 Hello, everyone. This is Stilio. Today, I'm going to talk about large precision homomorphic sign evaluation using few TFUG bootstrapping. It's a joint work with Daniela Michantro and Yuri Podekoff. So first, some background information. In this paper, we use functional or programmable bootstrapping. What that means is that basically, we can evaluate an arbitrary mega-cyclic function basically for free, while also performing some bootstrapping over L2B psychedelics. This is first introduced in DM-15. And it has two constraints. The first one is that it requires an exactly function. Exactly basically means that for some input x and zq, you need to satisfy that f of x equals to negative of f of x plus q over 2. You can see this is a relatively strong restriction. And the second restriction is that we can only evaluate functions for messages up to k bits of precision, and usually ks equal to 5 or 6. Basically, the runtime is exponential in the number of bits. So for more than 5 or 6 bits, the runtime will be relatively impractical. And there are two bootstrapping methods discussed in the paper. The first one is few, introduced in DM-15. It can efficiently support arbitrary secrets. But the bootstrapping key size is relatively large. And the second method is TFUG. It's introduced in CGGI-16. And it provides better runtime performance with more secrets like binary alternative and the bootstrapping key size is much smaller. Please see Mp21 for detailed comparison and see this new paper for a third method. This comes after our paper is published, so it's not included in the paper. In this paper, we use functional bootstrapping as a black box. So any of these methods will work for us. And the main result of this work is that we added some new powerful tools to the TFUG, a few schemes. And the mentors are the following. The first one is large precision comparisons. And why comparison first? Because comparisons are widely used in ML, vaccine inference, or DPP training, and data science. And comparisons are also considered hard to be used via homomorphic encryption. As you know, the current runtime is exponential with the number of bits. And we want to make the runtime linear scale. And the second tool we provide is the large precision arbitrary function evaluation. Now it only supports necocyclic function and supports only small precision. We have implemented our schemes and the implementation is open sourced and open FUG and passive libraries. Let me conclude this abstract talk by describe our very high level idea. And our first observation is that the comparison is equivalent to a sine evaluation. So to compare A and B, we just need to find the sine of A minus B. And the sine evaluation is equivalent to computing the most significant bits and the computing the MSB is the central part of UQP2. To evaluate it for large precision, our method is to remove the large number stitches gradually. For example, we have this 18 bits. We first remove the last six bits and then remove the central six bits, the middle six bits. And then when we have only six bits left, we can use a few extra bits to directly extract the most significant bit. However, directly remove these bits by things like emotional switching is not acceptable because they will introduce some errors to know what is exactly the technique we used and you should use it in the paper. Please come to the full talk. It will be on Tuesday, December 6th, 525 and it will be GMT plus eight time zone. Thank you for listening.