Secret sharing

Goal: share a secret $S\in F$ between $n$ people so that

• every combination from $t-1$ people cannot restore $S$
• every combination from $t$ people can restore $S$

For $t=n$: Choose randomly $y_1,y_2,\dots ,y_n$ with ${\sum }_{i=1}^n{y}_i=S$

The general solution ($|F|\ge n+1$)

• choose distinct ${\alpha }_0=0,{\alpha }_1,\dots ,{\alpha }_n\in F$
• choose randomly $a(x)\in F[x]$ from degree $\le t-1$ with $a({\alpha }_0)=a(0)=S$
• Person $i$ get $a({\alpha }_i)$ for $(i=[n])$
• For any $t-1$ shares, $a({\alpha }_1)\dots a({\alpha }_{t-1})$, every $a(0)=S$ is possible
• For any $t$ shares, $a({\alpha }_{i_1}),\dots ,a({\alpha }_{i_t})$ makes interpolation $a(x)$ possible $a(0)=S$