So I'll present today a few results I obtained while being at Janelia Farm in Virginia on how a few animals represent the sense of direction at the neuronal level. So recently I moved to Marseille, but that's the result I obtained in Janelia. So the first example I'd like to show you has been already introduced by Reiner. Sorry, on the left or on the right. So in this example, there is no predator, but still I'll summarize it quickly because I think it's fascinating. So the nest is here. And the bumble bee is released in the wild. And the experimentalist has placed five flowers, fresh flowers, in the environment. And the bumble bee here in blue is the first strip. Explore the environment. Sorry, that's a bit annoying. And so you see that the first strip is a bit random. It explores all the different places. And then in black, it's after 25th trips only. It manages to find the optimal path through the five flowers. And then after more trips, it still does some random exploration. But I find it fascinating because it means that the animal, which is an insect, managed to have a long-term memory of the places of the flowers and managed to do some computation on it with only 10 to the 5th to 10 to the 6th neuron in the brain. And another example from the desert ant. So also this desert ant managed to go out of the nest, also explore to forage, find a source of food. And it's very hot outside. And it managed to path integrate and go back on a straight line to the nest. And it's a desert. And in this case, there is no real visual landscape. And so it's really that there is a sense of direction in the animal that make them go to the nest. And so this is shown by placing obstacle on the trip back. So you see that here with the food, it tries to go in the direction of the nest. So it cannot see it. It goes against the obstacles and goes around it. Again, try to go directly, explore, and then go back to the nest in straight time. Also if you have question, please don't hesitate. So it means that really these animals manage to somehow integrate angles, also distances, but at least angles. And so I have to again change the animals. So I'll switch to the fruit flies, the Drosophila, because it's the animal where we really have the best genetic access to the neuron. And so the other experiment is the following. So it's the equivalent of the Maurice water maze that people do for mice, for flies. So this is performed also in the group of mycorrhizae at Janelia. So you have a Peltier array here in this arena so that the temperature can be controlled everywhere on this arena. But at this, and these flies, they prefer cold spots. Okay, so naturally they will tend to, they will try to stay in this square here. But at the same time, you have a visual arena around the arena and with some features. And so in the experiment, when the landscape is turned, the cold spot is turned as well, so that the animal can in principle associate a visual landscape to a particular place. Okay, and that's what it manages to do. And so here, this is the experiment. So you see a few flies. So there is no social interaction, but you will see that it's just better to see more at the same time. And so here is the visual landscape and the cold spot. And so you will see what happens. And here is the mean time to reach the target. So you will see first it's run down. Then they go to this spot. Okay, so now it turns and goes here. Some are more lost. I mean, it's like we could make some anthropomorphic comparison. And so you see that almost all of them are quicker and quicker. And so the time to reach a target decreases so that they really manage to associate visual landscape to a position. And what is great with the flies is that you can really do, you have good genetic access, and so you can screen many of the neuron. So you can inactivate some of the neuron and see the phenotype on this behavior. So that's what has been done. And they saw that to perform well the task, the flies need neuron in the mushroom body, which is a place for associative learning, but also in a place central to the fly, which is called the central complex, which has a ring-like structure in the middle here. And so the idea is now, if we want to understand how the circuit works, we will try to image the neural activity in this neuron during a task. And so yeah, so this organ is really like a central structure in the middle of the brain, something that is a place of consciousness for the flies. Anyway, I won't go that far. And in this structure there are about 3,000 neurons and again we have good genetic access to it. And the experiment is the following. So there is a fly tethered walking on a ball while we image with a two-photon microscope, some of the neuron in the central complex. And there is a possibility to display again a bar or a visual landscape around the fly. And that's what they see. So in this particular experiment, the fly is totally in the dark. So there is no absolute cues for a direction and it's walking on its ball. And so here you see the calcium activity in this central complex. And so you see that at the beginning there is a localized activity on the ring at this position. And so you will see what the fly does. So it's walking on the ball and okay, maybe I can let you observe what's going on. So you see that the fly walks in the ball. So what happened is that with its leg it manages to turn the ball. And when it does that, actually the bump of activity turns around the ring. And quantitatively in this graph, this is the, in blue, this is the accumulated rotation of the ball, okay, as a function of time. And in red, it's the same thing for the ring here. And we see that it agrees very well so that the bump actually follow the rotation of the, I mean, relative to the external environment without having any external cue at any time. So it's really integrating small movement of the ball. So from this, there is really two questions for me. First of all, I mean, there is no external cue to the fly. So how can you generate a persistent activity for such long time scales? Several tenths of seconds. And also how is the activity rotated around this ring? And so for the persistence of the bump, I mean a few models have been proposed before actually in mice whereby electrophysiology people observed at a single cell level some similar activities. But so they observed that for some neurons if you record the activity while the animal is turning its head, it will be active only when the animal has a particular orientation direction with respect to the environment. And so to model that, there is a typical recipe that you can use to generate a local activity. It's so you place a few neurons on a ring. So I mean on a ring because the connectivity of the network is supposed to have circular symmetry, but nobody really imagined that you had a ring in the brain of the mice. And then you connect as follow. You excite your neighbors and you inhibit at long range. And this way you can with a good nonlinearity, you can generate localized activity. So that's how you can model that. So you have an activity F as a function of time and as a position T down the ring, and you can couple it with this and with a nonlinearity phi and a convolution product for the connectivity so that you respect the ring symmetry. And so we'd like to check if that's really the mechanism at play in the fly. And so to do that, you need to perturb the activity of the neurons. The thing that was not possible to do in mice. So here is the ring, okay? And you see the chemograph here of the activity. So along the ring, so the position T times unwrapped here and you have the activity as a function of time. So the bump here is here first. And then there will be for a short amount of time an optogenetic stimulation of some of the neurons. So you will see here, so you have the bump here, then some optogenetic stimulation. And then the bump here, as you see, remains as a new location even after the stimulation. So it means that this activity here is not a mere readout of another more internal activity that we are really touching, the circuit that is responsible for the persistent activity. And then we can study it a bit in more detail. So we can explore potential connectivity of the network. So people have looked before our cosine connectivity where you excite broadly and you do any bit also at long range quite