Efficiently Fuzzy Controlling with Dynamic Window in Elliptic Curve Cryptography Sensor Networks

Transcription

1 Efficienty Fuzzy Controing with Dynamic Window in Eiptic Curve Cryptography Sensor Networks Xu Huang, Dharmendra Sharma and Pritam Gakumar Shah Abstract It is noted that the wireess sensor networks (WSN) based on the rapid progress of wireess communications and embedded micro eectro mechanica systems technoogies are becoming important part in our daiy ife. However, the security of the WSN becomes one of the maor concerns in its appications. Even eiptic curve cryptography (ECC) prominenty offers great potentia benefits for WSN security there is sti a ot of work needs to be done due to WSN has very restraint running conditions such as imited energy source, capabiity of computing, etc. Aso it is we known that scaar mutipication operation in ECC accounts for about 8% of key cacuation time on wireess sensor network motes. In our current paper we present an optimized dynamic window based on our previous research works. The whoe quaity of service (QoS) has been improved under this agorism in particuary the power consuming is more efficienty. The simuation resuts showed that the average cacuation time, due to fuzzy conditions decreased from previous 26 to current 9 as a whoe the cacuation time, decreased by approximatey 17% in comparison to our previous agorithms in an ECC wireess sensor network. environmenta monitoring e.g. vocano detection [2,3], distributed contro systems [4], agricutura and farm management [5], detection of radioactive sources [6], and computing patform for tomorrows internet [7]. Generay speaking, a typica WSN architecture can be shown in Figure 1. Contrast to traditiona networks, a wireess sensor network normay has many resource constraints [4] due to the imited size. As an exampe, the MICA2 mote consists of an 8 bit ATMega 128L microcontroer working on 7.3 MHz. As a resut nodes of WSN have imited computationa power. Radio transceiver of MICA motes can normay achieve maximum data rate of 25 Kbits/s, which restricts avaiabe communication resources. The fash memory that is avaiabe on the MICA mote is ony 512 Kbyte. Apart from these imitations, the onboard battery is 3.3.V with 2A- Hr capacity. Therefore, the above restrictions with the current state of art protocos and agorithms are expensive for sensor networks due to their high communication overhead. Index Terms Eiptic curve cryptography (ECC), scaar mutipication, non-adacent form, side window, fuzzy T I. INTRODUCTION HE high demand for various appications shows the fact that the rapid progress of wireess communications has become popuar in our daiy ife. With the growth in very arge scae integrated (VLSI) technoogy, embedded systems and micro eectro mechanica systems (MEMS) has enabed production of inexpensive sensor nodes, which can transit data over a distances with free media and efficient use of power [1, 22, 23]. In the WSN systems, the sensor node wi detect the interested information, processes it with the hep of an in-buit microcontroer and communicates resuts to a sink or base station. Normay the base station is a more powerfu node, which can be inked to a centra station via sateite or internet communication to form a network. There are many depoyments for wireess sensor networks depending on various appications such as Xu Huang is with Facuty of Information Sciences and Engineering, University of Canberra, Austraia. (e-mai: Xu.Huang@canberra.edu.au). Dharmendra Sharma is with Facuty of Information Sciences and Engineering, University of Canberra, Austraia. (e-mai: Dharmendra.Sharma@canberra.edu.au). Pritam Gakumar Shah is a PhD candidate at the Facuty of Information Sciences and Engineering, University of Canberra, Austraia. (e-mai: Pritam.Shah@canberra.edu.au). Figure 1: A Typica WSN architecture Recaed that the Eiptic Curve Cryptography was first introduced by Nea Kobitz [9] and Victor Mier [1] independenty in the eary eighties. The advantage of ECC over other pubic key cryptography techniques such as RSA, Diffie-Heman is that the best known agorithm for soving eiptic curve discrete ogarithm probem (ECDLP) which is the underying hard mathematica probem in ECC which wi take the fuy exponentia time. On the other hand the best agorithm for soving RSA and Diffie-Heman takes sub exponentia time [11]. In summary, the ECC probem can ony be soved in exponentia time and, to date, there is a ack of sub exponentia methods to attack ECC.

