The auctioneers use the submitted bids to compute the selling price
, which will be equal to the second highest bid submitted. This
computation is performed over d rounds, one digit per round, going
from most significant to least significant. We will use k to refer
to the number of the current round. The value of the digits of 
is not kept secret (from the auctioneers), but is known to all
auctioneers as it is computed.
Some shared polynomials 
will be computed from
the 
over the course of price determination. Intuitively,
these values are used to track the bid 
of bidder 
as
compared to the first k-1 digits of the selling price.
will be non-zero exactly when 
.
will be non-zero exactly when 
.
will be non-zero exactly when 
.
, the results
are determined entirely by the first k-1 digits, which are known
prior to round k. The 
are indicators for the winning
bidder. The value 
is non-zero when the first k-1 digits
of the bid 
indicate that the bid is greater than 
. Since
the selling price is the second highest bid, 
means that
bidder j has the highest bid. The 
are indicators for the
losing bidders. The value 
will equal 0 when the first
k-1 digits of the bid 
indicate that the bid is less than
. So, 
means that 
is at most the third highest
bid and that the particular bid 
will have no effect on the
outcome of the bidding.
The 
, which are computed in round k by
, indicate whether a bid is at least
as high as some specific test value. The value 
is
non-zero when the first k digits of bid 
(interpreted as a base
c number) is greater than or equal to the first k-1 digits of
appended by l (interpreted as a base c number). These 
are used in round k to
determine the 
digit of the selling price by testing all
possible values.
Since the 
and 
are computed from values 
,
, and 
in round k-1, we must establish
values for the 
and 
. For all j, let 
be the
constant polynomial 0 and let 
be the constant polynomial
1. In an implementation, computation using these initial values will
be trivial (i.e. addition to 0 or multiplication by 1).
The auctioneers will perform summations of the 
over certain
values of j (i.e. subsets of 
). Define 
to be a collection of subsets of
(the index set for the bidders). Let 
if
and only if the 
bit (in a 
bit binary
representation) of j-1 is zero. We will interpret these 
as
partitions which separate 
into two parts based on
inclusion in the set 
. The rationale behind the selection of
these partitions (the 
for 
) is
that for any pair of distinct bidders j and 
, there is at least
one partition which separates j from 
.