broadly. That you can also study analytically the behavior of the bump in such a model. But there is another possibility that you have actually connections that are extremely local and a flat inhibition at long range. So here this model is extremely local. And again, only the inhibition is long range and totally flat. So we explore these two possibilities. And so here is a model of how it works. So this is the equation. And you see that in this model, which is local, you still, thanks to the diffusion term, you manage to get a broad bump here. And here there are some input, but you don't need input. And again, here is also the chemograph. And so you see that you switch your position of the input, then you can switch your position. And once it disappears, you keep the same, I mean, a persistent activity similar to what we observe. You can quantify a bit of the network as a function of the parameter alpha, the diffusion D, and the strength of the inhibition gamma. And for some of this parameter, you have a bump that is here. And then you can study how you switch the position of the bump. So here, in black, you have a bump here at zero initially. And then you put some localized excitation at pi, and you try to see how you can make the bump switch position. And you record the strength of the input so that you make the bump switch. So there's a minimal input you need to make the bump switch. And you compare the amplitude of the bump with the stimulation, the limit stimulation that you need to make the bump jump, and the bump without any external stimulation. And this is what is represented here. So for most of this diagram, you find 1.4. So something not too different. So we made these experiments with optogenetics, and this is what we obtained here. So this is a distance from the bump, and we see that, first of all, the strength that you need to input to make the bump switch doesn't depend on the distance, so it's flat. And also, it's comparable to the values that we predicted. So we think that we really have a model of local interaction for this bump, and not something that was broad connectivity as was proposed before. But so now we'd like to understand a bit more how this bump rotates around the ring. And again, there was something already proposed for mice. And the circuit is as follows. So you have the main circuit with neurons, the ring circuit as shown before, but you connect it to two other rings. So one ring here that connects to the right in blue, and one ring here that connects to the left, okay? And the way to make the bump move is to stimulate one internal ring or the other, okay? I'll explain that better with the example for the fly. So in the fly, what is great is we have access to many of the neuron. We have access to their connectivity, also genetic access so we can measure the activity of these neurons very precisely. And indeed, this was published a bit, I mean, some years ago also at Genelia. There are at least one other cell types that looks a bit like the internal ring of the model I presented before. So this neuron here that I call the central ring neurons is the neurons that were measured before in the experiment of the optogenetic switch. But there is an additional neuron that connects as follow. I mean, so this is really how this is positioned in the fly. The connectivity is simplified here in this diagram. So you see that you have a ring of central ring neurons. They connect to two other lateral rings that I call left ring neurons and right ring neurons. And these neurons, they connect back to the central ring neurons. The left ring neurons they connect here to the right and this one they connect to the left, okay? So a bit similar to what was proposed, but just notice that here in this model there are no internal connection to the central ring neuron. So only the interaction between these two rings and this one is actually sufficient to generate them. So that's what is simulated here. And so, so again, this is a chemograph and this is a central ring activity that will move as you stimulate one left ring neuron or the right ring neuron. So you see that there is an imbalance between the two side. You manage to move the bump in one direction or the other. And just notice that there is a phase difference between these two activities. The activity in this ring or the activity in this ring. And so that's what we quantified a bit more precisely. So these are the bumps. And so if you look at this bump here and you stimulate this ring stronger, you will see that the input to this ring will be stronger on the right side compared to the left side. So basically the input to the ring has a phase advance compared to the bump. And now if you compare, if you look at the neurons here, it's actually the contrary. The neurons here have an advance to the activity in these neurons. And so you have opposite phase if you measure, if you compare the activity of the inputs and the activity in this ring or in this ring. And so that's what is plotted here. And the experiment was done and that's actually by two color calcium imaging. We managed to show that the phase are exactly like predicted by this model. So we think that this is really the architecture of the neurons that allows to do the integration along this ring. And one thing that is predicted by the theory but which was not really measured in the system is the fact that is a relationship between the input rotational velocity. So basically you give an input to the circuit and you try to see how, what is the speed of the rotation of the bump itself, right? And for a good integration, you need to have an agreement between what you give as the input and the rotation of the bump itself. And since the neuron has a finite time scale, as it leaves a bump, cannot travel at infinite speed, right? So we know roughly the time scale of the neurons and actually predicts a saturation of the velocity of the bump of about 200 degrees per second, okay? So anyway, it cannot go faster than this. And this architecture, but this is non-trivial. It's not true for any architectures, allowed to have a good agreement between the input and the bump velocity for a large range of velocity until the saturation. So we think that this architecture actually performs very well.  Okay, so this is it for the fly. I don't know if you have questions at this point, maybe it's good, but otherwise I'll continue. So I tried to extend this to navigation in 3D. So flies don't seem to rotate 3D in 3D when they fly. They seem to be still horizontal, except maybe I take off and landing, but during their flight they seem to travel quite horizontally. And from the circuit, it doesn't seem that, I mean it seems that they only integrate angle in a 2D plane, right? But maybe for some, from other animals is not the case. And here is for a bat. For a bat you see that they have a very complicated maneuvers. So you see that their head explore many angles in 3D. And that's also true during flights, right? And then if these animals are also able to angle, integrate, then mean that you need a more complicated model. And indeed in the group of Narum Ulanovski, they show that some neurons, some head direction cells in the brain of the bat are actually tuned not only to the 2D azimuth, but also to the pitch and the role of the head. So this is a parameterization in Euler angle that we don't like too much, but it shows that at least you have localized activity for these neurons in this space. So then how to generalize this 2D bump attractor to this 3D rotation. So there is no real theory for that. But so if we want to have a continuous attractor, I mean a bump of activity in this space, you need to understand really what is the space of all the 3D rotation. And it's not a completely trivial space, I mean also it's well known. But here is a bit, it's structured that I show here in what is called axis angle parameterization. So imagine an object or it can be a head, but it can be anything else that you rotate in 3D. So the way to obtain any kind of rotation of the object is to turn the object around an axis that is given by a sphere in 2D and an angle that is given by, I mean a number. And so this is parameterized as follows here. The direction, I mean the center here represents the reference orientation of the object and the direction from the center gives the axis of rotation and the distance from the center gives