# A Secured and Energy Efficient Wireless Sensor Networks (WSN) Utilizing CRT-Based Forwarding Technique.

1 IntroductionWSNs comprise of interconnected Sensor hubs which are sent in a huge number and are equipped for detecting, assembling, preparing and transmitting information (Sharma and Sharma, 2016). They are normally used to monitor regions, gather information and send to the base station or sink. There are considerable numbers of potential applications for sensor systems. For instance, they can be utilized in a combat zone, where they can identify and keep an eye on the adversaries or they can bolster the positive powers. Likewise, they can be utilized in shrewd security frameworks in structures and security of basic applications. They can also be utilized for habitat monitoring applications and for study changes in phenomena, for example, temperature and sound mugginess (Kianifar, Naji and Malakooti, 2015).

Wireless sensor networks are capable of sensing and forwarding the sensed data, and performing different reactions. It comprises of sensor hubs and sink hubs which for the most part have low costs, constrained vitality supply and restricted transmission; they are in charge of distinguishing occasions or detecting ecological information (Campobello, Leonardi and Palazzo, 2009). The base stations are asset more than extravagant hubs and at the same time possess rich vitality sources, higher correspondence and computation capacity, and the capacity to perform ground-breaking responses. At the point when a sensor hub recognizes information to be conveyed in its checking zone, it would transmit the occasion to neighboring hubs, which in turn would forward the occasion one bounce further.

Figure 1 demonstrates a Wireless Sensor Network containing sensor hubs and the sink. A sensor hub is a modest gadget that incorporates three parts: a processing unit used to process neighborhood information, a sensing subsystem utilized for information procurement from its condition, and a wireless communication transceiver that exchanges the detecting data to the sink. These sensor hubs perform information detecting and preparing errands and furthermore speak with each other (Kianifar, Naji and Malakooti, 2015).

WSNs are typically furnished with restricted vitality which makes them asset obliged. Thus, in this way, the most essential perspective is the means by which to limit the vitality exhaustion of hubs remembering the ultimate objective to extend the network lifetime. Sensor hub batteries of WSNs cannot be replaced or revived once deployed. Therefore, WSNs application should be designed in an energy efficient manner (Lee, Min, Choi and Lee, 2016). Also, hubs in WSN are ordinarily mass delivered and are regularly sent in neglected and unfriendly situations which now make them more susceptible to failure than other network systems (Kshirsagar and Jirapure, 2012). However, manual assessment of flawed sensor hubs after deployment is ordinarily unfeasible. In any case, numerous WSN applications are mission-basic, and therefore requiring continuous operation. Given the characteristics and with end goal to meet application prerequisites, an anchored, dependable and vitality productive philosophy is required in WSNs (Sachan, Imam and Beg, 2012).

In this paper, a secured and energy efficient WSNs based on Chinese Remainder Theorem (CRT) bundle part sending procedure is proposed. This strategy include breaking the detected messages into a few parcels (contingent upon number of hubs in the next hop) with the end goal that every hub in the system will forward just little sub-bundles to the sink. When the sink gets all sub-bundles effectively, it would reproduce the first message. The significant purpose behind this approach is to break the information that was sent by each sending hub in order to diminish the number of bits transmitted by each sending hub in the system, therefore power utilization is reduced. The proposed scheme outflanks traditional methodologies regarding security, vitality utilization, unwavering quality, straightforwardness, and diminishment in end-to-end delay.

1.1 Fault Tolerant in WSNs

A Wireless Sensor Network is appropriate in different fields, for example, information procurement in risky condition, observing of basic frameworks and military activities. The unfriendly environment affects the monitoring infrastructure of WSNs. Since sensor hubs are relied upon to work independently in unattended and conceivably threatening situations, they are powerless against deficiencies where issues are probably going to happen often and unexpectedly (Kshirsagar and Jirapure, 2012). WSNs are failure inclined because of any of the reasons like malignant assault, energy depletion, hardware disappointment and communication.

It is generally trusted that nothing is flawless in this universe; issues are likewise unavoidable in the sensor system and it is extremely important to recognize defective and working hubs (Manisha and Nandal, 2015). With the specific end goal to keep up the system nature of administration, it is necessary for WSN to have the capacity to distinguish the deficiencies and take fitting actions to deal with them. When outlining a mistake control plot for WSNs, vitality proficiency is most essential. The utilization of a particular fault tolerant system relies upon the prerequisites of the system and the requirements of the sensor network. Thus, it is important to pick an ideal fault correcting code for a sensor network where both the execution and vitality utilization are considered (Arutselvan and Maheswari, 2013).

