# How can a node establish pairwise shared key with other nodes using its own polynomial share together with other's public values?

A server has a symmetric bivariate polynomial $$F(x, y) = \sum_{{i,j}=0}^{t-1}a_{i,j}x^iy^j$$ $$\in GF(p)[X, Y]$$ of degree $$t-1$$. For simpliciy, $$F(x, y) = a_{0,0}+a_{1,0} x+a_{0,1}y+ a_{1,1}xy$$ mod $$p$$, where $$a_{0, 1} = a_{1,0}$$ and $$p$$ is a large prime number.

• From the origional polynomial $$F(x, y)$$, the server generates $$n$$ univariate polynomial shares as $$f_{x_i}(y)=F(x_i,y)$$ , where 1 ≤ $$i$$$$n$$, and $$x_i$$ is private (noboday knows it except the server) and distinct number for each node $$i$$.

• Each node $$i$$ has a distinct public number $$r_i$$, where 1 ≤ $$i$$$$n$$ and every node else knows this number as it's public

Now, Let's say $$node$$ $$i$$ has got its own share $$f_{x_i}(y)$$ and $$node$$ $$j$$ has got $$f_{x_j}(y)$$ , where $$x_i \neq x_j$$, as said earlier.

The question is:

Utilizing the symmetric property of the original bivariate polynomial and Using their own polynomial shares together with other's public numbers, how can $$node$$ $$i$$ and $$node$$ $$j$$ ,$$i \neq j$$, establish a pairwise shared key/value such that: $$k = f_{x_i}(r_j)$$ =$$f_{x_j}(r_i)$$ holds???

$$Hint$$: node $$i$$ evaluates its own polynomial share using node $$j$$'s public number and node $$j$$ evaluates its own polynomial share using node $$i$$'s public number. Is there any trick to make them get the same value? and How?

• Actually, I got stuck in making $f_{x_i}(r_j)$ =$f_{x_j}(r_i)$ holds. I couldn't make it hold with pure algebra since they are both generated using private value. So, I tried adding and using the approach of homomorphism property $(f + g)r = f(r) + g(r)$. However, it still doesn't solve the problem and moreover, the node becomes vulnerable to node capture attack – A. AZEMi Mar 18 at 9:35