the angle of the rotation. So if you see here in this example a rotation around the z axis, okay. So you see that the object turns here. So it goes up and then now if you rotate in the other direction, it goes down. But if you noticed well this object here is exactly in the same position as the previous one. So that actually this point on the ball and this point represent exactly identical rotation. So the structure of all the rotation is actually a field ball whereas the point that are opposite on the external sphere, sphere are identical, right. So we call them antipodal points. That's a bit how you can embed this space in 3D. And when you represent some of the tuning curves, localized tuning curves of the bat in this space, you obtain some activity like this where you have a small bump in this ball. So we'd like to build a continuous attractor for this network. And so the question is if each neuron represent a particular orientation in 3D, so you will feel the, I mean each neuron will be associated to a position in the ball, I mean parameterizing all the rotations. And you need to understand how you will connect the different neurons in this space. And since these neurons are on the sphere represent identical rotation, you need to also connect them. So you'll try to have local excitation and long range inhibition, but close to the border, you will also excite the other side, right. And so I'll summarize a bit how this continuous attractor were obtained in the ring. So one way with this cosine coupling that is shaped like this is you have a kernel here that you integrate with the activity. And so one way to decompose the kernel is to decompose it in a 4-year series, right, right. And what is nice in the 4-year series is that these harmonics are what is called ideals with respect to the convolution product. So what does it mean? Is that if you convolve a J0 plus J1 cos theta with any kind of function, you will obtain a function that is some constant plus another constant time cosine theta plus sine theta, right. You remain in the same space. And since here we use linear rectification non-linearity, it means that we solve exactly the stationary state here because here we know that we'll have a cosine function and so this will be just a three-shoulded cosine function. So it's easy to solve this system. And if we want to do the same with the space of all the rotation, you can integrate on this group of rotation and you also try to do the same but you try to find a good connectivity on this space. And to respect the symmetry, you can also write it as a convolution and so the connectivity between J and the rotation J prime will be a function of JJ prime minus one, like a good convolution. And actually you can also write Fourier series in this space and you decompose your kernel in this series, first a constant. And then this is a bit more complicated than on the ring but not so complicated. So the first harmonic is written as a trace of a matrix J1 times the rotation matrix of your element. And so again to have a bump, you can stop at this term at the constant plus first harmonic. And then you show that you have steady state. So you use this function and you have steady state and the space of steady state is a bit more complex in the ring, more diverse. You can have bumps of activities but also lines. You can't really have planes but for some reason they are unstable. But you have a greater diversity and so they are interesting to study for themselves. And we can try to compare these tuning activities that we obtain that are symmetric with what was obtained experimentally. And actually if we represent the symmetric states in the parametrization of Euler angles, you find a great diversity and actually they compare quite favorably with what you can find in the experiment. So that's a good sign. And recently, I mean even more recently, similar head direction cells in 3D were found in rats and also in mice. So you make the animal walk on vertical walls and you look at the tuning of some cells when the animal rotates on this plane or on what is called the west wall here on the side or on the other side that they call east wall. And you have, you record the tuning of the cell in the plane of the wall and you observe that the direction shifts as they change wall. And that's also what we manage to have with this parametrization of 3D. And that's non-trivial prediction in some sense. And so now we'd like to understand how we can integrate angles in this 3D space which is not trivial because as you might know, 3D rotation is a non-commutative structure so integrating in it is not necessarily trivial. So what we did for the 2D ring is to connect asymmetrically two side rings to a central ring. So we can do something a bit similar for 3D rotation. So instead of having one 3D bump attractor, you copy them several times and since this manifold is a 3D manifold, you need to copy them as much as the number of dimension. So you need six copies and you also couple them asymmetrically. So this bump here will couple a bit on the side and this bump will connect on this side so that if you have a higher input on this ring, you will tend in this space to go to the right. And that's what happens here. If you couple it, you manage to make the bump travel here. And since the whole connectivity is invariant by translation in this space, I mean in the space of 3D rotation, you are sure that you will take care of the non-commutativity of the group and integrate correctly the angles. And so that's what animals perform for navigation, but it's also related to a nice psychophysics experiment that was done in the early 70s. It's a fabulous experiment by Shepard and Messler. And so it's really for humans, so it's psychophysics and the experiment is the following. You ask a subject to identify a rotated version of an object. So you ask, is it the same object or not? So you can try for yourself. So for this one, I mean, are these two objects identical up to 3D rotation? So yes, I see. And for this one, no, OK. So there is a bit of variance of the reactivity time here. It's late and we are close to the break, so I'll try to be quick. But in reality, actually, the variance is very low. And what is fascinating is that the reaction time just depend on the angle of the rotation between the two objects and not the axis of the rotation. So it's really invariant to the direction of the rotation but only the angle. So it really means that the parametrization of the rotation in our brain doesn't seem to be like Euler angles but really like direct 3D rotations. And so we have a simple modelization of that. I mean, it's a bit trivial, but still it's really trying to show that in our brain we really have implementation of this rotation in 3D. And actually, when you ask a subject, I mean, how did you identify the object and maybe you tried yourself, you really tried to rotate the object in your head, right? So that's probably why the reaction time scales linearly with the angle. And so here, OK, we suppose that our object here is in one place in the space of 3D rotation. You have an object to another position. And so when you display the other object, the bump will travel to the new location and you can compute the time it takes for the bump to reach a new location. And this scales almost linearly with the distance between the bump. And if you do the same with Euler parametrization, the variance is huge because it depends on the point where, I mean, on the particular orientation of the object and it doesn't seem to be linear. So that's something that could be also studied at the neuronal level, but that's working in progress. I mean, in the mice, if you are curious, I can tell you about that. But I'll just thank you, my collaborator. So as I told you, I moved recently to Marseille. So the lab really is in its infancy. But so the work on the fly central complex was performed by Tsung-Suk Kim and then Evan Steerner and Stephanie Vegener in the lab of Vivek Jayaraman. And for the 3D head direction cell, I work with Alon Rubin from the Vahid Main Institute and Sandro Mani at Janina. So we have a few exciting projects starting in the lab. I mean, it might not be as nice as in Trieste, but it's still a nice place. It's not so far from the Calanque in Marseille. And we have PhD and postdoc positions available. So I'll stop here if you have questions. Thank you.