2 Literature Review

There have been a few examinations on limiting energy utilization in remote systems and particularly in Wireless Sensor Networks (WSNs). At the same time, a few vitality preservation plans have been proposed for limiting the energy utilization of the radio interface. This is because radio units consume the highest amount of power when transmitting information. Duty cycling (rest/wakeup plot) and In-Network information accumulation (Anastasi, Conti, Francesco and Passarella, 2007; Fasolo, 2007) are the most well-known preservations plans. The main approach includes putting the radio transceiver in the rest mode at whatever point communications are not required and wake when gotten a bundle from a neighboring hub. Be that as it may, energy reduction is gotten to the detriment of an expanded hub multifaceted nature and system idleness.

The second approach is aimed to merge routing and data aggregation techniques by reducing the number of transmissions. In this plan, multipath steering calculations are typically utilized. Nonetheless, numerous ways could surprisingly expend more vitality than the briefest way in light of the fact that few duplicates of a similar bundle could achieve the destination. Barati, Movaghar and Sabaei (2014) observed that redundant residue number systems (RRNSs) are appropriate for use in real time wireless sensor networks applications, because it enhances real time operations, strong error control capability, energy saving, and security. The RNS has been utilized as an instrument to lessen transmission vitality and increment unwavering quality in WSNs.

In addition, Arutselvan and Maheswari (2013) proposed an approach that depends on a parcel part calculation-based CRT described by a basic particular division between numbers. The structure has low overhead in calculation, correspondence and limit, and is immune to DoS attack. Campobello, Serrano, Leonardi and Palazzo (2010) researched a tradeoff between vitality effectiveness and unwavering quality of the CRT sending plan when obligation cycling strategies are considered. This was accomplished with a direct increment in the general unpredictability and with low overhead.

Additionally, Roshanzadeh and Saqaeeyan (2012) observed that the restricted vitality utilization prerequisites and the low many-sided quality in the sensor equipment require vitality effective blunder control and forestall high many-sided quality codes to be sent. Redundant moduli that plays no part in deciding the dynamic range was presented. This was utilized in WSNs to diminish reestablished information sending by means of any error occurring in information parcels which were centered around low multifaceted technique of error detection. The system was implemented with low data redundancy and efficient energy consuming in wireless sensor node using residue number systems. It is therefore necessary to develop proficient and secured WSNs while still conserving the limited energy of the network as well as end-to-end delay.

3 Chinese Remainder Theorem (CRT)

Chinese Remainder Theorem is a hypothesis of numeric theory, which expresses that, in case one knows the remnants of the division of an entire number n by a couple of numbers, one can choose curiously whatever is left of the division of n by the consequence of these numbers, under the condition that the divisors are pairwise coprime (Gbolagade and Contofana, 2009). The CRT is an outcome about congruence's in number hypothesis and its speculations in unique variable based math (Arutselvan and Maheswari, 2013). In its essential frame, the CRT will decide a number n that at the point when partitioned by some given numbers (divisors) leave given remnants (Omondi and Premkumar, 2007).

Assume [n.sub.1],...,[n.sub.k] be entire numbers more prominent than 1, which are consistently called moduli or divisors. Assume N implies the consequence of the [n.sub.i],. The CRT hypothesis declares that if the [n.sub.i], are pairwise coprime, and if [a.sub.1],..., [a.sub.k], are numbers with the end goal that 0 [less than or equal to] [a.sub.i] < [n.sub.i] for each i, at that point there is one and just a single whole number x, to such an extent that 0 [less than or equal to] x < N and the rest of the Euclidean division of x by [n.sub.i] is [a.sub.i] for each i.

