So thanks for the invitation and it's a pleasure to speak about fantastic trees. There were some fantastic trees presented already in the Vector Commitment Day by Chris and the Bridges Talk on the Speedlog based BCs. So my name is Aline. I just joined Apo's lab as a research scientist. We're building an L1 for smart contracts. We're looking to hire, so feel free to talk to me if you do research or engineering and join us. The goal of this talk is twofold. First of all, we're not going to go in depth on any of the constructions. That would take too much time and there's too many constructions to talk about. So instead, I kind of just want to paint a landscape of what's out there in terms of trees. And even I can't even paint a full landscape that some of the constructions were presented today actually, but I think it'll still be interesting for you guys. And really the goal of this talk is to spark curiosity and incite new research in this space. So I hope you'll come out inspired to work on tree-based BCs from this talk. Okay, so we're going to start by basically dismissing the BCs with constant size roofs as amazing as they are. Then we're going to talk about local trees and dismiss them too because they have some problems. And then we're going to talk about the three or four constructions, lattice-based constructions, polynomial-based constructions, and vertical trees, which Chris briefly touched upon. Okay, so let me start by dismissing constant size VCs. So there's a long line of work on constant size VCs started by Catalano Ture and Messina, I believe is the earliest reference, and then Libert Jung and KZG. They're great. You have a vector of, let's say, eight files. You can commit to them using your favorite constant size VC scheme. And then you can compute proofs for each individual position. So for example, you have Pi 8, which convinces anybody that F8 is the eight file under the commitment C, enable the laser pointer. And they are nice, and it takes you at least linear time to compute all the proofs because each proof is constant size, so you have to output 10 things. So you spend that least omega n time. But unfortunately, when something changes, like let's say the fourth file changes in the vector, you not only have to update the commitment, but you have to update all of these proofs. And now, since these proofs are constant size, they don't intersect at all, they don't share any information. So this really implies an omega n lower bound on the time to update all proofs. So I basically came claim that the constant size VCs are not maintainable because of this and this is quite inherent. And there's some things you can do about them and we do want to talk about this to some extent like you can do time space tradeoffs, where you defer the updates, and you only apply them when you serve the proof. And that can be nice for some applications but not for others where you just want to serve proofs on the fly, instead of spending computation time. Okay, so this is why we don't like constant size VCs and we like tree based VCs better.  This was a seminal work by Ralph Merkle which is, you know, everybody loves this stuff. And how do you prove something in a Merkle tree you give a path of sibling nodes and the verifier takes the leaf that it's verifying and recomputes the hash along the parents by using the siblings, and then eventually checks that the hash to verify got is the same with the root hash of the Merkle tree. That's the ground truth. So if you log in size proof it's very small, you're sending 32 bytes per sibling note for example it's very efficient so a proof of in a tree of four billion things will take you something like 960 bytes can be very small. The time to update all proofs is also really nice and Merkle trees because for example if age for changes. I only need to update this path of from age for along to the root right so I'm only doing logarithmic time work. That's why Merkle trees can be better than constant sizes is the time to maintain proofs is is much much more efficient, the only do logarithmic work they don't do linear work. All right, but Merkle trees are not perfect they they present several challenges and in this talk I want to talk about three of them, specifically the lack of stateless update which Chris's talk talked touched upon the lack of proof aggregation or I should say efficient proof aggregation and the lack of database friendliness when you actually want to store Merkle trees on this as is the case in many cryptocurrencies like Ethereum, and so on. So first, what do I mean by stateless updates Chris touched upon this but imagine you have a setting where a verifier just maintains the Merkle route the digest the commitment of your vector in other words, and you have people who tell this this verifier that anyway you know that the third file in the vector changed by Delta three can you please update your root hash. And if you use Merkle trees this verifier has this root hash only and doesn't have the tree can do nothing to update it. In fact, in order to update it, this verifier must be given the Merkle proof for the old F3. And what the verifier will do is it'll take the new F3 and hash up to obtain the new route, and then the verifier can update it through. Right, so this is not stateless. You need to give the proof for the modified file in order to update the vector commitment. Right, and in some applications this is problematic and Anka touched upon this in her talk when talking about stateless occurrences. So that's one problem with Merkle trees. Another problem is proof aggregation if you're proving one thing like F3 this is nice but if you're proving two things. Well, you kind of have to give two proofs, you do save a little like you can see you don't need to send this fashion more because you will compute it as you have shop from F5 so you do save a little, but not a lot really asymptotically if you're proving K things you still have to give K Merkle proofs. So it's K log n size, the aggregated proof size, and there's not much you can do about this in Merkle trees beyond just to throw a snark at the problem and aggregate these proofs using a snark which can be quite inefficient or might require you to use these new snark friendly hash functions that perhaps need more crypt analysis for us to get comfortable with them. And lastly, this is something that I was thinking about doing my PhD early on but I never got to work on Merkle trees are not very easy to store on this. It turns out so imagine I have a Merkle tree of n leaves, and I want to store it on this. I, at minimum have to store these n minus one internal nodes of the Merkle tree on this. Right. So I have a linear overhead in terms of in terms of this overhead. And it's a bit worse than that because a database has its own overhead like usually you put this stuff in a key value store where where the key is the position of the node and the value is the hash of the node, and databases use their own trees internally like LSM trees or B trees or whatever, and they have their own overhead so you're storing a tree inside of a tree and you're doing a lot of reads whenever you want to read a simple hash. There's a lot of right and read amplification. This is really a problem in practice and I think if you want to be convinced of this because you don't believe me, just ask anybody who's implementing a cryptocurrency and they'll tell you, writing and reading Merkle trees from this because actually a big slowdown in the validation during validation. In fact it's even worse than that because often you want to create a persistent Merkle tree where you can keep track of the version. So for example, you want to store both the path for h8 but also the path for an updated h8 using path copying, and that implies kind of a logarithmic overhead for update. And it's a bit worse than that. Okay, so now that I told you about all of these things that Merkle trees are not so good at, I want to go over a few constructions that address these things of course there's no single construction that addresses all of them. I'm going to do a little bit of future work and I'll touch upon that at the end of the talk. And I want to start with lattice based the seas. And as, as Chris pointed out the first work on lattice based PCs is by Papa month to at all. And the key ingredient in their construction is this algebraic hash function. Normally we use shot to 56 we input two hashes to shine we get a shot hash right in a lattice based PC by Papa month with all this hash function uses two matrices L and R and does a linear combination of the two inputs. Now, this is an extreme over simplification. I'm kind of pretending like you can take the output of the hash function and feed it back in. But in practice, in this construction, you can't actually do that. The co domain of the hash function is not the same as the domain and there's a layer of complication that I'm going to completely ignore due to lack of time. But I still think the key ideas will come through as we'll see in a second. Again, the goal of this talk is to inspire you and stir up your curiosity so that you can actually read this paper. After. So, how does the lattice based construction by public throughout all work. You have your leads in the tree and you apply the hash function to them hierarchy key so so you hash this this last level you get a new level. And now you continue on this on this level to so for example you take these two children to take L times left child R times right child, you can distribute the matrix product can do the same thing here on on on these guys and then finally you can take the root left times left child R times right child, again you can distribute the matrix product so it kind of looks like this. And it's pretty nice because the root is a linear combination of these of these children, as you can see. But of course, because of the oversimplification I made, you might notice that, for example, eight six and eight seven would collapse together into the same product because the matrix is should commute the matrix product should commute the matrix product, so this is the same matrix product, but in practice that's not actually what happens because the matrix product doesn't commute this is just a, an oversight of my oversimplification. So let's just pretend this doesn't happen and please read the paper if you want to learn more about how this actually works, but at a higher level. The idea is that the root digest is just a linear combination of the leaves. And this is very nice because now what you can do is if the second leaf changes by Delta two, you can really easily update the root like you can update this LLR times Delta two update to this part of the digest right, and even better like if you have a proof that you want to update you can do that on the proof to like you can update the proof here with LR Delta two, and you can update the proof here with our age to right, so this is what we call a homomorphic we see. I'm not showing it in the slide but what I mean by homomorphism is the property is actually is actually homomorphism like if you had two vectors, and you built a tree over each one of these two vectors, you could add up the trees, and you would get a tree for the sum of the vector. That's actually what this construction supports, which is really neat and actually has a lot of applications which have not been explored. And I encourage all of you to think about that. This kind of is all I want to say about lattice basis he's of course as much more to say. So let's go a little bit about the advantages and disadvantages of this construction. So one advantage is that it's homomorphic but again a subtlety here is that it's homomorphic for a bounded number of updates. So you cannot actually apply too many of these to the tree. You can only do a bounded number of updates which is polynomial in some security parameter or something like that. They have a public setup which is really nice you can just pick this hash function anybody can do it and just