Researchon Polynomial Interpolation Methods and their Usage for Implementing Key Management Schemes for MANETs

International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue- 6S4, April 2019

Abstract—A flexible, inexpensive and dynamic infrastructure of Mobile AdhocNETworks (MANETs) makes it significant for various applications. Taking into consideration its pervasive and ubiquitous popularity, MANETs pave the way for researchers to develop different security mechanisms to enable reliable communication. It is found from the review that the central aspect of security in MANETs is its key management which is highly restricted because of its features.Through this article, the computational complexities in key management have been addressed through polynomial interpolation which mitigates the burden of mobile devices to a large extent and enhances security. In this paper, some of these polynomial approaches and its applications in different scenarios are reviewed and how they can be used for enabling the secure exchange of key shares has been discussed. In the end, it is concluded that Chebyshev Polynomials are best among them. This interpolation method not only minimizes the liability of the moving nodes but also lessens the requirement of computation power and memory.

Keywords— Security, Polynomial interpolation, Key Share, Key management scheme

I. INTRODUCTION

The shift from wired devices to wireless devices has been a worldwide vogue since past few years. With the ongoing advances in computing and wireless technologies, MANETs have become an indispensable part of the computing environment. Unlike wired networks, MANETs are infrastructure less networks that contain wireless mobile devices. The popularity is because of their self-maintained and distributive nature. The mobile devices operate in a self-organized way through wireless channels and are desired to cooperate with each other because of the lack of infrastructure [1]. The most important thing here is free of cost communication through the unlicensed bandwidths and protocols like Bluetooth, Wi-Fi and ZigBee[2]. Because of its numerous applications, security is a more crucial issue in these types of networks as many passive or active attacks can be caused by inside or outside hosts[3].

In security services, cryptography is certainly a significant and dominant tool for providing integrity, non-repudiation, confidentiality and authentication. There should always be some elementary secure, efficient and robust key management system beneath any cryptosystems. It has two methods symmetric-key and asymmetric-key. Threshold Cryptography (k:n) given by Shamir in 1979 is quite a different approach where according to a random polynomial[4], a secret is fragmented into n parts. The secret

Revised Manuscript Received on April 12, 2019.

Chetna,Ph.D. degree in Computer Science & Engineering at Chitkara University Institute of Engineering & Technology, PunjabIndia

K.R. Ramkumar, Anna university, Chennai, Tamilnadu, India.

Shaily Jain,JUIT, Waknaghat

key is recalculated by collecting k parts using various interpolation methods.

In this article, the state of the artwork related to polynomial interpolations has been extensively analyzed. The remaining article has been sectioned as stated. Section 2 states the challenges incurred and motivational force that drives the researchers to explore more in key management schemes is specified. Section 3 presents an outline of the Key Management and how it can be implemented through polynomial interpolation methods. Section 4 depicts the brief review of the papers where these methods were used and how it can be used for implementing key management has been discussed. Section 5 offers obligatory conclusions.

II. KEY MANAGEMENT

Keying material includes initialization parameters, public/private key pairs and other parameters (secret as well as non-secret). When the keying material is shared by the network nodes, it is called “keying relationship” state. In key management schemes, the keying relationship between the authorized parties are supported, established and maintained [5].

A.Key management through Polynomial Interpolations

Polynomial interpolation is the process of estimating values of the unknown data point or knowing the value within the gap of the graph when the data is available on both sides of the gap or at particular points[6].

Alike bilinear mapping, polynomial interpolation requires few modular multiplication operations. Bilinear pairing is the pairing in which the sender and receiver perform pairing operations several times which results in high computation costs and shorter lifetimes [7][8]. These properties of the interpolation method make it useful in distributing keys in MANETs. To ensure the security of payload [9], to minimize the computation costs and to effectively utilize the limited battery power [10], different polynomial interpolation methods are used. These methods are Lagrange Interpolation method, Curve fitting, Spline Interpolation and Chebyshev Polynomials which can be used for distributing key shares among nodes. Evaluation, differentiation, and integration can be effortlessly done by these methods.

The foremost step in securing MANETs using these methods is to elect the desired number of Security Association Members (SAMs) whose responsibilities and authorizations are pre-stated in advance[11]. As all the

Researchon Polynomial Interpolation

Methods and their Usage for Implementing

Key Management Schemes for MANETs

Manets duties of implementing key management scheme are entitled to SAMs, very limited number of them are deputed to generate the key shares. The coefficients of the polynomial function are determined beforehand and known only to SAMs. Intruders cannot be able to determine these coefficients so the risk of breaching is least. All SAMs calculate its key share from the agreed polynomial using its unique ID. If a new node enters in the network, it accesses any random SAM for its key share. It is the responsibility of each SAM to decide the authenticity of the new node that whether it can be trusted or not. If the new node is trustworthy, a key share is allotted to it by the SAM. At least threshold „k‟ number of key shares must be collected from distributed SAMs by the new node. If the new node is efficacious in collecting the desired number, then it is able to generate the secret code using any polynomial interpolation method. The larger the intended security level, higher the value of „k‟.

