In Shamir's secret sharing scheme, Dealer performs the following steps
Choose a prime number $q$ such that $q > n$
Choose a secret $s$ from finite field $\mathbb{Z}_q$
Choose $t-1$ degree polynomial
$$g(x)=s+c_1x+c_2x^2+\cdots +c_{t-1}x^{t-1}$$
Compute shares $s_i = g(id_i) \mod q \text{ for } i=1,2, \cdots,n$ and sends secretly to participants
At least threshold number of participants can reconstruct secret by using Lagranges interpolation formula
My doubt is:
Instead of step 4 mentioned above, if we write without $\mod q$ as shown below then what will happen?
- Compute shares $s_i = g(id_i) , i=1,2, \cdots,n$.
Is there any advantage to use $\mod q$ over naive method (without using modulo)? If yes, is it security or computational complexity or any other?