  We study a general scenario where confidential information is distributed
among a group of agents who wish to share it in such a way that the data
becomes common knowledge among them but an eavesdropper intercepting their
communications would be unable to obtain any of said data. The information is
modelled as a deck of cards dealt among the agents, so that after the
information is exchanged, all of the communicating agents must know the entire
deal, but the eavesdropper must remain ignorant about who holds each card.
  Valentin Goranko and the author previously set up this scenario as the secure
aggregation of distributed information problem and constructed weakly safe
protocols, where given any card $c$, the eavesdropper does not know with
certainty which agent holds $c$. Here we present a perfectly safe protocol,
which does not alter the eavesdropper's perceived probability that any given
agent holds $c$. In our protocol, one of the communicating agents holds a
larger portion of the cards than the rest, but we show how for infinitely many
values of $a$, the number of cards may be chosen so that each of the $m$ agents
holds more than $a$ cards and less than $2m^2a$.

Cryptography and Security