III. A BRIEF REVIEW OF THE ARTICLES WHERE POLYNOMIAL INTERPOLATION METHODS

ARE USED& RESULTS

This special issue has sought manuscripts which focus on using polynomial interpolation methods in various scenarios. A thorough analysis of each method is done where its merits and demerits both have been discussed.

A.Lagrange Interpolation and related work

Lagrange polynomial is one of the polynomial interpolation methods for distinct points xjand yj where each

yj value is assumed for each corresponding value of xj and

the same polynomial can be generated through different approaches[12][13].Set of k+1 different data points are taken(x0,y0)....(xj,yj)...(xk,yk) for kth order polynomial in

which interpolation is done linearly [14]. Lagrange basis polynomial is L(x) = ∑yjlj(x); where,

𝐿𝑗 𝑥 = 𝑥0≤𝑚 ≤𝑘

𝑥 − 𝑥𝑚 𝑥𝑗− 𝑥𝑚

= 𝑥 − 𝑥0 𝑥𝑗− 𝑥0

… … … … .

((𝑥 − 𝑥𝑗 −1)(𝑥 − 𝑥𝑗 +1))/(𝑥𝑗− 𝑥𝑗 −1)))/

𝑥𝑗− 𝑥𝑗 +1) … … … … . (𝑥 − 𝑥𝑘))/((𝑥𝑗− 𝑥𝑘) (1)

The secret key is divided into multiple parts where some or all of the parts are required to regenerate the secret [15]. Suppose the secret „S‟ is 26 and it is divided into 6 parts (n=6) which are available with SAMs. Any subset of 4 data items is enough to regenerate the secret in 3rd order polynomial. Firstly, 3 integers (k-1 Random Number, where k is an order of Polynomial) are selected randomly. Suppose these are 12, 8 and 4 and the proposed polynomial used to generate secret shares (points) is:

𝑓 𝑥 = 12𝑥3_{+ 8𝑥}2_{+ 4𝑥 + 26 (2)}

Create 6 points from the polynomial by taking x from 1 to 6: (1, 50); (2, 162); (3, 434);(4, 938) ;( 5, 1746) ;( 6, 2930).Taking any 4 values out of that: (x0, y0)= (1, 50);(x1, y1)= (2, 162); (x2, y2)= (4, 938); (x3, y3)= (5, 1746)

Lagrange's Interpolation formula is used to reconstruct the original secret. Hereby, Secret key (26) is regenerated even if 4 parts out of 6 are known. So interpolation methods are used not only for key share generation (x, f(x)), but also

for its distribution to a new node when it enters in the MANETs. This sharing work is described by Shamir in [16].

𝑓 𝑥 = 𝑦𝑗

2

𝑘=0 . 𝑙𝑗 𝑥 (3)

= 50 ∗ − 1

12𝑥 3₊11

12𝑥

2₋19

6 𝑥 +

10

3 + 162

∗ 1 6𝑥

3₋5 3𝑥

2₊29

6 𝑥 −

10 3 +

938 ∗ −1

6𝑥 3₊4

3𝑥 2₋17

6𝑥 + 5

3 + 1746 ∗ ( 1

12𝑥 3₋ 7

12𝑥 2₊

7 6𝑥 −

2

3) (4)

𝑓 𝑥 = 12𝑥3_{+ 8𝑥}2_{+ 4𝑥 + 26 (5)}

S=26

The main advantage of the Lagrange polynomial over former methods like Newton‟s method is that the data may be unequally spaced. Uniform spacing is not required between knots and a new node can randomly access any SAM to collect its key share. After collecting desire number of key shares, a polynomial is regenerated at a new node and secret code is calculated.

Wan et al. [2016] used the concept of Lagrange‟s where they stated that the previous identity-based cryptography method mainly uses bilinear map technique which has high computational cost. As battery power of each node is limited, this leads to the shorter lifespan of sensor nodes. To overcome this issue, they proposed identity-based key management protocol called Identity-Based Key Management using Lagrange‟s Interpolation (IBKML).Instead of executing several energy-consuming pairing processes many times, the proposed technique requires less pairing processes and hence much more effective.

