 In our talk, we'll present new algorithms and analysis for the problem of some preserving encryption. This is joint work between myself and Dr. Scott Yellen. The problem of some preserving encryption and creating some preserving encryption schemes was first introduced by Tajik Edel in 2019. The idea is you want to encrypt vectors so that they have the same sum. So you start with a vector of integers and you want to encrypt it also as a vector of integers, but with the same sum as the original vector. In addition to preserving the sum will also typically have a bound on the components of the vector. So both the plain text and the cipher text, each point in the vector is going to be bounded between zero and a component bound of D. Tajik Edel presented this problem along with an application. So the application was the problem of thumbnail preserving encryption. The idea is that you want to encrypt an image so that the thumbnail of the image is preserved. So both the original image and the encrypted image have the same thumbnail. The thumbnail is created by dividing an image into B by B blocks of pixels. Each block of pixels is replaced by a single pixel whose value is the mean of the original pixels in the block. So if we take each block in the image and we imply some preserving encryption scheme. So we encrypt each block with some preserving encryption, then each block in the original image and the encrypted image will have the same thumb sum, thus the same mean, and we'll end up with both an original image and an encrypted image that have the exact same thumbnail. In addition to preserving the sum will also bound each pixel between zero and 250th up. Tajik Edel present this problem and an encryption scheme. So the idea is that we're going to again we start from a vector of integers, you're going to shuffle this vector of integers until we get a random shuffling. Then we're going to pair adjacent points in the vector to create a perfect matching. So we're just going to pair points that are next to each other to create a perfect matching. Then independently for each pair in the matching, we're going to replace it uniformly at random from one of the valid pairs of integers that we could replace it with. So we're going to look at all pairs of integers with the same sum, and that respect to the component bound and we're going to choose uniformly at randomly from each of these pairs of points. So we're going to look at all pairs with the same sum, and we choose uniformly at random. If you repeat this algorithm enough times, you'll end up with a sum preserving encryption scheme. In our talk, we'll go into more detail on the background of this problem and previous work, and we'll present our results. So we, we give three main results. The first is a formal mixing time bound. Essentially an upper bound on the number of times we need to repeat the algorithm given by Tajik et al. So how many times do we need to repeat this? How many times is sufficient? We give an upper bound on that. This formal proof uses a path coupling technique with which we'll also discuss at a high level in the talk. In addition, we'll present two algorithms for ranking and unranking these integers with a given sum. So these are practical algorithms that we can use. We can take the rank and unrank algorithms and combine them in a rank in cipher unrank approach to come up with new sum preserving encryption schemes. In addition, we'll give performance results comparing all three of these algorithms, both the shuffling based algorithm and the rank in cipher unrank.