 Hello everyone, I am Han Nyniu from Shanghai Jiao Tong University. Today I am presenting our paper, Push the Limit of Viennian's universal circles, simpler, tight and more compact. This is a joint work with Yu Yu, Su Yao Zhao, Jiang Zhang, Viennian and Zheng Kaihu. The universal circles can be seen as general-purpose circles. For any circle C of size up to n, we can efficiently encode C with control B's PC, such that the universal circle, configured under PC, is a functional equivalence to the original circle C. For any circle C of size up to n, and 5 in 5 to 2, Viennian proposed a construction of universal circles of size 19n log n. The universal circles can be seen as graph object called edge universal graphs, EOG in short, and the size of EOG is 4.75n log n. As we will see, the size of a UC is roughly 4 times that of the corresponding EOG. Before our work, the best size-efficient EOG. has size of 4.5n log n, and it is known to be the lower bound by 3.64n log n in the Viennian framework. In this work, we give a construction of edge universal graphs of size 3n log n, and we also lower bound the size of our EOG by 2.95n log n, and our framework. Notice that the size of our EOG is even smaller than the lower bound in Viennian framework, which is not contradiction, because our construction is not under Viennian framework. A natural application of the universal circles is to preserve functional privacy in multiparty computation. A multiparty computation can preserve input privacy. For example, anise has input x, and a bubble has input y. Multiparty computation can guarantee that anise and a bubble can get the result CXY, and link no further information about inputs. The protocol guarantees the input purposes of the bubble and anise. However, in some situations, we made one to preserve functional purposes which can be achieved based on universal circles. For example, bubble has an input xC, then bubble can encode in C into the configuration stream PC of the universal circles. The problems reduces to the multiparty computation of evaluation of a public function UC on input x and the configuration stream PC. In the end, the purposes of input x and the accuracy is preserved. The universal circles can be seen as H-universal graphs. Here I want to introduce H-inbounding and H-universal graphs. H-inbounding can be seen as injection from graph G to a larger graph G star. The H-inbounding includes two paths. First, it maps from the load site V to V star one by one. Then, it maps every H in G to a path in G star in a way that is H is joint. Meaning that every two paths in G star have no common edge. Notice that there are two types of loads in G star. The bigger one, called the polar loads, is mapped from a load in G and the smaller one that have no premises in V. Such nodes can be exact because the load, the map is only an injection, not a backjection. G star is called an EOG, if and only if any DAGG of size N and phi in file D can be H-inbounding into G star. We typically set phi in file of DAGD to one or two. We need framework including three steps. First, reducing universal circles to EOG2. Namely, edge universal graph for any DAG of phi in file 2. Second, reducing EOG2 to EOG1. Finally, reducing the problem of construction of EOG1 to itself but with a much smaller size and this reduction is recursive until the underlying EOG1 is smaller enough and can be constructed by hand. Here, we introduce the fourth step, reducing UC to EOG2. The left figure is a circle with two inputs, x, y and two gates, x or n and two outputs. The circle can be seen as a DAG. The right figure is a universal circle which can be seen as a DAG, an EOG. We showed the edge inbounding from the DAG in the left figure to the EOG in the right figure. In the left figure, the load side includes x, y, x or n. In correspondence, in the right figure, we can also see x, y and x or which are called polo loads. The EOG includes other smaller loads called control loads. First, the control load with one input and two outputs is called split which copies the view of the input to the two outputs. Second, the one with two inputs and two outputs we call it x-switch which can configure into two cases switch or on-load switch. Finally, the control load with two inputs and two and one output we call it y-switch which also work in two cases. Copy the view of the left input to the output or copy the view of the right input to the output. The x-switch can be implemented by four BASICAL GETS. A y-switch can be implemented by three BASICAL GETS and a universal GET can be implemented by three y-switch. For example, x-switch needs one control load BISC. If C is set to zero, then it becomes a long switch left in left out. Otherwise, it works in the switch models left in right out. Now, we introduce a second step, reducing EOG2 to EOG1. Every instance of DAG2 can be divided into two instances of DAG1 which can be as inbounded by two instances of the corresponding EOG1. We merge the two EOG1 into a single EOG2. For each DAG1, we're as inbounding it into an instance of EOG1. Finally, we combine two instances