Constructing Ideal Secret Sharing Schemes Based on Chinese Remainder Theorem

Abstract

Since (t, n)-threshold secret sharing (SS) was initially proposed by Shamir and Blakley separately in 1979, it has been widely used in many aspects. Later on, Asmuth and Bloom presented a (t, n)-threshold SS scheme based on the Chinese Remainder Theorem (CRT) for integers in 1983. However, compared with the most popular Shamir’s thresholdtn SS scheme, existing CRT based schemes have a lower information rate, moreover, they are harder to construct due to the stringent condition on moduli. To overcome these shortcomings of CRT based schemes, (1) we first propose a generalized (t, n)-threshold SS scheme based on the CRT for polynomial ring over a finite field. We show that our scheme is ideal, i.e., it is perfect in security and has the information rate 1. Comparison show that our scheme has a better information rate and is easier to construct compared with the existing threshold SS schemes based on the CRT for integers. (2) We prove that Shamir’s scheme, which is based on the Lagrange interpolation, is a special case of our scheme. Therefore, we establish the connection among threshold schemes based on the Lagrange interpolation, schemes based on the CRT for integers and our scheme. (3) As a natural extension of our threshold scheme, we present a weighted threshold SS scheme based on the CRT for polynomial rings, which inherits the above advantages of our threshold scheme over existing weighted schemes based on the CRT for integers.