This might be rewritten as follows in term of congruences: If the [n.sub.i] are pairwise coprimes, and if [a.sub.1],..., [a.sub.k] are any whole numbers, at that point there exists a whole number x to such an extent that any two such x are compatible modulo N.

x = [a.sub.1] (mod [n.sub.1]) x = [a.sub.2] (mod [n.sub.2]) . . . x = [a.sub.k] (mod [n.sub.k])

In any case, all arrangements' x of this framework are compatible modulo of the item, N = [n.sub.1], [n.sub.2],..., [n.sub.k]. In this way, x = y (mod [n.sub.1]) for all i[less than or equal to]i<k if and just if x [equivalent to] y (mod N). The customary CRT is characterized as take afters: for a moduli set {[m.sub.1], [m.sub.2], [m.sub.3],..[m.sub.k]} with the dynamic range M [mathematical expression not reproducible], the residue number ([x.sub.1], [x.sub.2], [x.sub.3], [x.sub.k]) can be changed over into the decimal number X, as takes after:

[mathematical expression not reproducible]

where [mathematical expression not reproducible], [M.sub.i] = M/[m.sub.i], and [mathematical expression not reproducible] is the multiplicative inverse of [M.sub.i] with respect to [m.sub.i]. The primary downside of CRT rises up out of the required modulo-M activity which, given that M is a somewhat vast number, this task can be tedious and fairly costly as far as territory and vitality utilization are concerned. The CRT is valuable in invert change and also a few different activities (Omondi and Premkumar, 2007).

However, the nonattendance of convey spread between the arithmetic block results in RNS fast arithmetic. This feature is beneficial for wireless sensor networks that need to perform run-time applications. RNS also has parallel operations that reduce power consumption and delay simultaneously (Barati, Movaghar & Sabaei, 2014; Gbolagade & Cantofana, 2009). Sangeetha and Pugazendi (2013) also observed that CRT likewise can be utilized to take care of issues in processing coding. In computing it can compete with short instead of large numbers and this will make the computing-process faster and easier. In coding it can be utilized for blunder seeking and mistake controlling. The calculation permits recreating a vast whole number from its remnants modulo, an arrangement of moduli. At the point when every one of the moduli are co-prime, CRT has a straightforward single recipe, which is notable not hearty, i.e., little mistakes from any remnants may cause a substantial recreation error.

4 Packet Forwarding Technique

The capacity to furnish separated administrations to clients with broadly changing necessities is becoming progressively imperative, and Internet Service Providers might want to give these separated administrations utilizing the same shared system framework. In a computer network, packet sending procedure is the handing-off of bundles starting with one system portion then onto the next by hubs (Anton, 2015). The least complex sending model, uni-throwing, includes a message (packet) being transferred from connection to interface along a chain driving from the message's source to its destination. Another sending model is communicating which requires a message (packet) to be copied and duplicated before it moves on different connections with the objective of conveying a duplicate to each gadget on the system. Numerous ways could surprisingly expend more vitality than the single most limited way on the grounds that few duplicates of a similar parcel must be sent.

Packet Processing involves a wide collection of calculations that apply to a parcel of data or information as it goes through the diverse framework parts of corresponding arrangement (Anton, 2015). Ordinarily, there are two broad classes of packet processing that line up with the organized network subdivision; these are control plane and data plane. Calculations are associated with either and each has control information in its bundle to ascertain safe transfer of the parcel of source to destination. The data content (as frequently as conceivably called the payload) of the package are used to give some substance specific change or make a substance driven move.

Packet Splitting involves breaking the first bundles into different sub-parcels before transmitting them towards the hubs (Anton, 2015). The first messages are broken into a few parcels with the end goal that every hub in the system will send just little sub bundles and reproduce them back to the original messages. The breaking technique is accomplished by using the packet splitting scheme as seen in Figure 2.

When all sub packets are gotten accurately by the sink hub, it will rearrange them, subsequently reproducing the initial message. This system is particularly useful for those sending hubs that are more requested than others because of their location in the system. The initial messages are breaking down into a few bundles with the aim that every hub in the system will forward just little sub parcels. The system is accomplished by applying the packet splitting algorithm. Subsequently, this gives an exhaustive systematic model that enables us to determine some precise outcomes in regards to vitality utilization and multifaceted nature (Fasolo, 2007).

