 Σας ευχαριστούμε! Σήμερα θα συνεχίσω για το μας εργασίο με το δημιουργείο του Βαμμάνθου, η εξογραφία εξογραφίας με αρκεσή, βοήθεια σε καρδιά εξογραφία, εξογραφία με δημιουργείο, νομίζω ότι Ταρικ έτσι έγινε ένα καλύτερο ενδραφείο σε εξογραφία. Έχει ένα κλίον, who owns a document collection, και θέλει να οδηγήσει την κλειογραφία σε έναν δικανατήριο κλειοδητο. και δεν έχουμε any search capability. So, in searchable encryption we create an index, we encrypt the index, we encrypt the document, and we upload everything to the untrusted cloud. Later the client can send encrypted queries and take back the encrypt results. In searchable encryption we have some unavoidable leakages in order to have efficient constructions which are the setup leakage, the total leakage prior to the query execution and the query leakage which is divided to search pattern whether a search query και το αξιωτικό παράδειγμα. Θα δοκιμηθούνται αυτές οι φιλές που δοκιμηθούνται στην CERD's Query. Βλέπουμε ότι η κρύψη της CERD's is secure. Αν η δημιουργία δεν δημιουργεί τίποτα τίποτα από τις αυκολογές της CERD's. Παρακολουθούμε τίποτα πιο στοιχείο. Οι δημιουργίες δημιουργούνται από τοιχείο της CERD's. Η CERD's Query δημιουργούνται από τοιχείο της CERD's Query. Εκείνονται και τίποτα από τις αυκολογές της CERD's. Αυτοκλήων μπορεί να διαπρεσsekτε επειδή τους εξηγέντες εξηγέντες. Θα δοκιμηθούνται αυτή την καθηγητηρισμότηteς. Ξεκ koόκρυψη της CERD's. Αν η δημιουργία δεν δημιουργεί τίποτα από τις αυκολογές της CERD's. Υπάρχει η δημιουργία από τις εξηγέντες εξηγέντες. To scale a sensible encryption for big data. The problem here is that we can achieve optimal locality just by reading all the time... ...the entire index which is not efficient. This is the reason why we need to define another metric which is called 3-deficiency. The number of memory locations that we read per result item. So in the first example, 3-deficiency was optimal... ...while in the second example, 3-deficiency is order of n. Επίσης, για τα σχεδιασμότητα, we need to maintain both locality and read efficiency as low as possible. Can we achieve optimal, optimal in both terms? There isn't a possibility result in EuroCrip 2014, which actually tell us that we cannot achieve at the same time optimal locality, optimal read efficiency in linear space. Let's explore the locality of where searchable encryption schemes, who have schemes that in order to achieve optimal or bounded locality, optimal read efficiency, they have to sacrifice and they have to blow up the space. In 2016, in stock, we have worked from Asarov et al, the one choice allocation scheme, which is the first scheme that achieves optimal locality, sub-logarithmic read efficiency, and linear space. In the same paper, the authors present an improvement. They propose two choice allocation algorithms, which reduces the read efficiency. They achieve sub-logarithmic read efficiency under the assumption that they don't have a keyword list in the data set with size more than n to the power of 1 minus 1 over log log n. In the next presentation, Edo will present an improvement of two choice allocation algorithms. In our work, what we did is we combined the benefits of both words. In particular, we proposed the first scheme that achieves optimal locality, sub-logarithmic read efficiency, and linear space without having any assumption about the maximum keyword list size that we can support. Let's build some intuition behind the searchable encryption schemes with good locality. Let's assume that we have three keyword lists with sizes 4, 5, and 3. Let's ask a fair question here why we don't store these keyword lists in a way that we can retrieve them later, all of them with optimal locality, optimal read efficiency, and have a linear space. The reason is the following. If we access the blue keyword here, we don't learn information only about the blue keyword, but we learn information about the input data set and the future queries. In particular, since we don't have enough space left and right, the server now knows that we will not have in the future any query with result size 5 or more. A second approach in order to solve this problem is to get rid of this deterministic allocation algorithm and for each keyword to decide where it starts in an array of super buckets uniformly at random. We can decide the starting point uniformly at random and we can throw the keyword list here. We can see now that by doing that, the keyword lists now have overlaps. In particular, we can see that there is a bucket that the blue, green, and red keyword have an overlap. This is promising in terms of security, but the question is what is the maximum bucket size here? Using the same intuition, one choice allocation actually proved that if we set the number of buckets to be n over log n times log log n, then they proved that the worst case bucket capacity is 3 times log n times log log n, and we can see overflows only with negligible probability. They perform the random allocation, they pad up to the maximum capacity, and every time now that they want to actually access the red keyword, they have to read all the buckets with optimal locality and logarithmic read efficiency. In the second scheme, in the scheme of two choice allocation, based on the assumption that there is a bound to the maximum keyword list size that they can support, they significantly improve their read efficiency and the algorithm works as follows. We first start the keyword lists, we start for the most frequent keyword lists, and for each of them we make two random decisions and we decide where to store this keyword list based on the decision that minimized the bucket load. In particular, in this example for the green keyword, we will choose to store the keyword list in the second choice. Now every time we want to access the green keyword, we have to read both choices, we have to read the buckets for both choices, for security purposes. So this is what we have from the related work. We have a scheme which works without any limitation and achieves logarithmic read efficiency. We have a scheme that works in this region and achieves sub-logarithmic read efficiency. This is the two choice allocation. And our goal is to provide a generic scheme that achieves optimal space, optimal locality with sub-logarithmic read efficiency. So