Case study one: Polynomial Auction Protocol

The independent variables to vary for the Polynomial Protocol are the number of goods under auction, number of bidders participating, maxi-mum bid price, threshold number of evaluators required to decrypt bids, number of mask publishers and the maximum coefficient. The different phases are bid generation, total evaluation time and its component phases:

7.2. PROTOCOL EVALUATION 127 optimal value and optimal path determination. Evaluation time makes up the majority of time taken to run an auction and this is provided for each experiment. Bid generation time is provided for each of the experiments, except where it is not relevant as bid generation is independent of other bidders. It is also not included in the mask publisher’s experiment, as mask publishers are involved only in evaluation. Optimal value and opti-mal path determination are divided for the number of goods and bidders experiments, as the ratio of work is consistent over the other variables.

The protocol specific default variables for the polynomial protocol that are held constant are:

Variable default Default value

Maximum bid 5

Threshold 1

Number of evaluators Varies according to other variables Number of mask publishers 2

Maximum coefficient 10

The default maximum bid and threshold are not sufficient for a real auction. They have been kept low because of the relationship between the maximum bid, the threshold and number of evaluators. This relationship cripples the scheme as was explained in section 4.4.1. A proposed modifi-cation to the protocol to reduce the effect of the relationship was provided in section 4.4.2. The number of mask publishers and maximum coefficient is also low, though there are currently no recommendations in the related work for suitable values.

Increasing the Number of Goods

This experiment shows that the number of goods under auction has a sig-nifiant effect on scalability. An increase in goods, as shown in Figures 7.2 and 7.3, corresponds to an exponential increase in evaluation and bid

generation time respectively. Both of these figures grow exponentialy due to the exponential increase in goods combinations requiring evaluation.

For this experiment, a maximum number of five goods has been used be-cause a number greater than this requires the JVM heapspace to be in-creased past 256mb for each evaluator. Increasing the heapspace past 256mb would be unreasonable for the number of auction resources re-quired to run for some of the experiments and, in addition, it would need to be greatly increased with more goods. It can be seen from the graph though that with an additional two goods, evaluation would take well over one thousand seconds per auction. Future work should investigate methods of reducing the memory usage of each protocol and providing a way in which a larger number of goods can be supported, such as caching intermediate results.

In the evaluation phase breakdown shown in figure 7.4, it is apparent that the growth of optimal path determination is less than that of value de-termination. Optimal path determination, though, is initially higher, due to the number of evaluations required during bidder search. The num-ber of edges to search in optimal path determination is significantly lower than that in optimal value determination, due to the nature of edge trace-back with dynamic programming. Therefore as the number of edges in-creases, the number of decryptions required during optimal value deter-mination begins to outweigh the number of decryptions required when determining optimal edges and bidders.

Increasing the Number of Bidders

The effects of increasing the number of bidders on evaluation time are provided by Figures 7.5 and 7.6, as the total evaluation time and the break down of evaluation phases respectively. Increasing the number of bidders increases the evaluation time a slight amount linearly. This increase is min-imal as all bids for a particular bundle are added together and evaluated together until the winning bidders need to be determined.

7.2. PROTOCOL EVALUATION 129

Polynomial auction evaluation time (increasing goods)

1 10 100 1000

1 2 3 4 5

Number of goods

Evaluation time (seconds)

Figure 7.2: Effect of increasing the number of goods on total evaluation time.

Polynomial auction bid generation time (increasing goods)

0.0 0.1 1.0 10.0

1 2 3 4 5

Number of goods Bid generation time (seconds, per bidder)

Figure 7.3: Effect of increasing the number of goods on bid generation time.

Polynomial auction evaluation time (increasing goods)

Determine optimal value vs determining optimal path

1.1

Determine optimal path Determine optimal value

Figure 7.4: Effect of increasing the number of goods, optimal value deter-mination vs optimal path deterdeter-mination.

The difference between optimal path and optimal value determination varies erratically, most likely due to demand of system resources by other users. As the evaluation times are short, they are greatly susceptible to other process demands.

