 So hello everyone, my name is Nati, and I'm going to talk about the paper secret sharing schemes for general and uniform access structures, a joint work with Benny Eppelbaum, Amos Beymel, Oriol Faras, and Oded Neu. So I will start this talk by telling you about what we do in this work. So in our paper we construct new improved secret sharing schemes for the two types of access structures. The first is secret sharing schemes for any access structure, and the second is the secret sharing schemes for specific family of access structure that are called uniform access structures. I will define you later in this talk what are uniform access structures. So secret sharing schemes were defined by Shamir and Blakely for the threshold case, and later by Ito Saito and Nishizeki for the general case. So we have a set of N parties, P1 to PN, and an access structure gamma, which is a family of subset of parties that contains the authorized set, which I will explain in a minute. For example, one of the best known access structures is the threshold access structure that contains all subsets of parties of size at least K. So the dealer holds the secret S, and it creates random shares S1 to SN, according to the secret and the random schemes, and privately communicates each SY to the party PI. So in the secret sharing scheme we want that set of parties will learn the secret if and only if it is in the access structure gamma. So more precisely, the correctness requirement of a secret sharing scheme is that any set of the access structure is an authorized set that can reconstruct the secret S from the shells by applying some reconstruction function, and the security requirement is that any set that is not in the access structure gamma is an authorized set that should not learn any information about the secret. So I will show you a simple example of secret sharing schemes. Consider the following set of two parties, P1 and P2, and the one-bit secret S, and the access structure will be the set of both parties P1 and P2, which means that parties P1 and P2 should learn the secret, and the party P1 by itself or party P2 by itself should not learn any information about the secret. So in a secret sharing scheme for this access structure, the dealer choose a random-bit R, give it to party P1, and it gives to party P2 the bit R XOR S. So it is very easy to see that both parties P1 and P2 can reconstruct the secret S by XOR in their shells, and each one of parties P1 and P2 cannot learn any information about the secret since it's all the random bits. So in our work, we consider linear secret sharing schemes. So I will define you linear schemes. So generally, we say that the secret sharing scheme is linear if the shares that the dealer gives to the parties are generated by a linear function. So more formally, the secret is an element for a finite field. And to create the share, the dealer choose random elements from the finite field, and each share will be a vector over the field in which each coordinate is a linear combination of the secret and the random elements. For example, the previous scheme we saw in the previous slide is a linear scheme. In this scheme, party P1 gets a bit R, and party P2 gets the bit R XOR S. So this scheme is linear over the field F2. So we have some good reasons to study secret sharing scheme. The original application of secret sharing scheme is to store sensitive information and to protect data that should be restricted from some parties. And now the secret sharing scheme are using many cryptographic protocol, such as multi-party computation and attributes-based encryption, and many more. So now I will tell you about some of the known results for secret sharing schemes. So we measure the efficiency of secret sharing schemes by the size of the shares. So for this, we define the share size of secret sharing scheme as the size of the largest share. So the upper bounds for secret sharing scheme for realizing any X structure. First, Ito Saito and Nishizaki showed the linear scheme realizing any X structure with share size of O of 2 to the power of N. More than 30 years later, this was improved by Liu and Vaikun Tanatan. They showed a non-linear scheme realizing any X structures, X structure with share size O of 2 to the power of 0.994N. And they also showed a linear scheme realizing any X structure in which the share size is O of 2 to the power of 0.999N. So let's see what we know about the lower bounds on the share size. So, SIRMAS show the existence of an X structure that requires share size of omega of N over log N in any secret sharing scheme that realizing it. And by counting argument, it can be shown that there exists an X structure that requires share size of omega of 2 to the power of N over 2 in any linear secret sharing scheme that realizing it. So as you can see, for general non-linear secret sharing scheme, there is a huge gap between the best known lower and upper bounds. And for linear scheme, we know much better. So the next slide that we tell you about our main results, which is a secret sharing scheme realizing any X structure. So for any X structure gamma over a set of N parties, we show a non-linear secret sharing scheme realizing gamma with share size of O of 2 to the power of 0.892N. We also show a linear scheme realizing gamma with share size of O of 2 to the power of 0.942N. So to construct this scheme, the following schemes, we show a general technique that enable us to construct a better secret sharing scheme realizing any X structure. So next I will tell you about some of the ideas of our schemes. So you and Vankun Tanatan show that for a parameter delta, we can decompose any X structure gamma to free X structures. The first is gamma top that contains the large sets of size at least half plus delta N. The second is gamma bot that contains the small sets of size at most half minus delta N. And the first is the X structure gamma mid that contains the medium size sets of the size between the above two values. So you and Vankun Tanatan show that any X structure gamma equals gamma top intersection gamma mid union gamma bot. So to construct secret sharing scheme for gamma, it is sufficient to construct secret sharing scheme for gamma top, gamma bot, and gamma mid. This follows for non-closure properties of secret sharing schemes. So in our work, we construct improve secret sharing schemes realizing the X structure gamma top and gamma bot. And this scheme uses combinatorial covers. So now we go over the ideas of the construction. We start with a secret sharing scheme realizing gamma with share size of O of 2 to the power of 0.994 N, the scheme of you and Vankun Tanatan. We choose set B of side 0.99 N. When the value 0.99 is arbitrary just for the explanation, it can be replaced by any value between half and one. And we consider the X structure gamma B that contains all subsets of parties from the X structure gamma bot that are contained in the set B. So by the result of you and Vankun Tanatan, there is a scheme realizing the X structure gamma B with share size of O of 2 to the power of 0.984 N. In the next step, we define a cover of the X structure gamma bot as a family of sets B1 to BL of size 0.99 N, such that every X structure of gamma bot is contained in at least one of the X structure, in at least one of the sets B1 to BL. So we can decompose the X structure gamma