# Shamir Secret Sharing - How to reconstruction secret? Example

I try implement Shamir Secret Sharing.

In this video: https://www.youtube.com/watch?v=j7Ngtl1cCpY is example.

But for another three share holders is can't reconstruction secrete.

For example: a) a(1) = 15 a(2) = 6 a(4) = 10

b) a(1) = 15 a(2) = 6 a(5) = 6

• Welcome to Cryptography! Could you turn this big images into nice $LaTeX$/MathJax? – kelalaka Oct 21 '19 at 16:17

But for another three share holders is can't reconstruction secrete.

Sure they can.

In the field $$GF(17)$$, we have:

$$\frac{43}{3} = \frac{9}{3} = 3$$

because $$43 = 9 \pmod {17}$$, and $$3$$ is the value that, when multiplied by 3, gives us 9 modulo 17.

And, we have:

$$\frac{23}{2} = \frac{6}{2} = 3$$

beause $$23 = 6 \pmod {17}$$, and $$3$$ is the value that, when multipled by 2, gives us 9 modulo 17.

Remember that, when we work in a finite field, division isn't exactly the same as it is when we're working with in $$\mathbb{R}$$; for example $$\frac{1}{2}$$ is not 0.5 (which is meaningless in $$GF(17)$$, but instead it is 9.