Increasing the Maximum Bid

As shown in figure 7.7, evaluation time for the Polynomial Protocol does not scale well as the size of the maximum bid increases. This is due to the relationship between the number of goods, evaluators and the thresh-old (see section 4.4.2). For this experiment the number of goods and the threshold have been maintained at the defaults, so to increase the maxi-mum bid the number of evaluators has to be increased. One consequence of this is that the security is reduced as the ratio of threshold to evaluators (t, n).

The exponential increase in evaluation time is due to the large number

7.2. PROTOCOL EVALUATION 131

Polynomial auction evaluation time (increasing bidders)

0

Figure 7.5: Effect of increasing the number of bidders on total evaluation time

Polynomial auction evaluation time (increasing bidders)

Determine optimal value vs determining optimal path

7.8 8.0 8.3

8.9 9.5

10.2 10.2 10.3 10.8 11.3

0.0

Determine optimal path Determine optimal value

Figure 7.6: Effect of increasing the number of bidders, optimal value de-termination vs optimal path dede-termination.

of evaluators which have been spread accross a consistent number of ma-chines. Not only does the number of messages and amount of processing increase with the number of evaluators but, as the machines are limited, the resources must contend for system resources.

Polynomial auction evaluation time (increasing maximum bid)

1 10 100 1000

10 20 30 40 50

Maximum bid

Evaluation time (seconds)

Figure 7.7: Effect of increasing the maximum bid on total evaluation time.

7.2. PROTOCOL EVALUATION 133

Polynomial bid generation time (increasing maximum bid) Bid generation time (seconds, per bidder)

Figure 7.8: Effect of increasing the maximum bid on bid generation time.

Polynomial auction evaluation time (increasing maximum bid)

Determine optimal value vs determining optimal path

493.1

Determine optimal path Determine optimal value

Figure 7.9: Effect of increasing the maximum bid, optimal value determi-nation vs optimal path determidetermi-nation.

Increasing the Threshold Number of Evaluators

Exponential growth occurs when increasing the threshold number of eval-uators, as depicted in Figure 7.10. Similarly to increasing the maximum bid, when increasing the threshold the number of evaluators used also has to be increased because of the circular dependency between evalua-tors, threshold and maximum price. As in the maximum bid experiment, increasing the number of evaluators introduces an exponential cost.

Polynomial auction evaluation time (increasing evalautor threshold)

0.00 5.00 10.00 15.00 20.00 25.00

1 2 3 4 5

Threshold number of evaluators Evaluation time (seconds, per bidder)

Figure 7.10: Effect of increasing the threshold number of evaluators on total evaluation time.

Increasing the Number of Masking Agents

Figure 7.11 shows the effect of increasing the number of mask publishers on total evaluation time and Figure 7.12 provides the breakdown of evalu-ation phases. Increasing the number of mask publishers has a linear effect on the total evaluation time. This is because of the time each evaluator must wait for the additional masking values from the added mask

pub-7.2. PROTOCOL EVALUATION 135

Polynomial auction evaluation time (increasing number of mask publishers)

0

Figure 7.11: Effect of increasing the number of mask publishers on total evaluation time.

lishers. It is not due to the additional masks themselves, as they require only an extra addition operation which is trivial compared to the commu-nication required to send the mask.

The number of mask publishers has a greater effect on determining the optimal path than on determining the optimal value. The minimal effect when determining the optimal value is due to a much smaller number of node evaluations required, compared to the number of bids which must be evaluated when determining the optimal path (hence a reduced number of masks required).

Increasing the Maximum Coefficient

There is no effect on evaluation or bid generation when increasing the maximum coefficient, therefore it is not shown. This is because increas-ing the maximum coefficient increases the pool that polynomials can be generated from, but polynomial operations remain constant irrespective

Polynomial auction evaluation time (increasing number of mask publishers)

Determine optimal value vs determining optimal path

8.3

Determine optimal path Determine optimal value

Figure 7.12: Effect of increasing the number of mask publishers, optimal value determination vs optimal path determination.

of polynomial size.