of EOG1 into one instance of EOG2 by eliminating the overlapping polar loads. Therefore, the size of EOG2 is two times of the size of EOG1. And minus the number of overlapping polar loads which is n. The final step, reducing EOG of great size n to EOG of smaller size typically divided by a factor k. We can repeat this step many times until the size is smaller enough. Before introducing this reduction, we showed two important notions used in this reduction. Augmented DAG and superload. First, the augmented DAG can be seen as a super-DAG. For example, DAGG has k-loads, P1 to Pk and 3 edges. The augmented DAG can add k-input loads and k-output loads. Then, the augmented DAG can add 3 edges either from an input node to a load in the middle or from a load in the middle to the output node. As long as the resulting augmented DAG is still of the grid 1. Now, with the augmented DAG, we can define the notion of k-way superload. A k-way superload, or SNK in chute, is a special EOG that can add any augmented DAG k. For example, the above figure is an augmented DAG with 4 polar loads, 4 input loads, 4 output loads, and the below is a corresponding 4-way superload. As defined previously, edge-inbounded mapping includes load mapping and edge-mapping to a path. For example, edge-in-b mapping to a path from edge-in-b5 to edge-in-b9. Now, we reveal variance-recursive reduction from EOG of size n to EOG of size n over k, which is achieved based on the building block k-way superload. First, we put n over k superloads in rows. Then, we merge that the output nodes of each superload with the input nodes of the next immediate superload component rows. These merge nodes can be divided into k-sides. For each side, for each side, we use a smaller EOG1 to combine these combination nodes and the polar loads of the smaller EOG1, one by one. The resulting merge nodes have two inputs and two outputs, which can be implemented by x-witch node. It can be proved that any DAG of size n can be edge-inbounded into this graph. And therefore, it gives a construction of EOG, when it proposed two kinds of superloads, two-way superloads of size 5 and four-way superloads of size 19. Note that the input and output nodes are not counted into the size of superloads. The result into the corresponding EOG of size 5 n log n and 4.75 n log n respectively. Our construction does not follow VINNian framework. The left hand is a VINNian framework and the right hand is our intermediate construction. The difference between the two constructions is that in VINNian framework, we combine the output node of the above superload and the input node of the next below superload into a single node, which will then match it with the polar of the smaller EOG. Well, in our intermediate construction, we combine the input node and output node of the same superload into a single node and merge it with a polar of the smaller EOG. It is easy to say that both the VINNian framework and our intermediate construction are about the same size and both achieve edge inbounding. But we don't seem to accomplish any improvement with this intermediate construction. We have two observations about our intermediate construction. First, node A is nx-wage, which have two options. But it is unnecessary that a pasta and the smaller EOG slew edge number 4 that leave it intermittently. In other words, if a pasta has edge number 4 then its next edge must be edge number 3, not edge number 2. Second, if a pasta passes through the node A from inside the smaller EOG then it will end the corresponding superload. In other words, edge 1 will be followed by edge number 2, not edge number 3 on the pasta for edge inbounding. Therefore, the option for x-wage node is always fixed. Our end construction take these redundants into the account and remove the x-wage node The node A- is the previous node of the node A in the smaller EOG then we delete edge number 1 and edge number 2 and the edge from node A- to the corresponding superload. Node A- is the next node of the node A in the smaller EOG then we delete edge number 3 and edge number 4 and the edge from the corresponding superload to node A- plus. This leads to a more compact construction of EOG without affecting the function of the universal edge inbounding. Based on our introduction, the size of EOG2 is 2 times of the size of EOG1 minus N In our construction, the size of EOG1 is k times of the size of the smaller EOG1 plus N over k times of the size of k-way superload minus N combination nodes The advantage of our construction over venous framework is minus N highlighted in red. This leads to the size of EOG2 of our construction. Moreover, we prove that a low bound of 2.95 N log N for our construction. When instantiated with a two-way superload of size 5, we can get a construction of EOG2 of size 3 N log N. The size of our construction 3 N log N is very close to the low bound 2.95 N log N. Finally, we visualize the comparison with the previous works. Our construction improves upon the best previous work by about 33%. There are some related works about universal circles. Thanks for listening.