For Shamir's secret sharing scheme (doi 10.1145/359168.359176), one obtains a random polynomial $q$ of degree at most $n-1$ (over $\mathbb{Z}_p[x]$).  The constant coefficient of this polynomial is defined to be the key.  Then the scheme sends to player $i$ the value $q(i)$ (for $i \leq k$ for some fixed $k$). The interpolation theorem says that if $n$ players come together, then I can use interpolation to reconstruct the polynomial (there is a unique polynomial $p$ of degree at most $n-1$ such that $p(i)=q(i)$ .. in otherwords $p=q$.  So that I can grab the constant coefficient of $p$ and it must be the same as the constant coefficient of $q$, and hence must be the key).The question I have, is that what if the polynomial is degree less than $n-1$.  There seems to be nothing in the construction as described by Shamir that prevents this.  However, Shamir states that $n-1$ players cannot reconstruct the polynomial.The problem I am having is that there seems to be a most of the time clause missing. If I happen to pick a degree one polynomial then any two players can reconstruct the polynomial.  So is it that the players do not know the degree of the polynomial, or that the probability of picking a small degree polynomial is low enough that the probability of two participants guessing the key is still the same as random guessing, or am I missing something else.
Condition in Shamir Secret Sharing Scheme
probability theory;cryptography