Hi everyone. Today I'm going to talk about our work where we construct new non-interactive zero knowledge for NP based on travel hash and NPN assumption. Maybe the bottom line and consequence of such a result is the first music for NP from NPN and EDH, which are both standard assumptions that were not known to imply this before. This is a joint work with Svika and Venkata and let's begin. I'm going to start with defining NISIX. So an interactive zero knowledge for an NP language is simply a proof system where both a prover and a verifier have an access to an honestly generated CRS. And the protocol is non-interactive, so it consists of a single message that the prover sends to the verifier. And we ask that the verifier is efficient, of course, but we also want that the prover is efficient given an NP witness W for the statement. And we consider the standard completeness, soundness and zero knowledge properties. So NISIX are interesting by themselves, but they also have lots of applications in crypto, including CCA security, signatures and new applications in the region of the protocol. So since the ultimate goal is constructing NISIX from standard assumptions, let's quickly go through the known approaches from the literature that do that. So the textbook approach is called the hidden bits model, and it can be instantiated using either type of mutations or VRFs or VPARGs. But this approach is a bit limited since the only known constructions of such building blocks are either based on factoring related assumptions or assumptions over bilinear groups. Another way to get NISIX was discovered by Roth, Ostrowski and Sahai, and it works in the setting where we have a bilinear group, and we're assuming some different style assumptions over it. But this approach is very non-generic. It uses properties of such groups, so it doesn't seem that NISIX from new standards assumptions will come from this direction. The last approach I'm going to discuss is an approach where we start with an interactive zero knowledge protocol, and we use the Fiat-Chamille transform to turn it into an uninterrupted protocol. And in order to instantiate the Fiat-Chamille transform in a provably sound manner, we use a primitive or correlation-interactable hash. And a long and nice line of work has eventually led to a construction of correlation-interactable hash under the LWE assumption. So to summarize things up, we have NISIX under factoring related assumptions or assumptions over bilinear groups or LWE. In our paper, we extend the set of standard assumptions from which we have NISIX, and we follow up on this last approach and instantiated assuming a tabular hash, which in particular can be constructed from the edge, and jointly assuming the LWE assumption. So how do we really get NISIX from Fiat-Chamille? So we start with the base protocol, which consists of three messages, and we assume it's a public-point protocol. So in the beginning, the prover sends his first message A, and then the verify replies by a random challenge E, and then the prover sends his third message, and the verify either accepts or rejects. So the Fiat-Chamille transport using a hash function edge looks as follows. So we add a randomly sampled hash key to the CRS, and now the prover computes his first message A as before, but in order to simulate the verify challenge, he uses the hash function. So now the verify challenge is computed as the hash of A under the key which is found in the CRS. Once he computed the first message A under challenge E, he can continue and compute the third message Z, and then he sends the entire transcript to the verifier at once. So it's easy to see that this protocol is interactive, and also that if we start with a public-point protocol, then the non-interactive protocol is also complete. We also know that if the base protocol is honest verifier zero knowledge, then the Fiat-Chamille transform preserves zero knowledge. So the only missing piece in the puzzle here is soundness. So it's not clear that even if the base protocol is sound, that the non-interactive protocol is also sound. And the reason is that the prover has some control over the choice of the verifier's challenge. And in particular, he can choose a first message A such that the verifier's challenge H of A falls inside the soundness error of the protocol. So this leads us to consider a specific kind of base protocols which we call sigma protocols. So sigma protocols are free message public-point honest verifier zero knowledge protocols which also enjoy this unique bad challenge property. And this property basically says that for any first message A, there exists at most one value of E that may possibly allow a cheating prover to cheat. And we call such value of E the bad challenge E. So more formally, a fixing a CRS and X which is not in the language and the first message A, there exists at most one bad challenge E for which there exists a transcript A, E, Z that the verifier accepts. So this property allows us to define the bad challenge function which is defined using a CRS and X and gets as an input a first message A and outputs the corresponding bad challenge E. And now we can claim soundness of the non-interactive protocol as follows. So if we can say that it's hard for the cheating prover to find the first message A such that H of A is the bad challenge, then it's computationally hard for a prover to cheat. Because the only way he can cheat is by choosing the first message such that H of A is that bad challenge corresponding to A. And a hash function that satisfies this hardness requirement is called a correlation intractable hash for the bad challenge function F. Okay, so now we know how to construct any ZX using FHM here and all we need is two ingredients. So first of all we need a sigma protocol to start with and then we need a hash function that is correlation intractable for the class of bad challenge functions corresponding to the sigma protocol. So now we try to instantiate this recipe and the guiding rule that we should keep in mind is the following. The simpler the bad challenge function is, then the easier is the task of constructing correlation intractability for such a class of functions. So our goal is kind of dual here. So first of all, we need to describe the bad challenge function for some sigma protocol as simple as possible. And on the other hand, we need to construct correlation intractability from standard assumptions. So now we can start asking questions. For example, it's not clear how simple can the bad challenge function be and it's not even clear whether it's efficiently computable. And on the other hand, we can ask about constructions for correlation intractability under standard assumptions. So the first work that tries to prove the sounds of the HMM under cryptographic assumptions is this work by Kanate et al. Where they observe that the bad challenge function for a sigma protocol for NP is efficiently recognizable. So if I give you a bad challenge, you can efficiently say that this is a bad challenge. Consequently, the correlation intractability task becomes to construct CI for all efficiently recognizable functions. And they are able to do that under sub-exponential IO and some other strong notion of obfuscation and these assumptions are clearly not standard. There have been many follow-up works, each trying to construct correlation intractability for a different class of functions that is sufficient for NISX. But again, they all do that under exotic assumptions. So for example, some assume something to match IO or fully exponential AKDM security. But very recently, last crypto, or even a bit before, this kind of work led to constructions under standard assumptions. And in particular, this last work by Paikard and Shrihan, they use the observation that the bad challenge function is efficiently computable for some sigma protocol. And therefore, all they need now is correlation intractability for all efficiently computable functions and they can get that under LW. So let's say our goal is to extend this paradigm and base it on other standard assumptions. And there seems to be a barrier here. So notice that these last two works, they consider the complexity of computing the bad challenge function and they say that the bad challenge function is an habitually polynomial time computation. And therefore we need correlation intractability for any polynomial time computation and it seems that we need homomorphism for any polynomial time computation in order to do that. But in other words, we need for homomorphism and we know that we can get that only under LWE, as far as we know of course. What we do differently is we consider the complexity of approximating the bad challenge. And using such an approach, we can show that it is sufficient that we use partial homomorphism. And in particular, we can use homomorphism that we can get from other standard assumptions, such as DDH or QR. So more specifically, we observe that the bad challenge for some sigma protocol for a complete language can be approximated by constant degree polynomials. And therefore it's sufficient now to get correlation tractability for all functions that can be approximated by constant degree polynomials. And this is exactly what we do. And we use Traktor hash and the LPEA assumption. Good. So let's draw a more elaborate comparison between a period work and our work. So