2 An eiptic curve E over GF(p) can be defined by y 2 = x 3 + ax + b where a, b GF(p) and 3 2 4a + 27b in the GF(p). The point (x, y) on the curve satisfies the above equation and the point at infinity denoted by is said to be on the curve. If there are two points on the curve namey, P (x 1, y 1 ), Q (x 2, y 2 ) and their sum is given by point R(x 3, y 3 ) the agebraic formuas for point addition and point doubing are given by foowing equations: 2 We have: x3 = λ x1 x2 y = λ ( x x y 3 1 3) y2 y1 λ =, x x x + a λ =, 2 y 1 1 if P Q if P = Q showed that 85% of execution time is spent on scaar mutipication. Scaar Mutipication is the operation of mutipying point P on an eiptic curve E defined over a fied GF(p) with positive integer k which invoves point addition and point doubing. Operationa efficiency of kp is affected by the type of coordinate system used for point P on the eiptic curve and the agorithm used for recoding of integer k in scaar mutipication. This research paper proposes an innovative agorithm based on one s compement for representation of integer k which acceerates the computation of scaar mutipication in wireess sensor networks. The number of point doubing and point addition operations in scaar mutipication depends on the recoding of integer k. Expressing integer k in binary format highight this dependency. Figure 3: Diffie-Heman protoco based on ECC (a) Addition: P + Q = R (b) Doubing: P+P = R Figure 2: Point addition and point doubing on eiptic curve. Here, the addition, subtraction, mutipication and inverse are the arithmetic operations over GF(p), which can be shown in Figure 2. II. ELLIPTIC CURVE DIFFIE-HELLMAN SCHEME (ECDH) PROPOSED FOR WSN Before we get into our innovation method, we need to have a coser ook at the popuar egacy scheme for WSN. As per [13] the origina Diffie-Heman agorithm with RSA requires a key of 124 bits to achieve sufficient security but Diffie Heman based on ECC can achieve the same security eve with ony 16 bit key size. The cassica Eiptic Curve Diffie Heman scheme operates as shown in the Fig. 3 Initiay Aice and Bob agree on a particuar curve with base point P. They generate their pubic keys by mutipying P with their private keys namey K A and K B. After sharing pubic keys, they generate a shared secret key by mutipying pubic keys by their private keys. The secret key is R = K A *Q A = K B *Q B. With the known vaues of Q A, Q B and P, it is computationay intractabe for an eavesdropper to cacuate K A and K B which are the private keys of Aice and Bob. As a resut, adversaries cannot cacuate R the shared secret key. In ECC two heaviy used operations are invoved: scaar mutipication and moduar reduction. Gura et. a. [14] The number of zeros and number of ones in the binary form, their paces and the tota number of bits wi affect the computationa cost of scaar mutipications. The Hamming weight as represented by the number of non-zero eements, determines the number of point additions and bit ength of integer K determines the number of point doubings operations in scaar mutipication. One point addition when P Q requires one fied inversion and three fied mutipications [13]. Squaring is counted as reguar mutipication. This cost is denoted by 1I + 3M, where I denotes the cost of inversion and M denotes the cost of mutipication. One point doubing when P = Q requires 1I + 4M as we can negect the cost of fied additions as we as the cost of mutipications by sma constant 2 and 3 in the above formuae. Binary Method Scaar mutipication is the computation of the form Q = kp, where P and Q are the eiptic curve points and k is positive integer. This is obtained by repeated eiptic curve point addition and doubing operations. In binary method the integer k is represented in binary form: 1 = k K 2, K = {,1} The cost of mutipication when using binary method depends on the number of non-zero eements and the ength of the binary representation of k. If the representation has k - 1 then binary method require ( 1 ) point doubings and (W 1) where is the ength of the binary expansion of k, and W is the Hamming weight of k (i.e., the number of nonzero eements in expansion of k). For exampe, if k = 629 = (111111) 2, it wi require (W 1) = 6 1 = 5 point

