Shamir's secret sharing homomorphism for different degree polynomials

Question

The $(t,n)$ Shamir’s polynomial based secret sharing scheme is $(+,+)$-homomorphic in which the addition of two polynomials secrets equals the Lagrange’s interpolation of the sum-of-shares for the same subset of shares.

My question is: Does the two polynomials need to have the same degree to satisfy the SSS $(+,+)$-homomorphic property? Specifically, suppose that polynomial $P_1$ defines secret $s_1$ and polynomial $P_2$ defines secret $s_2$. The first polynomial $P_1$ is of $(t-1)$-degree and the second polynomial $P_2$ is $(m-1)$-degree in which $m\ge t$. Suppose I have $m$ shareholders. In this case, can I still say that SSS is $(+,+)$-homomorphic for the same subset of $m$ shares?

Answer 1

Yes. The polynomial $P_3(x)=(P_1+P_2)(x)$ is of degree $m-1$ and we can see that if we have shares $P_1(x_i)=s_{1,i}$ and $P_2(x_i)=s_{2,i}$ for $1\le i\le m$ then $P_3(x_i)=s_{1,i}+s_{2,i}$ and we can write $s_{3,i}$ for these recovered values. We now have $m$ points $(x_i,s_{3,i})$ through which $P_3$ passes and so can recover $P_3(x)$.

In general SSS works if we replace the words "polynomials of degree $m$" with the words "polynomial of degree at most $m$", though secrets that are known to be associated with a polynomial of smaller degree can be recovered by a smaller set of colluders. Note that if $F_1$ and $F_2$ are two polynomials of degree $m$ it is possible (albeit unlikely) that $F_1+F_2$ is a polynomial of degree less than $m$ (if the leading coefficients are sign changes of each other).