I have tried to look online in various sites and texts and tried to use several youtube tutorials but I still cannot grasp it correctly.
could someone please help me understand a simple case including mathematics so I could better understand it?
For simplicity's sake, there are 3 trust levels: 1, 2, 3. to recover S you need either two users from group 1, OR one user from group 1 along with three users from group 2 and five users from group 3 (offsets of 2 for simplicity).
how a secret can be shared in such a case?
if you can, please elaborate mathematically so I can better understand the words and then begin to exercise with it to understand it better.
EDIT: my attempt, i don't understand how to it, but this is what i got:
To recover a secret S:
Assume that group 1 consists of j participants, group 2 of k participants, group 3 l participants. and groups are denoted as a,b,c accordingly for clarity.
1)2 users from group 1: we needed 2 out of j so we choose one positive integer, so $y=s_0+a_1x+a_2x^2+...a(j-2)x^{j-2}$. to recover we use a straight line(threshold of 2) for $(a,f(a)), (b,f(b))$
2)A user from group 1, 3 from group 2 and 5 from group 3: $y=s_0+a_1x+a_2x^2+...a(j-1)x^{j-1}+b_1x+b_2x^2+...+b(k-3)x^{k-3}+l_1x+l_2x^2+...+l(k-5)^{k-5}$. it is really complicated for me, but i think that choose the following: $(a,f(a)),(b_1,f(b_1)),...,(b_3,f(b_3)),(c_1,f(c_1)),...,(c_5,f(c_5))$, then since $f(x)=\sum_{i=1}^ty_i\prod\frac{x-x_j}{x_i-x_j}$, we set x=0 to calculate the secret, $f(0)$, and we get: $\frac{f(a)*b_1*b_2}{(a-b_1)(a-b_2)}+...+\frac{f(c_5)*c_3*c_4}{(c_5-c_3)(c_5-c_4)}$
would appreciate corrections so i can fix my mistakes. thank you very much