# A Key Management Scheme for Mobile Wireless Sensor Networks
## 4 Evaluation and Comparison
### Size of the Network and Storage Efficiency  
The proposed scheme, like FPWK, requires an amount of memory that is proportional to the quantity of nodes in the network. Therefore, the limit to the memory that can be used for the cryptographic material also limits the number of nodes. Table 1 shows the memory required by the considered approaches. Some parameters are used to calculate the total memory required. For FPWK and PCAT, which have in common the presence of a memory upper bound, the adopted parameters are as follows:  
* \(l_{k}\) is the size of the symmetric keys,
* \(l_{a}\) is the size of the indexes for the authentication table of PCAT. In FPWK, the only data stored correspond to a symmetric key for any other node. However, the size of a symmetric key represents a considerable overhead. In PCAT, the quantity of memory stored for any other node corresponds to a row of the authentication table (\(m+l_{h}\)). However, PCAT also requires a fixed memory area to store the public and private keys, the bit mask, and an area of memory proportional to the quantity of connected nodes, in order to store the symmetric keys. The upper bound to the quantity of nodes in FPWK can be calculated as:  
\[N_{EG}=\frac{M}{l_{k}}+1\, \tag{1}\]where \(M\) represents the size of the memory. The upper bound to the quantity of nodes in PCAT can be calculated as:  
\[N_{PCAT}=\frac{M-m\cdot l_{a}-l_{pu}-l_{pr}-v\left(l_{k}+l_{ID}\right)}{m+l_{h}}+1 \tag{2}\]  
Considering \(l_{k}=16\) bytes, \(l_{ID}=\left\lceil\log_{256}n\right\rceil\) bytes, \(m=4\), \(l_{a}=\left\lceil\log_{256}l_{pu}\right\rceil\) bytes, \(l_{pu}=l_{pr}=32\) bytes and \(l_{h}=20\) bits, Figure 4 shows the upper bound to the size of the network according to the memory available to store the secret material and to the density of the network. Since the content of a row of the authentication table is smaller than a symmetric key, the upper bound rises faster for PCAT than for FPWK. However, the proposed approach requires a fixed memory area. According to the formulas presented in Table 1, it is possible to calculate when both the approaches require the same memory area:  
\[v=n\frac{l_{k}+m+l_{h}}{l_{k}+l_{ID}}+\frac{m+l_{h}-l_{k}-m\cdot l_{a}-l_{pu}-l _{pr}}{l_{k}+l_{ID}} \tag{3}\]  
Considering the previous example, Figure 5 shows the ratio between the quantity of nodes in direct communication and the size of the network at which PCAT and FPWK require the same memory area. Therefore, if the ratio \(\frac{v}{n}\) is under the curve (i.e., a node is directly connected with a percentage of nodes lower than that ratio), the proposed approach can be applied to networks composed of a larger number of nodes with respect to FPWK. In the chart, the ratio normally increases, with the only exception of a small decrease when the increasing number of nodes requires an additional byte to store ID\({}_{i}\). However, it is observed that PCAT is generally better than FPWK if the nodes are in direct communication with less than 70% of the nodes in the network. The size of the memory area used for the secret material is also significant for the other state-of-the-art approaches. The schemes based on the random predistribution of secret material provide a higher level of security if they store a large quantity of secret material. The additional parameters are:  
* the quantity of keys per ring stored by each node (\(r\)),
* the size of the identification of the keys (\(l_{k_{ID}}\)),
* the size of the seeds used by RSDTMK (\(l_{s}\)),
* the size of the identification of the seeds (\(l_{siD}\)), and
* the total quantity of keys stored by RSDTMK (\(\varrho\)).