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I try implement Shamir Secret Sharing.

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

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

For example: a) a(1) = 15 a(2) = 6 a(4) = 10 enter image description here

b) a(1) = 15 a(2) = 6 a(5) = 6 enter image description here

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

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