we have to provide a solution to this particular range. How we did that, our first step was easy. We used the analysis of two choice allocation in a way that we set the parameters that now two choice allocation works in a bigger range, and that still achieves sub-logarithmic read efficiency. This is how we handle what we call small keyword lists. The problem is also trivial when we want to access huge keyword list sizes. The keyword list sizes that have size bigger than and overlook the power of gamma. In that case, just by reading the entire index all the time, we can achieve sub-logarithmic read efficiency. So the real problem boils down to this particular region. In order to solve this problem, we actually divided this region to two sub-brances. In particular, we will focus in this talk only to medium keyword list sizes due to time constraints. But you can see the paper for how we can solve the problem for large keyword list sizes, since we're proposing a multi-level keyword size compression approach. So let's focus on the medium keyword lists. So our starting point was a paper from 2003, from Sanders et al, for offline two choice allocation. What is offline two choice allocation? We have m-balls and 10 bins. Now for each ball we make again two random choices, and we store the ball in both choices. After we finish with all balls, we run a max-flow algorithm, which has the goal to minimize the bucket load. And Sanders et al proved that the maximum load is m over n plus 1 with very high probability. But this works only for single-size keyword lists. So our idea was to use one offline two choice allocation per size, and then somehow merge. So let's explain how this works. Let's assume that s is the smallest keyword list size that we have to handle with medium keyword lists. We organize this in an array of m-super buckets. Assuming that all the keyword lists have a size power of 2, we create another array for size 2s with m over 2 buckets, et cetera. And for each of them we run an instance of the offline two choice allocation. Let's assume now that this is the output of the offline two choice allocation per size, and let's see how the merging works. We create this gray array here, and we start the merging procedure just by storing the entire column here in the first super bucket. Similarly, we fill the remaining super buckets. The question now is, what is the maximum size of this super bucket, because this is what will define the redeficiency. In order to see that, we define as ks the number of keyword lists with size s and as bs the number of super buckets for size s. We know from the analysis of offline two choice allocation that the maximum load for each size and for each bucket will be ks over bs plus 1. If we take the summation and set carefully m to be n over log to the power of gamma n, we prove that the capacity of each bucket is at most log to the power of gamma n. But what is the catch here? The overflow probability for each bucket is not negligible, which is essential for the security of searchable encryption. In particular, it's 1 over bs. In order to tackle this problem, we had to extend the analysis of offline two choice allocation. In particular, we proved new probability bounds for OTA, and we proved that the probability that more than log square n lists of sizes will overflow is negligible. If we know that, now we can handle this overflow list differently. In particular, we create one stash per size and we store this overflow list in these stash. So if we do that, now we can guarantee that the maximum capacity of each super bucket is log to the power of gamma n and the overflow probability is negligible. So how we access this complex infrastructure? For this keyword, we have to access two things in the gray array, the two choices which are outputs of the offline two choice allocation, and we also have to access the stash without revealing which position we want to access. This is essential. Otherwise, we leak information about the input distribution as well. So how we can do that? One solution is to read the entire stash, but this actually blows up the read efficiency. It requires log square n read efficiency. So this is not a solution. So the second solution was to actually create a new RAM with the requirements that I will describe below and solve the problem. The requirements are the following. We need a new RAM that has optimal locality. We need a new RAM that has zero failure probability. Since we will apply this RAM only to the overflow list, which are log square n elements, and finally, we want a new RAM with bandwidth overhead, a small square root of n, a small omicron square root of n bandwidth in order to achieve sublogarithmic read efficiency. We did that. We created a new RAM which meets these requirements using ideas of square root of n RAM, hierarchical RAM, and the amortization technique from Goodrich et al. One last thing is that we cannot use this idea for any size bigger than this, and the reason is the following. If the maximum size is bigger than n over log square n, then the stashes are starting becoming having size super linear, which is like, and then we'll have overall super linear index size. This is the reason why we had to handle the keyword list which are bigger than this bound here differently. This is why we proposed the large allocation algorithm in our paper. Let's conclude the work. In this work, we proposed the first scheme that achieves sublogarithmic read efficiency. It goes closer to the non-lower bound. In order to do that, we used the analysis of two-choice allocation. We proposed our own approach for medium keyword list sizes, which is based on offline-to-choice allocation. We proved new probability bounds from this well-known problem. We created a new RAM that we are using in both for medium and large keyword list sizes. We didn't have the time to cover it, but we proposed a multi-level keyword size compression approach for large keyword list sizes. Finally, for huge, we just performed a sequential scan. As a future work, we believe that this problem is very interesting and intriguing. An open question is, can we go closer to this lower bound? A second thing that is very interesting for us is to try to extend this locality analysis for dynamic searchable encryption schemes. Thank you.