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3.4. Sampling Technique and Sample Size
It is beyond the scope of the study to gather data from the total populations. Hence, sampling techniques were used; both systematic random sampling and purposive sampling was employed. In Sodo town, there are three sub-towns i.e. Arada, Mehal and Merkato. Out of these three sub-towns for this study two sub-towns are selected. Mehal and Arada sub-town is purposively selected for this study because the kebeles under these sub-towns selected for the study are newly developed parts of the town and faced with scarcity of water.
In two sub-towns there are eleven kebeles. Data was collected from five kebeles (Damota and Ofa Gendeba Kebels from Mehal and Waja kero, Dil Betigil, Selam kebels from Arada sub towns) by employing systematic random sampling. The total population of the five kebeles is 11,267 and there are 1,485 total households (CSA, 2014). Out of these, 147 households were used for the collection of sample data. Therefore, for this study the sample size is 10% of the total households (with the following formula as given on page number 15). In order to determine sample households the investigator selected every item on the list of households.

The sample size taken from each sample kebeles is comparative to their total number of households. The total households selected for this study was 147 households residing in the town. There were 310, 290, 285, 295 and 305 total households in kebeles Damota, Ofa Gendeba, Waja kero, Dil Betigil and Selam respectively. From these 30, 29, 28, 29 and 30 households respectively selected. The sample size for this research is determined by using the formula, as indicated in Burt et al, (2003). This study used the following formula to calculate sample size as follows:
µ=n/N x µ¡
Where, µ = sampled sizes of each kebeles
n = sample size
N = total household size of sampled kebeles
µ¡ = number of households in each sampled kebele

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3.5. Methods of Data Collection
3.5.1. Procedures of data collection
In order to have general understanding of the specific locality, the investigator first organized a general survey of the study area. The data gathering tools was intended on the bases of objectives, research questions and review of related literatures. During the initial stage of questionnaires administration, the investigator was prepared the objectives of the study clear to the respondents in order to avoid any confusion. Before distributing questionnaires, the time convenience for the respondents was considered in order to maximize the rate of return of the questionnaires. In addition, taking into consideration the non response rate of questionnaires, the investigator added some extra questionnaires were distributed.

3.5.2. Instruments of data collection
Collecting data through different tools leads to the accurate research findings. Having this in mind, the investigator used the following data gathering instruments: questionnaire, semi- structured interview and observation.

3.5.2.1. Questionnaire
Questionnaire was used as the major tool for collection of data from the randomly selected sample respondents. Due to resource and time constraint, the research could not entertain large number of people in case studies, in depth interviews and wider focus group discussions. Therefore, questionnaires were used to fill the gap and support the representative sample to address as many individuals as possible to help gather relevant first hand information. Two different sets of questions were prepared: close-ended and open-ended questions.

For those respondents who could not understand English, the questionnaires were prepared and translate in to Amharic, so that the respondents were easily understood.

3.5.2.2. Key informants interview
The purpose of interview was to collect supplementary information, so as to stabilize the questionnaire response. Semi-structured interview were conducted to a purposively selected group of informants. The interview was conducted face to face. The interview were used to dig out information on issues like water supply, health condition that is related with water particularly accessibility, consistency, and adequacy to assess water supply and distribution condition in the town. The reason behind using a semi-structured interview is the advantages of flexibility in which new questions can be forwarded during the interview based on the responses of the interviews.

Moreover, the tool also has been instrumental in generating recommendations. Accordingly, key informant interview were used to gather more of qualitative data explains and narrates the study population rather than expressing it in terms of numbers. The recorded data were categorized based on similarities of response and then translate into English language during the transcription. This assisted the researcher to collect relevant and more reliable information to this study.

3.5.2.3. Field Observation
Among the primary data collection techniques, observation is crucial to understand peoples’ activity in the basis of how, what and why they are doing something. This allowed the investigator to develop confidence to speak and analyze what is being said and what is really going on the actual setting. Further, this participatory observation has the benefit of becoming part of the selected group and observes how they get; fetch water and asking clarification on their actions. The field observation was used by the researcher to get additional information to validate the information received from other sources. A checklist was prepared for systematic observation.

