# A problem involving Commitments

Suppose there is a set $$P=\{p_1, p_2, ..,p_l\}$$ of stock buyers who can make commitments to a share $$s_i$$ in a set $$S=\{s_1,s_2,...,s_m\}$$ of shares for an amount $$a_i$$ in a set $$A=\{a_1,a_2,...,a_n\}$$. They can make commitments only till a certain time period, say $$T$$. But people in this group are nasty and will try to break the scheme. They may send wrong commitments, may commit to shares not in S or may not open their commitments later.

The commitments are provided to a person $$\mathcal{B}$$.

Intent of $$\mathcal{B}$$ is to find, after time $$T$$, the total amounts commited to each individual share $$s_i$$ in $$S$$ without knowing which person in $$P$$ commited to which share in $$S$$. $$\mathcal{B}$$ wants to ignore (and may be report) any incorrect commitment or the ones which are not opened even after expiry of time $$T$$.

I thought Pederson commitments can be used here since they have homomorphic additive property. But, I have few queries.

1. How can we do homomorphic addition for individual shares in $$S$$?

2. When we homomorphically add two commitments, we assume they are correct and will be opened later. But, in this case, people in $$P$$ may not do any of them. How to tackle this part?

3. Or, should I look for some other commitment scheme?

• In KZG commitments, which are also homomorphically additive, if $C_f$ is commitment to polynomial $f$ & $C_g$ is commitment to polynomial $g$, then commitment of polynomial $h$ defined as $h = f + g$ would be $C_f + C_g$ - this is easily proven by the definition of the commitment itself Commented Apr 28 at 11:26