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For security in Shamir secret sharing we need that the coefficients of the polynomial are independent and uniformly random in the field. Multiplying the polynomial by a constant from the field does not change this, so yes, you can do it and still be secure. In fact, in multiparty computation, something akin to this is done when we want to privately multiply ...


If you have $t-1$ shares of an $(t,n)$ system, you have a chance at learning of the roots of the system... as long as some of your shares have a 0 for the $y$ coordinate. You can't learn any other roots. Demonstration that the possession of a share with a $y$ coordinate of 0 gives you knowledge of a root (and forgive me if this is too obvious): A share ...

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