Let $N$ and $p$ be very large positive integers with $p$ a prime and $p>N$. The Chair of the Committee is able to inscribe pairs of very large integers on discs. The Chair wishes to inscribe a collection of discs in such a way that any Committee member who acquires $r$ of the discs and knows the prime $p$ can deduce the integer $N$, but owning $r-1$ discs will give no information whatsoever. What strategy should the Chair follow?

[You may use without proof standard properties of the determinant of the $r \times r$ Vandermonde matrix.]