Communication efficient and strongly secure secret sharing schemes based on algebraic geometry codes

Secret sharing schemes with optimal and universal communication overheads have been obtained independently by Bitar et al. and Huang et al. However, their constructions require a finite field of size q > n, where n is the number of shares, and do not provide strong security. In this work, we give a general framework to construct communication efficient secret sharing schemes based on sequences of nested linear codes, which allows to use in particular algebraic geometry codes and allows to obtain strongly secure and communication efficient schemes. Using this framework, we obtain: 1) schemes with universal and close to optimal communication overheads for arbitrarily large lengths n and a fixed finite field, 2) the first construction of schemes with universal and optimal communication overheads and optimal strong security (for restricted lengths), having in particular the component-wise security advantages of perfect schemes and the security and storage efficiency of ramp schemes, and 3) schemes with universal and close to optimal communication overheads and close to optimal strong security defined for arbitrarily large lengths n and a fixed finite field

Unifying notions of generalized weights for universal security on wire-tap networks

Universal security over a network with linear network coding has been intensively studied. However, previous linear codes used for this purpose were linear over a larger field than that used on the network. In this work, we introduce new parameters (relative dimension/rank support profile and relative generalized matrix weights) for linear codes that are linear over the field used in the network, measuring the universal security performance of these codes. The proposed new parameters enable us to use optimally universal secure linear codes on noiseless networks for all possible parameters, as opposed to previous works, and also enable us to add universal security to the recently proposed list-decodable rank-metric codes by Guruswami et al. We give several properties of the new parameters: monotonicity, Singleton-type lower and upper bounds, a duality theorem, and definitions and characterizations of equivalences of linear codes. Finally, we show that our parameters strictly extend relative dimension/length profile and relative generalized Hamming weights, respectively, and relative dimension/intersection profile and relative generalized rank weights, respectively. Moreover, we show that generalized matrix weights are larger than Delsarte generalized weights.Comment: 8 pages, LaTeX; the current version will appear in the Proceedings of the 54th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA, 201

Rank error-correcting pairs

Error-correcting pairs were introduced as a general method of decoding linear codes with respect to the Hamming metric using coordinatewise products of vectors, and are used for many well-known families of codes. In this paper, we define new types of vector products, extending the coordinatewise product, some of which preserve symbolic products of linearized polynomials after evaluation and some of which coincide with usual products of matrices. Then we define rank error-correcting pairs for codes that are linear over the extension field and for codes that are linear over the base field, and relate both types. Bounds on the minimum rank distance of codes and MRD conditions are given. Finally we show that some well-known families of rank-metric codes admit rank error-correcting pairs, and show that the given algorithm generalizes the classical algorithm using error-correcting pairs for the Hamming metric