3 additions and 1 = 1 1 = 9 point doubings operations. The binary method scans the bits of K either from eft-toright or right-to-eft. The binary method for the computation of kp is given in the foowing agorithm 1, as shown beow: Agorithm 1: Left to right binary method for point mutipication Input: A point P E (F q ), an 1 = k K 2, K = o {,1} bits integer Output: Q = kp 1. Q 2. For = 1 to do: 2.1 Q 2 Q, 2.2 if k = 1 the Q Q + P. 3. Return Q. Signed Digit Representation Method The subtraction has virtuay the same cost as addition in the eiptic curve group. The negative of point (x, y) is (x, y) for odd characters. This eads to scaar mutipication methods based on addition subtraction chains, which hep to reduce the number of curve operations. When integer k is represented with the foowing form, it is a binary signed digit representation. k = = S 2, S {1,, 1} When a signed-digit representation has no adacent non zero digits, i.e. S S +1 = for a it is caed a nonadacent form (NAF). The foowing agorithm 2 computes the NAF of a positive integer given in binary representation. Agorithm 2: Conversion from Binary to NAF Input: An integer 1 = k K 2, K = {,1} k = S 2, S {1,, 1} = Output: NAF 1. C 2. For = to do: 3. C +1 [(K + K +1 + C )/2] 4. S K + C 2C Return (S S ) NAF usuay has fewer non-zero digits than binary representations. The average hamming weight for NAF form is (n 1)/3.. So generay it requires (n 1) point doubings and (n 1) /3. point additions. The binary method can be revised accordingy and is given another agorithm for NAF, and this modified method is caed the Addition Subtraction method. III. DYNAMITIC WINDOW WITH FUZZY CONTROLLER IN ECC PROPOSED ALGORITHM BASED We are going to use the agorithm based on subtraction by utiization of the 1 s compement is most common in binary arithmetic. The 1 s compement of any binary number may be found by the foowing equation [19-22]: a C1 = (2 1) N (1) where C 1 = 1 s compement of the binary number, a = number of bits in N in terms of binary form, N = binary number From a coser observation of the equation (1), it reveas the that any positive integer can be represented by using minima non-zero bits in its 1 s compement form provided that it has a minimum of 5% Hamming weight. The minima non-zero bits in positive integer scaar are very important to reduce the number of intermediate operations of mutipication, squaring and inverse cacuations used in eiptica curve cryptography as we have seen in previous sections. The equation (1) can therefore be modified as per beow: N = a ( 2 C 1 1) (2) For exampe, we may take N =1788 then it appears N= ( ) 2 in its binary form C 1 = 1 s Compement of the number of N= (111) 2 a is in binary form so we have a = 11 After putting a the above vaues in the equation (2) we have: 1788 = , this can be reduced as beow: 1788 = (3) So we have 1788= As is evident from equation (3), the Hamming weight of scaar N has reduced from 8 to 5 which wi save 3 eiptic curve addition operations. The above recoding method based on one s compement subtraction combined with siding window method provides a more optimized resut. Let us compute [763] P (in other words k = 763) as an exampe, with a siding window agorithm with K recoded in binary form and window sizes ranging from 2 to 1. It is observed that as the window size increases the number of pre-computations aso increases geometricay. At the same time number of additions and doubing operations decrease. Now we present the detais for the different window size to find out the optima window size using the foowing exampe: Window Size w = = ( ) 2 No of precomputations = 2 w 1 = = [3] P 763 = The intermediate vaues of Q are P, 2P 4P, 8P, 11P, 22P, 44P, 47P, 94P, 95P, 19P, 38P, 76P, 763P Computationa cost = 9 doubings, 4 additions, and 1 precomputation.