again, like I said, period work uses the fact that there exists a sigma protocol for a complete language where the bad challenge is efficiently computable. And therefore a correlation tractability for all efficiently computable functions suffices for music. And they can get such correlation tractability using homomorphism for all efficient functions or for homomorphism in other words, under LWE. So homomorphism is a very vague notion, but you can think about if it's in such a case or fully homomorphic commitments. So at a high level, what they do is they use homomorphism for a function class F, which is in this case, any polynomial term computation to get correlation tractability for F for any polynomial term computation. So again, I will go to start with other standard assumptions in particular assumptions that do not give us full homomorphism. So our starting point is a work also from last year by looking at our when we show how using many standard assumptions, in particular at the age, we can get some form of well structured homomorphism. Again, this is very vague, but the formal abstraction for that was called table hatch. So we get some sort of well structured homomorphism for all constant degree functions. And we show that using well structured homomorphism for such a weak class of functions, we can get a great flexibility for a much stronger class of functions. So other than starting with homomorphism for F, we start with homomorphism for a much weaker class C. But using some structure of the underlying homomorphic primitives, we get relation tractability for the strong class F. And more specifically, we get relation tractability for all relations that can be approximately by constant degree polynomials. And if we want to compare this notion to the usual relation tractability notion, then in the standard relation tractability, we ask that it's hard to find an X such that H of X is equal to the back challenge of F of X. But what we want now is a hash function such that it's hard to find an X for which H of X and F of X are even close to each other in some metric. Okay? So this is a stronger notion, but we're considering a much weaker class of functions, in particular constant degree polynomials. And then we show that using CI for approximate relations, we can get collection tractability for all functions that can be approximately computed by constant degree polynomials. And then we show that under LPM, this is a subdivision in every physics. More specifically, we show a sigma protocol for an NP language in which the back challenge function can be approximated using constant degree polynomials under the LPM assumption, of course. And this is how we get our result. Good. So following this outline, we can divide our results to three parts, and I'm going to go through one after the other. Okay? So let's first define what correlation tractability is. So we say that the hash function H is correlation intractable for a function F. If for any polynomial time adversary, it's hard to win the following game, which we call the correlation tractability game. So the game goes as follows. The challenger first samples a random hash key, K, and then he gives K to the adversary, and the adversary's goal now is to find the correlation with respect to K. Namely, his goal is to find an X, such that H of X is equal to F of X. So H is correlation tractable for F, if no PPT adversary can win this game. And we say that H is a correlation tractable for a function class F, if it is relation tractable for every function in the class. It will be useful at this point to talk about a technique that was used in period work to get CI and this called summer statistical correlation tractability. So we say the following. Assume that for every function F, so given F, we can sample a fake hash key K sub F, which is distinguishable from the real hash key, and further that the CI game when we sample a fake hash key rather than a real hash key is statistically hard to win. So again, we consider the game which is identical to the game we've seen before, but now rather than sampling a real hash key, we sample a fake hash key using the fake generation algorithm. And what does it mean for the game to be statistically hard to win? It basically means that the probability that there exists a correlation between H under the fake hash key and F is negligible. So assuming we have these two properties, then using a standard distinguishability argument, we can claim that H is actually correlation tractability for the entire function class. Good. So let's start with a very trivial case towards our final construction. So we consider the case where our function class F contains only length expanding functions. Specifically, every function in the class takes a small domain and expands it to a large range. And notice that in particular, in such a case, the image of any function F is some small subset of the range. So our candidate for correlation tractability in such a case is a hash function that simply samples a random vector R in the hash key and outputs it regardless of the value of X. So notice that once we fix a hash key, then our hash function is constant. So what happens now? For every function in the class, the probability that there exists a correlation is equal to the probability that R is even in the image of F. So if the image of F is sufficiently small, or more precisely, if it's exponentially sparse in the range, then the probability that R is in the image or equivalently that there exists a correlation is negligible. Good. So this simple construction is actually correlation tractable for expanding functions. And our idea is to impose the same image in the general case, even when our class contains shrinking functions. So when our response contains shrinking functions, we cannot simply sample random R and assume it's going to be outside of the image because the image of the function can be the entire range. So what we do is that we define our hash function H using some other hash function we call H prime. So our hash construction will look as follows. We will use a hash function H prime, and then H of X will be equal to H prime on X, so a random R. So again, R is part of the hash key and is sampled randomly. So what we require from H prime in order for it to be a correlation tractable. So we require that for every function in the class F that exists a fake hash key which is indistinguishable from a real hash key. We require that the correlation function H prime of X under the fake hash key XOR F of X. So I'm calling this the correlation function because it represents the correlation of H prime under the fake hash key and F. We require that this correlation function has exponentially sparse image. And in such a case we say that H prime with a fake hash key and F have a sparse correlation. So assuming we have a sparse correlation between the fake hash function and F, then we can say that with the overwhelming probability we have that a random R is not in the image of this correlation function. And therefore, with a negligible probability that exists a correlation between H and F. And therefore H is a somewhat statistical relation. Good. Now that we have reduced our goal to constructing sparse correlations, I'm going to show you real quick how to do that using trapdoor hash. So like I already said, trapdoor hash allows us to construct a homomorphic encryption scheme with very strong structural properties. So in general, in a homomorphic encryption scheme, we can take an input X and given an encryption of a function F, we can homomorphically evaluate and obtain a ciphertext encrypting F of X. In a homomorphic encryption scheme, which we construct from trapdoor hash, the post evaluation ciphertext can be divided into two parts. One part we call the rate one ciphertext and I'm going to refer to it as the encryption of F of X. And another part is a small hash H of X. And the decryption in trapdoor hash goes as follows. So the first stage takes the hash of X, H of X, and produces a long vector E. And then one can recover F of X simply by exploring E and the encryption of F of X. So the second stage of the decryption is public. So how can we use such an encryption scheme in order to get sparse relations. So I'm going to define the fake hash key for every function F to be the encryption of F, and then given an input X will compute H prime of X simply as the ciphertext or the rate one ciphertext of F of X. Okay, so see we saw the function at prime is simple homomorphic evaluation. So notice that the correctness of the decryption of trapdoor hash, the decryption function is simply this vector E, which is computed only given a small H of X. And if H of X is sufficiently small, then we can say that E is sampled from an exponentially sparse image and therefore the correlation between H prime and F is sparse. So I didn't say what the real hash key is and why is it indistinguishable from the fake hash key, but notice that the fake hash keys are simply the encryption of the functions they correspond to. So the real hash key can be an encryption of any arbitrary fixed function, and we can claim indistinguishability based on the security of the encryption scheme. So good. The work of Dachling et al, we know that we can get such well-structured homomorphism for constant degree functions from many standard assumptions, including DDH for instance. And