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.