In [this][1] paper of **Asmuth–Bloom SSS** 

**Shares Distribution**

To distribute n shares of a secret $K$ among the set of participants
$P = \{ p_i : 1 ≤ i ≤ n\}$, the dealer D does the following:
 
A set of integers $\{ p, m_1 < m_2 < · · · < m_n \}$, where $0 ≤ K < p$, is chosen subject to the following:

$\gcd(m_i , m_j)=1$ where for $i!=j$
    
$\gcd(p , m_i)=1$ ,for all $i$,
    
$\prod \limits_{i=1}^{t}m_i > p \prod \limits_{i=1}^{t-1}m_{n-i+1}$
     
$M =\prod_{i=1}^{t}m_i$
     
Dealer computes
     
     y = K+ap
     where a is a positive integer generated randomly subject to the condition that 0 ≤ y < M
	
The share of the $i^{th}$ participant,$1 ≤ i ≤ n$, is
 
 $y_i = y~ mod ~m_i$

**Secret  Construction**

Assume $C$ is a coalition of $t$ participants to construct the secret. 
Let $M_C =\prod_{i=1}^{C}m_i$ 
$y ≡ y_ i \mod m_ i $ 

for $i ∈ C$, solve y in $GF(M_C  )$ uniquely using the CRT.
  Compute the secret as
     K = y mod p

According to the CRT, y can be determined uniquely in $GF(M_ C)$ . Since $y < M ≤ M_C$ , the solution is also unique in $GF(M)$.


How to prove formally that the number of participants lesser than threshold number $t$ of participants cannot get secret?

[1]: http://pdfs.semanticscholar.org/440d/fa8ab99301d4ea3a8b19c0748575915aef15.pdf