like in a Merkel tree. It's somewhat snark friendly there's some previous work by Jean get all that explores using snarks on this PSUI 13 construction. So unlike more countries, for example, and their post quantum secure as disadvantages. The proof size can be quite large. So one paper estimates that it's 70 kilobytes using. I want to say polynomial ring lattices which are supposed to be more efficient. And they're slow in practice to so updating the digest or verifying the proof or updating the proof all take hundreds of milliseconds. Nonetheless, what was really cool about the PSUI 13 work is that it showed for the first time that you can have a homomorphic vector commitment or formal homomorphic mercury. And that's the key thing you should take from this and you should think about ways to leverage this homomorphism. And I think as open problems here of course improving the efficiency is crucial getting sub vector proofs, including proof aggregation like Chris mentioned is also extremely interesting. And the subsequent work by Chris pie card at all that that was just presented shows how to get smaller proofs using a trusted setup, and also shows how to get a homomorphic lattice based vertical tree from lattices so I encourage you to to check out this paper as well. I will have references at the end of the talk so you can see what these papers actually are. Okay, so this is a lattice based VCs. Now I want to talk about polynomial base VCs. So again lattice base VCs kind of address the stateless updates with polynomial base VCs, we're hoping to address the proof aggregation and the stateless updates. So here I'm going to talk about two lines of work the address line of work CPC 18 and the hyper proofs line of work SCP 22 the key ingredient in both of these works as a multi linear polynomial commitment. I will not spend time talking about what this is. But I encourage you to check out the references and instead I'm going to give you an idea of how the address and hyper proofs tree works. In both of these works, hyper proofs follow the tracks so it was inspired from the tracks and, in fact, hyper proofs added this tree construction that I'm about to present to you. Adrax didn't have it. So what's the idea the idea is that let's say you have a vector of eight elements h1 through h8, you can pick a multivariate polynomial of three variables that encodes the vector as follows it takes the index I. It splits it into bits so the index is from one to eight that has three bits and using the bits in as as the variables of the polynomial you can get the value of the sector back so we all know how to do this well maybe not all of us but but it's not complicated to do this. So let's say this polynomial, and then you can actually build a tree and unlike Merkel trees, you kind of build this tree from the roots to the bottom, rather than from the bottom to the root and you proceed as follows. You start with this multivariate polynomial in the root, and you do a division you divided by x three minus zero x three is the first variable of the polynomial. And you get a quotient and a remainder, as you always do when you divide a polynomial by something. You get a quotient in the root in the parent and commit to it. And the remainder you remember it. Now, in the right side, you also divide by something slightly different x three minus one. You also get a quotient and a remainder. The quotient will actually be the same as the quotient they got here, and you've already committed to it in the root. The reason being that these these monomials just differ by a constant, but the remainder will be different.  So you will now recurse on the remainder you'll take that remainder from this node, and you'll divide it by x to minus zero and x to minus one.  And then you get a remainder and you will curse.  And once I have x one I will be done actually the remainders that I will get will be the evaluations themselves will be the vector elements. Why is that well intuitively you're doing five module X three minus zero module X two minus zero module X one minus zero, which you can prove that it's actually just five evaluated at zero zero zero, which is just five of one, which is each of one as defined here. And, and the same holds for for this guy and all of these other guys. So, now this construction. This is all I want to say about this construction the ideas that you do.  Unlike in the Merkel tree approval consists of the quotient commitments along the path to the leaf that you're proving so for each one, for example, the proof would be the quotient commitment here, here and here. And that will be approved. And in a miracle tree if you recall you gave sibling that so this is quite different in some sense. And another thing that that's not visible here is that this construction is also homomorphic like the lattice space, and that's really one of what I want to touch upon next like the advantages of this construction is that it's homomorphic and it's actually for an unbounded number of updates, unlike the lattice based one. It has the same proof size as Merkel trees. The reason being that these quotient commitments that you put in these notes can just be 32 bytes. So it's quite efficient in practice in terms of proof size, and it's efficient in practice in terms of computation now it depends who you ask you know if you ask a lattice based researcher. It's definitely much faster than lattice is probably 100 times faster. So if you ask a blockchain developer this is too slow. And trust me, I work with blockchain developers these days and they think it's too slow. So what what a researcher thinks is reasonably fast in practice doesn't hold for engineers. The other advantage is that it's friendly to inner product arguments for those of you who know what these are and they were mentioned earlier in the discrete lock talk. And in fact in hyperproofs we leverage this IPA friendliness to do proof aggregation. And to also build sub vector groups. And additionally because of the unbounded homomorphism, you can get a property called on still ability, which is again something we exploring hyperproofs. One of the things that you can find still ability is that when you compute one of these trees, which again if you ask the blockchain engineers they'll tell you it's very expensive to do. When you, when you've done all of that expensive computation you can actually watermark your tree with your identity to make proofs unstealable, meaning that every time you serve a proof from the tree that you computed. It's watermarked with your identity and everybody can see that you computed it and nobody can change your identity in this group. Right. I believe this could have interesting applications and cryptocurrencies this type of on still ability. Of course, there are disadvantages it's not post quantum unlike the lattice space construction. It does require a trusted setup unlike the lattice space construction, and even even worse it has linear size public parameters so your vectors if you want to commit to and leaves, you need to have and public parameters that are generated by this trusted setup here. And another disadvantage is a sub vector proofs are quite large 54 kilobytes so aggregation will only start making sense once you aggregate about 100 proofs or so. I think what is cool about this line of work is that it added an unbounded homomorphism and it showed that the lattice space construction can be efficiently instantiated in practice, using of course a bunch of caveats, like the trusted setup in the public parameters. And then what the what the hyper proofs line of work did is it showed that, well first of all that you could build this tree you don't have to compute proofs individually you can build a tree efficiently. You can do proof aggregation using IPAs and then the instillability. And here as open problems I think it's very interesting to start and reduce the proof size, either by improving the IPA or coming up with new strategies, and to do it much much faster the aggregation, whether again by improving the IPA or coming up with new strategies.  I have links here to a video presentation in the slides, and I recommend all of you watch Shravan stock at using security in August on hyper proofs. Ali, can I. So, just for curiosity, can you give like at least an intuition of how you do some vector opening proofs in the tree. Yeah, so, because I guess you aggregate right so you you don't. Yeah, I think I can but but I have to go to a different presentation. I don't want to. No, no, no, it's all good so so it's the idea will be that proof verification. It looks kind of like this.  PSD of FR is the vector commitment. These are the quotient commitments that I mentioned earlier. VI is the value of proving so you just have a bunch of pairing equations when you when you prove a bunch of different positions, and using an inner product argument you can prove that all of these equations hold very very succinctly without giving all of these quotient commitments you can just prove knowledge of quotient commitments that satisfy these verification equations. Is that makes sense. Yeah, now it's pretty good. Yeah, and what we use is the BMM 19 inner product argument that generalizes bullet proofs to not just. Yeah, that generalizes bullet proof to parents. Okay. Thanks for the question. Okay, I want to tell you a little bit about another construction that was kind of in between edgars and hyper proofs this work occurred in between edgars and hyper proofs. It's called an authenticated multi point evaluation tree. This is work I did with my high school students. A few years ago, while I was a graduate student. And here is a univariate permanent commitment like KZG 10, rather than a multivariate one. And again so this work was done after edgars and in some sense was very inspired from the edgars construction, and then hyper proofs was inspired from both of these works so the idea of this tree from hyper proofs that I just presented was really kind of inspired from the tree in this amt work, just applying that to the edgars construction and of course everything is rooted in the lattice based construction from Pokemon to adult. So what's the idea here, it's very similar to hyper proofs or edgars it's you encode the vector in a polynomial but the polynomial is univariate so you use an index directly to access the vector elements inside the polynomial. And you're also going to do repeated divisions you're just dividing by different things so let's say you have a vector of eight things you have your interpolated polynomial that it encodes the vector. You divided by an accumulator over all of the positions so an accumulator is a polynomial that has roots at all of the positions.  And you recurse on the remainder you have one remainder from here you're going to divide it by the left half of this accumulator and by the right half of this accumulator.  Like on the remainder that you got from this division, you divided by the left and right house on the remainder that you got from this division divided by the left and right house.  