. Start focusing on the things of God. You know, become wise in those things. Get more and more knowledgeable and, you know, get more and more wisdom regarding those things. Don't keep chasing after the earthly things. Rather, when it comes to the earthly things, be innocent about them where you're, you know, ignorant. Where you don't, where you almost don't even know about those things. No, because there are people, you know, who are living with the Lord in such a wonderful way. You talk to them about certain sinful things. They don't, they didn't even realize that such things existed on the earth. They're like, oh my goodness, such things are also going on. Is it? They're so innocent about what is evil. And they have been growing in their wisdom regarding the godly things of the heavenly realm. And that is the way we should be, you know, rather than setting our affections on the things of the world and being so knowledgeable about all the sinful things of the world. Rather, we should be people who have become more and more, you know, heavenly minded in the sense that now our interest lies over there. Our ambitions are, you know, in reaching those heights.  So we are very, very wise in those things. And we are like almost innocent about evil things. We don't even know about these sinful things, you know, let alone indulge in them. So that is the kind of attitude that we should begin to develop. Because when we do that, verse 20, very important, the God of peace will soon crush Satan under your feet. Because you see, you're someone who has been focusing on the Lord. You have set your affections on the heavenly realm. You are getting wiser and wiser about things which are good. And it comes to evil, you're innocent. You almost know nothing about these things, about these things which the world does and the world is running after.   He will, you know, because we already saw that, right? Satan has no power over us. So Satan, so God will have Satan crushed under our feet. You know, he will give us the strategy on how to overcome sin. We looked at all of that in the previous verses. So even as we are using our spiritual giftings, even as we are depending on him, even as we are abiding in him, he will show us what to do. He will see to it that Satan gets crushed under our feet because we are wise in the things of God and we have chosen to stay innocent regarding evil things. So we are in a position of power. On the other hand, if we are very knowledgeable about the world and the evil in it and all the different kinds of sins and, you know, that is our speciality, that's our specialization, then we are in a position of weakness because we know very little about the heavenly things. We know so less about the things of God. So our hearts have not got excited about the things of God because we don't even know about those things. How are we going to get excited about them? And when we are living in that way, it is so easy to get swayed by Satan. Forget about Satan getting crushed under our feet. He will crush us, you see? So it is so important for us to be in a position of strength. That is, we choose to be very wise about the things of God and when it comes to the things of the world, we have such little contact with the world that we are in fact not even aware of certain of the things which are going on. As far as the evil deeds and the evil sinful actions of the world is concerned, we stay innocent regarding such things. I'm not saying that we should be ignorant about current affairs. We need to be very aware of what's going on in the world because only then we can pray and intercede. We can use the word of God against the things which Satan is doing in the world and break those things. So yes, we should be very aware of current affairs and what's going on in the world around us. But when it comes to sinful deeds, sinful actions, the things that people of the world indulge in, we should maybe stay very innocent about such things and not even be aware of them. So that is the good place of strength to be in because then this is what happens for us. John 14.30 where Jesus is talking about himself. What does he say? John 14.30. John chapter 14 verse 30. I will not speak with you much longer for the Prince of this world is coming. He has no hold on me. So the Prince of this world Satan had no hold over Jesus. Why? Because Jesus was wise in the things of God. He was innocent when it came to evil and because he was in that position of strength and he never gave into temptation and sin, the evil one had no hold over him. So when we start living like Christ, we are not only seated in the heavenly realm in our spirit, but even we are renewing our mind and causing our mind to dwell on spiritual things when we are in that position of strength. Satan starts losing his hold over us. He can't control us like he used to earlier and then he will find himself getting crushed under our feet on a daily basis because God will do that for us. He will enable us. He will help us. So these are things that we need to remember regarding our identity in Christ. Now moving into this next section, using the word of God, we all are very familiar with this. We know that we are supposed to use the word of God as a spiritual sword to fight Satan and we also always refer to Jesus Christ, how Jesus Christ when he was tempted, he overcame temptation with the word of God. But just to kind of dwell upon aspects of this which we generally don't seem to be talking about, so we will try to focus more on those areas which are kind of neglected regarding this usage of God's word. So let's get started off. We don't have much time, so let's kind of rush a little bit. 1 John chapter 2 verse 14, if someone could read out. 1 John 2, 14. 1 John chapter 2 verse 14, I write to you, Father, because you have known him who is from the beginning.  Okay, so here it's talking about spiritual maturity. Okay, so when it says fathers, it's not talking about people who are older in age. It's just talking about people who have been in God so long that now they have reached a level of spiritual maturity where they are mentoring other people. They are being spiritual fathers towards other people. So it's talking about those who have kind of arrived in spiritual life. They've reached a very high level of maturity. But the beautiful thing is what he says about these youngsters, the ones who are still growing in the Lord, and he says something so, I mean the highest compliment that you could possibly give a person who's still growing in God. He says, I have written to you young men because you are strong and the word of God abides in you and you have overcome the wicked one.  