5 Research Methodology

Packet splitting forwarding technique is the basic method for sharing information across systems on a network. Packets are transmitted between a sender interface and a receiver interface, usually on two different nodes. With the CRT for information bundling, a hub begins at an arbitrary position and assigns prime numbers to the information parcels for security reason. For these reasons interlopers would not recognize the information packet because the original message is not sent but the prime numbers are. CRT-based part is more productive than a straightforward part (Campobello and Leonardi, 2015). The major reason for using forwarding technique is to split the messages sent by the source node so that the maximum number of bits per packet that a node has to forward is reduced, in this way the network lifetime can be expanded (Campobello and Leonardi, 2015). CRT can be planned as follows:

Given that N primes [p.sub.i]>1, with i [member of] {1...N}, and considering that their item M = [[product].sub.i] [p.sub.i], at that point for any arrangement of any given whole numbers {[m.sub.1],[m.sub.2],...,[m.sub.N]} there is always a novel whole number m < M that understands the arrangement of synchronous congruences m = [m.sub.i] (mod [p.sub.i]), and it can be gotten by m = ([N.summation over (i=1)][c.sub.i][m.sub.i])(mod M). The coefficients ci are given by [c.sub.i] = [Q.sub.i][q.sub.i] where [Q.sub.i]= M/[p.sub.i], and [q.sub.i] is its secluded backwards, that is, [q.sub.i] explain [q/sub.i][Q.sub.i]= 1(mod [p.sub.i]).

For outline reason, assuming that the moduli-set {3, 5, 7} with buildup portrayal (1, 2, 3), subsequently are:

m = 1 (mod 3) m = 2 (mod 5) m = 3 (mod 7)

Then by CRT, we have m = 52 (decimal value). However, it is worth mentioning here that in the above example 7 bits are needed to represent m, while no more than 3 bits are needed to represent each mi. Therefore, if instead of m, mi numbers, with [m.sub.i] = m (mod [a.sub.i]), are forwarded in a wireless sensor network, the maximum energy consumed by each node for the transmission can be substantially reduced. In the event that hubs M and N need to send packet, to the base station, consider Figure 3:

On the off chance that there are n bits for every bundle, the most extreme number of bits transmitted by a hub having a place with the set {O, P, Q} is n/3 bits. Specifically, when O, P, and Q get a message (packet), they break it and forward to the base station just a part (e.g. n/3 bits each). For this situation, O, P and Q need to send at most 2/3 n bits each.

It can be inferred that this strategy diminishes the most extreme number of bits forwarded by a hub having a place with the set {O P, Q} if contrasted and other sending systems (like ordinary sending with various next-jump or typical sending with the same next-hop). Be that as it may, utilizing part computation, it is ensure that most extreme number of transmitted bits per hub is lessened, and in this manner the vitality that a hub expends for the transmission is diminished. The splitting scheme is accomplished by using the CRT which applies a low intricacy approach requiring just a secluded division amongst whole numbers and thus it can be done by extremely basic gadgets as node hubs without underestimated organized dependability.

Besides, from Figure 4, if hubs O, P, and Q get a message [M.sub.s] communicated from hub M, every one of them, applying the methodology appeared above, can forward a message [m.sub.i], with i[member of] {1, 2, 3} (called CRT segments), to the base station rather than [M.sub.s]. Besides, the base station, knowing [a.sub.i], with i[member of] {1, 2, 3}, and utilizing the CRT approach, will have the capacity to reproduce [M.sub.s]. Be that as it may, as indicated by the CRT, the number m can be on the other hand related to the arrangement of numbers mi gave that [a.sub.i] are known.

For the sake of illustration, let expect that N = 11 messages of n = 90 bits are sent. Clearly without splitting, no less than one of the hubs O, P and Q will forward four messages (i.e. 90*4 = 360 piece). In a similar way, while utilizing a breaking, each information can be parted into three segments of 30 bits every, so 30*11=330 bits are sent.

Thus, when utilizing part, the most extreme number of forwarded bits per hub is diminished by around 8% (30/360 * 100). In addition, there will be lessening increments if the proportion "message length over number of parts" diminishes (i.e., if the quantity of accessible next-hop hubs are increments).

5.1 Proposed Packet Splitting Forwarding Algorithm

Considering the system in Figure 6, the clusters are acquired in instalments. Initialization sorted out the system in bunches and furthermore limited the quantity of jumps expected to achieve the base station. During initialization, it is expected that the sink knows the prime numbers [p.sub.i] with the specific end goal to remake the original packet and furthermore extraordinary [p.sub.i] are picked by each next-jump of the source. In any case, initialization is acknowledged through a trade of initialization messages (IMs) beginning from the sink that should have a place with the group 1, i.e., CLID = 1, where CLID distinguishes the cluster number.

