Perfectly secure data aggregation via shifted projections

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 $2m^2a$.