The word of God is abiding in them and because the word of God is abiding in them, they are strong and they are able to overcome the wicked one. These are the kind of believers that we should be imitating that we should be emulating. So what does it mean over here when it says that the word of God is abiding in them? It's like the word of God has taken permanent residence in them. These are people who are really renewing their minds on a daily basis. The scripture is like it's there in their hearts. It's there in their system. So whenever there's a situation that needs to be handled, it's the word of God which comes to their minds first. They approach the situation in a biblical manner, in a word manner, in accordance with the word of God. So because these people are literally having the word of God abiding in them, dwelling in them on a permanent basis, these people are strong and they are able to overcome the evil one. So there are three simple things that we too can do so that the word of God abides in us in this powerful manner. The first one obviously is that we would have to feed ourselves with the word of God. And this is something that's taken for granted and so we all know that we need to spend time in God's presence reading His word. And even as we are feeding our inner man with the word of God, our inner man becomes stronger. This is something that we are aware of. But let's look at this in slightly greater detail. If someone could read out 1 Peter chapter 2 verses 2 and 3. 1 Peter 2, 2 and 3. 1 Peter chapter 2 was 2 and 3. If indeed you have tasted that the Lord is gracious versus 2 and 3. As newborn babies desire the pure milk of the word that you may grow thereby. If indeed you have tasted that the Lord is gracious. So it says like newborn babies crave the pure spiritual milk. That's the wording that is used in our NIV. So we look at this in slightly greater detail. I think all the things that I wanted to say for this particular chapter are not going to happen this class. So yeah, no props. We'll just continue next class. But at least this first one bit. Let's kind of grasp that in the little time that we have left. It's talking about newborn babies. The way they crave. The way they desire the spiritual milk. The way they desire physical milk. We should also be craving for spiritual milk. A newborn baby when it first comes out of the womb. It doesn't know anything about milk. Because you see when it was inside the womb it was not having milk. So a newborn baby when it first comes out. It's not even aware of such a thing as milk. But then gradually as the days go by it kind of begins to realize that there's a direct connection between milk and that uncomfortable feeling inside going away. So whenever that feeling of that hunger comes it realizes oh if I open my mouth and cry loudly then they give me milk and when they give me milk that empty feeling inside goes away and I feel satisfied. The baby begins to understand this slowly. Once the baby gets the hang of this idea you know what happens right? Then day out the kid screams to the top of its lungs. And you see this is not just a desiring the milk. It's literally that's the actual word used in your Greek. It's that word epipotheo which means a deep craving. So a baby the way it craves for the milk the way it screams for the milk you know when it wants the milk. That is the way we are supposed to crave the spiritual milk the word of God. Why do we crave in that way? Because we have tasted that the Lord is good. This baby which knew nothing about milk kind of got the hang of it. You thought ah this is good stuff. Whenever I have this stuff I feel satisfied. And so now that the baby has understood that milk is a nice thing it craves it. I mean it doesn't get it just screams. And we are supposed to be like that. Once we get a taste of the Lord and how his word feels what his word can do for us. You know how it draws us closer to him into his presence. Once we have tasted that we will begin to crave it. We will long for it. And that is something that we would have to consciously cultivate in us. There are some very very important practical things that I wanted to say regarding this matter. But then with two minutes let's not even go there. We will talk about that next class. But there are some things that we are neglecting to do regarding this whole thing about feeding our inner person with the word of God. And I see this a lot of people doing that and it kind of affects their being able to live a victorious life. So we will talk about that next class. So please do log in next class. For today we will just end with this that at least we should be having a craving the deep craving for the word of God and that craving starts once we taste and see that the Lord is good. If that has not happened for us we will not even feel any craving first of all. So yeah. So we kind of will just end on that incomplete note but we will of course resume next class once again. So yeah. Let's just close with the word of prayer. Thank you a lot for all the things that we could see from your word today. We pray a lot that we will always remember the finished work of the cross. Just like those believers in Revelation who were able to overcome the evil one by the blood of Christ and the word of their testimony we pray that we too will be people who will always remember the blood of Christ the finished work of the cross and how that has placed us in a place of victory how we now have a new legal status in you we pray that we will always remember that and we pray a lot that we would declare that as our testimony and by declaring it we pray that we will overcome Satan and we will overcome temptation and that Lord we will be able to live in a way which honors you. You enable us a lot. You a lot turn these lessons and this theory into practice on a day to day basis for us so that we can really be people who are honoring you. You do that for us a lot. Thank you a lot. In Jesus name. Amen. Amen. Thank you so much. And yes, we'll meet again next week. Thank you Pastor. Thank you Ma'am so much. Thank you.