3.6. Methods of Data Analysis
After the completion of data gathering, the data was coded, tabulated, analyzed, described, interpreted, and descriptive statistical technique (percentages, ratio, average, using table, frequency distribution charts etc.) were employed as methods of data presentation. The data was analyzed both quantitatively and qualitatively. The Statistical Package for Social Science (SPSS), v.20 software and MS-EXCEL was used to process the data. Moreover, qualitative data was collected through questionnaires, interview, and observation was rationally interpreted and analyzed to strengthen the quantitative data.

3.7 Ethical considerations
Ethical issues in research are concerned mainly in balancing the right of people for privacy, safety, confidentiality and protection from dishonesty with the pursuit of scientific endeavour (Pilot and Hungler, 1998). The Sodo City Administration was engaged from beginning as the current water services provider of the city of sodo. In addition consent was sought for each household before the interview and the purpose of the interview was clearly explained beforehand.

3.1. Screening and identification of bacterial producing lipase

Lipase producing bacteria were screened in enrichment culture medium supplemented with olive oil as a sole source of carbon. Furthermore, methanol (30%, v/v) was also used to acquire the methanol tolerant lipase. The clear area around the colonies on the tributyrin agar plate was evaluated as lipase production. The greatest lipolytic strains were also examined on the olive oil plate complemented with phenol red, as a pH indicator. Results showed this isolate was a strain which displayed the maximum pink area around the colony. The 16S rDNA gene of MG isolate was amplified and sequenced (Genbank Accession No. MF927590.1) and compared by BLAST investigation to other bacteria in the NCBI database. The results proposed a near relationship between MG10 isolate and the other members of the Enterobacter genus with a extreme sequence homology (99%) to Enterobacter cloacae. The phylogenetic tree (Fig. 1) designated that the strain MG10 was associated with Enterobacter species and used for the following study.

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3.2. Purification and immobilization of the lipase
Cell free supernatant of MG10 stain was exposed to ammonium sulfate precipitation (85% saturation) and Q-sepharose chromatography. Lipase MG10 was eluted from the Q-Sepharose column with a 19.5-fold purification and a 38.1 % yield, and it displayed a specific activity of 442.6 U/mg. This yield of MG10 lipase was analogous to the lipase of S. maltophilia (33.9%) (Li et al., 2013) and lower than lipase from P. aeruginosa PseA (51.6%) (Gaur et al., 2008), but greater than lipase of B. licheniformis (8.4 %) (Sharma and Kanwar, 2017). SDS–PAGE analysis of the MG10 lipase shown that it has a single band about 33 kDa, which it is dissimilar with the other Enterobacter cloacae.
Results of protein measurement with Bradford technique displayed that protein loading on these coated magnetite nanomaterials was succeeded. Moreover, the results of determination of protein loading on these nanomaterials shown that, immobilization efficiency was achieved about 73%. mGO-CLEAs lipase was spread in phosphate buffer. After a magnet was positioned sidewise, mGO-CLEAs Lipase showed fast response (60 seconds) to the peripheral magnetic field. It incomes that the magnetic CLEAs-Lip particles were shown suitable magnetic concern even though layers of CLEAs-Lipase were covered on their surfaces, wherein it is significant in term of lipase immobilization.

