 Welcome to this Icelandic session. So I guess we can start. So we have one thoughtful offer in this Icelandic session. So it's a pleasure going to do a talk with you. We'll reward you on your work with Christian Meissen, Tegel, Sean Hallgren, Gustav Lauer, and David Meissen on super simple Icelandic graphs and the work that brings relations and so on. Thank you, Oger. So you may have heard a bit about the disciplines. There's something around the corner up here, and this is a great poll in the repertoire we're using today. So one of the directions that has been suggested for our business is this fact, is the whole SDG discipline, right? So this is a quite interesting approach by me. The problems we're using there are natural from a preliminary point of view. The first problem we have is the possibility of initial time, even in quantum autism. But of course, this big value, it has the inverse value. And the inverse is above that. So our results need more, need more angles. So we show some relations between natural problems, variants in this area, some problems that have appeared in the inverse form. And we also show efficient solutions to problems that we require to initial time and now our SDG for unit time. All right, so this is the outline of the talk. I'm not assuming there's... that the audience is very familiar with SDG, so I'll first give an overview of a very big overview of SDG and related problems. Then I'll talk about the main motivation for this work, which was the hatching machine by Charles Verlin and others that uses SDG problems. And then I'll very briefly describe our results and very, very briefly, as if it works for us, to ease the values to obtain them. Okay, starting with our keys. So the key to prime, and it is both old curves, oddity curves, pursuit of curves, so we can define right this way. And as with E, from curve E1 is essentially along to the whole portion. So it's codified here, from E1 to E2, to part of the E to part of the SDG. In very stress coordinates, you'll write this as a rational mass in this way, right? So those are all variables. The degree is, or the degree is rational mass, so this will be the mass of this guy and this guy. There's a special case to be addressed when E1 is equal to E2. So in this case, instead of putting this in nitrogen, E2 is an amorphous, so it's an nitrogen from E2 itself. And there were no examples of that, scarving indications for these. So this is it for the interaction. So SDG problems, as you see them, they all, and in critical, they all will continually in nitrogen ease, whatever that is, or some variance in this problem. So it's actually a tricky to find. So if you would like this problem to be hard, you must be large. But if you remember this question before now, it came to a great start, and this was usually, you can't even represent that efficiently. So there are ways to get around it, but what I want to say here is that the natural distribution for the output is not even quite right. So you need some kind of tricky way there. If you're interested in amorphous, then it's also very tricky to find. So it is more in general, at least we hope so for UTO, but there are curves for each. It's very easy. You do know all the amorphous, and there are curves, this is from the SDG, and there are small curves. Also, whatever the curve is, you always know scarving indications. So if you want this to be hard, you can have to define this problem for random curve stress, and you can do this from the output. Okay. So let's not go ahead of amorphous and rain if you don't care about mutation. So we know that this has, well, a rain structure, as I said, an operation of a curve and a position like an SMI. Right? So for surface stretchers, the amorphous and rain is what is called a maximum order in the Gaussian natural. The Gaussian natural are provided via infinity. And in fact, there is a variation between all surface-unit curves on one side up to ghetto application, and all maximum orders in the Gaussian natural up to some application top. Okay. So this is our first response that this variation is currently response. This is a paper by Derey in 1931 already. So the correspondence essentially sends the curve, or j-parade, to a maximum order isomorphic to a set of arms. Right? That's what Derey responds. I'll come back to this. So one more slide of general context. You can often see 9 to 3 graphs in those papers. So over here, very close to the end ocean. So this guy is responding to the other materials. They're at this one second circle there, and each of them will respond to an isology. So each of them will be kernel of 3D L isology. So they're only at this one isology starting from an epic curve. So you can draw a graph where essentially every epic vertex is once per senior an epic curve, or j-parade, and every edge will respond to one isology. So there's an edge, two curves, and there's an isology. Those graphs are undirected because if you have an isology, it's still essentially taking the first graph. In the first case, those graphs are very nice. Well, all j- and isology satisfy every square, which leads to efficiency improvement. But also, the graphs are an addition. So if you don't know what this is, this is an essential attention graph that applies like a very greedy version of the random order of the justice and the unilaterals as well. So the actually problem that we just did can be confused into finding paths that describe this graph. We want to find a short path. Okay, so that's it for the introduction. Now let me describe the last question and explain how it was motivated this work. So how should a large history and a short history be very interesting in this graph? So this is nice looking graph. Make sure it's taken from a science magazine. So every work is essentially a geo-reference. And there's an action to where this is exactly. So you start from one here and then depending on you catch your message into bits, depending on the next bits you go either one side or the other side. So in this case, it's a three-year geo-reference, so I'll have to make one step. So every bit of it is the next choice of the issue. So give me a message this way you can choose and the other half of this is essentially the final geo-reference. Now if