# Find the product of two sums via SMPC

I'm currently working on a distributed threshold DSA scheme that requires to find the product of two sums via secure multi-party computation. Specifically speaking, every one of $$n$$ parties $$P_i$$ possesses a DSA key pair $$(sk_i, pk_i)$$, where $$sk_i=d_i \in \mathbb{Z}_q$$ and $$pk_i = g^{d_i}$$. I want to collectively generate a signature $$S_{\Sigma} = k_{\Sigma}^{-1}(m+r_{\Sigma}d_{\Sigma})$$, where$$k_{\Sigma}=k_1+\dots k_n$$, $$r_{\Sigma}d_{\Sigma}=(r_1+\dots+r_n)\cdot(d_1+\dots+d_n)$$. My prior question is that is there a proper paradigm to compute $$r_{\Sigma}d_{\Sigma}$$ without leaking information about secret keys $$(d_1,\dots,d_n)$$? For computing $$k_{\Sigma}$$, I'm using the BGW Protocol and Shamir threshold secret sharing scheme. Is it possible to compute $$r_{\Sigma}d_{\Sigma}$$ using BGW protocol as well?

PS: I'm new to SMPC, and English is not my first language. Sorry for the troubles. Thanks!

Yes it is, see for example this page. Note that multiplication requires $$O(n^2)$$ additional communication, in contrast to addition (which is free).