3.3. Analytical characterization
Lipase MG10 was immobilized on the surface of magnetic functionalized graphene oxide, in which aldehyde groups of glutaraldehyde making linkage between amine of lipase and amino coated magnetite nanomaterials (Xie and Huang, 2018). Fig. 2a and b display SEM images of magnetic functionalized graphene oxide and mCLEAs-Lipase on magnetic graphene oxide, respectively. The SEM analysis of graphene oxide on Fig. 2a shown an irregular circular structure which was similar to the earlier reports (Wang et al. 2015; Dwivedee et al. 2017), given that a bulky specific surface zone of the nanomaterials. Results of SEM image in Fig. 2b shown that lipase immobilization seem to diminish the construction of stacked GO structures. These results designated that the glutaraldehyde linkage successfully have been occurred between the amine surface of magnetic functionalized graphene oxide and amino groups of lipase.
Elemental EDX investigation from particular part of SEM image of magnetic CLEAs-Lipase for elemental plotting obviously specifies the existence of associated atoms of support containing C, N, O, Si, P, S and Fe which displays the effective functionalization of APTES, particularly by noticing Si atom (Heidarizadeh et al., 2017). Furthermore, the remarkable attendance of phosphorous atom can intensely endorse the effective lipase immobilization (Fig. 3).
Presence of functional groups on graphene and lipase immobilization onto these nanoparticles were investigated by FTIR spectroscopy. FTIR spectra of graphene oxide (A), magnetic functionalized graphene oxide (B) and magnetic functionalized graphene oxide-CLEA lipase (C) have been shown in Fig. 4. The peak around 532-614 cm?1 could be evaluated to the stretching vibration of Fe–O in Fe3O4 nanoparticles (Fig. 5B, C), representing the presence of Fe3O4 in the graphene oxide which focused that the preparation of Fe3O4-graphene oxide nanoparticles was effective (Thangaraj et al., 2016; Xie and Huang, 2018).
Moreover, peaks at 1635 and 1636 cm?1 resemble C=O vibrations of the present carboxyl and carbonyl functional groups on the mGO and presence of amide link between glutaraldehyde with Fe3O4 nanoparticles and CLEAs (Cui et al., 2015; Xie and Huang, 2018). Additionally, a characteristic adsorption band achieved at 3447 cm?1 equivalent to the adsorbed H2O and OH group on the surface of mGO (Paludo N, 2015), which shown excessive absorbance in all of these nanoparticles and the magnetic functionalized graphene oxide-CLEA (Mehrasbi et al., 2017). FTIR spectrum of magnetic functionalized graphene oxide shows the presence of a peak in 2922 cm?1 spreads to aliphatic chain of coated APTES (Heidarizadeh, et al., 2017).
After lipase immobilization on the mGO (Fig. 5c), the 614 cm?1 band owing to the stretching vibration of Fe–O in Fe3O4 nanoparticle was practically vanished, which signifying the covering of Fe3O4 by lipase. Moreover, FTIR spectrum of magnetic functionalized graphene oxide-CLEA lipase also shown two absorption peaks at 2840 and 2922 cm?1 mentioning C-H stretching in -CH3 and -CH2-, which demonstrate the immobilization of enzyme on the support. In addition, the appearance two new FTIR absorption bands at 1404 and 1514 cm?1 owing to the lipase immobilization were discovered, which specified that the enzyme was covalently bounded to the mGO nanocomposites via amide links.

3.1 0th Iteration

The 0th iteration mainly consists of the basic patch with dimensions 11.33×11.33mm. The feed position is taken at 1.9mm from the x-axisas shown in the g.3.1 The R.L vs. frequency graph is shown in g.3.2. The g.3.3 shown below represents VSWR graph. The Smith Chart for change in Width is as shown in g.3.4.

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10

Figure 3.1: 0th Iteration

Figure 3.2: R.L Vs Frequency Graph

Figure 3.3: VSWR Graph

Figure 3.4: Smith Chart

3.2 1st Iteration

The 1th iteration includes removing the slots having centre at 4.265mm from the axis and slot width of 2.6mm to increase the bandwidth of the antenna. In rst Iteration,the feed is given at a distance of 1.65mm from the centre.This is because the feed should be such that, the smith chart should passes from the centre, so that the impedance is 50 which is matched with probe feed connector leading to high power transfer.VSWR graph gives information about the frequency resonance and bandwidth. The R.L vs frequency graph is shown in g.3.6. The g.3.7 shown below represents VSWR graph. The Smith Chart for change in Width is as shown in g 3.8.

Figure 3.5: 1st Iteration

Figure 3.6: R.L Vs Frequency Graph

Figure 3.7: VSWR Graph

Figure 3.8: Smith Chart

Chapter 4

Optimized Structure

The proposed antenna is designed by using concept of Minkowski fractal structure, which originates from the plane square patch and subsequent fractal antenna. Minkowski iterations produce a cross-like fractal patch with even more ne details at the edges. The antenna is designed by using the square patch and iterating rst iteration at the center of each side. Iterated polygons (indentation) in the shape of square are created. The square patch fractal antenna is based on minkowski square shape with a ground dimension of 38mX38mm and patch dimension of 11.33mmX11.33mm . The FR-4 material is used as substrate. The thickness of the substrate is 1.59 mm. The dielectric constant (r) of the antenna is 4.4.To design the fractal antenna a square shape structure is designed on the simulator.Square indentation is cut down from the each side of the square, by doing this the path for the current ow also increases and the e ciency of the antenna also increases.

14

Figure 4.1: R.L Vs Frequency Graph

Figure 4.2: VSWR

Figure 4.3: Smith Chart

Figure 4.4: Gain Vs Frequency

Chapter 5

CONCLUSION

A simple Square Microstrip patch antenna using air and FR4 dielectric is designed and optimised to operate over 4.5 GHz to 5.875 GHz. The parametric study, such as e ect of change in slot Width, Feed position on antenna radiation pattern is carried out to optimize the antenna structure.

