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 ${\displaystyle \mathbb{Z}/n\mathbb{Z}}$, with ${\displaystyle n > 0}$ 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.

## Secret sharing schemes: several types

Main article: Secret sharing

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

Main article: Chinese remainder theorem

Let ${\displaystyle k \geqslant 2}$, ${\displaystyle m_1,...,m_k \geqslant 2}$, and ${\displaystyle b_1,...,b_k \in \mathbb{Z}}$. The system of equations

${\displaystyle \begin{cases} x \equiv & b_1 \ \bmod \ m_1 \\ & . \\ & . \\ & . \\ x \equiv & b_k \ \bmod \ m_k \\ \end{cases} }$

has solutions in ${\displaystyle \mathbb{Z}}$ if and only if ${\displaystyle b_i \equiv b_j \bmod (m_i,m_j)}$ for all ${\displaystyle 1 \leqslant i,j \leqslant k}$, where ${\displaystyle (m_i,m_j)}$ denotes the greatest common divisor (GCD) of ${\displaystyle m_i}$ and ${\displaystyle m_j}$. Furthermore, under these conditions, the system has a unique solution in ${\displaystyle \mathbb{Z}/n\mathbb{Z}}$ where ${\displaystyle n = [m_1,...,m_k]}$, which denotes the least common multiple (LCM) of ${\displaystyle m_1,...,m_k}$.

## 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 ${\displaystyle m_1, m_2, ..., m_k}$, given that ${\displaystyle S < \prod_{i=1}^k m_i}$, then, the idea is to construct a scheme that will determine the secret S given any k shares (in this case, the remainder of S modulo each of the numbers ${\displaystyle m_i}$), but will not reveal the secret given less than k of such shares.

Ultimately, we choose n relatively prime integers ${\displaystyle m_1< m_2< ...< m_n}$ such that S is smaller than the product of any choice of k of these integers, but at the same time is greater than any choice of k-1 of them. Then the shares ${\displaystyle s_1, s_2, ..., s_n}$ are defined by ${\displaystyle s_i = S \bmod \ m_i}$ for ${\displaystyle i=1,2,...,n}$. In this manner, thanks to the CRT, we can uniquely determine S from any set of k or more shares, but not from less than k. This provides the so-called threshold access structure.

This condition on S can also be regarded as ${\displaystyle \prod_{i=n-k+2}^n m_i < S < \prod_{i=1}^k m_i}$. 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 ${\displaystyle n \geqslant 2}$ be an integer, and k be an integer such that ${\displaystyle 2 \leqslant k \leqslant n}$. A (k,n)-Mignotte sequence is a strictly increasing sequence of positive integers ${\displaystyle m_1 < ... < m_n}$, with ${\displaystyle (m_i,m_j)=1}$ for all ${\displaystyle 1 \leqslant i < j \leqslant n}$, such that ${\displaystyle m_{n-k+2}...m_n < m_1...m_k}$. We call this range the authorized range. Now, the scheme works as follows: We build a (k,n)-threshold secret sharing scheme. We choose the secret S as a random integer in the authorized range. We compute, for every ${\displaystyle 1 \leqslant i \leqslant n}$, the reduction modulo ${\displaystyle m_i}$ of S that we call ${\displaystyle s_i}$, these are the shares. Now, for any k different shares ${\displaystyle s_{i_1},...,s_{i_k}}$, we consider the system of congruences: ${\displaystyle \begin{cases} x \equiv & s_{i_1} \ \bmod \ m_{i_1} \\ & . \\ & . \\ & . \\ x \equiv & s_{i_k} \ \bmod \ m_{i_k} \\ \end{cases}}$

By the Chinese remainder theorem, since ${\displaystyle m_{i_1}, ... , m_{i_k}}$ are pairwise coprime, the system has a unique solution modulo ${\displaystyle m_{i_1}...m_{i_k}}$. By the construction of our shares, this solution is nothing but the secret S to recover.