4 We continue to derive the remaining cacuations for Window Size w = 6, Window Size w = 7, Window Size w = 8, Window Size w = 9, and Window Size w = 1. The resuts for a cacuations are presented in Tabe 1. The effects of doubings and additions as shown in Tabe 1 are further considered in the next section. size is obtained by tradeoff between pre-computation and computation cost. Agorithm for siding window scaar mutipication on eiptic curves. 1. Q P and i -1 2.whie i do 3.if n i = then Q [2]Q and i 4.ese 5. s max ( i - k + 1,) 6. whie n s = do s s for h = 1 to i - s + 1 do Q [2]Q 8 u (n i... n s ) 2 [n i = n s = 1 and 9. Q Q [u]p [u is odd 1. i s return Q i -1 i - s + 1 k] so that [u]p is precompute TABLE 1 WINDOW SIZE VS NO OF DOUBLINGS, ADDITIONS AND PRE COMPUTATIONS Window Size No of Doubings No of Additions It is interesting to note that Figure 4 shows the charts obtained from tabe 1(in order to see the reation ceary, we used so-caed ampified chart because of the range is ony imited to 1 in window size). As we woud ike to know the further detais about the reations between Doubing and Addition, from Figure 4 it ceary shows that the Pre-computation and doubing are the dominated factors hence we need to focus on this factor in our discussion of window size. IV. FUZZY CONTROLLER SYSTEM IN ECC It is cear, from above description that there is a tradeoff between the computationa cost and the window size as shown in Tabe 1. However, this tradeoff is underpinned by the baance between computing cost (or the RAM cost) and the pre-computing (or the ROM cost) of the node in the network. It is aso cear that, from above description that the variety of wireess network working states wi make this contro compex and cacuations coud be reativey more expensive. Therefore, we propose a fuzzy dynamic contro system, to provide dynamic contro to ensure the optimum window d] No of Pre computations Fig. 4: ampified chart (focus on imited range) for the Tabe 1 The fuzzy decision probem introduced by Beman and Zadeh has as a goa the maximization of the minimum vaue of the membership functions of the obectives to be optimized. Accordingy, the fuzzy optimization mode can be represented as a muti-obective programming probem as foows [21]: Max : min{ μ ( D)}& min{ μ ( U )} s S & L such that A C s r RP x rs = 1 L, p P & s S, xrs = or 1 r R & s S In above equation, the obective is to maximize the minimum membership function of a deays, denoted by D, and the difference between the recommend vaue and the measured vaue, denoted by U. The Fuzzy contro system is extended from and shown in Figure 5. For accurate contro, we designed a three inputs fuzzy controer. The first input is storage room, which has three statuses, showing storage room in one of the three, namey (a) ow, (b) average, and (c) high. The second input is pre-computing working oad (PreComputing) in one of three states, namey (a) ow, (b) average, and (c) high. The third input is Doubing, expressing how much working oad for the cacuation doubing which has three cases, namey (a) ow, (b) average, and (c) high. The output is one, caed WindowSize, to express the next window size shoud be moved in which way, which has three states for the window sizes, namey (a) down, (b) stay, and (c) up. Figure 5: Three inputs fuzzy window contro system There are ony 9 Fuzzy Rues isted as foows (weight

5 are unit) due to StorageRoom in Figure 5 can be ignored due to the resuts of Figure 4 athough it is a factor needs to be appeared in terms of this contro system. 