A new technique of implementing Fractal Antenna can be evolved from this project. Plan of action for this semester was to complete the 0th and 1st iteration of simple Microstrip Patch which we have carried out successfully.

It is found that this structure with an indentation in the border length o ers considerable miniaturisation compared with a conventional square patch antenna. For this iteration the resonance frequencies decrease to lower side which indicates size reduction as compare to non fractal structure.

Our plan of action for next semester would be simulating the 2nd iteration and optimizing it to achieve maximum gain. The fabrication of the optimized antenna will be carried out followed by testing of the antenna.

3.0 METHODOLOGY

3.1 Introduction
This chapter shows the speculative and realistic approach that is employed to provide a clue to objectives stated in this research. This chapter will show the standard that will be used to explain government expenditure and its effect on the economic growth of Tanzania. It is extended to capture theoretical context and also provide insight on where the data was acquired and methods employed in analyzing data.

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3.2 Conceptual framework
The theoretical relationship between economic growth and government expenditure is well acknowledged in many literatures. This study follow the structure of Cobb douglous production function that was exploited before by Kweka (1999) and Ketema (2006) on their reports on Tanzania and Ethiopia economy respectively in the same study which was prolonged to include government expenditure reaching growth function. In the model, output is assumed to depend on capital (K), government expenditure (G) and labour (L).

Y = f (K, G, L) …(1)

But in this period of globalization, export (X) is very vital in defining variations of output but not captured by the general model. The model can be improved to include export value as follows;

Y = f (X, K, G, L) …(2)

The increase in capital can be characterized as investment, which is later explained as an objective of government to increase development expenditures of which part of it establish investment capital. Government development expenditure can be disintegrated into three sections that are health expenditure (H), education expenditure (E) and also defense expenditure (D). Health expenditure and education expenditure can be termed as expenditure on human capital as it increases productivity and or output through finding of new technology, maintenance of good health for the people increases efficiency and productivity in delivering services on required amounts. Thus, the model above can be inflated with these variables.

Y = f (1, H, E, D, X) …(3)

Where as, the output is the function of value of export of a certain country, investment and (health, education and defense) expenditure.

3.3 Model Specification and Variables Definition
3.3.1 Model Specification
Different researchers have discussed about the effect of government spending on economic growth using diverse variables relying on the availability of the data, the literature they reviewed and the country resources. This study combines various variables used by Ketema (2006) and that of Kweka and Morrissey (1999) in Ethiopia and Tanzania, respectively the selection of this variables best ensemble the literature reviewed and also due to data accessibility. The basic equation can be presented as follows;

LGDP = ?0 + ?1Lp + ?2Lh + ?3Ld + ?4Lx + ?5Le + ?t …(4)

Where; L denote logarithms, ?t is the error term which follow all the rules of classical linear regression, LGDP is the logarithms of GDP, Lh is the logarithms of health expenditure, Lp is the logarithms of government investment expenditure, Le is the logarithms of education expenditure, Ld is the logarithms of defense expenditure and Lx denotes logarithms of export. The attachment of logarithms on both sides of the equation helps in standardizing variables under the study. Independent variables could be expressed as a ration of Gross Domestic Product but this could lead to simultaneity partiality and multiple correlation problems.

3.The framework of This Study:To analyze the security and performance implications of different consensus and network layer protocol author has prepared a quantitative framework to carry out this study. Author’s framework is a combination of two key elements.

Figure:6 Components of Study Framework
** Pictures taken from ETH Zurich Research Report.

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They are (i) POW Blockchain and (ii) Security Model. A blackchin instance is a proof of work blockchain instantiated by consensus layer and network layer parameter. As discussed earlier a consensus mechanism is what all the blocks in the network follow to validate a transaction. For example, Bitcoin uses a POW consensus layer mechanism which searches for a nonce value such that the current target value should be lesser than the hash value. In network layer two most important parameters for POW blockchain is
Block size: This defines how many transactions can be put into each block. If the block size is bigger then block propagation speed decreases. On the other side, it increases the stale block rate.

Information Propagation mechanism: This shows how information is delivered in peer to peer network. There are four types of standard information propagation mechanism:

Relay Networks: It enhances the synchronization of miners of the common pool of transaction.

