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

Perfect difference systems of sets and Jacobi sums

AbstractA perfect (v,{ki∣1≤i≤s},ρ) difference system of sets (DSS) is a collection of s disjoint ki-subsets Di, 1≤i≤s, of any finite abelian group G of order v such that every non-identity element of G appears exactly ρ times in the multiset {a−b∣a∈Di,b∈Dj,1≤i≠j≤s}. In this paper, we give a necessary and sufficient condition in terms of Jacobi sums for a collection {Di∣1≤i≤s} defined in a finite field Fq of order q=ef+1 to be a perfect (q,{ki∣1≤i≤s},ρ)-DSS, where each Di is a union of cyclotomic cosets of index e (and the zero 0∈Fq). Also, we give numerical results for the cases e=2,3, and 4

On balanced complementation for regular t-wise balanced designs

AbstractVanstone has shown a procedure, called r-complementation, to construct a regular pairwise balanced design from an existing regular pairwise balanced design. In this paper, we give a generalization of r-complementation, called balanced complementation. Necessary and sufficient conditions for balanced complementation which gives a regular t-wise balanced design from an existing regular t-wise balanced design are shown. We characterize those aspects of designs which permit balanced complementation. Results obtained here will be applied to construct regular t-wise balanced designs which are useful in Statistics

Further combinatorial constructions for optimal frequency-hopping sequences

AbstractFrequency-hopping multiple-access (FHMA) spread-spectrum communication systems employing multiple frequency shift keying as data modulation technique were investigated by Fuji-Hara, Miao and Mishima [R. Fuji-Hara, Y. Miao, M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inform. Theory 50 (2004) 2408–2420] from a combinatorial approach, where a correspondence between frequency-hopping (FH) sequences and partition-type cyclic difference packings was established, and several combinatorial constructions were provided for FHMA systems with a single optimal FH sequence. In this paper, by means of this correspondence, we describe more combinatorial constructions for such optimal FH sequences. As a consequence, more new infinite series of optimal FH sequences are obtained

Perfect Hash Families: The Generalization to Higher Indices

Perfect hash families are often represented as combinatorial arrays encoding partitions of kitems into v classes, so that every t or fewer of the items are completely separated by at least a specified number of chosen partitions. This specified number is the index of the hash family. The case when each t-set must be separated at least once has been extensively researched; they arise in diverse applications, both directly and as fundamental ingredients in a column replacement strategy for a variety of combinatorial arrays. In this paper, construction techniques and algorithmic methods for constructing perfect hash families are surveyed, in order to explore extensions to the situation when each t-set must be separated by more than one partition.https://digitalcommons.usmalibrary.org/books/1029/thumbnail.jp