like I showed you this gives us correlation activity for all constant degree functions, but unfortunately this is insufficient for physics. So now we ask whether we can get something stronger out of our idea of sparse correlations. And remember that we claimed that a random R is with overwhelming probability not in the image of sparse correlation. But we also observe that the overwhelming probability R is actually far away from this image. And almost immediately leads us to defining a stronger notion of correlation interactivity, which we call CIAPX, correlation interactability for approximable relations. So we say that the hash function H is correlation interactability, is correlation interactable for relations approximable by F. If it's hard for adversity, not only to find the correlation, but rather to find an X such that H of X and F of X are close in hamming distance. And we can similarly extend the CI notion to a CI for a class of functions. Good. So now we know that we can get CI for approximable constant degree relations based on trapdoor hash from DDH, QR, LWE or VCR. And I'm not going to go through the details of this transformation, but you will have to believe me that using such notion of CI we can get relations interactability for all functions that we can approximate using constant degree. And let's be a bit more formal. I just want to tell you what notion of approximability we're considering. So we say that the function F is probabilistically computable by constant degree polynomial. If there exists a distribution over such polynomials, such that for every input X, the probability that F of X and P of X are far from each other in hamming distance is negligible. And where P here is this polynomial sampled according to the distribution which probabilistically computes F. Good. So now all is left to show you is how under LPM we have a sigma protocol for an NP complete language such that the batch challenge function has a probabilistically computed using constant degree polynomial. But I'm going to first recall period work where all they had to do is show that there exists a sigma protocol with an efficiently computable batch range. So they start with the classic sigma protocol for Hamilton City, which in particular uses some commitment scheme. And they use the observation that if this commitment scheme is efficiently extractable, meaning that we can extract the values underlying some commitment, only by seeing the commitment and possibly using some trapdoor. Then the batch on each function is efficiently computable. And more specifically, the batch of function given a, it simply consists of first extracting the values underlying the commitment a and then applying some polynomial time computation, which we denote here by V. So clearly if both the extraction algorithm are efficiently computable, then so is the batch on each function. So for us, we need something stronger. We need to show that the same protocol has that challenge, which is for list of constant degree. So we have the following observations. First, we observe that there exists an LPM based commitment scheme where the extraction algorithm can be from list of computable by linear function, which are a constant, which are special case of constant degree functions, of course. The problem is that this polynomial time verification is of high degree. Okay, at least in the protocol for Hamilton City, and it's not clear how we can probabilistically computed using a constant degree polynomials. But then we remember that using the cocleven transform, we can transform any polynomial time verification to a three CNF formula.  And why is this useful? This is useful because three CNF formulas actually have probabilistic representations as constant degree polynomials. So, again, using a cocleven we can adjust a bit the Hamilton City protocol into a protocol where the batch on each function consists of first extracting the values underlying the commitment a formula rather than an arbitrary polynomial time computation. And now since we know how to probabilistically compute both the formula and the extraction using constant functions, we can probabilistically compute the batch challenge function using constant degree. And this completes our result. So let's conclude. So we show a new notion of CI for approximate relations. We show that we can get physics from standard assumptions through this notion. In particular, we get the first physics under the H and the PM another many natural questions we can ask. First of all, whether we can use this new notion to get applications, maybe it's apps or in complexity or hardness results. We can also ask questions related to the applications itself. So it will be interesting to see whether we can minimize the set of standard assumptions. So maybe we can get music only from the edge or only from LBM. It will be interesting to see whether we can extend our result to get statistical zero knowledge or statistical soundness soundness, which we don't get either as opposed to a period work. That's it. Thanks for listening.


Today, soy is intensely cultivated around the world, with the U.S. and South America dominating production. Worldwide soy cultivation takes up an area the size of France, Belgium, Germany and the Netherlands combined. But is soy production bad for the environment? With its widespread expansion, soy producers often convert huge areas of natural land into soy plantations. Soy is the second largest agricultural driver of deforestation after beef, particularly in tropical countries. This wide-scale deforestation creates more environmental effects, including biodiversity loss, carbon emissions, soil erosion, strained and contaminated water resources. However, counter to the myth that increased soy production is a result of demand from vegan and vegetarian diets, the real driver of widespread soy cultivation is animal livestock. 80 to 90% of the world's soybeans are fed to farmed animals. Pigs, cows, poultry, farmed fish and most livestock are grown on soy-based feed because it's cheap and high in protein. Only 6% is turned into soy products for our consumption. But could soy as a protein substitute be a more sustainable solution? When we consume soy directly rather than have it filtered through the livestock chain, soy can actually be incredibly land efficient. To produce the same amount of protein, soy requires much less land than any animal product. Chicken requires 3 times the area of land, pork requires 9 times the area of land, and beef requires 32 times the area of land. If the world were to swap meat protein for soy protein, agricultural deforestation would decline by as much as 94%. Of course there are other considerations to bear in mind. But what do you think? Would you swap your meat protein for soy protein?


People are tricked, are forced, are abducted, are threatened into situations against their will. Victims can be anyone. They can be men, women and children of any age, any nationality, they could even be a U.S. citizen. Human trafficking is a very profitable industry and it's one of the fastest growing crimes in the world. You can only sell a kilo of coke once. With these victims, you can make money off them again and again and again. These are somebody's kids, somebody's parents, somebody's sisters or brothers that are being held in conditions that no American and no reasonable person in the world would find to be suitable. As we all know, slavery has been outlawed for a very long time but in fact this is modern day slavery. I came here when I was 17 years old and I came here with a family from my country as a babysitter. When I got to this country, things were nice but after a few months, things changed. Many trafficking victims don't know that they've been trafficking victims. It's very hard for them to name what the problem is. They start treating me bad and I don't have explanation for that. A lot of the clients that come into a non-governmental organization for services may actually be trafficking victims. They're never easy to identify. However, I think it'll be easier to look for something when you know what to look for. So, how do you recognize a trafficking victim? I think there are some red flags. One is that the person seems to be under constant surveillance. They're watched, someone's always with them. I couldn't go nowhere. I couldn't have friends. I couldn't go to school. So it's critical to get that person alone because even if they're with somebody who may say they're a friend they may in fact be the trafficker or an affiliate of the trafficker. When you talk to them you have to ask them questions because they might not tell you anything at first and when you start asking them questions you realize they were forced into doing this activity. Other indicators, red flags of trafficking would be they've been working and yet they don't really know their way around. They live and work with their employer on the same premises. I was scared about to leave from the house to try to escape because I thought when I'm going after the house when I'm going to go I have no place to go. I don't know anybody. There are some visible signs that you could maybe look for. A person who looks abused, maybe some bruises, a lot of medical neglect, maybe you have untreated medical condition. Somebody who's very scared, who looks very intimidated not looking at you in your eyes. People may think that they're being attacked when they're not so they may become irritable or inappropriately angry. They may have lost their ability to manage their anger emotions, their defensive emotions. You need to try to find out whether or not they were held in a condition of compelled service in the past or there's somebody who's recently been liberated who you know was in that situation and then you have to open your toolkit to respond to their needs. They took me to a shelter for three months. They were great people. That's why I'm here now because they really, really helped me. When a big oil pipeline manufacturer in Tulsa, Oklahoma advertised job openings in India, dozens of skilled metal workers jumped at the chance. But instead of training for permanent employment, the men were treated like slaves, forced to live on factory premises and paid less than $3 an hour. Fifty-two men managed to escape and sought shelter and legal assistance with help from members of a nearby church.  