In the left-hand side, POW blockchain takes consensus and network parameters as input and gives output like block propagation time, throughput. To realistically capture the output of this POW based blockchain authors have put this blockchain on the simulators they have developed. These simulators take input parameter such as block interval, mining power as well as block size, propagation protocol, the location of miner’s etc. Stale block rate is an important output from this POW based blockchain because it gives the efficiency of peer to peer connection of an honest network. This Stale block rate is taken as an input to Security model. This model also takes different security parameters as input such as adversarial mining power, mining cost, number of required confirmation. The main objective of this model is to holistically compare the security and performance of different POW blockchain with different parameters as input. This security model is based on Markov decision Process and provides an optimal adversarial strategy for double spending and selfish mining as an output.
3.1Security Model:
Parameters for the Security Model:
Stale Block Rate: Stale block rate captures information propagation mechanism.

Mining Power: This is typically used in the study model to capture the fraction of the total mining power possessed by the adversary.
Block Confirmation Number: Total number of blocks required to confirm a transaction.

Impact of Eclipse Attack: This study model accounts for eclipse attack as well.

3.2 Markov Decision Process: (MDP)
The right tool for a problem which deals with “states” and “discrete events” with a certain probability is a Markov Decision Process (MDP). MDPs are a mathematical model which decides the best policy means in what sequence the actions should be implemented to maximize a goal. An MDP model has multiple states and actions. Actions are the transitions between states. In MDP each transition can happen with some probability. In this model, some actions might provide a reward or loss to occur. Figure 7 shows a graphical depiction of a Markov Decision Process. In the intended security and performance of POW study, MDP is based on four tuples. It is represented as follows M:=<S, A, P, R>. Where S represents state space, A is for representing actions, P is the stochastic transition matrix and R is the reward matrix.

Figure 7: A graphical depiction of MDP with states s_0, s_1, S_2 and action a_0, a_1.The two rewards are -1 and +5. (Figure created by MistWiz on WikiCommons).

In this model an adversary can perform the below actions:
Adopt: If an adversary thinks it can never win over an honest miner then it performs this action.

Override: If adversaries chain is longer than the honest miner then it overrides the honest mining chain.

Match: if the length of adversarial chain and honest chain are same then adversary perform this action.

Wait: If an adversary has not yet found a block then it continues mining until it finds one.

Exit: This action is performed during the double-spending attack.
Now state space S also has four-tuple namely length of honest chain, length of adversarial chain, blocks mined by eclipsed victim and fork. In the research, paper authors built MDPs for a rational attacker and asked what the attacker should do to successfully double-spend or selfish mine.

Selfish Mining vs Double Spending: Main goal in selfish mining is to increase the relative share of the adversarial block in the main chain. In double spending, the adversary is more focused on earning maximum revenue. It is also found in the study that selfish mining is not always rational. Following an adversarial strategy for mining 1000 blocks with 30% hash power, an adversary can mine 209 blocks, but an honest miner can mine 300 blocks. In honest mining, an adversary can earn by mining a block. It also loses it’s reward if a block is adopted by the main chain. As the main chain poses maximum hash power, the probability is always high for an adversary to lose the competition.
Eclipse Attack: In this type of attack attacker takes control of peer to peer network and obscure target node’s view of the blockchain. The researcher has found attacker can saturate the connection to a target victim. It means all the connection to the victim would be bottlenecked and passed through attacker nodes so that it can manipulate the connections. Following eclipse attack scenarios are captured by our model:
No Eclipse Attack: This study model captures this case.

Isolate the Victim: This captures those cases where total mining power decreases. In return, it increases the fraction of mining power possessed by an adversary.

Exploit the eclipsed victim: Adversary uses victims mining power to expand its own chain.

3.3 Selfish Mining MDP:
As discussed previously the main goal of a selfish miner is to increase the relative number of adversary block in the main chain. In this study, the model author has captured that by optimizing the relative revenue. But there is a problem of applying single player MDP in this particular case because selfish miner deals with relative revenue. To overcome this problem the author has applied Sapirshtein el. Sapirshtein el proposes that an adversary with less than 33% of total hash power can make a profit from the network. This model captures various parameter such as block propagation time, block generation interval, block size and eclipse attack.

3.3.1 Optimal Strategies For Selfish Mining :
Authors have used MDP solver for finite state space MDP’s. The output author received from the model is below. Here the author tries to find the impact of stale block rate on selfish mining.