### Asmuth-Bloom's threshold secret sharing scheme

This scheme also uses special sequences of integers. Let ${\displaystyle n \geqslant 2}$ be an integer, and k be an integer such that ${\displaystyle 2 \leqslant k \leqslant n}$. We consider a sequence of pairwise coprime positive integers ${\displaystyle m_0 < ... < m_k}$ such that ${\displaystyle m_0.m_{n-k+2}...m_n < m_1...m_k}$. For this given sequence, we choose the secret S as a random integer in the set ${\displaystyle \mathbb{Z} / m_0\mathbb{Z}}$.

We then pick a random integer ${\displaystyle \alpha}$ such that ${\displaystyle S + \alpha.m_0 < m_1...m_k}$. We compute the reduction modulo ${\displaystyle m_i}$ of ${\displaystyle S + \alpha.m_0}$, for all ${\displaystyle 1 \leqslant i \leqslant n}$, these are the shares ${\displaystyle I_i=(s_i,m_i)}$. Now, for any k different shares ${\displaystyle I_{i_1},...,I_{i_k}}$, we consider the system of congruences:

${\displaystyle \begin{cases} x \equiv & s_{i_1} \ \bmod \ m_{i_1} \\ & . \\ & . \\ & . \\ x \equiv & s_{i_k} \ \bmod \ m_{i_k} \\ \end{cases}}$

By the Chinese remainder theorem, since ${\displaystyle m_{i_1}, ... , m_{i_k}}$ are pairwise coprime, the system has a unique solution ${\displaystyle S_0}$ modulo ${\displaystyle m_{i_1}...m_{i_k}}$. By the construction of our shares, the secret S is the reduction modulo ${\displaystyle m_0}$ of ${\displaystyle S_0}$

It is important to notice that the Mignotte and Asmuth-Bloom (k,n)-threshold secret-sharing schemes are not perfect schemes, in the sense that a set of less than k shares contains some information about the secret. Nevertheless, by a suitable choice of the sequences and the parameters (${\displaystyle \alpha}$ in the Asmuth-Bloom case), one can get a reasonable security factor. This is why the Asmuth-Bloom scheme is more secure, for it involves more random parameters.

### 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 ${\displaystyle m_0 =}$3, ${\displaystyle m_1 =}$ 11, ${\displaystyle m_2 =}$ 13, ${\displaystyle m_3 =}$ 17 and ${\displaystyle m_4 =}$ 19. They satisfy the Asmuth-Bloom required sequence because ${\displaystyle 3.17.19 < 11.13.17}$.

Say our secret S is 2. Pick ${\displaystyle \alpha = 51}$, satisfying the required condition for the Asmuth-Bloom scheme. Then ${\displaystyle 2+51\cdot 3=155}$ and we compute the shares for each of the integers 11, 13, 17 and 19. They are respectively 1, 12, 2 and 3. We consider one possible set of 3 shares: among the 4 possible sets of 3 shares we take the set {1,12,2} and show that it recovers the secret S=2. Consider the following system of congruences:

${\displaystyle \begin{cases} x \equiv & 1 \ \bmod \ 11 \\ x \equiv & 12 \ \bmod \ 13 \\ x \equiv & 2 \ \bmod \ 17 \\ \end{cases}}$

To solve the system, let ${\displaystyle M = 11 \cdot 13 \cdot 17}$. From a constructive algorithm for solving such a system, we know that a solution to the system is ${\displaystyle x_0 = 1 \cdot e_1 + 12\cdot e_2 + 2\cdot e_3}$, where each ${\displaystyle e_i}$ is found as follows:

By Bezout's identity, since ${\displaystyle (m_i,M/m_i) = 1}$, there exist positive integers ${\displaystyle r_i}$ and ${\displaystyle s_i}$, that can be found using the Extended Euclidean algorithm, such that ${\displaystyle r_i.m_i+s_i.M/m_i = 1}$. Set ${\displaystyle e_i=s_i\cdot M_i/m_i}$.

From the identities ${\displaystyle 1 = 1\cdot 221 - 20\cdot 11 = (-5)\cdot 187 + 72\cdot 13 = 5\cdot 143 + (-42)\cdot 17}$, we get that ${\displaystyle e_1 = 221, e_2=-935, e_3=715}$, and the unique solution modulo ${\displaystyle 11\cdot 13\cdot 17 }$ is 155. Finally, ${\displaystyle S=155\equiv 2\mod 3}$.