They need food, they need shelter, they need clothing. Those things have to be provided. Fifty-two people, they were living in the... four or five people living in the one room, you know, different rooms. And people were donating the clothes. People were donating the TVs and the air conditioners and the, you know, bad sheets and the blankets. And you can imagine, fifty-two people needs. Everybody has a different need, everybody has a different requirements. Traffic persons have an overwhelming need for services and the key is that each individual has such unique needs. Another important aspect is the physical health. So the person needs to generally get a complete checkup, make sure that they're okay physically, that they don't have any communicable diseases. Most trafficking victims have come here to make a better life. And they need to work, for example. So if that part is not taken care of, then it's going to be very difficult for them to make progress. So you have to teach them things like taxes, how to fill out an application for employment, how to fill out a lease to rent a house, how to open a bank account, how to go food shopping, how to get a driver's license. At my home in India, my wife, she got the threatening calls and this says, your husband left the John Beagle company. He has to go back and join the work. Otherwise, when he will become back, he will have a big trouble, a big problem in India. Trafficking victims are most often unaware that they have rights in this country and through the Trafficking Victims Protection Act, which was passed in October of 2000, they do have rights extended to them and they are eligible for certain benefits. The Trafficking Victims Protection Act of 2000 has three main goals, prevention, protection and prosecution. The Trafficking Victims Protection Act gave us those tools to do victim-centered prosecutions where we can catch the bad guys but also protect their victims. There is a process that traffic persons go through called certification and that comes when the law enforcement federal agents have deemed the individuals a victim most severe form of human trafficking. Once they are certified, that individual is eligible for all sorts of benefits, same benefits as a refugee or a political assailee which means cash assistance from the government, food stamps, they are work authorized. They can access Medicaid, Medicare services. And everybody was scared because everybody paid a lot of money to come here in the United States and we cannot go back to India under their circumstances. Law enforcement has the option of filing for something called continued presence which is a statement saying this person is a victim of trafficking and they are assisting me with my investigation. It's critical that they remain in the U.S. during my investigation or prosecution. Then they will have a temporary legal status while you're working to get a more permanent form of relief. TVCES are a longer term immigration status that happens typically after the criminal case is done. It allows the victim to stay in the country and even bring their family to join them to reunite the families. It's a very good way of making sure that the victims are not retaliated against by being deported back to the home country after they've cooperated with the investigation. And it was a very excited moment in my life. I've seen my family, I mean my wife and my son. Last year my wife should go to the boy here in the United States and he's like, now he's an American citizen. While it's important to give the victims hope you also can't tell them something that you can't deliver on. I think the most important thing to do is to give them the tools to make the decisions in their lives. Many of the early trafficking NGOs, the specialists who are now seen as the leaders in this field started off as battered immigrant women's service providers. SAFER hasn't expanded its mission to include human trafficking. We started to see a number of young women initially coming forward with these types of complex cases where they were undocumented, where they were being brought to New York or to the United States, having experienced crimes that would typically be described as trafficking. People get scared when they hear about a new subject and think, I don't know how to do that. I wouldn't know how to work with that kind of person. And we had a discussion as any agency would about whether this was an issue we felt we could tackle, whether we were equipped to do this and we thought it was unnatural for us since we were a crime victim service agency. They were able to take the tools that they were using for one group of exploited people and immediately switch those over to trafficking victims. I'm from Mexico. When I was 16, I met someone who was older than me. I fell in love with him, so we married. He forced me to work as a prostitute. He told me a lot of lies about the police and his country, that we were legal. If I could say police, they're going to put me in jail. I believe him. In many of the cases, the traffickers have often recruited people from their hometowns, from their homes, so there's a real intimate knowledge about who they are, where they're from, and their family's situation. And there were a lot of threats made against her, against me, her child. I always want to quit, to run away. And he knew it because he always was watching me. He never left me alone. He don't let me carry any money, nothing. If you're in another country and you don't speak the language, if your documents are withheld, who do you believe, where do you go for help? Who's going to believe you? You can't even explain what you want. And that was the last day that I worked. Here at Safe Horizon, we have a counseling center. Lucia was referred to us by a law enforcement agency, and so we were able to respond to her needs. You have to know who to call. You don't want to call somebody for the first time when you hear that there's 50 people in a case. You want to already have your relationships built. Even though we have a very large agency here and a number of programs that we use, we have many partners in the community that add pieces to our program that we don't do. Project Reach's mobile response team, we offer initial crisis mental health services to trafficking victims and also support and technical assistance to providers. We'll work with the victims and work with the providers to address the immediate mental health needs of trafficking victims. In some ways they're very similar to other trauma survivors of domestic violence, but this other dimension of being in a context of a complex legal system, being out of the country and away from the families adds another level of stress that makes it very hard to hear from the specific types of trauma that they've gone through. It's definitely important to be aware of someone's body language, their fear level. The fact that for many people who are trafficked, there's a very deep shame factor. We need to help victims achieve a sense of safety. And when we think about safety, we sometimes think about safety in the outside. But I think we also need to remember that people need to learn how to feel safe inside. I think we have very effective services and I think that's the key that we've worked very hard on the relationships that we have in the community. In Austin, we started our Central Texas Coalition against Team and Trafficking after we had our first federal case. Three young girls were being made to prostitute and had gotten away from the traffickers. The victims were minors, and so all of a sudden we had a few social service providers and a few law enforcement saying, what are we going to do with these minors? At least here in the Austin Police Department, there wasn't a unit specialized to handle that type of situation. It took a lot of manpower and training to understand, to decipher what was going on. And we learned the hard way because we did not know what we were doing in the beginning, taking this case on. Who are you going to call at 2 o'clock in the morning where we need to house a victim, whether it be a human trafficking or a sexual assault? I just called every agency in town, and one person would tell me about somebody else. But I didn't know about the refugee service programs because we don't work with them. So that was eye-opening. None of our agencies are all to everything, and so we