## 4 Modified Blockchain Model
### The Proposed Algorithm
#### 4.6.3 Preliminaries and Notations
For simplicity, in this experiment, we built a blockchain of ten nodes. The block creation was fairly distributed among the ten nodes, as every cycle-one node proposed a new block of size 50 transactions (every block contained 50 transactions). The block also had a hash number which represented the content of the block, including the identity of the creator node, for authentication. After the block creation step, it was sent to the other nodes for verification. Actually, a traditional blockchain treats all transactions with the same procedures, while a modified blockchain relaxes the validation of all _TrustedT1_ and the read transactions from _TrustedT2_. The update transactions from _TrustedT2_ and all \(T\) were treated as in a traditional blockchain. Consequently, the execution time of _Block1_ was the minimum, the execution time of _Block2_ was average, and the execution time of _Block_ was the maximum. Figure 5 shows the total execution time for the traditional blockchain and modified blockchain, where the modified blockchain reduced the execution time by about 49% since about 33% of blocks were the _Block1_ type, about 33% of blocks were the _Block2_ type, and about 33% of blocks were the _Block_ type, while all blocks of the traditional blockchain were the _Block_ type. In addition, Figure 6 illustrates the detailed execution time for the validation process of the read and update transactions for _Block1_, _Block2_, and _block_. It shows how relaxation reduced the execution time. Actually, for _Block1_, it only checked the hash for the block, since the transactions came from trusted executers. The same thing applied to read transactions in _Block2_, but for update transactions, it validated each transaction, then checked the hash for the block. For _block_, it validated each transaction (both the read and update ones) and the hash for the block. Figure 5: Total execution time for the traditional blockchain and modified blockchain. ### Numerical Analysis for the Number of Messages

The decentralization characteristic of a blockchain requires a large number of messages. In fact, the miner node sends the proposed block to all other nodes for validation; say \(n\) nodes require \(n\) messages. Then, every node validates the proposed block and sends the vote to all others (excluding itself), which costs \(n(n-1)\) messages. After that, every node calculates the majority of votes and send the consensus decision to all others (excluding itself), which costs another \(n(n-1)\) messages. The number of messages for this blockchain process is presented below:

\[b(n+(2(n(n-1)))) \tag{1}\]

where \(b\) is the total number of blocks and \(n\) is the number of nodes. The modified blockchain reduces the validation process as well as the number of validators for _Block1_ to the minimum--say, only three nodes (in our proposal). It also reduces the number of validators for _Block2_ to the average--say only six nodes (in our proposal)--while the number of validators for _Block_ will be ten for both the traditional and the modified blockchain. Using our experiment's specifications, there were 4000 _TrustedT1_ (both reads and updates), 4000 _TrustedT2_ (both reads and updates) and 4000 (both reads and updates). Since the block size was 50 transactions, there were 80 _Block1_, 80 _Block2_, and 80 _Block_. Figure 7a shows the relationship between the number of messages and the number of nodes where the number of blocks was constant (50 blocks). It highlights the total number of messages for _Block1_, which was 1200 messages; 5280 messages for _block2_; and 15,200 messages for _Block_. By increasing the number of nodes, the number of messages increased exponentially. Figure 7b demonstrates the relationship between the number of messages and number of blocks where the number of nodes was constant (10 nodes). By increasing the number of blocks, the number of messages increased linearly. Figure 8, illustrates the total number of messages the modified blockchain which includes 80 _Block1_, 80 _Block2_ and 80 _Block_, 240 total.