1. If (PreComputing is ow) and (Doubing is ow) then (WindowSize is Up) 2. If (PreComputing is ow) and (Doubing is average) then (WindowSize is Up) 3. If (PreComputing is ow) and (Doubing is high) 4. If (PreComputing is average) and (Doubing is ow) then (WindowSize is Up) 5. If (PreComputing is average) and (Doubing is average) then (WindowSize is Up) 6. If (PreComputing is average) and (Doubing is high) 7. If (PreComputing is high) and (Doubing is ow) then (WindowSize is Up) 8. If (PreComputing is high) and (Doubing is average) 9. If (PreComputing is high) and (Doubing is high) The number at each fuzzy condition with a bracket is the weight number, currenty it is unit. Later we sha change it with different number according to the running situations as described in the next. The three inputs with 9 fuzzy rues in Mamdani mode running fuzzy controer part. The three inputs are StorageRoom, PreComputing and Doubing. The output is WindowSize. It is noted if we did not take the advantage of Figure 4, there wi be at east 26 fuzzy rues need to be considered as shown in our previous paper [23]. This is because that the StorageRoom has ow, average, high with other two parameters s combinations. The output with StorageRoom and PreComputing is shown in Figure 6. The surface StorageRoom vs. Doubing is shown in Figure 7. is getting big, it woud be nice to have arger window size for the doubing. Now if we change the weight for above fuzzy rues as such the rues 1,5 1, 13, 14, 15, 16, 18, 2,21, 22, 23, 25, and 26 are set in.5 (the rest wi keep the same) due to the maor functions are controed by the storage room, and doubing wi rapidy increasing by the window size arger. The outputs wi changed as the average storage room wi increased.4% and the other two inputs are decreased by.2% the output become window staying a itte wider side by.3%. Figure 7: The output of the surface for the StorageRoom vs. Doubing. It is cear that this fuzzy controer for the dynamic window is aso invoved a tradeoff between accuracy and contro costs. For exampe the same system may go further for the second order parameters, not ust check the changes about the input variabes but aso check the change tendencies of the variabes, which wi be discussed in another paper. Figure 6: The output of the surface for the StorageRoom vs. PreComputing. From above figures, it is ceary observed that in the ow window size side, if the storage room is ow the dominated function of doubing wi pay roe as Figure 6 shown but if the window size is at the high side, the storage room wi be fairy stay at the midde either for PreComputing or Doubing, which is the doubing wi sharpy increased when window size a itte bit arger that aso can be shown from Tabe 1. From Figure 7 it is ceary to show when the storage room Figure 8: The output of the surface for the StorageRoom = constant (.4) and PrecComputing vs. Doubing. If we keep the StorageRoom constant, e.g. StorageRoom =.4, and the output of the PreComputing vs. Doubing is shown in Figure 8. Now in order to show the StorageRoom is not dominated factor as we discussed above, we took another vaue, StorageRoom =.8 and the output of the PreComputing vs. Doubing is shown in Figure 9. It is observed that the output of the PreComputing vs. Doubing is not much difference between Figures 8 and 9.

6 From Figures 6 and 7 we can observe that the dominated factor is not the members of the StorageRoom in Figure 8 due to the fact that the StorageRoom goes to arge than 4% (or.4) the factors PreComputing and Doubing are dominated in Fig. 6 and Fig. 7 separatey. The StorageRoom cannot pay a roe in the process of the dynamic window, which is underpinned by the fact that has been shown in Figures 4. The simuations of the exampe described in above were impemented. Aso it is noted that with equation (2), the computationa cost has been reduced from 3 additions as in the binary method to ony 1 addition in one s compement subtraction form. The number of precomputations has remained the same. Figure 9: The output of the surface for the StorageRoom = constant (.8) and PreComputing vs. Doubing. In our simuations, the proposed method together with a fuzzy window size controer makes the ECC cacuation in the current agorism is about 17% more efficient than the methods in [23] with the same QoS eve. V. CONCLUSION The positive integer in point mutipication may be recorded with one s compement subtraction to reduce the computationa cost invoved in this heavy mathematica operation for wireess sensor network patforms. As the NAF method invoves moduar inversion operation to get the NAF of binary number, the one s compement subtraction can provide a very simpe way of recoding