Structured ramp secret sharing schemata over rings of real polynomials

Abstract

Two new ramp secret sharing schemata based on polynomials are proposed. For both schemata, the secret is considered to be a polynomial created by the dealer. The participants are separated into ℓ⩾2, groups, that are specified by the dealer's levels Li for i=1,2,…,ℓ and each level Li, i⩾2, is separated into subsets. The shares of the secret are given to participants in the form of polynomials. For the first proposed scheme, the dealer creates ℓ polynomials one for each level. Specific participants from every subset of each level have to cooperate all together in order to construct the polynomial of their level. Next all the authorized participants cooperate for computing the greatest common divisor of the polynomials in order to retrieve the secret. In the second scheme, the authorized participants cooperate per two levels using a bottom-up procedure. In both schemata the greatest common divisor can be evaluated by implementing numerical linear algebra methods, and precisely factorization of matrices of special form such as Sylvester matrices. The triangularization of these matrices can be obtained by exploiting their special structure for the reduction of the required floating point operations. The innovative idea of the paper at hand is the use of real polynomials in secret sharing schemata. This is particularly useful since the greatest common divisor can always be computed with efficient accuracy using effective numerical methods. New theoretical results are proved and provided that support the error analysis of our approach.