is also non-zero. The probability that such a sum is 0 is at most
. Such errors may occur in the computation of the
or 
.
In the case of computing 
, such a chance will introduce an
error into the result of the bidding only if it occurs for the l
that are relevant to the selection of 
(i.e., 
or 
).
Also, 
will never be non-zero for any bidder other than the
winner, so this error will be possible only for the winning bidder.
The non-zero 
are uniformly random (from
). They are also independent of 
for
, and thus are independent of the 
and 
(which are functions of the 
with 
). Therefore, the
value 
is uniformly random (from
). So the sum of 
and 
(for any particular k and l) can equal zero with
probability at most 
.
Since the non-zero 
are uniform and
independent, the 
, if non-zero, will also be uniform and
independent of any 
which is non-zero (for 
). In other words, the only dependency is contained in the property
of being non-zero.
In the case of the 
, an error in the result of the
bidding will be caused only for the one relevant l value 
.
The chance of incorrectly generating a 
with value 0 at 0 is
at most 
. To see this consider the values of 
as
we compute it over increasing subsets of the bidders. Note the change
in 
when we include the final bidder whose bid is as high
as the test value (i.e. for whom 
).
We can bound these sources of error by 
from the
and 
for the 
, totalling 
. We must choose p large enough that 
is
negligible. Since 
, where V is the number of bidding
points, its value in any implementation would be small (certainly
less than 40). Making a tradeoff between speed and possibility of error,
p should be in the range of 64-128 bits.