And in the leaves, you will get the evaluations themselves. Again, the idea being that you have five module low this huge guy, which contains x minus one then five module this other huge guy which can take six minus one, this and this. Again, you're just doing five module X minus one, which by the power one and remainder theorem is five one, which we know is equal to H of one. And that's that's the idea for why this works. Again, it's a different approach and Merkel because you're starting from the top and you're going all the way to the bottom. And again, this is homomorphic to. So it actually has the same advantages and disadvantages as a Jackson hyperproofs, except this construction. As far as we can tell has a quadratic time trusted setup. If you want to be able to support homomorphic updates. And it has n log n size public parameters rather than n size public parameters. So open problems here are to fix the setup and the public parameters and wager one who just gave a talk earlier, I think, is doing some ongoing work on fixing this and the n squared time trusted setup I think you already fix the n log n size public parameters I'm not sure if you did but but feel free to to discuss with him. I think both of these are very addressable. And in that sense, once you address all of these. If you think about amp these and address their kind of dual constructions to one another. One uses multivariate other uses univariate. But I think really the holy grail of the same deconstruction would be to figure out a way to take the same deep proof, which is just the quotient commitments along the path to the leaf they are proving and figure out a way to compress it to a case G proof. Actually like algebraically do that, because if you could do that then case G proofs are aggregatable and that would imply aggregation for amp peoples. And I haven't been able to do this and embarrassingly like this could be something very simple like maybe it's just a bunch of arithmetic tricks that I'm not seeing. So if this is possible that would be amazing. If this is impossible then somebody should prove it I tried proving that it's impossible but I couldn't do that either. So I think this is a really nice open problem to try and work on. We have an in depth blog post on amp that I encourage you to read we have slides and we have a bunch of video presentations to, they can look at. Good. So that kind of covers polynomial base PCs which address status updates and proof aggregations and have other properties like on steel ability and can be reasonably efficient in practice depending on who you ask. Lastly, let's see what we can do about database friends. And here I want to point to this vertical line of work. So, in the literature. There is actually the idea of having a large area tree with Eric K for arbitrary K, you can find work as early as the work now or in 98 who use this for a stateless digital signature scheme. As early as Katalina Fiori 2008 in Messina, Libertiung, Papamantuta, Masia, and Triandopoulos, and then my high school student John Krusmal and I in 2018. Also, we started investigating this under the name vertical trees. So the idea the idea is as Chris pie card explain earlier, you have a higher branching factor in your tree, rather than binary, you go to carry. This implies a smaller height. And interestingly, of course it also implies fewer internal loads in your trees, which implies a lower overhead on the database that you store the tree in. So if you, in addition to using a higher branching factor you commit to the children using the constant size VC, then you can actually get lucky and size proofs because you no longer have to give the siblings along the path. So that's very nice and I'll show you in a second how all of that works. So first of all, why don't you want to do a high area Merkel tree rather than a vertical tree why is that a bad idea. Again, if you want to prove something like H4. This is what the proof would look like you would be giving all of these siblings at every level. You have locked KN levels and K minus one siblings per level so the proof size would be K minus one lock KN, which is always greater than log two and which is what you would have in a binary tree. So if, if your goal is to optimize for proof size. Hi already Merkel doesn't make sense. But if your goal is for example to be database efficient, then I already Merkel might make sense if you use let's say case equal to 16. I think you get a proof size is not that much bigger it's like four times bigger or something like that. So it could could make sense as practice. Now if you use a high already vertical, which I'm depicting here. I can't obviously depict all of the sub trees in the tree so these are meant to be sub trees here. It would look kind of like this, and you would commit to each of the children using a vector commitment scheme in this fashion. So what would be the advantages of this of course you get lucky and depth as with the Merkel. But additionally what you can do because you've used a vector commitment is you can now compute these proofs for every child with respect to a parent so for example this proof argues that why seven is the seventh child of Z. There's a efficient API you can use for this and efficient algorithm and a lot of vector commitment scheme so so you can basically do a proof all call that in K log K time for for these K children, gives you all of these K proofs here. So this can be very efficient that's in product with at least, and you do this for every every note in the tree, right you can put proofs for every note in the tree. So even here even here even here and here. And as a result, in addition to committing to the tree using the vector commitment scheme, which will take you order and time, you're doing this extra and lock to K computation for getting these proofs. So you're doing extra computation in vertical trees. And this, it's all concretely much more expensive to this is much more expensive than computing hash functions, the vector commitments themselves are much more expensive. So you're giving up a lot of computation, just to get a smaller proof size and to get database friendliness. And what is a vertical proof look like for example if you want to prove position h4. What you do now is you no longer have to give the siblings right you just give the path with the proofs. So if you want to verify or can check, for example, that h4 is the fourth child of parent w3 using this proof by h4 that you gave, then you can check that w3 is the parent of why six at position sorry that w3 is the child of why six at position w3 using this proof. Same thing for why six with respect to Z. And the proof is locked in size. Which is much smaller than than the lock K minus one lock in in a miracle. And then another Vitalik Buterin from Ethereum pointed out that these proofs in a vertical tree can be cross aggregated into a single proof. So then actually what you would have to give are just these pink commitments along the path, and one group element for the cross aggregated proofs in schemes like KZG or Libertium or point proofs. So it can be quite efficient in practice that proof size can be concretely locked K and plus one group elements. So it's much smaller than Merkel trees. Of course there's no free lunch, if you want to update a vertical tree you're going to be doing a lot of work. So you have to update the commitments a lot the change path. So here you have to update the w3 to w2 prime by accounting for the change at position for in h4. Here you have to update why six by accounting for a change at position three in w3. You have to change the root by accounting for a change at why six right, but you're not really done you also have to update the proofs, because these proofs are no longer valid they all need to change the proofs along the pack. Really need to change right so, for example, this proof here at position eight, you need to update it to account for the change at position six in why six right and you need to do this for for all of the proofs on this level. So that would be the basis of this of this node here and for all of the proofs of this node. So that implies actually K log K and time for the proof updates, because you have locked in levels and at each level you're up in K proofs. It's the same asymptotically in a miracle tree because you need to rehash this parent and you have to do order K work to account for all of the children and have locked in levels, but it's concretely slower in a vertical, because the proof updates are much more expensive in a vertical. So if you look at vertical trees you have a bunch of advantages the proof sizes locked to K times smaller than in a in a miracle. They're reasonably fast in practice, their database friendly, they have public setup if you use the right the right vector commitment like bullet proofs, they're unstealable. They have some high proofs, but they have some disadvantages they're much slower than Merkel trees of course because you use internally, not a hash function but a case G vector commitment let's say they're not homomorphic unless you use these new techniques that Chris by card at all proposed. And they're not friendly to snarks or inner product arguments, either, and they even require trust a setup with other VCs like Casey G the vacuum, and so on. So the things here that are worth working on is efficiency. In terms of updating the vertical tree serving proofs fast, if you're not going to be maintaining proofs. Coming up with a sub vector proof for a vertical trees is super interesting open problem. And even better, can you aggregate person to sub vector proofs. What I'm going to do right now is something that Chris pointed in his stock. It's just because of this mapping of commitments to two vector elements that you have to do via hash function, which breaks any friendliness towards snarks or IPAs. And there was subsequent work on vertical trees by the folks at Ethereum they propose this bullet proof based verco. And again by Chris by card at all they propose the homomorphic lattice based vehicle tree. So, okay, so to conclude kind of the stock what is the most fantastic tree that we hope to get in that I encourage all of you to to to explore in your research. Well obviously it has to be maintainable because it's a tree, and I put this here because you can come up with some tree based constructions that have lock size proofs were actually not maintainable but the proofs are not easy to update. So it has to be maintainable. It has to have aggregation homomorphism is a good property to have for status with occurrences database friendliness is also a good property to have in those applications. Small proofs concretely speaking computationally efficient concrete is speaking constant size public parameters ideally. Of course other nice properties, if you can get post quantumness that would be great and still ability, perhaps, maybe compressible proofs this idea of taking a lock size proof and compressing it down to a constant size proof. Those will be nice things to have. And in addition to all of the open problems I mentioned in the previous talk here are a few concrete directions for future work that that I've been keeping in the back of my head, but never had enough cycles to work on. So first of all, can we get three basic is from the speed log assumptions, and there was an earlier talk today on the speed log assumptions and trees. So there's some progress in that, or from our say assumptions, and we have some ongoing work on the RSA assumptions line. In particular, we believe the catalog of your construction can be turned into a tree, but computationally it doesn't work out it's too efficient. So feel free to talk to me if you're interested in that. Second of all, can we build the authenticated multi point evaluation trees from different polynomial commitment schemes like dark polynomial commitment schemes in RSA groups. That could actually allow us to do proof aggregation name these because these polynomial commitments support a multiplicative homomorphism, in addition to an additive homomorphism. And if you have a multiplicative homomorphism in the MTS, you can aggregate as far as I can tell, but there are challenges there to vertical tree aggregation would be interesting to explore using chains and cycles of elliptic curves. And finally, vertical aggregation could also be explored by using this new homomorphism proposed by pie card at all, combined with the snark friendliness of the lattice based constructions. There's also some work on building VCs from subgroup hiding, but they're constant size VCs. So can you do VCs that are tree based from subgroup hiding, what kind of things can you get there so you'd be using composite order groups. They wouldn't be so efficient, but maybe new ideas can be sparked. Lastly, there's a connection with digital signature schemes and a lot of these vector commitment scheme. So Catalano Fiori has some connections with the water's digital signature