Shamir Secret sharing - Can share generator keep x values secret? I'm wondering, in Shamir secret sharing, can generator of the shares, keep the x values which are used in evaluating the polynomial to obtain y values (i.e., the shares) secret, and whenever the original secret is needed, he retrieves the t shares and reconstructs the secret? In this setting the share holders should not be able to construct the secret even all of them collude. Is there any secure way of doing so?
Each share in Shamir-type secret-sharing schemes has two parts: the share value
$f(x_i)$ which is the value taken on by a certain polynomial $f(\cdot)$, and the share "location" $x_i$. The "dealer" who constructs the shares and hands them
out to individual shareholders is assumed to have handed out $(x_i, f(x_i))$
to the $i$-th share holder. In this scenario, any $k$ (or more) shareholders in a
$(k,n)$ scheme can get together and reconstruct the secret, and fewer than $k$
shareholders cannot reconstruct the secret.  

If the dealer
hands out only $f(x_i)$ to the $i$-th share holder and merely makes a note
in his records that the $i$-th share holder's share location is $x_i$,
then only the dealer can reconstruct the secret when $k$ or more shareholders
make an application to the dealer; the $k$ shareholders cannot 
reconstruct the secret by
themselves because while they all know their share values, they do not know the
location of those shares.  This scheme is, of course, fraught with danger in
case the dealer's list is lost or is corrupted in various ways: the secret
might be irretrievably lost.

See also this answer for
how enemies can attack a Shamir-type secret sharing scheme by tampering
with share values and by tampering with share locations, and how to
get around such problems.