Suppose I have a k-dimensional secret $\langle x_1,\cdots,x_k \rangle$ which I share using a packed Shamir's secret share $(t,k,n)$ where $t$ is the threshold and $n$ is the number of shares as follows:
Construct a polynomial $f$ of degree $t+k-1$ such that $f(-1)=x_1, \cdots, f(-k)=x_k, f(-k-1)=r_1, \cdots, f(-k-t)=r_t$ where $r_1,\cdots,r_k$ are randomly sampled from the field. Now the n shares are generated as $(1,f(1)),\cdots,(n,f(n))$.
Let's say every party $i $ has share $(i,f(i))$ and two $k$-dimensional public vectors $\langle a_1,\cdots,a_k\rangle, \langle b_1,\cdots, b_k \rangle$. Is it possible to compete linear operations on the packed shares, i.e., generate the share for $\langle x_1\cdot a_1+b_1,\cdots,x_k\cdot a_k+b_k\rangle$?
I want to be able to do it non-trivially, i.e., not by reconstructing the secret and then computing on the clear. Basically, is it possible to perform SIMD linear operations on the packed shares locally? Note that when $a_1=a_2\cdots=a_k:=a, b_1=b_2\cdots=b_k:=b$ it is possible to obtain the shares as $(i,a\cdot f(i)+b)$. But I am interested in the more general case where $a_i/b_i$s are different.