Every hub that gets an IM from its neighbours with a grouping number SN = h, will have a place with cluster h and will resend the IM with an expanded SN together with its own particular address and the rundown of the hubs that will be utilized as senders. Based on the gotten IMs, toward the finish of the system every hub in the system will know its own particular next-hops, which different hubs will utilize it as a next-jump, and into what number of parts the gotten bundles can be parted (split). In any case, initialization is generally enacted just once. The initialization phase algorithm is given below:

Initialize SN=1 // To reset an initialized message While message IM arrives at a nodei do //IM is initialization message If CLID = 1 //base station is the only node in C[L.sub.ID] = 1 Then transmit IM with SN=2 to the next C[L.sub.ID] at startup Increase SN=SN+1 Else Transmit IM to the next C[L.sub.ID] Increase SN=SN+1 All nodes that receives the IM with SN=i assume to belong to C[L.sub.ID]=j

However, the packet splitting forwarding rule is given as follow:

While message arrives at Nodei do If next cluster next-hop = 1 Then send packet without splitting Else Use CRT to split the packet into numbers of next-hop Use Local Closet First (LCF) // To determine the best destination Send mi = m(mod pi) if node = sink Then reconstruct using m = (mod M) //where M is the product of the prime numbers related to the received components. Else Send mi = m(mod pi)

6 Discussion of Results

Consider the Figure 6, packets sent by every hub when the sending hub K communicates message specific m to the base station S. As per the initialization system, hub L realizes that it is the main next-jump of hub K and thus it should forward the bundle without splitting. In any case, it is not fundamental for K to indicate the rundown of the destination that tends to be {M, N, O, P} in the bundle. In the instatement stage, hubs {M, N, O, P} have effectively gotten the IM message IM:[SN = 5, L, { M, N, O, P }], and along these lines they realize that hub L has 4 next-jumps and that every one of them needs to part into nodeL = 4 sections the messages gotten from L. Along these lines, when M, N, O, and P get the bundle, they continue as follows:

a. From the parcel size, n, and also the quantity of next-bounces, nodeL, they freely acquire the arrangement of prime numbers;

b. One of the prime numbers is selected, every one of them based on their situation in the rundown of addresses {M, N, O, P} indicated in the already specified IM;

c. Then they forward the segments [m.sub.i] = m (mod [p.sub.i]) one each, together with a legitimate veil, to one of the conceivable next-jumps (Q or R). Clearly just hub Q is in the scope of hubs M and N and just hub R is in the scope of hubs O and P. Hubs Q and R just forward the gotten CRT segments to the sink since they realize that the gotten messages were at that point split.

d. Lastly, when the base station S gets a segment [m.sub.i], it distinguishes the quantity of expected parts based on the cover, and in this way it computes the arrangement of prime numbers, and the coefficients [c.sub.i] is expected to reproduce the original message.

When the base station receives the parts of the sent message, it can then rearrange the message by utilizing m = [[summation].sub.i][c.sub.i][m.sub.i] (mod M) where M is the product of the prime numbers related to the parts that were received.

7 Performance Evaluation

WSN exhibitions are assessed by utilizing the following framework:

Parcel Lost = Number of Packets send - Number of Packets Received

Throughput: It can be defined as the total sum of bundles conveyed divided by the total number of reproduction time.

Throughput= N/1000

Where N is the quantity of bits received effectively by all receiving nodes.

Energy Efficiency: It is characterized as the aggregate unused vitality level of hubs in the system. Node Energy consumption is defined as the communication (transmitting and receiving) energy the network consumes; the idle energy is not counted.

8 Simulation Results

Simulations were used to assess the execution of the hubs. The simulations were performed on prowler (Probabilistic Wireless Network Simulator) running under MATLAB. Different numbers of sensor nodes were deployed randomly starting from 10 to 50 in a sensing field of 150 x 150 square meters. All the sensor hubs were related to a unique id and it was expected that every sensor hub is stationary after deployment. The simulation was kept running on irregular system shows, where the hubs arrangements were changed haphazardly in consistently square territory. Every hub had constrained battery vitality, while the accessible energy at the sink might have been moderately boundless. At initial stage, 10 Joules of vitality was doled out to each hub and afterward infusion of the system with 500 arbitrarily created message packets. The reenactments results are in figures underneath.

