# Accumulation of elements with a bilinear (pairing-based) accumulator

My question concerns accumulation of new elements in bilinear (pairing-based) accumulators.

Suppose you have a pairing $$e:\mathbb{G}_1\times \mathbb{G}_2 \longrightarrow \mathbb{G}_T$$ with generators $$g_1, g_2$$ for the cyclic groups $$\mathbb{G}_1$$, $$\mathbb{G}_2$$. The accumulated state is defined as $$A = g_1^{\prod\limits_{d\in D} (s+d)}$$ where $$s$$ is the private key (i.e. the trapdoor). What is the most efficient way to add the elements of a new set $$D_1$$ to the accumulator?

It seems that without the trapdoor, one would need to keep track of the coefficients of the polynomial $$\prod\limits_{d\in D} (s+d)$$ and keep updating this set of coefficients which grows linearly with the size of the accumulated set.

More generally, given an element $$a\in \mathbb{G}_1$$ how does one compute $$a^s$$ without using the polynomial $$h(X)$$ such that $$a =g_1^{h(s)}$$?