scheme and KZG 10 is connected to Bonnet Boyan short signatures. In general it'd be interesting to think of whether there's a transformation here that gives a VC given a digital signature scheme, and what does that imply for tree based VCs. And lastly, I have this overview of all of the work is stable that kind of surveys the field. So I'm not going to present everything in it but I'll leave it here for you guys to look at. And I think with that, I want to conclude and take questions. Thank you so much. I hope you're all inspired to work on trees after this talk. Thank you, Aline. I don't see any questions in the chat. I have one question, one quick question when you talked about the dark. So everybody said, one of the things about the dark point on a commitment is that we could do it even without the trust to set up with the, with that other assumption like the the class groups right. How efficient is that, compared to our side you know, or the class group. Yeah, there are implementations out there I think she, which is a cryptocurrency company implemented class groups. I remember asking some folks at some point and they were not too concerned about the inefficiency I think it was even a claim that it's roughly on the same order of efficiency. But I suspect it's a little bit less efficient. The complication with class groups is that usually like the assumptions are a bit different when you instantiate these things from class groups so there are things that don't hold in class groups that hold in RSA groups and you can run into problems. So you have to be really careful when you use class groups. But yeah, to avoid the trust to set up in that part of the commitment you probably want to use class groups and you might lose even more efficiency in addition to RSA, and you might be to be, you might need to be extra careful when proving security of your scheme. I mean, this is the idea. I have a question about the your vertical aggregation question. So, like in the, I mean, when you do build this vertical tree. So even if there is a way to aggregate the openings along the path. I mean it seems to me that there is an inherent problem that you should provide commitments and this might not be aggregatable and you know that's right. Are you do you think that this is solvable or yeah. Yeah, so I claim this is solvable. I mean, I claim. I want this to be solvable. So you're right like you can aggregate these proofs, but what about the commitments. Right. So that's what what Vitalik pointed out, let's let's cross aggregate these. We still have to send the commitments. Now, if you think about these new techniques by pie card at all. They, they do not need to. What's the problem, the problem is that these are group elements, you have to hash them into a field element so that you can compute the commitment here at Z. So this is not really why one I'm kind of lying here, this would be hash of why one here. And that's where you lose the start friendliness or the idea friendliness and that's where you kind of lose the ability to aggregate the commitments themselves efficiently. But but pie card at all showed that that if you use lattices, then you don't really need to hash right you actually get a homomorphic vertical tree, which I claim that could be much more friendly for aggregation I don't know how but I have a, I have a hunch. So, so that's the idea. And of course it's going to be lattice based which we don't know how to make suit to efficient practice. But hey, you know, at least we can aggregate vertical proofs and we'll let other people instantiate the lattice stuff more more efficiently. So that would be the second, the first idea with vertical aggregation it would be sorry in this future work here, vertical aggregation the pps 21 homomorphism and snarks or IPAs I should say right. Something like that. But the other idea would be, you know, can you use some chain cycle of elliptic curve thing to remove this need of of hashing the group elements to the field element so like when I if if when I compute the vertical here. What if I don't need to hash these group elements, because I can maybe use their x coordinate, which is a field element, and just use that as as the input to the vector commitment on the next level which uses a chain of elliptic curve something like that. It's actually something that we have explored a little bit, but we didn't see the advantages because even if you can do use the cycles like the way you encode things still break the homomorphic properties of the let's say the other. The way you encode things you mean. So let's say that you know you want to commit to group elements. Right, so you want to complete commit to group elements that are each group element is actually a pair of field elements that are in your message space, but then the problem is that I mean you can do it right and these probably avoid to make an expensive ashing. But many other properties like breakdown, for example, you lose any homomorphic property of the commitment because committing to the points of the elliptic curve. I mean, let's say adding over the field the pairs of elliptic curve points does not correspond to the group operation, for example. Maybe from an engineering perspective of analyzing, you know, avoiding ashing from groups to fill maybe this could make sense. Right. Yes. Yeah, unfortunately I don't I'm not a master in elliptic curves so if I knew more I could think a little bit about like.  But that's why I'm leaving it as an open problem. Thanks. Thank you. I see that this is a question. Great talk. Lots of really cool stuff to think about. I was curious about the techniques involving these like repeat these, you know, sequential division by polynomials either linear or higher degree and then moving to lower degree. So any way in which the operations here are commuted or commuting. So, you know, divide by x minus one and then divide by x two minus zero or something. Are these these commute in a way. I wasn't able to work it out into my mind. Let me bring up the slide just so. So, what what exactly is the question about commuting so what do you think should commute. Well, I guess, yeah, it's not entirely well posed question but suppose I, you know, here I'm dividing by x three minus something and then I'm dividing by x two minus something. Yes, and suppose I flip, you know, do it in the exit order. The question really is, is this actually a tree, or is it more like a tensor where the order, you know, where you can maybe do do things in any order right or permute the order in which you do these and is there any advantage to that. Yeah, so they do commute. So you could, I don't think so you could compute this tree by starting with x one minus x one minus one, then doing x two at this level and x three at that level, you could do it that way to it would be a different tree, but it would still have security. So, so does that for example, answer your question. Yeah, I guess I'm what I'm trying to get at is whether. Yeah, I mean the order in which you do these is arbitrary, you know, for each at each level, but it's more that, you know, if I divide by. So if I think about the doing two steps here, like dividing by x three minus zero and then by x two minus zero. Okay, that's dividing by x two x three, right, and the order, you know, x two and x three commute. The overall quotient will be the same, regardless of which, you know, whether I do two first and then three or three first and then two. And the individual, the individual remainders, maybe won't be the same. But the quote the ultimate quotient will be the same. Right. So I actually I believe that if you divide by extra minus zero first to get, let's say q three, and then you divide that remainder by x two minus zero to get q two, let's say, if you were to flip if you were to start with x two minus zero, you'd get a q two prime. You wouldn't get the same q two that you got here. For sure but then after I divide wrong but then after I divide by x three minus zero I'll get the same q two three, let's say. The same. If it's a quotient. So in this order, like you get q three q two. And if you flip I think you get q two prime q three prime. Right, but what I'm I guess what I'm trying to see is the quotient after, sorry, so you take the remain. Maybe I misunderstand the first step so I get a quotient and a remainder from the first step. And then I take. Oh, I remainder and divide by x two. Yes. Yes. Yes. I see. Okay, not the quotient divided by x two. Yeah. What I was thinking is if you divide each quotient by the next thing then they kind of commute. Okay, right. Yeah, so there's some intrinsic order here. Yes. Hmm. I'm wondering, is there a way to transform, you know, if you change the order of these things, can you more efficiently transform the tree to the to its, you know, prime values, right, to its updated values. Is there any use of that to it's how do you mean update the tree to its updated value.  So, if I if I knew q two and q three, could I then efficiently transform them to to three prime q two prime something like that. Is there some use. It gives me the feeling that there's no inherent real order here and that right somewhat somewhat arbitrary and so yes, maybe this is like a tree where you can access any level cheaply or something like that, which might be useful. Hmm. Interesting. I put in the chat a reference to the paper that that first explored this multi linear polynomial commitments and these repeated divisions for everyone. What can I say something. Sure. Are you done or sorry. Yeah, that's all I was interested in thank you. No, so I want to say that I thought about something like what preset in a different context. And so I for update proving for updating proves I'm not sure because like you already pre computed them in some order so I don't know how this would help. But what I was thinking is for subset openings. If you open certain positions then you can build maybe the tree in a specific order so that like you can group some of the openings in a certain way or something like that. Yes, yes. Yeah, indeed like, I think, I don't think we prove this but I think like if you just gave this quotient and this quotient that could serve as a subject or proof for these positions. We haven't written the security reduction for that but I think that should work. I think that works in amp is at least I'm not sure about hyper proofs, but they're quite analogous. So then, to what extent, I don't think, I don't know if you could reorder these in any arbitrary way though. Like for example I don't know how you could put this leaf next to this leaf when their divisions are by different things. Well, the way I think about these trees usually is like, you have leaves, and you number them in binary. And then, yeah, like, not like they, they are labeled in by a lawyer. Well, they have a label of binary beats. And then like what I was thinking is something like you want to open up positions. And you look at which coordinates they coincide. And then you could like somehow build a tree so that the maximum number is like grouped in a certain way. But but then like if you like the proofs are already pre computed then I think like somehow this doesn't help. Like if you had to compute them on the fly like you could do something. Yes, we've had a lot of difficulty coming up with aggregation for these two construction hyper person MPs without back boxes like IPA and snacks. My only hope is that that for the empty construction, we can take a proof and turn it into a KZG proof that would be amazing, but I, it doesn't seem to it doesn't seem possible it seems to require a homomorph, a multiplicative homomorphism and if you read the post that I have that I have here, you can kind of tell how how you can turn an MP proof into a KZG proof although I don't spell it out specifically. But they're quite related algebraically. It's just that you seem to need a multiplicative homomorphism. Maybe there's other things you can do. Thanks for all the questions everyone by the way.


