# Combining secret keys into a single shared key

Let's say there are $$n$$ participants each with their own secret key $$x_i$$. Is it possible for them somehow to build a shared key which is an arithmetic sum of all their secret keys ($$x_0 + x_1 + ... + x_n$$) but without revealing their individual keys to each other.

Also, ideally, this would be done non-interactively. Meaning, each participant shares something about their secret key with everyone else, and once everyone has shared this "something", the shared key can be constructed.

• If you use summing, parties can collude to reveal a key by subtracting all their keys from the shared secret, it is best to use multi-party DH, see crypto.stackexchange.com/questions/1025/… Commented Sep 27, 2018 at 1:38
• For my purposes collusion is not a significant problem. Also, the number of parties is really large (10K+) - so, I'm not sure DH would be practical. Lastly, for my purposes the shared key must be an arithmetic sum of all secret keys - I don't think that would be the outcome with DH. Commented Sep 27, 2018 at 1:51
• why must they be a sum? that is a very unusual requirement Commented Sep 27, 2018 at 2:31

In a more detailed manner, illustrated with additive secret sharing, every party $$P_i\ \forall i\in Z_{n+1}$$ generates $$n$$ random values in some valid group denoted by $$m$$ which will serve as shares of the secret $$x_i$$ for all the other player i.e. $$[x_i]_{\mathbb{Z_m}}^{P_j}\in_R\mathbb{Z}_m \ \forall i,j\in Z_{n}$$ and the final share satisfies the following equation $$[x_i]_{\mathbb{Z_m}}^{P_n}=x_i-\sum_{j=0}^{n-1}[x_i]_{\mathbb{Z_m}}^{P_j} \mod m$$. Then every party $$P_i$$ sends $$[x_i]_{\mathbb{Z}_m}^{P_j}$$ to all other parties such that $$i\neq j$$. Finally, the parties all locally compute the following summation locally in which $$x_\Sigma$$ denotes the summed secrets. $$[x_\Sigma]_{\mathbb{Z}_m}^{P_i} = \sum_{j=0}^{n}[x_j]_{\mathbb{Z}_m}^{P_i}\mod m$$
Now each party holds uniformly random and information theoretically secure shares of the sum of their individual secrets and all parties must cooperate to reveal the underlying value. Specifically they would have to communicate all their individual shares to one another then they could each locally calculate: $$x_\Sigma = \sum_{i=0}^{n}[x_\Sigma]_{\mathbb{Z}_m}^{P_i} \mod m$$
• Thank you! If I understood correctly, this requires 2 rounds of communication: (1) to distribute all but one shares of each secret $x_i$, and (2) to distribute all locally summed up shares from the first round - right? If so, is there any way to reduce communications to a single round? Commented Sep 27, 2018 at 4:25