On the Information Ratio of Non-perfect Secret Sharing Schemes

Abstract

A secret sharing scheme is non-perfect if some subsets of players that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes and the construction of efficient linear non-perfect secret sharing schemes. To this end, we extend the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information on the secret value that is obtained by each subset of players. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, access functions whose values depend only on the number of players, generalize the threshold access structures. The optimal information ratio of the uniform access functions with rational values has been determined by Yoshida, Fujiwara and Fossorier. By using the tools that are described in our work, we provide a much simpler proof of that result and we extend it to access functions with real values.

Optimal Uniform Secret Sharing

“…Related Work: After our work in [21], the same results were independently achieved in [8] and [9]. There the problem is placed into a wider context: not only uniform secret sharing schemes with rational values, but also nonuniform 0018-9448 © 2018 IEEE.…”

“…Note that by Eq. (8) our lower bound is equivalent to n−1 l=0 ( l − l−1 ) + H(S), which is presented in [8] and [9]. Eq.…”

“…We note that in [8] and [9], optimality is constructed to the same class as in this work, including [21], i.e., the uniform rational-number class. For the uniform real-number class, [8] and [9] proved that the lower bound is achieved by taking the limit of a number of ramp schemes. That is, an infinite number of ramp schemes are required to achieve the lower bound.…”

“…For the nonuniform class, optimal secret sharing is an open problem. In comparison with our lower bound, the lower bound in [8] and [9] is given by summing every positive change in the gradient.…”

“…In this paper and most of those in the literature, including [8], [9], and [21], the leakage is defined by the entropy. However, in many cryptographic applications of secret sharing, such as secure storage, the definition of leakage only via the entropy is not enough [11].…”