Applications of finite geometry in coding theory and cryptography

We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory

On Ideal Secret-Sharing Schemes for kk-homogeneous access structures

A $k$-uniform hypergraph is a hypergraph where each $k$-hyperedge has exactly $k$ vertices. A $k$-homogeneous access structure is represented by a $k$-uniform hypergraph $\mathcal{H}$, in which the participants correspond to the vertices of hypergraph $\mathcal{H}$. A set of vertices can reconstruct the secret value from their shares if they are connected by a $k$-hyperedge, while a set of non-adjacent vertices does not obtain any information about the secret. One parameter for measuring the efficiency of a secret sharing scheme is the information rate, defined as the ratio between the length of the secret and the maximum length of the shares given to the participants. Secret sharing schemes with an information rate equal to one are called ideal secret sharing schemes. An access structure is considered ideal if an ideal secret sharing scheme can realize it. Characterizing ideal access structures is one of the important problems in secret sharing schemes. The characterization of ideal access structures has been studied by many authors~\cite{BD, CT,JZB, FP1,FP2,DS1,TD}. In this paper, we characterize ideal $k$-homogeneous access structures using the independent sequence method. In particular, we prove that the reduced access structure of $\Gamma$ is an $(k, n)$-threshold access structure when the optimal information rate of $\Gamma$ is larger than $\frac{k-1}{k}$, where $\Gamma$ is a $k$-homogeneous access structure satisfying specific criteria.Comment: 19 page

Linear Codes from Some 2-Designs

A classical method of constructing a linear code over \gf(q) with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over \gf(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented

Ideal Secret Sharing Schemes: Combinatorial Characterizations, Certain Access Structures, and Related Geometric Problems

An ideal secret sharing scheme is a method of sharing a secret key in some key space among a finite set of participants in such a way that only the authorized subsets of participants can reconstruct the secret key from their shares which are of the same length as that of the secret key. The set of all authorized subsets of participants is the access structure of the secret sharing scheme. In this paper, we derive several properties and restate the combinatorial characterization of an ideal secret sharing scheme in Brickell-Stinson model in terms of orthogonality of its representative array. We propose two practical models, namely the parallel and hierarchical models, for access structures, and then, by the restated characterization, we discuss sufficient conditions on finite geometries for ideal secret sharing schemes to realize these access structure models. Several series of ideal secret sharing schemes realizing special parallel or hierarchical access structure model are constructed from finite projective planes.Comment: This paper was published in 2009 in the "Journal of Statistics and Applications Vol 4, No. 2-3", which is now inaccessible and has been removed from MathSciNet. I have decided to upload the paper here for those who wish to refer to i

Algebraic Techniques for Low Communication Secure Protocols

Internet communication is often encrypted with the aid of mathematical problems that are hard to solve. Another method to secure electronic communication is the use of a digital lock of which the digital key must be exchanged first. PhD student Robbert de Haan (CWI) researched models for a guaranteed safe communication between two people without the exchange of a digital key and without assumptions concerning the practical difficulty of solving certain mathematical problems. In ancient times Julius Caesar used secret codes to make his messages illegible for spies. He upped every letter of the alphabet with three positions: A became D, Z became C, and so on. Usually, cryptographers research secure communication between two people through one channel that can be monitored by malevolent people. De Haan studied the use of multiple channels. A minority of these channels may be in the hands of adversaries that can intercept, replace or block the message. He proved the most efficient way to securely communicate along these channels and thus solved a fundamental cryptography problem that was introduced almost 20 years ago by Dole, Dwork, Naor and Yung

A remark on hierarchical threshold secret sharing

The main results of this paper are theorems which provide a solution to the open problem posed by Tassa [1]. He considers a specific family Γv of hierarchical threshold access structures and shows that two extreme members Γ∧ and Γ∨ of Γv are realized by secret sharing schemes which are ideal and perfect. The question posed by Tassa is whether the other members of Γv can be realized by ideal and perfect schemes as well. We show that the answer in general is negative. A precise definition of secret sharing scheme introduced by Brickell and Davenport in [2] combined with a connection between schemes and matroids are crucial tools used in this paper. Brickell and Davenport describe secret sharing scheme as a matrix M with n+1 columns, where n denotes the number of participants, and define ideality and perfectness as properties of the matrix M. The auxiliary theorems presented in this paper are interesting not only because of providing the solution of the problem. For example, they provide an upper bound on the number of rows of M if the scheme is perfect and ideal

A Remark on Hierarchical Threshold Secret Sharing

The main results of this paper are theorems which provide a solution to the open problem posed by Tassa [1]. He considers a specific family Γν of hierarchical threshold access structures and shows that two extreme members Γ and Γv of Γν are realized by secret sharing schemes which are ideal and perfect. The question posed by Tassa is whether the other members of Γν can be realized by ideal and perfect schemes as well. We show that the answer in general is negative. A precise definition of secret sharing scheme introduced by Brickell and Davenport in [2] combined with a connection between schemes and matroids are crucial tools used in this paper. Brickell and Davenport describe secret sharing scheme as a matrix M with n+1 columns, where n denotes the number of participants, and define ideality and perfectness as properties of the matrix M. The auxiliary theorems presented in this paper are interesting not only because of providing the solution of the problem. For example, they provide an upper bound on the number of rows of M if the scheme is perfect and ideal

Sharing Computer Network Logs for Security and Privacy: A Motivation for New Methodologies of Anonymization

Logs are one of the most fundamental resources to any security professional. It is widely recognized by the government and industry that it is both beneficial and desirable to share logs for the purpose of security research. However, the sharing is not happening or not to the degree or magnitude that is desired. Organizations are reluctant to share logs because of the risk of exposing sensitive information to potential attackers. We believe this reluctance remains high because current anonymization techniques are weak and one-size-fits-all--or better put, one size tries to fit all. We must develop standards and make anonymization available at varying levels, striking a balance between privacy and utility. Organizations have different needs and trust other organizations to different degrees. They must be able to map multiple anonymization levels with defined risks to the trust levels they share with (would-be) receivers. It is not until there are industry standards for multiple levels of anonymization that we will be able to move forward and achieve the goal of widespread sharing of logs for security researchers.Comment: 17 pages, 1 figur