This method has several disadvantages too. In this all of the work must be redone for each degree polynomial. As it is enormously expensive to evaluate the polynomial, regenerating the polynomial for any change in degree just adds to its cost of implementing. This method also suffers from Runge‟s problem where if the polynomial of high degree is confronted, the issue of oscillation starts at the boundaries of an interval and it starts giving erroneous results (depicted in Fig. 1).Fig. 1 depicts interpolation of equally spaced 11 points of function u(x) = 1/1+x2 with a 10 degree polynomial, Q10(x).

Fig. 1.Runge’s Problem [6]

B. Curve fitting and related work

International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue- 6S4, April 2019

fitting [17][18]. Suppose the x points are 1,2,3,4,5 and corresponding y are 320, 153, 456, 526, 638. The effort of fitting starts from lower order polynomial and order is incremented till trends are matched. With the above stated data points, it is observed that 1st, 2nd and 3rd order do not fit in the curve, but 4th order gets more accurate fit covering all the data points( as shown in Fig. 2). So the result is 4th order polynomial.Equation 6 is derived from this method.

y = 40.75x4 - 524.67x3 + 2364.3x2 - 4198.3x + 2638 (6)

Similarly to build the secret key for key management schemes, this interpolation method is used to generate the polynomial function where the required key shares are interpolated. „k‟ out of „n‟ shares arecollected from SAMs and tried to be able to fit in the curve. The polynomial is derived from this technique and hence secret key is generated. Unlike Lagrange‟s method, curve fitting consumes lesser computation time.

Fig. 2.Fourth order polynomial curve fitting for the given y data points

Kurundkaret al. (2017) tried to find out the scenario of statistical parameters for an AODV routing protocol where it performs its best.The curve that fits best, gives the exact values of quantified parameters. Minor modifications in the preferred range of attributes affect the operation of the network, therefore marking the appropriate values of the parameters are very crucial to give the best performance. They concluded that the regression analysis equation helps in depicting the performance of MANETs after choosing the properties of the goodness of fit with respect to speed, area, packet transfer rate and packet size. Their simulations findings using Curve fitting approach proved that their empirical model is proficient at creating accurate estimates when statistical values lie within bounds[13].

It is clear from the review that since Curve Fitting is based on an approximation approach, sometimes it may not interpolate all the points and again more computation efforts are required to make it fit. This method also suffers from Runge‟s phenomenon. To avoid a high order of the interpolating polynomial, other interpolation methods are considered like Spline curves or Chebyshev polynomials which rectify these problems.

C.Spline Curves and related work

Each pair of successive data points is connected with lines in linear interpolation which results in a piecewise interpolation (shown in Fig. 3). There is a need for generating functions that interpolate the data and are smoother than linear piecewise functions [6]. When a polynomial of high order is to be interpolated, there also remains no control over its oscillatory behavior. This makes

them debarred for using in any practical applications. So the effort is always made to generate functions that are the smoother version of previous and simultaneously interpolate all data points which do not lose continuity. These are called Splines. Affluent mathematical organization and strong algorithms make Splines to be preferred in geometrical design and for approximating the data in software packages [19].

Fig. 3.Pieces of Linear Spline [6].

Cubic Splines is one of the most commonly used spline curves. It is defined as the curve in which on every subinterval, the degree of the polynomial function is always ≤ 3 and it has overall two continuous derivatives (shown in Fig. 4).

Fig. 4.Cubic Spline [6]

In Fig. 4, the polynomial function has degree ≤ 2 for every subinterval [ti−1, ti.For a spline curve to have a third order continuous derivative, all these polynomial functions from different subintervals are linked in that way to each other. Curve segments should have the same slope where they are joined together. As the shape of these curves can be controlled locally, this is the most preferred reason for approving the idea of Splines. Changing the position of few nodes does not affect the curve globally and has limited distal effects in it. Runge's problem is also rectified by the spline method.

Martínezet al. [2015] used B-spline functions by carrying the feature of fitting in high order and knot multiplicity usage while performing Dynamic Metabolic Flux Analysis (DMFA), hereby extended the property of linear method [20]. Dynamic fluxes are computed by authors using B-splines representation and designed without any constraint of fitting in particular polynomial order. They proved that their performance is better as compared to previous interpolation methods.

D. Chebyshev polynomial and related work

Chebyshev polynomials, the name coined after PafnutyChebyshev, are associated with De Moivre's formula, a series of orthogonal polynomials that can be determined recursively. Chebyshev polynomials are divided

Manets

categories: the First kind denoted by Tnand symbol

Undenotes the Second Kind. These polynomials are

important in approximation theory as their nodes are used in interpolation; these nodes are called Chebyshev nodes. Runge's problem is also minimized by the resulting interpolation polynomial and under the maximum norm it gives the most accurate estimation to a continuous function[6].

