 Okay, so let's talk in this session is on secret sharing schemes for very dense graphs by almost by male oil for us And you will mince the talk is be given by way on Okay, so I'm Muriel for us and I will present this young work with almost very male and you are means on Secret sharing schemes for very dense graphs First I will give an introduction to secret sharing then I will give an introduction to graph secret sharing And finally I will present our results So secret sharing a scheme is a method to protect the secret and the way to protect the secret is The following the one that holds the secret which is called the dealer of the scheme Generate some pieces of information from the secret that are called shares and then the dealer sends privately One share to each participant These shares are generated in such a way that some subsets of participants can recover the secret these subsets are called authorized and Some other subsets cannot obtain any information about the secret these subsets are called forbidden In any scheme we define the access structure as a family of authorized subsets The first secret sharing a schemes were represented by Shamir Blackley and later it was a tonish Jackie presented a way to construct secret sharing a schemes for every access structure The schemes we consider in this work are unconditionally secure and are perfect Which means that the subsets not in the access structure are forbidden that these subsets not in the access structure Cannot obtain any information about the secret So secret sharing is a cryptographic primitive that has many applications as for instance multi-party computation threshold cryptography access control attribute-based encryption oblivious transfer But for using secret sharing for these applications We need of efficient schemes and in particular shares have to be small In this work in order to measure the efficiency of the schemes We use the total the total share size of a scheme Which is the sum of the size of the shares divided by the size of the secret? So we know that there exist efficient schemes for certain access structures For instance for threshold access structures on n participants Shamir and blackly presented the schemes with total share size n where n is the number of participants And we know that for every secret sharing a scheme n is the best possible total share size So it's the lowest total share size for a secret sharing a scheme Therefore the schemes with total share size n are called ideal But not every access structure admits an ideal a scheme and in general it's not known Which is the most efficient scheme for an access structure? There are some methods to construct the schemes for every access structure For instance the ones presented by been a law and like to Simone's et al Brickle Kashmir and Big Derson, but these methods in general are not efficient Are not efficient because for most access structures The total share size of the schemes constructed by these methods is 2 to the big one After these constructions we have that in general the best upper bound on the total share size of the best scheme for an access structure Is 2 to the big one? We have also some lower bounds So sir must prove that there's a family of access structures for which the total share size of any scheme is big omega of n Square divided by log n And this is the best known lower bound So it's the highest lower bound for sharing a secret with respect to an access structure And so these are the best upper and best lower bounds on the total share size for sharing a secret For an access structure and so there is a huge gap between upper and our bounds And one of the main open problems in secret sharing is to bridge this gap In fact the main motivation of our work was to bridge this gap And in particular we want to we want it also to study this open problem That is which access structures are hard, which means which access structures require large shares to be realized We have studied these problems for graph access structures and we have fine New upper and lower bounds on the total share size and we have extended The techniques for finding upper bounds to homogeneous access structures and to the general case So now I will talk about graph secret sharing So an access structure is a graph access structure if the minimal authorizes subsets Out of size 2 in this case The access structure defines a graph in which the Participants are the vertices of the graph and the edges are the minimal authorizes subsets therefore given this graph a Set is authorized if it contains an edge for instance a set one two and a set is forbidden If it does not contain any edge for instance a set one five six Graph secret sharing a scheme is a scheme with graph access structure So in a graph secret sharing a scheme all minimal authorize subsets out of size two It is a simple case, but it is also interesting and it has been studied in many previous works And in some of them it has been the first step for obtaining more general results Every graph admits a very simple construction that works as follows so given a graph we can Share the secret independently for every edge in the graph for instance this graph has four edges And so we should share the secret for every edge And so for the edge one four we should pick a random and Number and set it to participant one and then send to participant for the secret plus the random number And we can do the same for every edge And in this way we we obtain a scheme that realize these graphs and The total share size of this scheme is two times the number of edges There are some special graphs that admit ideal schemes and some of them has been used in our works So a click is a graph in which every pair of vertices is an edge and Observe that the click defines a threshold access structure of threshold two therefore it can be realized by the Shamir secret sharing a scheme also complete bipartite graphs admit ideal schemes and Stars that are a particular case of complete bipartite graphs also admit ideal schemes We have some results. There are some results on the The total share size for graph access structures The total share size of the best scheme realizing a graph access structure is upper bounded by big O of M because of the trivial construction I presented before But also, it's upper bounded by big O of n squared divided by log n This derives from works of buble it's blue to et al Erdos and Piper and we have also some lower bounds It's known that there exists a family of access structures for which the total share size of The scheme realizing them is big omega of n log n This derives from works by Van Dijk blue to et al and Sirmas And also we know that there exists a family of access structures for which the total share size of the linear schemes Realizing them is big omega of n to the three over two this derives from a work of Bamel Gallon Patterson So our main motivation was to bridge the gap between upper and lower bounds and the total share size for graph access structures and so we want to look for efficient construction for graphs in this way we can improve the upper bounds on the total share size and we want also to look for hard graphs Since the total share size of As of the schemes realizing a graph are up around it by big O of n On n squared divided by log n. We try to solve the following question Is there any graph with total