Say hi to YouTube live. Hello YouTube live So it's day one. We just got all this set up. It looks like this. Why are you so blown out? Cuz life It's like there. I'm regular now. So this is Rudy's Rudy's y'all know what our stuff looks like and then also our stuff and then Coasters we'll put the how-to we did like 800,000 and one sitting what 800,000 how many was it? 1630 He doesn't know so we did half of these all at one sitting It was bananas and oh wait There's a lot going on we're trying to what's up Kelly? Hey boss rundown of where we're at you can fix it enable flash new my room. I don't know why it's not oh Okay, good Judy a little tour of where we are now. Let's go this night. There's so much artwork. I can't even tell you guys Also super bright if you're in Dallas deep Ellen area. I almost dropped you. I didn't drop you though. It was close This is the intersection There's art everywhere all the way down there take you guys on a tour of all the things later. I almost just died Everybody saw me almost just died It's part of my life anyways We'll be out here from 11 to 11 today and Tomorrow everybody says hi So we out here today and tomorrow till 11 and then on Sunday. We'll be out here Till eight who knows But we're here if you guys are in the area Come see all our work in real time If not, I'll post We have videos to post still right Anyways, this is mainly to test to see if I could do live stuff so that we could do live giveaways and Etc in the future Oh and Q&A's we're gonna do a Q&A coming up Next week next week we're gonna do Yeah, I think a week from Sunday, we're gonna do a live Q&A so you finished your questions or Just tune in save them up. Anyways, thank you guys for being so loyal to us We love seeing all the questions comments the shares the hashtag ADT ART So we'll see you guys a little bit later Bye


What's up everybody Jason here for Jasper's case breaks calm 2017 18 and 2020 tops definitive baseball three box case break Or three case breaks are just sold out total of nine boxes guys three boxes in a case of 2017 definitive three boxes in the case of 2018 definitive and three boxes in the case of 2020 definitive 27 total spots. There's three combo teams which include the d-backs and race Marlins Rangers Padres pirates in three cases Just in time for baseball as it's coming back guys number if you hit an expired redemption in this break We will send you a hundred dollars in break credit. This is not to apply though to active redemptions So there should there could potentially be an expired redemption in 2017 So if you hit one, we'll send you a hundred dollars in break credit. So there you go Here's a dice. Well, there's a customer nasal from Chris to Joseph then to Aaron in there Then to Christopher to David or Charles a nick you guys got those spots straight up and then from series one fat pack number one Corey Nick and Matt got it Richard Corey and Richard got it again And then of course Robert Danielle and Matt Arnold got in series three All right there at the top So there you go guys and then there's a teams bucks or sorry d-backs raise Marlins Rangers and Pirates Padres So roll it We got ourselves a four and a three for eight times good luck. Oh, sorry four and a three seven times One two three four five six and seven So four plus three seven Christopher down to Christopher. So four five six seven seven let's go here seven one two three four five six and seven D-backs raise at the top none of the Mets remember guys. This is three years worth of product So Christopher you got the d-backs and raised Charles with the Mariners and Nationals Richard Miller from series one two Yankees Christopher with the Royals Nick from series one one Marlins Rangers combo Aaron with the St. Louis Cardinals Cory from series one two Boston Red Sox Matt Arnold from series one three Pirates Padres Joseph with the Orioles David with the Angels Chris with the Phillies Charles with the Brewers Jason L with the Dodgers Cory from series one one Indians Cory your straight-up spot braves Robert Uncle Oakland A's from series one three Travis Tigers Travis with the Tigers are Chris Cole with the Astros Danielle with series one three Cubs Alex with the Reds Richard Miller from series one two with the Giants Matt Arnold from series one one Rockies Nick last Bob module blue Jays I with the twins Christopher with the White Sox as well as the Mets Alrighty guys alphabetize it, and I'll give you guys a couple minutes to trade As I know that this is a big three caser. I remember three years worth of stuff guys So that's the cool thing um while you would have to Potentially have Jason L if he's even here to potentially want to trade So there you go Jason is there Dodgers up for trade. Would you guys like to trade Dodgers in for blue Jays? Looking for the author. I didn't see you in the comment Jason looking for the Yankees or angels All right you guys I think I'm gonna close the trade window then Jason's looking for specific teams denied that trade no one's answering for the Astros No one's asking for the Tigers So Let's close with them. Here you go guys. Good luck So let's go with 17 definitive first guys. It's been a while since I ripped this 2017 definitive. Oh, that's right The cars were just out there They weren't like even in a box back in the day Usually it's supposed to just like kind of take them out Now they put them in a little little slider box Back in the day that we're just thrown in there like that All right, guys. Good luck first one. We got It's Hunter Renfrow one out of 50 autograph our podgers are combled up and That'll be going to Matt Arnold Pirates and Padres combo That's what we got little pinstripes 24 to 50 that is Noah Cinnigar Have to have to play a little surgery Charles. I don't want to damage these cards, you know, it's like damaging organs that is for the Mets and What's the Mets Christopher? All right, how about a little Clint Kershaw 12 out of 40 Dodgers Jason had him for trade couldn't trade mojo How about some more Dodgers how about a little kentamaita and Julio Urias eight out of ten That's pretty strong Never kentamaita. I believe was also rookie with Julio that year I want to say another back-to-back Dodgers We have World Series champ Roberto Alomar try to trade couldn't trade mojo blue jays Nick with that one 435 another try to trade couldn't trade mojo Both for the Houston Astros that's Jose Altuve and Craig Bijio out of 35 There you go, man We have for the Cleveland Indians Carlos Santana Remember guys they have this little holographic code in there if you go to the MLB website You can see what Jersey what game was this Jersey used in? But you know if you like had a horrible game if you had a great game Pretty strong we have Nice little patch here of Carlos Gonzalez 29 out of 30 for the Colorado Rockies It's been a mad arm And the last one here. We have a one of one How about a Greg Maddox one-of-one laundry tag Russell athletic? That is awesome. Remember guys machine wash separately use warm water gentle cycle Only non chlorine bleach when needed. There's a little one-on-one on the top right It's awesome Sweet sweet nice boxer here guys That was awesome They're just giving you the directions you need you know open up box. What are you doing? Whoa? So has a frame Come on carts Come on come out. I managed to get the cards out next one We have Yuli Jury L To 25 11 out of 25 Houston Astros try to trade couldn't trade mojo. We have to 50 Weaver San Luis Carlos Luke weaver Cardinals Aaron Pettit, so no nice little patch auto to 25 Minnesota Twins I with that one we got Trevor stories of five out of ten nice three color patch another one for the Colorado Rockies and That's Matt Arnold. We got a frame of Noah Cindergarten five out of ten for the Mets When a Christopher It's pretty strong Johnny Bench two out of five that is Cincinnati Reds Going to Alex. We have Joe Panic for the Giants When a Richard Miller five out of fifty last one here is a another color sent out patch two out of fifteen believe them Parsons Alrighty last box. Good luck Tyler Glass now rookie autograph to 50 for the Pittsburgh Pirates Pirates combo is Matt Arnold Cork Luber 11 out of 50 Cleveland Indians Going to Corey We got a little Kenta Maeda five out of 50. I think Kenta was actually a rookie the year before with Cork or see if I want to say but who you know might have been a rookie though. Oh Speaking of them Well, of Corey Seager and Julio Urius to 35 There you go. More Dodgers Trade couldn't trade Moja. We have Dream big Anthony Rizzo Inscription out of 35 Chicago Cubs Danielle with the Cubs We got a frame patch autograph Kyle C 24 out of 30 nice what four color patch Seattle Mariners with that one Charles with that one. There you go, man. We got for the Yankees Carlos Beltran Forgot he played for them good for the Mets and the Yankees, right? Both New York teams Carlos Beltran with that one Yankees on the board Richard and the last one here from 2017 is a Justin Verland 16 out of 30 That's three color patch Detroit Tigers Travis with the Tigers. Alrighty. There you go guys 17 is in the books Let's move it along go to 18. Oh, is that a hundred bucks for me? As you can see the improve the boxes in the cases from one year later And I'm sure they got a lot of complaints of the car. It's just being loose.  I'm sure they got a lot of heat Oh, I kind of thought that was a joke. I thought I was trying to say like Tiger Woods or something Alrighty nice patch here Jackie Bradley six out of 30 for the Boston Red Sox We have out of 25 Julie Jury L That's three color patch for Houston if we're busy bill, of course, they're gonna be late night I'm here as long as you guys want to break We got Orlando Sapeida for the Giants new 100s Have to come back for that. I'll put that over here we have For the Chicago White Sox Nikki Del Monaco Chicago White Sox that is going to Christopher We have Jim tommy for the Indians We got a dual autograph Barry Larkin and Joey Votto 28 out of 30 Cincinnati going to Alex We got big else to know 19 out of 30. I did not watch the game though. The game ended up being seven to six now I didn't last one here in this box.  