needed to really bring in all the experts within the community to be able to work with these victims and to make it less traumatic for them. Before, I would have to worry about, how do I get a hotel room to put these people and how do I get clothing? Many times, we were pulling money out of our own wallet to satisfy those needs. When I called Katrina, she was over in my office within an hour and just said, we'll help you out here. We know all these different services. We'll find shelter. We'll find this. And ultimately, what happened from that is that we dropped the seeds for the beginning of the Central Texas Coalition against human trafficking because there was no way any of us could handle this on our own. That was three victims. What if we'd had 60 or 100 victims? That would have been unfair for the victims. We met over at some church hall one afternoon and we called as many people as we knew from the various agencies and organizations in town and asked them to please come and let's put our heads together and let's start talking about what needs to happen. Immediately, we had law enforcement that had our coalition meetings. We communicate on a very regular basis and we built up a level of trust. Robert Sanchez, Austin Police Department of Victim Services. Mike Miller with the Refugee Services of Texas. Our mission is to find human trafficking victims and to rescue them. The more we work with each other and learn how we could take advantage of each other's services, we've found out that we could reach the objective much faster by working together as a partnership versus against each other. And as we grow, we also become more confident and we become more at ease with each other to actually bring up some controversial subjects. And she was making cell phone calls from the shelter we had her in in the city where she was rescued from to one of the traffickers in another city because he was her boyfriend. When I first got into this, I didn't quite realize the safety issues involved. We have to, one, be very concerned about the client and the victim. These traffickers are organized. The NGOs need to be aware that when they're dealing with the victim, that they're also dealing with the traffickers. We do not want to identify, well, our agency is the one who handles all the victims. That wouldn't take much, then, for someone to find out where their victims are located. We work with them, whether it's the Austin Police Department, the FBI or the immigration, you know, we work with them about these issues. And since the victim and the traffickers are both in the same city, that's what we're a little concerned about. We might be able to move one victim to a different part of the city or even to another city and that would help alleviate the threat level. Our goal is to take that intimidation factor away from these traffickers and keep these victims safe and knowing that they are going to be safe in our custody. Not only are our victims, but also, too, our providers, our NGO partners, are going to be safe. Having that partnership with the non-governmental organizations is vital to these cases. Being prosecuted successfully into the victim's safety and to them being able to get all the benefits that they are entitled to. The crime of trafficking is not a crime against our borders. It's a crime against humanity. Through interacting and networking with other agencies, we've been able to learn better how to identify victims of trafficking. Our efforts are to continue to do outreach, to continue to educate every member of society from law enforcement to social services agencies to different ethnic communities in the area of trafficking. And there's a lot that we can do about it, but we have to be willing to extend ourselves, extend our missions, learn, and get involved.


Welcome to Laptop Media, before us today we have the Acer Aspire F15 F5573G. This model offers not only a prominent visual upgrade but also packs the latest hardware, Intel and Nvidia can offer right now. You can check the price in the description below. The new changes in the Aspire F15 allows for a big performance leap and will surprise you with some excellent screen properties for a TN panel. It is no longer plastic and instead now offers sturdy black brushed aluminum plate. As for the bottom piece of the machine, it's mostly the same except for the added service covers for easier access to some of the hardware. Opening the laptop we see brushed aluminum surface around the input devices. The keyboard offers comfortable ergonomic keys with long travel and typing on them is a pleasure. The touchpad is clicky, stable, accurate and has good gliding surface. Acer Aspire F5573G uses a quality full HD TN panel with good properties. The display offers poor viewing angles which is considered normal due to the nature of TN panels. The maximum recorded brightness in the middle of the screen is 290 nits and 213 across the surface, which means that it is more than optimal for well-lit indoor use but not very much for outdoor use. The color temperature is exactly 7500 kelvins or in other words 1000 kelvins more than the optimal so colors will appear slightly bluish or just colder. Quite expectedly the display covers 100% of the sRGB gamut. That's an excellent result especially for a notebook at this price range featuring a TN display. We develop unique profiles tailored for each individual display model. We will include the purchase link in the description of every video. We recommend you the profile so you would have a better usage of your laptop. Firstly, for working with Photoshop, CoralDraw or just overall usage of the laptop, we'd recommend you the Officework Webdesign profile that will maximize the F5573G's color accuracy. You can see main and additional colors inside the sRGB gamut pre and post calibration. The Officework Webdesign profile has been created with a target color temperature equal to 6500 kelvins, gamma set to sRGB mode and 114 nits luminance. We tested the accuracy of the display with 24 commonly used colors. You can check out the results at factory condition and also with the Officework Webdesign profile. The next figure shows how well the display is able to reproduce really dark parts of an image which is essential when watching movies or playing games in low ambient light. The left side of the image represents the display with stock settings while the right one is with our gaming and movie nights profile activated, which you can find and purchase in the description below. You can easily check for yourself how your display handles the darkest nuances. And here is where this particular panel differs from the previous ones we've tested. While the others had PWM from 0 to 99% brightness, the aspire F5573G's display flickers even at 100%. At least the frequency of the emitted light is slightly higher than usual at 1 kHz, which significantly reduces the negative impact of one's eyesight. However, you can always use our health guard profile that eliminates PWM, which would be of great help in this particular situation. Again, the purchase link of all of our profiles can be found in the description of this video. The new Aspire F15 offers not only the usual 2.5 inch hard drive but also has a M2 SATA SSD slot supporting 2280 sticks. Under the service cover you can find 2 RAM slots as well. In our case they were both occupied by SK Hynex 8GB DDR4 chips. The 62Wh battery gave pretty satisfactory results. For average use and web browsing it withstands for around 9 hours and 35 minutes. The toll taking gaming tests went pretty good nonetheless with 3 hours and 3 minutes of playtime. The Intel Core i7-6500U processor gives a slightly better performance than its predecessor with an emphasis on the power efficiency feature. We compared F5573G with other notebooks with the same CPU. You can check the full test and review of this notebook in the laptop media website. Our F15 uses a GTX 950M graphics card and can be found in 2 variants with standard DDR3 VRAM and with GDDR5. The GTX 950M with GDDR5 memory should perform much better than its DDR3 counterpart and 15% slower than the GTX 960M, making it an ideal choice for budget friendly 50 inch or 17 inch laptops. During the extensive temperature test we saw relatively high temperatures giving that the hardware is not so powerful. The interesting part is that there is no drop in productivity or the so called thermal throttling, taking consideration that these tests were made in extreme conditions with 100 CPU and 100 GPU load. As a final summation hands down one of the best mid range devices we've tested so far. The build quality is good with an excellent choice of materials. All in all the new Acer Aspire F15F5573G succeeds where the previous model failed while standing on some solid key selling points beating the competition with ease. Still there are some similarly priced alternatives on the market like the Acer's G55VX and then there's the Lenovo IdeaPad 510 which offers a superior IPS panel but fails to deliver the Acer Aspire's F15's performance. Thank you for watching laptop media, we have many more reviews waiting for you in our channel. Also consider subscribing so you'll always be up to date with the latest laptops, test tops, VR technologies, phones and everything our technological future will provide us. For more extensive information about this model and many more visit laptopmedia.com