share size big omega of n squared divided by polylog n? Since every graph admits a scheme with total share size two times the number of edges in a hard graph The number of edges has to be big. Therefore. We study very dense graphs That are graphs in win in which n choose to mine that have n choose to minus l edges with l small therefore, we ask ourselves if Is there any very dense graph with total share size big omega of n squared divided by polylog n Now I will present our results Our main result is the following theorem If a graph has n choose to minus n to the one plus bit edges for some beta between zero one Then it admits a scheme with total share size big O of n to the five over four plus three bit over four log n Since the best total share size of Secretaring a scheme is n we can look at this bound as n times n to the one over four plus three bit over four log n In this in this work We do not take into account the logarithm factors And so we will use the big O till the notation and we will say that The total share size is big O tilde of n to the five over four plus three bit over four So there are some direct consequences of this theorem We asked ourselves if there was a very dense graph with total share size big omega of n squared divided by Polylog n so after this theorem the answer is no Therefore in a hard graph both the number of edges and n choose to minus the number of edges must be big For obtaining this result we use Coverings of graphs by easy graphs that these graphs that are easy to realize as clicks by prototype graphs and stars We use the probabilistic method and colorings of graphs So how we use coverings for constructing secret sharing a schemes so given a graph G We cover it by easy graphs that these in but in this case We consider graphs that admit ideal schemes as clicks complete by prototype graphs and stars so given a graph We obtain a covering in this case the covering is made by two clicks It is a covering because every edge in this graph is in one of these two Graphs and every edge in this graph and in this graph are in G Therefore, this is a covering of G and Once we have the covering the scheme for G consists on sharing the secret independently for every piece of the covering So we should share the secret independently for the two clicks in this way, we look for a small coverings in order to obtain efficient schemes and We have found a new method to obtain Small coverings for very dense graphs and the method is the following So we describe the graph G as a click minus the excluded graph G prime So given G this is the graph we describe but that we describe it as a click minus the excluded graph G prime Observe that an edge is in G prime if and only if it is not in G Therefore once we have the excluded graph we We notice that each coloring of the excluded graph G prime used to subtract of G That is if we have a coloring of G prime observe that If two vertices have the same color it means that in the excluded graph There's no any edge between these two vertices and so it means that in the original graph. There's a vertex There's an edge between them Therefore every click between vertices of the same color will be a sub graph of G So if we take many random colorings of the excluded graph G prime we end with a covering of G So now we will present some corollaries of the main result So if a graph has inches two minus one plus be the edges for some beta smaller than one third Then the graph admit the scheme with total share size small o of n to the three over two We also have another corollary for graphs with Many edges, so if a graph has n choose two minus l edges and L is a smaller than N over two then it admits a scheme with total share size n plus b code of L to the five over four Moreover, if L is a smooth ease of smaller order than n to the four over five Then the graph admits a scheme with total share size n plus small o of n We have also considered a more general Question which is deleting minimal authorizes subsets from an access structure And we study first this problem for graphs So let G be a graph and let sigma be a scheme for the graph with total share size are We know that if we add a ledges to the graph G by using the trivial construction We saw before we can construct a scheme for the new graph with total share size R plus 2 L Where is the number of edges added? But what happens if we delete edges from G so in general it is not known and We have obtained this result if we delete L edges from G the new graph So if we delete L edges from G which is a Graph that that meets a scheme with total share size R The new graph admits a scheme with total share size big O tilde of square root of L times n times R If L is bigger than R over n and R plus 2 times L times n if L is a smaller or equal than R over n There's a direct consequence of this theorem that is the following if G admits an efficient scheme Then the graphs that are close to G are not hard That is if G admits an efficient scheme and we delete or we add some edges to G The neck the the resulting graph will also admit an efficient scheme We have extended these techniques To homogeneous access structures that are access structures in which all minimal authorizes subsets are of the same size and to general access structures and We have provided new techniques and constructions and we have given answers to the following problems that is Deleting minimal authorizes subsets in a threshold access structure and deleting minimal authorizes subsets in any access structure We have obtained also lower bounds on the total share size all the previous results were upper bounds on the total share size So for every L between 2 and n There exists a family of graphs with interest 2 minus L edges whose schemes have total share size at least n times n plus L This should be compared with the previous result I show In that for these graphs there exists a scheme with total share size n plus Bco of L to the 5 over 4 and We have obtained also these results that for every beta between 0 and 1 there exists a family of graphs with N choose 2 minus n to the 1 plus beta edges whose schemes have at least big omega of beta n log n and That there's a family of graphs with interest 2 minus n to the 1 plus v the edges whose linear schemes Have at least big omega of n to the 1 plus beta half This should be compared with the main result of this work that says that for these graphs There exists a scheme with total share size big omega of n to the 1 plus beta half times n to the 1 over 4 plus beta over 4 and ends if beta increases both the both the lower bound and the upper bound increases and So for very dense graphs we have reduced the distance between upper and lower bounds on the total share size so This is the last slide So in this work we have studied secret sharing schemes for very dense graphs We provide new upper and lower bounds on the total share size for the scheme realizing realizing these graphs We answer to the following question. Does exist any hard very dense graph? The answer is no We provide new techniques for the construction of secret sharing schemes and we extend them to homogeneous and general access structures and The open directions after this work are to find hard graphs to search new techniques for finding Lower bounds and the total share size and to bridge the gap between upper and lower bounds on the total share size