All right, sweet First box Next one. Yeah, the Jollogerman is an auction as well. That one's at Golden auctions and then the The LeBron James is at Leland's auctions and I think they're both ending around the same time Was it really? Oh, Bennington. Whoa How about a one-of-one? Pittsburgh Pirates Andrew McCutcheon The N in McCutcheon. It's pretty awesome Pittsburgh Matt Arnold nice patch here six out of 25 Eric Hosmer Kansas City with that one Christopher Gerard with that one. Oh, love that he inscribed this door. Look at that 17 out of 25. No, Cindergarten. That is awesome You need to get some 100 so just put that in a stack over here Well, lower Reese Hoskins 17 out of 50. I forgot he was a rookie in this too Philly's But a Chris Walker beautiful autograph beautiful We got a frame Corey Kluber 22 out of 30 that Cleveland has gotten a lot of nice hits so far too Sunny Gray 10 out of 25 nice autograph sunny the wookie man is a sucker Sunny was right Yankees Going to return. How about this guys? What is that third one-to-one already? Manny Machado, I guess you could call this a little logo man, right? one-of-one and autograph For Baltimore Joseph Bart very nice man, and we have a redemption. Remember this is expired We'll be giving you a hundred dollars in break credit and Actually, we pulled this dual autograph earlier in 2017 About Craig B. Joe Jose al-Tube. I want to say it's like right here Dual autograph relic collection, and yeah, you did man. You got one in 2017. I guess in 2018 they decided not to sign and 632,020. So yes, this is expired So I'll be sending you a hundred dollars in break credit and that's a Chris, right Chris K. Yeah, there you go, man So you got one live got a redemption You know, I've heard tops and upper deck say that They could redeem it within a year of the expiration date. So you can always try. I mean it hasn't been a full year yet As we are barely in 2021 in February pretty much last day March So you might still be able to get this who knows maybe give them a give them a car email. Maybe they can work with you It's not too long ago So that's kind of the cool thing, but at least with us, you'll be getting a hundred bucks There you go. Yeah, that sounds awesome. See there you go. They do something at least something right, you know Well, no, it's okay. I mean because we don't know if it's a for sure thing But of course wants to buy us a bunch of dinner than it. It's all good Like yeah, I got a refund was back to a hundred bucks Chris Now it doesn't matter if they accept it great if they don't then All right, another Nikki Delmonico to 50 That's her second one. I think Alright sweet our last box of 2018.  It's one of those guys Dabra tours are rookie in here, too. This is a really strong draft class Or I shouldn't say draft class, but rookie class Wow, so nice two frames That's how low to this last box is gonna be two frames Let's go, let's go, let's go whoa and this this That's like our fourth one of one. I should have had a one-on-one bounty or something one-on-one Harrison Bader rookie the H That looks like an H, right? I Don't know that would be better.  I don't know. It looks like an H though. Oh Let's say it could be a B Could be a B same those carnals Yeah, it looks like a B now I'm thinking about it right here. I think it starts curving Either one nice one. I know this is a R&B one-on-one. Whoo. Yeah All right. Um, so yeah, same those carnals that is Aaron Pettit. Nice yellow patch. This is awesome Looks like maybe some socks Eric Thames time Thames 6 out of 20 Brewers with that one Charles with that one Got a little George Springer Houston Astros Astros times are very sad. He's not an astro anymore There you go another one for Houston Clint Frazier rookie 18 out of 25 Thames Yankees without one Anthony Rizzo two out of 30 Jared Dudley that dude's a chicken Three out of five Buster Posey. That's nice Very very nice That was a really really nice pass to 20 out of 40 Steve Carlton. Who did Oliver one of these players here Phillies when a Chris Walker and last one here guys It is a framed patch auto Corey seager for the Dodgers man Dodgers is raking in this break Jason with the Dodgers.  Imagine the Dodgers were traded away. Alrighty guys last case. Maybe Oliver walks Korean mass singer Oliver's like Wikipedia plus 130 point together in this chat. Good luck guys last case Chris Davis 14 out of 30 We have Michael Chavis for the relics 35 for the Boston Red Sox Now Boba Sheva would be nice Louise Robert any one of those top rookies. How about another one of one? How about that? Vladimir Guerrero senior one-of-one Hall of Fame inscription 2000 sorry Hall of Fame 18 Very very nice Another one-to-one Montreal Expos will be going to the Washington Nationals going to Charles There you go. Nice man. Cool. I'll put it on the schedule Primo This is not gonna be a rookie, but how about a second year? Fernando Tatis, Jr 4 out of 15 for the San Diego Padres going to Matt Arnold Three-color patch That's noise Was that like our fifth one-of-one already? Philly fans. Don't look Two out of ten Reese Hoskins and Bryce Harper dual autograph man. This is a box. All right This is a box Rockeroo I think that says 32 out of 50 angels California Angels Edition Rockeroo There you go Angels going to David there is a redemption, but this was just this past year It's not gonna be able to be a hundred dollars and break right, but I hope it's a nice one How about a autograph relic? collection Green Parallel Glaber Taurus for the Yankees going to Richard Miller. I Don't know what Green Parallel is guys. You guys want to go look that up But it is not expired. Of course you still have another year pretty much a whole year left a year in a month To redeem that so nice here for the Yankees Last but not least how about a little Cal Lewis On-card rookie autograph four out of 50 Very nice Seattle Mariners with that one. It's one of Charles. Oh, did I have some green? Maybe I'll look it up After a damn Hansel so hot right now. He's absolutely right One of ones would have just been raking good Lord Hunter Renfrow for the Padres and the Pirates Padres combo is killing it Look at that Five is the most one of ones I've seen Xander Bogarts time. You should have had a 101 bounty. Everybody would have been eating little Yelly Christian Yellich 11 out of 25 Beautiful autograph Christian Hardly don't really see too many Yellich autographs at least I don't maybe I just don't pull Birds without one Charles Nice patch. Look at this Of course, I was a Yellich patch, but Keston is a pretty nice one and it's had numbered six out of eight What a beautiful patch Look at that Had numbered six out of eight beautiful patch back to be against for the brewers Blake Snow and Brenna McKay both for TB out of 35 Tampa Bay. That's a combo team Christopher with that one Roberto Alomar for the Blue Jays look at that beautiful patch right there. It's like a old school patch. You could totally tell two out of five another one for the Blue Jays Nick with last Bob Mojo This has been an insane three case break Gavin Lux the last one guys is a Beautiful hit good Lord another one for the Dodgers Another logo man, and it's a one-of-one See look at this thing. It's coming out of the car. It's like legit bending over. It's like dangerous CC's of Accia The MLB logo man logo batting batting logo one-of-one For the Yankees That's sick That's seven one-of-ones Yankees is Richard Miller Good one, Phil. Oh my make that eight make that eight Oliver make that eight And we're like getting some old-school players, too One-of-one majestic tag Ryan Zimmerman for the Washington Nationals Charles with that one. We got Trevor story to 50 Colorado Rockies Robin you out The 25 for the Milwaukee Brewers We got a framed Pete Alonzo 10 out of 30 for the Mets dual autograph and patches Xander Bogarts six out of 35 Red Sox that is going to Corey We got Dale Murphy piece of the lumber and Otto to 50 Atlanta Braves Jordan Alvarez 24 to 25 rookie autograph for the Astros Houston had a great break Chris. There you go, man. And last but not least about a little Fernando Tatis, Jr 22 out of 50 Padres Again, I'll top load all these 100 points as I get top loaders after this, but what an amazing break guys. Honestly, this was pretty fun I mean crazy crazy stuff guys one-of-one one-of-one I Should have been a one-of-one one-of-one Tatis frame, which is pretty nice Corey see your frame out of five Buster Posey another one-on-one Harrison Bader one-on-one Manny Machado One-of-one here Andrew McCutcheon Like I said, there was a lot of a lot of one-on-ones in here guys And it'll be in a pretty sweet case I Hope you know everybody did get something. I'm not sure I mean there is customers here that had multiple teams That kind of does help, but I understand that there may have been some teams of course. I didn't hit but really appreciate it guys really strong That's this Russell tag was awesome, too We had like eight one two three four five six seven must be missing one over here. I think Vlad June of lat senior was a one-on-one. I got a top load all these hits though guys, but Really strong guys appreciate it guys. So Chris, I'll send you your hundred bucks for that That expired Actually, no spitballs angels actually got a rock crew It was like a patch auto on there California angels edition But appreciate it guys. There was no randomizer here. No, nothing everything good guys. Appreciate it. Jasper's case breaks calm


I recently purchased the Algato Stream Deck to enhance my online training. After spending a few days with this device, my only reaction is wow. Why did I not purchase this earlier? Let me show you how you can take your online training to the next level with the Algato Stream Deck. Let's get started. First, let's understand what is a Stream Deck. It's a USB device connected to your laptop or your computer. It has a series of physical buttons and as you press each button, you can trigger an action on your computer. The Stream Deck comes in three sizes. The small, which has just six buttons. This one which is the medium size with 15 buttons and then there is a larger version with 32 buttons. You program the device using the companion app. There are hundreds of actions that you can drag and drop onto the appropriate key. In addition to that, you can create your own custom functions. Everything that you can do with the Stream Deck, you can do by pressing the appropriate keys on your keyboard. So what makes the Stream Deck such a great productivity tool? Usually when I do an online training, I will have four to five windows open on my computer. For example, right now I have OBS Studio on one monitor and I have a website on which I am doing the training on the other monitor. In addition to that, I might have PowerPoint open. Let's say it's on monitor number two. And if I'm doing my training via Zoom, then I would have Zoom window open on this monitor. It's always a challenge trying to figure out which window is open on which monitor and switching to the right window at the right time. The problem would be even worse if you had only one monitor. For example, if I'm delivering my training right now and I want to switch to the next slide in PowerPoint, I have to first find out where is the PowerPoint window. It's on this monitor. Move my mouse to the corner of the screen and then click the next button. That's how I would change. On the Stream Deck, I have programmed this key called Next. If I press this key, it will automatically find where is the PowerPoint window and switch to the next slide. Switching to different scenes in OBS is equally easy. For example, if I want to switch to the main camera, I just hit the button and my view changes to the main camera. Or if I want a scene with my picture and PowerPoint in the same scene. If I want to demonstrate something on the desktop, I have created a scene for that which I can change with the press of a button. In my future videos, I will show some of the cool actions that I have programmed on the Elgato Stream Deck. In this video, I only wanted to highlight the benefits that you can gain as an online trainer by including the Stream Deck in your workflow. I hope you found this video useful. Please subscribe to my channel to get more content like this.


