Secret sharing consists of recovering a secret S from a set of shares, each containing partial information about the secret. The Chinese remainder theorem (CRT) states that for a given system of simultaneous congruence equations, the solution is unique in some Z/nZ, with n > 0 under some appropriate conditions on the congruences. Secret sharing can thus use the CRT to produce the shares presented in the congruence equations and the secret could be recovered by solving the system of congruences to get the unique solution, which will be the secret to recover.
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Secret sharing schemes: several types
There are several types of secret sharing schemes. The most basic types are the so-called threshold schemes, where only the cardinality of the set of shares matters. In other words, given a secret S, and n shares, any set of t shares is a set with the smallest cardinality from which the secret can be recovered, in the sense that any set of t-1 shares is not enough to give S. This is known as a threshold access structure. We call such schemes (t,n) threshold secret sharing schemes, or t-out-of-n scheme.
Threshold secret sharing schemes differ from one another by the method of generating the shares, starting from a certain secret. The first ones are Shamir's threshold secret sharing scheme, which is based on polynomial interpolation in order to find S from a given set of shares, and George Blakley's geometric secret sharing scheme, which uses geometric methods to recover the secret S. Threshold secret sharing schemes based on the CRT are due to Mignotte and Asmuth-Bloom, they use special sequences of integers along with the CRT.
Chinese remainder theorem
Let
has solutions in Z if and only if
Secret sharing using the CRT
Since the Chinese remainder theorem provides us with a method to uniquely determine a number S modulo k-many relatively prime integers
Ultimately, we choose n relatively prime integers
This condition on S can also be regarded as
Since S is smaller than the smallest product of k of the integers, it will be smaller than the product of any k of them. Also, being greater than the product of the greatest k − 1 integers, it will be greater than the product of any k − 1 of them.
There are two Secret Sharing Schemes that utilize essentially this idea, Mignotte's and Asmuth-Bloom's Schemes, which are explained below.
Mignotte's threshold secret sharing scheme
As said before, Mignotte's threshold secret sharing scheme uses, along with the CRT, special sequences of integers called the (k,n)-Mignotte sequences which consist of n integers, pairwise coprime, such that the product of the smallest k of them is greater than the product of the k − 1 biggest ones. This condition is crucial because the scheme is built on choosing the secret as an integer between the two products, and this condition ensures that at least k shares are needed to fully recover the secret, no matter how they are chosen.
Formally, let 2 ≤ k ≤ n be integers. A (k,n)-Mignotte sequence is a strictly increasing sequence of positive integers
By the Chinese remainder theorem, since
Asmuth-Bloom's threshold secret sharing scheme
This scheme also uses special sequences of integers. Let 2 ≤ k ≤ n be integers. We consider a sequence of pairwise coprime positive integers
We then pick a random integer α such that
By the Chinese remainder theorem, since
It is important to notice that the Mignotte (k,n)-threshold secret-sharing scheme is not perfect in the sense that a set of less than k shares contains some information about the secret. The Asmuth-Bloom scheme is perfect: α is independent of the secret S and
Therefore α can be any integer modulo
This product of k − 1 moduli is the largest of any of the n choose k − 1 possible products, therefore any subset of k − 1 equivalences can be any integer modulo its product, and no information from S is leaked.
Example
The following is an example on the Asmuth-Bloom's Scheme. For practical purposes we choose small values for all parameters. We choose k=3 and n=4. Our pairwise coprime integers being
Say our secret S is 2. Pick
To solve the system, let
By Bézout's identity, since
From the identities