you want to rate this for example in terms of drainage resistance the problem with this thing is that if you give it two curves and you have to find a 90 degree L to be some sort of connecting this curve to this curve. If you want to find collisions then you want to find 12 L to be the same curve and that is equivalent to finding a 90 degree L to this curve. So if you want the original solution was to break this, to study a lot of securities and a natural approach to that is to use these correspondence between super-spirited there is one side and maximum order is the question and it decides that all the geo-reference can be translated into nice actually right so just to remind you this is where it ends, this is the A&E to the end of the spring so the goal was to kind of translate all those drainage resistance properties into equivalent problems for the question after that to solve those problems seems to be a question after that I don't think you can save the effect to the journey, etc. Now the first process research programmer only solved it in my own paper they called this modern and feel in ads for me so the situation stated that those problems continue to set and we show that there everything I can solve right it should be quite a time and actually I'll I mean one way to detail but this is just not solving some diagonal equation and that kind of so this is what we did here and at the time there were a few more results that we didn't do to figure out not as to be good as much as it is in this book there it was kind of an issue to publish that so here we go results that are actually in its favor to do this learning so they are a task about the optimal solve process and then also we dodged 50 different outcomes so at least there are a few words about that so an attack on the hash machine in a property so when the initial curve that is chosen to define the hash machine is a special curve or some curve we actually don't remember from the beginning so when it's very special we have a different attack on the hash so coming on how special is this this is very special but don't give a hand there is no way today we don't know how to compute this for so many years if not by starting from one of those special curves so in a sense a natural way to implement this especially is to use one of those special curves and if you don't our attack was still 5 back in terms of the backdoor attack so this is a more effective than the ancient special curve the second result that we had is when we were constructing during correspondence in one direction maximum order in BPD I can confuse the JnR and other correspondence so I can compute a curve that has an enormous domain that is maximum order so there were a lot of that to see and there were a whole exponential time or I was missing a sheet by the time there were certain uses on the other hand we also considered a lot of problems to still erase one of them is the opposite of this one so you give the curve and you want to compute maximum order corresponding to that curve and I still believe this is hard so essentially the two results together can give a one way of functioning life so you can compute this in one direction and another in the other one the second problem that we considered is JnR so here you give the curve and you want to compute the rational mass that generates this JnR so it is more significant to this it says that both graphs comprise the description of the JnR except that in this one the description is more abstract so it is in terms of maximum order in BPD and here you actually require the rational mass as a model and finally the problem we considered the security of the energy of operation and energy distance are equivalent to any of those problems now there is actually a result in the paper where I had a target that some of the suggestions can also be achieved if you have many parts to solve in this problem that you define ok I want to say a few words about the tools that I used in the paper one of them is now going to convert question and ideals to hydrogen so to remember these problems so if you know a curve for which you know the hydrogen ring so if you have these three problems set upon points and you'd like to press move the problems move the information through the path and state ideal from easier so this can be done and this is what this is what it is so what you'd like to do is to look at the kernels and identify the kernels so the algorithm is there it's not efficient less the norm of the ideal which is to create the energy is the local persons so by persons I mean if you write this so these are part of the EIT you want each EIT to be all of this so the algorithm is only efficient when it can be all of these persons but it doesn't have a solution now you can go into that we have this algorithm first which essentially is to use this ideal by using an ideal with persons so to do together if you use an ideal and you want to compute the answer please you first use this ideal like another one with a person and then you can write as much as you can get just to you want to write this and say a few words about the effect that we have so first we start here from a very special curve easier for which I don't get an argument my goal is to compute the equation so this means an argument of easier of working with me and that will give the equation a good point of this I see the cycle is like fraction of okay so this is the algorithm so first you compute an algorithm well an algorithm was here we've known that to be so that would correspond to an algorithm of to be out to be and because it's special the normal equation that you have to solve this is manageable and this was actually done already in my calculator then once you have this guy you actually have a collision the legal question is saying so the collision has to form a sequence of ideals which should be right to see it as close so you just take you wish the ii modulo mgi oh sorry you take both your alpha modulo mgi this and then you like to use this transition algorithm from ideals to ideologies but the problem is that this guy is very big when e becomes bigger you can just apply it out to the transition algorithm so the trade we have here is that essentially for each i in sequence you're going to