Figure 7 indicates 10 sensor hubs that were consistently appropriated over a 150mx150m territory. Parcel conveyance proportions were ascertained in the light of the number of bundles sent and bundles gotten. The message breaking was performed just a single time by the hubs that are the nearest to the source, though the other node hubs in the system would simply send the sub bundles. Additionally, just the sink hub would remake the original message.

The Figure 8 shows packet delivery ratio for three different approaches. The proposed CRT-based forwarding techniques approach achieved a high packet delivery ratio compared to the existing approaches (shortest path and Normal forwarding techniques).

Table 2 shows the delay in receiving the sent message when different numbers of nodes were deployed using different techniques. Figure 9 also shows the delay in receiving the sent message when message packets were sent from source to sink. The CRT-based forwarding technique reduced the delay in packet forwarding compared to the existing approaches which increase the delay in packet forwarding.

Table 3 displays the percentage packet lost calculated for different numbers of nodes. Figure 10 gives the percentage packet lost when packets were sent over different approaches. Existing approaches achieve more packet loss but the CRT-based approach avoids this much of packet loss.

Table 4 displays the energy consumption by different numbers of nodes using three different techniques. Figure 11 displays the energy consumption by each node using three different approaches. In proposed CRT-based forwarding technique energy efficiency reach the level of 0.17.

9 Conclusion

An enhanced fault tolerant in wireless sensor network using Chinese Remainder Theorem is proposed. The proposed algorithm is compared with normal forwarding techniques that utilize different next jumps, next hop as well as shortest part. The simulated result shows that the proposed CRT-based forwarding technique outperformsother techniques in term of power utilization and the delay in receiving the sent message.

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Kamaldeen Ayodele Raji and Kazeem Alagbe Gbolagade

Department of Computer Science, Kwara State University, Malete, Nigeria kamalayour2004@gmail.com and kazeem.gbolagade@kwasu.edu.ng

Table 1: Number of Sensor Nodes Deployed vs PDR No of Node/PDR 10 15 20 25 30 35 40 45 CRT 0.73 0.735 0.739 0.75 0.76 0.78 0.79 0.80 Shortest Path 0.76 0.765 0.761 0.78 0.795 0.80 0.81 0.82 Normal Forwarding 0.80 0.81 0.84 0.85 0.88 0.885 0.90 0.91 No of Node/PDR 50 CRT 0.82 Shortest Path 0.85 Normal Forwarding 0.92 Table 1 shows the packet delivery ratios with respect to numbers of nodes. Table 2: End-to end Delay Traffic Data (kb)/Delay (ms) 50 100 150 200 250 300 450 CRT 3.1 3.6 3.8 3.6 3.81 3.9 4.5 Shortest Path 4.2 4.7 5.0 5.2 5.4 5.5 5.7 Normal Forwarding 4.0 4.55 4.8 5.0 5.25 5.35 5.5 Traffic Data (kb)/Delay (ms) 400 450 500 CRT 4.6 5.1 5.7 Shortest Path 6.3 6.5 7.2 Normal Forwarding 6.1 6.3 7.0 Table 3: Percentage Packet Lost Packet Size / %Lost 75 150 250 350 450 600 700 800 900 CRT 10.6 7.1 3.2 2.9 2.68 2.5 2.3 2.1 2.0 Shortest Path 16 12 5.0 4.8 4.4 3.2 3.0 2.6 2.45 Normal Forwarding 14 10 4.8 4.6 4.2 3.0 2.8 2.4 2.35 Packet Size / %Lost 1,000 CRT 1.8 Shortest Path 2.4 Normal Forwarding 2.3 Table 4: Energy consumption by each Node No of Nodes/Energy Spent(joule) 5 20 30 50 60 70 80 CRT 0.07 0.15 0.17 0.135 0.15 0.15 0.15 Shortest Path 0.09 0.22 0.23 0.2 0.223 0.224 0.224 Normal Forwarding 0.12 0.25 0.26 0.228 0.275 0.275 0.275 No of Nodes/Energy Spent(joule) 90 100 CRT 0.15 0.15 Shortest Path 0.224 0.224 Normal Forwarding 0.275 0.275

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Title Annotation: | Chinese Remainder Theorem |
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Author: | Raji, Kamaldeen Ayodele; Gbolagade, Kazeem Alagbe |

Publication: | Computing and Information Systems |

Article Type: | Report |

Date: | Mar 1, 2019 |

Words: | 5676 |

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