The recursive relation of the first kind given in Equation 7 is satisfied by Chebyshev polynomials.

Ta+1 (x) = 2xTa(x)-Ta-1(x),a>=1 (7)

where T0(x) =1, T1(x) =x and in Ta(x), „a‟ is order of

Chebyshev polynomial. For example,

T2(x) = 2xT1(x) −T0(x) = 2x2−1 (8)

T3(x) = 4x3−3x (9)

Equation 8 and 9 are calculated from recursive relation using Equation 7. Plot of polynomials T1(x), T2(x) and T3(x)

are shown in Fig. 5. The linear arrangement of Chebyshev polynomials is depicted by any N degree polynomial and is shown in Equation 10. Essentially U(x), any real function which is continuous can be well estimated by a linear arrangement of Chebyshev polynomials. Given below are the Chebyshev polynomial equations of the second kind:

Ua+1 (x)-2xUa(x) +Ua-1(x) =0

U0 (x) =1

U1(x) =2x (10)

Fig. 5. T1(x), T2(x) and T3(x) Chebyshev polynomials [6]

Computation effort is minimized to a large extent if the evaluation is done using the Chebyshev method as it uses recursion method which is its most captivating feature. These polynomials of any base number are generic and can construct the next polynomial sequence. Any order Polynomial can be resolved by using recursive calls in the Chebyshev method. Property of solving by substitution rather than by computation make Chebyshev sequences best for key management in networking [21].

The quantity of data transfer required by the network has decreased considerably to achieve the consensus in both fixed and ad hoc topologies by the proposed algorithm. Montijanoet al. [2013] have considered the features of Chebyshev polynomials to project a rapid and steady algorithm of distributed consensus [22]. It has no constraint on the degree of the polynomial and awareness of the second maximum.

IV. CONCLUSION

In MANETs, it is very easy to intercept the payload because of the open nature to the snooping elements. Accurate precautions have to be taken to assure about the security of payload and key shares. The emerging trend is to avoid attacks in the physical layer. Key management is considered to be a base in providing secure communication and cryptography through key management schemes becomes an effective solution. Underlying efficient, secure and robust key management makes the system work

effectively. In Lagrange interpolation

method, the set of well-defined points xj, there exist some

distinct yj, and for each random xj, corresponding value yj

can be ascertained. The polynomial could be regenerated through this method using the required number of secret shares and hence secret key. This method is extensively used as there is no requirement of transmitting secret key directly through communication links and only shares are transferred. Any new node has to approach „k‟ number of SAMs to regenerate the secret key. The higher the value of „k‟, the more the level of security is.

Curve fitting is another interpolation method used for key management where key shares are tried to fit on the plot so that it covers all the points to make a graph. The polynomial that generates from that curve gives the secret code. Since it is based on the approximation technique, the curve fitting algorithm does not give satisfactory results sometimes. When high order polynomials are used for a set of interpolation points, oscillation at the fringes of an interval starts coming.

Both the above mentioned methods start giving erroneous results. This phenomenon is called Runge‟s phenomenon.These errors could be rectified by the use of Spline curves and Chebyshev polynomials. Spline curves are preferred over Curve fitting and Lagrange if large set of closely spaced points is to be approximated and energy consumption to be minimized with desired precision. It also enables one to move few control points to get exactly the curve one requires instead of moving a large number of curve points. This local controlled nature makes Spline Curve superior as compared to the previous interpolation methods.

The Chebyshev polynomials are also unique as these polynomials are defined recursively in a series or a random way. Besides the advantages offered by Spline curves, the Chebyshev method reduces the calculation complexity because of its substitution nature. Rather than relying on large calculations, this method works on the principle of substitution.

International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8, Issue- 6S4, April 2019

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CHETNA received her B.Tech. degree in computer science & Engineering in 2001. She is pursuing her Ph.D. degree in Computer Science & Engineering at ChitkaraUniversityInstituteofEngineering& Technology, Punjab. Her research interests include dealing with security apprehensions of Mobile Adhoc Networks.

K.R.RAMKUMARis PhD in Computer Science and Engineering from Anna University, Chennai, India, having 15 years of Teaching and Research Experience. His areas of expertise are Network Security, Key Management and Relational Database Management Systems with advancements. He is currently working on the practical implementations of Mobile Ad-hoc Networks using Field Programmable Gate Array (FPGA). His research includes solving the routing issues, dealing with security and node failure apprehensions of wireless sensor networks.

SHAILY JAIN is PhD from JUIT, Waknaghat.

Having total 12 years of teaching and research

experience, she has published 25 research papers in