Good day. My name is Mupota Pela. I am based in Botswana, which is on the southern part of Africa. And I'm a specialist in the disability sector. I have worked a lot with people with disabilities, different kinds of disabilities, particularly the deaf. And I've also been in a space of civil society organizations where I was working for a council or a federation for people with disabilities where I have gained a lot of insights. But I haven't really had any lived experience of disability. So my insights are purely from a professional environment and the engagement that I've had both nationally and internationally through various engagements. I was invited at CBA 17 to help the IED team in terms of assessing the event itself in terms of accessibility, especially to people with disabilities and to give feedback on what the next CBA could look like in terms of accessibility and engagement of people with disabilities. So in terms of my approach of the whole event, I looked at it in terms of inclusion, accessibility and accountability. In terms of inclusion, I was generally looking at how inclusive the event is holistically and how inclusive. I also considered how inclusive IED is as an organization. So I did some smaller interviews with the staff of IED to find out in terms of inclusion of accessibility in policies and programs and structures. So I also looked at it in terms of accessibility to say how have we made reasonable accommodations for people with disabilities? Is information accessible to people with disabilities? Do we have equitable opportunities in terms of access to physical spaces? So I was looking at accessibility holistically. And then for accountability, it was a matter of is IED accountable? And this one, it speaks to different levels, but in terms of CBA itself, I think that it was accountable in the sense that they had people that could have been assisting people with disabilities. It's just that they were not in attendance. After assessing the entire event, I then wrote a report in terms of what could be done at the next CBA. And I made quite a few recommendations. You know, some of them include partnering with an organization or federation for people with disabilities, because I feel that when you do that, they will better advocate for the issues that you're trying to address. They will better give you access to people with disabilities in different localities. So a partnership like that would be mutually beneficial in the sense that IED would also be giving the organization and platform to be able to participate and give people a chance to participate who have disabilities. And the other thing is that I think it would also be very good if at the next CBA we included a topic that includes disability, like an aspect of disability as one of the thematic areas, and actually have a person with a disability to speak to that topic. Because a lot of times when we make decisions or when we plan and we make the reasonable accommodations without the people with disabilities themselves, we tend to overlook some other things. The other recommendation was to do more in terms of making information accessible to people with disabilities, like maybe if we have sign language videos instead of just text on the website or videos that do not have a sign language interpretation. Or if we also focus on making reports that are online to have screen reader options such that blind people can have access to them. So I feel that a lot is being done, but so much more can be done at the next CBA. Thank you very much.


Every single rocket that has ever launched anything into space, orbital mission, deep space, manned, unmanned, big, small, every one of those rockets has one thing in common, staging. Staging is a fundamental part of any mission launching, whether you're talking about a heavy lift vehicle like the Saturn V or the Falcon Heavy, or something much smaller like an Atlas. Staging is quite simply when one part of the rocket fires before another, so that the cumulative thrust on that payload is what gets it up into orbit. The reason rocket stage is because of weight, or more specifically mass, because a rocket would have to stage leaving another planet just like it does on Earth. Let's take my favorite rocket, the Saturn V, as an example. This was the only part of the payload that actually had to reach Earth orbit. The rest of the rocket, all this down here, was just designed to get that rocket up off the ground and into space. So when designing the launch vehicle, engineers had to take into consideration the mass of the payload that had to reach orbit. But that wasn't it. As soon as you start adding lower stages that have to provide the thrust to get that payload into orbit, that adds mass. And it's not only the mass of the structure of the rocket, it's also the mass of the propellant that has to fire to lift that mass off the ground. Do you see where this is compounding to make things a little bit difficult? It becomes a very interesting balancing act. That only is the payload part of the mass. The rocket structure and the propellant also add mass. But as the rocket burns that propellant, the mass changes. It's that changing mass that opens the door for staging. So let's look back to our Saturn V. As the Saturn V is launching and everything's perfect and beautiful and lovely, all of the propellant in that first stage is being used and burned up. Well once it's done, now the Saturn V is carrying that extra dead weight of the spent stage. There's no point in using more propellant to get more of the rocket off the ground if you don't need it. So what do you do? You ditch the spent stage. That's where staging comes in. Staging is really just a clever way to get rid of dead weight when you're doing something complicated, like putting a payload into space. Now all that being said, there are different ways a rocket can go through staging. Let's start again with the Saturn V. That uses a kind of staging called serial staging. The stages are stacked one on top of the other. The first stage fires first and then separates from the rest of the payload once its propellant is exhausted. Explosive bolts and retro rockets create clear separation between the spent stage and the active second stage, which then has room to fire. Now the Saturn V in this specific case was a three stage rocket, but the exact same thing happens for a two stage rocket or a four stage rocket. It's just that sequence of staging all the way up the rocket's body. Another type of serial staging is called hot staging. It's what the Titan II that launched the Gemini program used. If you look closely at this rocket, you notice, like I was shocked to find the first time I've really looked at a picture, that there are holes in the rocket's body. While it turns out that unlike the Saturn V, which gained distance between the rocket stages when one was finished before the other one fired, the Titan II fired while the stages were still connected. Those holes are vents for when the next stage fired to separate from the spent stage. There are rockets that marry serial and parallel staging. A great example is the Titan III. The main core stage of that rocket called the sustainer stage was a Titan II, and it had solid rocket boosters on either side. Those two boosters separated, parallel staging, before the main stage went through serial staging. Another kind of staging is parallel staging, and we're all familiar with that one because that's what the space shuttle used. The space shuttle's main tank fed the engines on the orbiter, but it needed a bit of an extra boost to get into orbit. Enter the solid rocket motors that flanked the side of the main tank. The boosters as well as the main engine fired to get the shuttle off the launch pad, and then once the SRBs had exhausted their fuel, they were detached and left to fall into the ocean. Now, there's another kind of staging that's pretty rare, but also familiar, though we may not realize it. This is stage and a half staging, and we know it because the Atlas D rocket that launched John Glenn, Scott Carpenter, Wally Shira, and Gordon Cooper into orbit on the Mercury program used stage and half staging. With the Atlas D, the main engine acted as the sustainer. It was the only one that burned to get the payload into orbit. Around the sustainer engine was a sort of booster stage with two extra engines. Now these were side by side, so sort of parallel, but they also fell away vertically, sort of like serial. So basically all three engines would fire to get the rocket off the launch pad, and then once the first stage or half stage was done, it fell away, and that sustainer stage continued on to put the payload into orbit. Of course, the dream, because it would be much simpler and potentially more efficient, would be a single-stage-to-orbit rocket. That is exactly what it sounds like. One single stage would be enough to power a payload into space. That is not something that exists yet, although there are companies that are trying to bring that dream to reality. If you guys like this video, be sure to subscribe so you never miss an episode, and also I want to hear from you. Do you have other questions about rockets, about staging, about the Saturn V, the Titan, the Atlas? Let me know all of your thoughts in the comments below. And of course, don't forget, you can follow me across social media for daily vintage-based content on Facebook, Instagram, and on Twitter. All right guys, thank you so much for watching.


