Given this verification model above, a bid submission consists of m
messages (one to each auctioneer) from the bidder, each of size
points. The auctioneers then verify the submission by
an all-to-all communication with messages of size 
points.
Determining a digit of the selling price causes two occasions for
degree reduction: the first when computing the 
in phase 1,
the second when computing the summands of 
in phase 2. This
is a total of 
multiplications over
4 communication rounds. There is one additional communication round
where each auctioneer broadcasts c-1 points to reveal the 
shares.