the integer. There is aways a decision between pre-computing and computing, the former is reated to the storage and the atter is associated with computing capabiity and capacity. The window size may be the subect of trade-off between the avaiabe RAM and ROM at a particuar instance on a sensor node, which can be controed by fuzzy controer. The fina simuation in a sensor wireess network shows that about 17% more efficient than our previous method [23] can be obtained with an ECC sensor network. [3] G. Werner-Aen, K. Lorincz, M. Ruiz, O. Marcio, J. Johnson, J. Lees, and M. Wesh, "Depoying a wireess sensor network on an active vocano," IEEE Internet Computing, vo. 1, pp , 26. [4] B. Sinopoi, C. Sharp, L. Schenato, S. Schaffert, and S. S. Sastry, "Distributed contro appications within sensor networks," Proceedings of the IEEE, vo. 91, pp , 23. [5] P. Sikka, P. Corke, P. Vaencia, C. Crossman, D. Swain, and G. Bishop-Hurey, "Wireess ad hoc sensor and actuator networks on the farm," 26, pp [6] D. L. Stephens, Jr. and A. J. Peurrung, "Detection of moving radioactive sources using sensor networks," Nucear Science, IEEE Transactions on, vo. 51, pp , 24. [7] Z. Feng, "Wireess sensor networks: a new computing patform for tomorrow's Internet," 24, pp. I-27 Vo.1. [8] I. F. Akyidiz, Weiian Su, Yogesh Sankarasubramaniam, and E. Cayirci, "A Survey on Sensor Networks in IEEE Communication Magazine. vo. 4, August 22, pp [9] N.Kobitz, "Eiptic Curve Cryptosystems," Mathematics of Computation, vo. 48, pp , [1] V. S. Mier, "Use of Eiptic Curves in Cryptography," in Advances in Cryptoogy - CRYPTO '85: Proceedings. vo. 218: Springer-Verag, 1986, pp [11] J. Lopez and R. Dahab., " An overview of eiptic curve cryptography," Technica report,institute of Computing, Sate University of Campinas, Sao Pauo, Brazi, May 2. [12] K. Lauter, "The advantages of eiptic curve cryptography for wireess security," Wireess Communications, IEEE [see aso IEEE Persona Communications], vo. 11, pp , 24. [13] H. Wang, B. Sheng, and Q. Li, "Eiptic curve cryptography-based access contro in sensor networks," Int. J. Security and Networks,, vo. 1, pp , 26. [14] N. Gura, A. Pate, and A. Wander, "Comparing eiptic curve cryptography and RSA on 8-bit CPUs," in Proceedings of the 24 Workshop on Cryptographic Hardware and Embedded Systems (CHES) August 24. [15] [16] D. J. Maan, M. Wesh, and M. D. Smith, "A pubic-key infrastructure for key distribution in TinyOS based on eiptic curve cryptography," in 2nd IEEE Internationa Conference on Sensor and Ad Hoc Communications and Networks (SECON 24)2nd IEEE Internationa Conference on Sensor and Ad Hoc Communications and Networks (SECON 24), 24, pp [17] I. Bake, G. Seroussi, and N. Smart, Eiptic Curves in Cryptography vo. 265, [18] D. Hankerson, J. L. Hernandez, and A. Menezes, "Software Impementation of Eiptic Curve Cryptography over Binary Fieds, CHES," 2. [19] Angeo C Giie, Binary Arithmetic and Booean agebra, McGRAW-HILL Book Company, pp53. [2] H R. Beman and L.A. Zadeh, Decision-making in a fuzzy environment, Management Science 17 (197), [21] Xu Huang, Shirantha Wiesekera, and Dharmendra Sharma, Fuzzy Dynamic Switching in Quantum Key Distribution for Wi-Fi Networks, 6 th Internationa Conference on Fuzzy Systems and Knowedge Discovery, August 29, Tianin, China. Proceeding pp [22] Xu Huang, Pritam Gakumar Shah and Dharmendra Sharma, Muti- Agent System Protecting from Attacking with Eiptic Curve Cryptography, the 2 nd Internationa Symposium on Inteigent Decision Technoogies, Batimore, USA, 28-3 Juy 21. Accepted to be pubished. [23] Xu Huang and Dharmendra Sharma, Fuzzy Controer for a Dynamic Window in Eiptic Curve Cryptography Wireess Networks for Scaar Mutipication APCC 21. The 16 th Asia-Pacific Conference on Communications, 31 October 3 November 21, Langham Hote, Auckand, New Zeaand, ISBN: pp REFERENCES [1] I. F. Akyidiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, "Wireess sensor networks: a survey," Computer Networks, vo. 38, pp , 22. [2] C. Chung-Kuo, J. M. Overhage, and J. Huang, "An appication of sensor networks for syndromic surveiance," 25, pp