be replacing ii by another ideal which is and is portion strong then you can apply a transition algorithm to the transition as such from easier to ei not yet, ei and you can recompute the world path from easier to itself to all these sequences first so that's that's a sketch for partial attack so again a sketch I want to provide this is both deductions I'll just get one of them so remember that we just didn't break this ii in terms of collision of our brainages and we have all these problems about completing animal treatments so what we show you in the favorites is a minstrel-to-stune problem so constructing dirty guidelines in reverse operation so even the jameron can do the responding by small orders this is a problem which essentially the same line description of maximum order you actually want to rationalize SNL boost and collision permissions so here in this book I'll just look at the reduction from here to here which was kind of maybe it was over in previous publications I'll just start so as we go here to just stress what we achieved is to have a curve and somehow you've even it's a morphosome ring already but if you only do this in terms of the in terms of the Z-basis for a maximum order of 18 to 15 the question is from this representation of the aneurysm can you re-confuse an efficient description of the of the masses of the aneurysm okay so here's a sketch of how you can achieve this with the aneurysm ring then we hope it's an ideal that connects all zero to this guy to this guy so that's quite efficient and then the theory of proton algebra you know that this O will be included into this order here or this Z is here okay so the trait will need to express O as elements of this form now if you can say this in terms of isochism you get any element in this aneurysm ring well I've heard if you look at arbitrary basis for this guy the degree of the very arbitrary amount to this is not but if you can search for a basis that can be expressed as elements of this form so if you see that the aneurysm ring is included into here you can return the basis for the aneurysm ring in the form of phi composed of the aneurysm ring in this guy composed with the yellow phi divided by the degree of phi okay so this is not natural representation for an aneurysm ring because of this fraction here but what we show in the paper is that this representation of the aneurysm ring will nevertheless allow you to do everything you like to do if I really think about the arbitrary part of this aneurysm ring this is a representation of both of us and those who return the answer in this guy in an aneurysm ring now in order to do this representation what you need to apply this is the ultimate so the whole circle you will look at a special curve to compute this ideal and then you will apply this out of the maths to get a person's ideal and then you express a basis for this guy in this form of phi and this is the ultimate person so that's it okay so to conclude if you take a random each of your curve prove that the chart of our aneurysm ring is secure if and only if the aneurysm ring is a part of the circle so that's what's been completed already but not read it down and this requires special tools to prove actually so for later so there was some confusion there's quite a need to return the aneurysm ring do you know what it means in the aneurysm ring and we show that and you can go from one to the other we also have a tax effect we have a fish nozzle and we show that if you use an initial curve it's very special then you can actually compute the relation to this function so if only more so a special curve remember those curves are kind of the natural ones used in the function and also even if you don't use that there's a risk that the aneurysm ring is a bit harder if you want to conclude as to keep all the results of your states there so there is room for improvement there although I would partially call solving a very complicated project first and many people are interested in an ICDH so this is not really able to ICDH so this is personally a ICDH it is partially in the sense that if you can compute an aneurysm ring so if you can compute an aneurysm ring then you can also break ICDH as we showed in the paper which is pretty obvious because the aneurysms ring are shiny but this may not be enough it was a paper a different class that she showed that the problems we used in ICDH could be easier to solve than those ones so it is relevant but maybe it is easier to solve so thank you for your questions in the collusion that you showed it is a collusion with what you call the void message so I was wondering isn't that like a particular collusion and maybe you could get around that what do you call the void message so what I need is that the message just doesn't include any disk but in fact you can generalize this but you have a collision so that is the cycle there so you take the message that corresponds to this cycle and you bag it with an arbitrary message and you are able to do the same so if you take it so let's say this is bad and then you have an arbitrary message and then there is a rush and then there is a rush do you consider the problem of like a higher degree like a higher degree into curveball or in a variety of cases during correspondence of the result does that make sense I have my thought back in water and he has a little bit and that's what people have been looking at and that's natural to consider so it's not like a it is a big problem so it's not a big issue by resulting in that but that's why I know that and there are people who fully can generalize this but that's why this is what you see here and I had a final question so you mentioned this back on the CTL hash function the initial curve is not random so as any of this people and it's not how we see that we can do that if you take another curve if you take an instant patient on CTL on the CTL hash function if you take another random patient like this if those patients that is what you have in mind I did demand that you have in mind well no sorry I did demand the everything that you have to say I did not implement stuff that transcends the traditional game but I think that that would be considered well used so I think it's a decision of the mind thank you thank you