Let's bring in Simon Michelle now from Fig Securities, check in on how the bond markets are tracking and we are seeing yields a little lower Simon certainly after the Fed held off on that September hike but it all becomes I guess listening to Janet Yellen this week with a couple of speeches from her and looking towards December. Absolutely Ingrid, good afternoon and we've definitely seen a bit of a drop right across the curve in Australia, our 10 years back below that 2% level. Interesting really because before we get the December meeting we also get the election and we get the first debate obviously Monday night US time so it's really going to be interesting as those polls tighten we are starting to see a little bit of volatility emerge and we could actually see markets go a little quiet and that could be a key focus before we get to that December meeting I think. Simon it's Tony Davison from Henderson Maxillum just interested we've had a big mean reversion rally in the bonds particularly longer data bonds do you think that the backup that we were seeing two weeks ago is that now completely done as the market reversed its view or do you think we're likely to see that come back again? Well it's really interesting isn't it I mean what we do know is that we are seeing less demand for those longer data bonds but yet we are seeing people take advantage of that kick up in yields and pushing those yields back down you know not dissimilar to what we saw you know 12 months ago as we you know had the same situation approached a possible September move to December so I think while we're still seeing people happy to step up and take demand at the long end of the curve and we are seeing that sort of drift within a fairly tight band we are seeing especially from offshore investors both in the US and in Australia starting to pull back on their holdings of US Treasuries and Sovereign Debt and so it's really going to be interesting you know if that demand keeps decreasing then that's certainly going to have an impact on those yields in the long end. Simon could I just ask on that what do you make of it you know the fact that the demand is reducing particularly in those longer dated issues what should we read into that? Well it's interesting in Australia here what we've noticed is as the government deficit increased and the volume of bonds that the government's had to issue offshore investors have actually pulled back you know so we've gone from about 74% holding to around about 58% holding at the moment so you know they could be aware of that aware of that increasing amount we know the Aussie government's got to issue more over the 2017 year as well so you know it could be a case of them just pulling back as the volume of issuance increases substantially over in the US well you've definitely had you know through Saudi Arabia they've been pulling back China we know been pulling back a lot of those countries that hold US Treasuries have been decreasing theirs and that's generally been because they've needed to utilise those funds elsewhere more so than a pullback in demand for all holding those sovereigns really. Just quickly Mike touch on a domestic bond issuance I know you're watching Virgin Australia looking to issue some potential US dollars what are you making of that? Yeah so Virgin Australia they're a high yield issuer they've got an existing 2019 bond issue about 400 million US dollars that was issued at 8.5 currently trading around six and a half so look I think don't think they'll have a problem with the US dollar issue out there and on the back of that domestically here we've got AAI they're fully owned by Suncorp Group they've been a fairly regular issue they're looking to do a tier two issue and I think that'll see very strong demand. Alright so I'm Michelle always appreciate your time thank you. Thank you. The bond market check.


Hey y'all, welcome to what Chelsea eats. Thank you so much for joining me today. In this video, I'm gonna share with you some information about feminine products that are safe to use. Someone hit me up on Instagram I guess a few days ago and I told her I had a video already posted but I looked at that video. I was like, oh my goodness, that video is terrible. It's so old and outdated. So I decided to do another one. So follow me into Whole Foods Market as I share with you what they have on stock and then I'll show you some other things that I have used that have been successful for me. Start off by saying that I have not used all of these brands that you see here in the cart. I have used the 365 brand and the Natural Care brand over the years. Now I have a favorite but it's not in this picture. I do have experience using Natural Care and 365 and when my favorite brand is not available, I use 365 options and they have everything from the panty liner all the way up to the super pads which you see here. I prefer to wear pads because they help your vagina to breathe, so to speak. Like you're not plugging it up. I don't prefer anything where you stick anything in there like tampons, I don't like tampons and I don't think I would like the diva cup because you have to stick it in there. I've never tried it. I don't have experience with it. There are people who love it, swear by it and some who hate it. So I think if you want to use something like that it's something that you're gonna have to try on your own to see if it's something that you like. Now this one I believe is one of those ones that you can wear for a 12 hour period and it's supposed to hold a lot of blood. I have no idea whether or not it holds enough blood to help someone who is a heavy bleeder but I can say that if I was a heavy bleeder I would be using probably a pad and I'd probably couple it with some type of period underwear on top of that that would have be a absorbent. But if you follow what I teach you to do and you do it the way I tell you to do it you won't be a heavy bleeder for long. All of these brands listed are available online and they are available in Whole Foods Market and probably other natural food stores and these brands are clean. Some of them are organic and some of them don't have the organic seal on them which you don't need something to have an organic seal on it for it to be clean. It's just sometimes it's so expensive to get the organic seal on your particular product that some companies opt not to do it but they do things higher than organic at times. You will find that with food as well especially if you go shop at a farmer's market. A lot of these farmers can't afford to buy the organic seal but they grow their foods organically and that's the same thing that happens with some products. All of these products here I will list in the description box below this video in case you cannot find them locally where you live. But let me just say this to it is important to wear clean sanitary products. It is important for your healing journey because when you put those toxic pads on what they do is cause more problems. They have chemicals in them, they have bleach in them and who knows what in them. Plastic in them at times and we need to stay away from those things so that we can heal reproductive issues. If you don't stay away from these, wearing these toxic pads can actually be the difference between you bleeding heavy or not or shrinking a fibroid or not or healing cysts or not. So it is important for you to use clean sanitary napkins. And one of the things that I found too is that people are afraid to change to the more natural supplies because they think that they won't hold them like say, like with the traditional pads will but what you have to do is just keep on trying them until you find the one that works for you. So that's what I have to say about that. So hopefully that's helpful. This next product I'm gonna share with you is called La Lova. I did a review on this product several years ago. There's a link down in the description box with the video and the reason that I like these is because they provide extra protection if you're a heavy bleeder. And also some people like to wear them if they're not a heavy bleeder, they just use them as their period solution. Like they don't buy pads, they wear them and if that's something that you're interested in trying, then you might benefit from these. I can tell you that whenever I tested them, I used them alone and I was not a heavy bleeder at the time I was trying these. So I have no reference point on whether or not they help heavy bleeding or not. But what I would say is if I was a heavy bleeder, I would buy these to have in addition to my pads so that you could avoid those accidents that can happen. I have had so many of those situations where I was at work in the office and all of a sudden I got blood all over my dresser, over my pants or something like that because I'm bleeding heavy and then you become a member of the Black Clothes Museum because that's all you can wear because you never know when you're gonna break out into a heavy bleeding episode. And these types of period panties help for that type of a situation. I'll put a link to this particular product down in the description box so you can check them out for yourself. So some people love wearing these reusable waterproof bamboo charcoal menstrual pads.   And they have instructions on how to clean them, how to get the best out of them. They're inexpensive and one of the other things is they have four and a half stars out of five and there's 4,800 ratings so they get a high review. I have never used these in my life but I know women who love to use these reusable pads and they swear by them. And so I don't have experience with them so I can't tell you what to do but they're worth a shot, they're cheap. Now these next pads I'm gonna show you are called Rayelle and they are my favorite. Everybody has their own preference. These happen to be my preference. I have been using these for years and when I don't use these I will use the 365 pads but I still love these so much better. These have a range of all kinds of options, everything from the panty liner to the disposable period cover. They even have incontinence pads as well if you struggle with that. And I just love this brand. They're not the cheapest and they're not the most expensive but for me they've been reliable and they are clean. That's what I love about them. And normally you can find these in your everyday store like your Target, your Walmart, especially in this area where I live in Raleigh, North Carolina but if you can't find them you can always get them online so I will link them below just like the rest of the products that I have shown you here on this video. So that's it for today. There are many, many more options out there. I just showed you a few. When I first started changing over my menstrual cycle pads there weren't that many to choose from. These days there's so many options and everybody is not gonna like the same. So I'm telling you that my favorite is Raelle but you may hate Raelle.   I would love to hear what you use and what you like as you have transitioned to the clean pad space. I don't know what to call it, clean feminine hygiene products. I guess that's what you would wanna call them. I would love to hear what you have to say about them. And if you like this video please like it, share it as well as comment below and let me know your thoughts. If no one told you that they love you today Chelsea in North Carolina loves you very much and I'll talk to you all soon. Take care.


With the first selection in the 2001 NFL draft, the Atlanta Falcons select, Michael Vick, quarterback, Virginia Tech.


One of the tendencies that I think we as Catholics have when something new comes out it has replaced the old. It's a horrible way to do theology. It's a horrible way if you've ever tried to study scripture of just you know saying well it doesn't matter what all these these old dead guys said here's what I think. Well no that's not a proper way to study it. What we've done is we've also annotated it with the 1992 Catechism so that when you as a reader are going through this you could say there's something that might be a little bit different here. I wonder if it's the same or I wonder if anything's changed or how has it changed how is the wording changed. You'll be able to quickly see how a priest explained the Catechism of the Council of Trent and where to go to see how the church explains it today.