bot as a union of X structure gamma B1 to gamma BL. And by the non-closure properties, we can realize independently each of the X structure gamma B1 to gamma BL and get a secret sharing scheme for gamma bot with share size of O of 2 to the power of 0.993 N. So for the X structure gamma top, we show a similar secret sharing scheme with the same share size. And using the secret sharing scheme of you and Vankun Tanatan for the X structure gamma mid, we get a secret sharing scheme realizing gamma in which the share size is O of 2 to the power of 0.993 N. So we already get a better scheme than the scheme of you and Vankun Tanatan. So in the next step, we can do the same, but starting with the X structure with the new share size and get an improved share size for the X structure gamma bot in which the share size is O of 2 to the power of 0.992 N. So to get our desired scheme, we choose the optimal parameter and use this construction a constant number of time recursively and we get the scheme with the desired share size. So we also analyzed the way we choose the optimal parameter to get the optimal share size. In this figure, you can see the optimal choice of the parameter. And overall, we get our scheme with share size of O of 2 to the power of 0.892 N and linear scheme with share size O of 2 to the power of 0.942 N. So our analysis can be applied to any secret sharing scheme for the X structure gamma mid, which means that any improvement in the X structure gamma mid will lead to an improved secret sharing scheme for any X structure gamma using our scheme for gamma top and gamma bot. So next, I will talk about secret sharing schemes for uniform X structure, uniform X structure were first defined by insane and she and later by Beymel Kushilevich and Nissem and Applebaum and Arches. So uniform X structure is defined by a parameter K between 1 to N and we call such an X structure K uniform X structure. In K uniform X structure, any set of size greater than K is authorized and can learn the secret also some of the sets of size exactly K are authorized and also can learn the secret and any sets of size smaller than K is unauthorized. So K uniform X structure is specified by the set of size exactly K. In other words, K uniform X structure contains all the sets of size greater than K and some of the sets of size exactly K. For example, we can consider the next X structure gamma which contains the sets P1, P2, P3, P2, P4 and P7 and additionally all the sets of size at least 4. So we can see that this X structure is free uniform X structure. So we have some good motivation of studying secret sharing schemes for uniform X structures. So the first reason is that it is an interesting family of X structure. It looks very simple family of X structure and also is more efficient scheme than the best know secret sharing scheme for general X structures. Uniform X structures can also serve as a test ground for studying secret sharing scheme for any X structures. By recent work, we know that such X structures can be used to construct secret sharing schemes for general X structures. And finally, it equivalents equivalent to conditional disclosure of secret's protocol or shortly CDS protocol which is a cryptographic primitive that enables a set of parties to disclose a secret to a referee, undersecretary to a referee under some conditions. So now we'll tell you about our main results dealing with secret sharing schemes for uniform X structures. So in this work, we construct three secret sharing schemes for K uniform X structures. The first scheme is a secret sharing scheme for long secrets of size T and the shell size of this scheme is O of K squared times the size of the secret T. In this case, T equals to at least omega of 2 to the power of n to the power of K squared. The previous best known schemes for such a secret as shell size of O of e to the power of K times the size of the secret T. The second scheme we show for uniform X structure is a secret sharing scheme for secrets of 1 bit and this scheme has shell size of 2 to the power of O of square root of K log n. The previous best known scheme for such a secret has a shell size of 2 to the power of O of square root of n. Our third scheme for uniform X structure is a linear scheme for 1 bit secret and the shell size of this scheme is O of 2 to the power of H of K over n times n over 2. So in this case, the function H is the binary entropy function and the previous best known scheme that is linear has shell size of O of 2 to the power of n over 2. So for this scheme, we also show a matching lower bound of omega of 2 to the power of H of K over n times n over 2 and this lower bound shows that our scheme, that our linear scheme is optimal up to a very small polynomial factor. So to construct all the above three schemes, we use implicitly or explicitly in a transformation for a multi-partite CDS protocol to uniform X structure. So to conclude this talk, in this work, we study secret sharing schemes for general X structures and uniform X structures. We show an improved secret sharing scheme realizing an X structure and new secret sharing schemes that realize uniform X structures. The open problems that arise from our work is first try to show an optimal linear secret sharing scheme realizing any X structure. And another question is closing the huge gap on the shell size for nonlinear scheme. So there are already some results that uses our work. The first one is a linear secret sharing scheme realizing any X structure in which the shell size is O of 2 to the power of 0.762n and there is a nonlinear secret sharing scheme realizing any X structure in which the shell size is O of 2 to the power of 0.64n. And both of these schemes uses our construction for the X structure gamma top and gamma bottom. So that's it. Thank you. We have time for one or two questions. Yes? So when you say you construct a cover of gamma bottom with sets B1 to BL, so what kind of sets are those and what exactly is the number L and what is exactly their purpose? The number L we show a probabilistic proof about the number L. It depends on the size of the sets in the cover. So this decomposition is meant to reduce allow you to combine secret sharing schemes for less general access structures so I don't understand the goal of this cover set. About the composition of the gamma bottom? Yes. So there is a non-closure properties that show that if you have secret sharing schemes for a few access structures there is a secret sharing scheme with the same shell size for the union or the intersection of those access structures. For key uniform access structures, does it matter? Should it matter for efficiency the size of the specification? So I don't know if you said you can specify your access structure with the number of sets. So does that matter? In uniform access structure the access structure is defined by the parameter k. Yes, it's the parameter k. So the shell size will depend on this parameter. Yes, but it will also depend on the because the specification is on the because you said it's not all sets on k but it's a few sets. It contains some of sets of size k and all sets of size bigger than k. So it doesn't matter which sets of size k do you have in this access structure. So you can realize it with the desired shell size. Thank you. Any more questions? If not, then let's thank the speaker again.