Figure 8: Selfish mining (Relative revenue vs Adversarial mining power)
** Pictures taken from ETH Zurich Research Report.

In Figure 8 author tries to understand how adversarial mining power influences the relative revenue of an attacker. For this he has put the adversarial mining power is in X-axis and relative revenue in the Y axis. The graph is drawn for a stale block rate of 1% and 10%. It is seen from this diagram that relative revenue increase with the increase of adversarial mining power. An upper bound is also taken in this diagram to understand the cases when the relative revenue of a selfish miner maximized by overriding a block of an honest chain. Figure 8 shows the upper bound exceeded when network delays and parameters are captured.

Figure 9: Relative revenue vs Stale rate
** Pictures taken from ETH Zurich Research Report.

In Figure 9 author tries to understand the relationship between stale block rate and relative revenue. He compares relative revenue in Y axis with stale block rate in X-axis for a mining power ? of .1 and .3 respectively. This diagram suggests a nonlinear relationship between relative revenue and stale block rate.

Author has also studied the impact of the eclipse attack in selfish mining. Figure 9 explains the relationship between eclipsed mining power ? and adversarial mining power ?. In this study the cases considered are
1. where adversary uses victims mining power ?
2. When an adversary uses honest miners blocks to advance its own chain.

It is seen for higher ? values selfish mining capability also increases. In this graph, an exceptional case is also observed for ?=.3 and ?=.38. For this situation, it is more profitable for an adversary not to include some of the victim’s blocks. Here victim’s blocks are accounted as a reward for the honest chain. This, in turn, reduces the block share of an adversary.

Figure 10: Eclipsed mining power vs Adversarial mining power
** Pictures taken from ETH Zurich Research Report.

3.4 Double Spending MDP: As discussed earlier in the double-spending rational adversary tries to maximize its profit. In double spending, it is assumed that loss in operational cost is less because the adversary can earn some goods or money in exchange for a transaction. In double spending, exit state can only be reached if the length of an adversarial chain is at least a block longer than the honest chain (la ; lh ) after k block confirmation for an honest chain with 1?? mining power. This is described in the below table 2. A question can arise during this study as the adversary is rational it is hard to reach an exit state. But it is found that in exit state adversary can earn a reward of

blocks.

** Pictures taken from ETH Zurich Research Report.

3.4.1 Optimal Strategies for Double Spending: To create optimal strategies author has used the pymdtoolbox library and applied PolicyIteration algorithm. By this block confirmation value, k is received which is sufficient to make a safe transaction in presence of rational adversary in the network. To decide in a certain scenario if a rational adversary would do double spend or selfish mining, a minimum value of double spend vd must be determined. For achieving that author start with high double spending value so that exit state is reachable in optimal double spending strategy. Author has done this because the presence of exit state in policy ensures high profitability for doubles spending strategy otherwise honest mining is more profitable. In this below Table -3 an example is shown for optimal strategy.

Table 3: Optimal Strategies for double spending.

** Pictures taken from ETH Zurich Research Report.

Here ? = 0.3,? = 0,rs = 0.41%,cm = ?,? = 0 and vd = 19.5. Length of adversary chain is la, taken as rows. Length of honest chain is lh. Three values of each entry are irrelevant, relevant and active. * means unreachable and w, a, e represents wait, adopt and exit respectively. In this example cut off value for honest chain and adversarial is taken as 20. This suggests both this chain length cannot be greater than the defined cut-off value. So what is the main goal of this analysis? The attacker must exceed a threshold if it successfully wants to double spend for a fixed number of block confirmation k. Otherwise, honest mining is more profitable. This result is illustrated in Figure 10. The x-axis shows how the adversarial mining power is influencing the threshold. Different values of k (the desired number of confirmations) lead to different curves.

The y-axis in Figure 10 shows how many successive blocks are needed to be mined before a double spending attack to be successful. For an adversary, around 30% mining power needs 6 block confirmation and the expected number of blocks is roughly 100.

An adversary with mining power of more than .25 needed less than 1000 blocks to successfully carry out double-spending attack.

Figure:10 Expected blocks for double spending rs = 0.41%, ? = 0, cm = ? and ? = 0.

** Pictures taken from ETH Zurich Research Report.

Here stale block rate is represented by rs. ?, cm represents the propagation parameter and maximum mining costs respectively.

Impact of Propagation Parameter: Propagation parameter signifies the connectivity efficiency in an adversarial chain. It suggests if connectivity increases in the adversarial network then adversarial mining power also increases. Author has put adversarial mining power in the X-axis and shown double spending transaction should have a threshold value. If transaction value is more than the threshold value, then only double spending is profitable. It can also be seen from Figure 11 that higher the propagation parameter ? lower the transaction value an adversary expects to double spend.

Figure:11 Impact of propagation parameter ? with respect to double spending transaction value.

** Pictures taken from ETH Zurich Research Report.

In this graph double spending value(vd) is taken in Y-axis and adversarial mining power(?) in the X-axis. If ? increase vd decreases.

Impact of mining costs: From the study, it is found that mining cost has a negligible impact on adversarial strategy. It is shown by the below Figure 12.

Figure 12: Impact of mining cost.

** Pictures taken from ETH Zurich Research Report.

Value of double spend (Vd) is in the Y-axis and adversarial mining power(?) in the X-axis. rs = 0.41%, ? = 0, ? = 0 Cm represents maximum mining cost ?vd is the difference in costs.

Impact of Stale Block Rate: In Figure 13 impact of stale block rate is explained for double spending. This below experiment is carried out for a mining power of .1 and .3 respectively. It can be seen if stale block rate grows the value of double spend decreases. Author has found double spending value of an adversary decreases from 9.2 to 6.4 block reward with mining power .3 and a stale block rate of 10% and 20 %.

Figure:13 Impact of stale block rate.

** Pictures taken from ETH Zurich Research Report.

Here Vd is the value of double spend in the Y-axis, Stale block rate in X-axis and adversarial mining power is represented by ?.

Impact of Eclipse Attack: The impact of eclipse attack is represented by Figure 14. It is assumed that an adversary attacks an honest block with ? eclipsed mining power. It can be observed eclipsed mining power increases with the increase of adversarial mining power. So eclipse attack is beneficial for an adversary. For example, an adversary with an adversary with ?=.025 and ? =.1 reduces the double spending value (vd) from 880 block reward to .75 block.

Figure 14: Full eclipse attack
** Pictures taken from ETH Zurich Research Report.

In Figure 14 eclipse mining power ? is in Y axis and adversarial mining power is in X axis and , rs = 0.41%, ? = 0 and cm = 0.

Bitcoin vs Ethereum: Figure 15 shows the reward required for a double spending attack to make a profit. The y-axes show the reward required from fraudulent behavior as multiples of the block reward, i.e. multiples of the reward of non-fraudulent behavior.
The figure also contrasts between Ethereum and Bitcoin. As a consensus algorithm both this chain uses proof of work, but the key difference is the block time. i.e. the duration between the generation of two blocks. Stale block rate increases because of shorter block times. It means the time gap between finding two blocks is much shorter in Ethereum. Thus, participant blocks more often return finding the same block which increases the stale block rate in the network.

Below points are observed by the author in the study.
First: Figure 15 shows 6 Bitcoin block confirmation is more resilient to double spending than that of 12 Ethereum block.

Second: Ethereum’s double spending resilience is better only for an adversary with less than 11% hash power.

Third: If block reward goes up blockchain is more resilient to double spending attack.

Figure 15: Double spending resistance of Ethereum vs Bitcoin
** Pictures taken from ETH Zurich Research Report.

Block reward is in the Y-axis and Adversarial mining power in the X-axis. Ethereum (k ?{6,12}) vs. Bitcoin (k = 6).

Author has also tried to compare both this block chains by equalling their stale block rate. It is observed that Ethereum’s security is lower in caparison to bitcoin Figure 16 explains the following.

Figure 16: Comparison between Ethereum and Bitcoin.

** Pictures taken from ETH Zurich Research Report.

Value of double spend is on the Y-axis and Adversarial mining power is in the X-axis. Here k is 6, rs = 6.8% and their difference is ?vd.

4.Blockchain Simulator and Results: The simulator author has developed for this study capture parameter like block size, block interval, propagation mechanism by measuring stale block rate, block propagation times. In this simulator point to point connections are established between nodes. Global IT latency statistics of Verizon are used to capture latency in the network. Regular nodes and miners are distinguished in this network. Bitnode’s geographical node location is adopted and used for the nodes in this simulator. Author has also used blockchain.info’s mining pool distribution and used it in this simulator. In Table 4 all the parameters which are captured by the simulator are listed.

Table -4 Parameters of Simulator
** Pictures taken from ETH Zurich Research Report.

4.1 Evaluated Result:
Simulator Validation: In order to validate the performance of this study author has adjusted the parameters of table 4 with the real world deployed blockchain. For determining the stale block rate author has crawled 24000 Bitcoin,1000,000 Litecoin, and 240,000 Dogecoin blocks. The performance achieved from this model is quite like the real world blockchain. Stale block rates of Dogecoin and Litecoin are particularly close and Bitcoin’s stale block rate falls in some cases like where relay network and unsolicited block push is not used by the miner.

Figure 17: Geographical Location of Bitcoin miner’s in study simulator.

** Pictures taken from ETH Zurich Research Report.

Block Interval: Author has tested block interval with a range of .5 sec to 25 minutes in the simulator. It is tested for four different block request management system namely 1. Standard block request management 2. Standard block request management enhanced by unsolicited block push from miners 3 Standard propagation mechanism with relay network 4. Send header mechanism with unsolicited block push and relay network.

For standard block request management system with 10 minutes block interval study simulator produces stale block rate 1.85 % in comparison to 1.69 % reported by Wattenhoffer.

Stale block rate reduces significantly after the introduction of unsolicited block push for miner because of two main reason—a. miners profit most out of unsolicited block push because they are interconnected b. propagation method is crucial to reach most of the network rapidly. To measure the impact of the block interval author has fed the resulting stale block into MDP models. It is found for an adversary with 30% of total mining power relative revenue is inversely proportional to consensus time.

Impact of Block Size: From the study, it is found block propagation time has a linear relationship with block size. But this linear relationship is valid up to 4 MB block size. From 4 MB to 8 MB stale block rate increases exponentially with propagation times. If block size increases the relative revenue of selfish miner also increase but double spending value decreases. Author has also found an efficient block propagation mechanism to increase the security of the blockchain. The results of this study for four previously discussed block request management system is shown in table 5.

Table 5: Impact of the block size on the median block propagation time (tMBP) in seconds
** Pictures taken from ETH Zurich Research Report.

The stale block rate is rs, vd, and rrel, given the current Bitcoin block generation interval and an adversary with ? = 0.3 and k = 6.

Throughput: Author has varied block size (.1 MB-8 MB) and block interval (.5 second-25 Minutes) to capture different blockchain throughput. Throughput is calculated in transaction per second (tps). Stale block rate and infer are represented with vd and rrel. The result author has got is shown in the below table -6.

Table 6: Impact of throughput for K=6 and 16 mining pool with 30% adversarial mining power.

** Pictures taken from ETH Zurich Research Report.

From this table, it can be seen 60tps throughput can be achieved with existing security in the bitcoin by changing the input parameters like block size and block interval.

5. Related Work: There were many who have worked on the double-spending attack, but no one has worked on adversarial strategies before the author. Eyal and Sirer in their study show relative revenue of a selfish miner can be increased by not publishing their blocks directly. Courtois and Bahack study is related to subversive mining. Author’s work is similar to Sapirshtein et al’s study. The only difference lies between their study is the author captures the optimal double spending strategies by considering the mining cost of an adversary, number of block confirmation and double spending value which Sapirshtein did not.

6. Conclusion: In this study author has proposed a quotative framework to measure the security and performance of different POW based blockchains. The impact of network level parameters on the security of blockchain is evaluated in this study. From the study, it is found 37 Ethereum block confirmation equals 6 Bitcoin block confirmation. It means Bitcoin blocks are more secured than Ethereum’s. It is also proved that 60 tps of Bitcoin throughput can be achieved without sacrificing the existing security by varying input the parameters.

7.Reference:
1. On the Security and Performance of Proof of Work Blockchains by Arthur Gervais, Ghassan O. Karame, Karl Wüst, Vasileios Glykantzis, Hubert Ritzdorf, Srdjan ?Capkun.

2. https://vimeo.com/1271955253. https://steemit.com/blockchain/@cryptonik/what-are-orphaned-and-stale-blocks-blockchain-techology-explained4. https://www.cryptocompare.com/coins/guides/what-is-bitcoin-selfish-mining/5. https://medium.com/@chrshmmmr/a-guide-to-dishonesty-on-pow-blockchains-when-does-double-spending-pays-off-4f1994074b526. https://www.youtube.com/watch?v=dTdXljsLiUs

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