AU2010101117A4 - An apparatus and method for selecting dynamic window size based on the fuzzy controller in elliptical curve scalar multiplication on wireless sensor network platform - Google Patents

Description

S Editorial Note N 2010101117 There are 9 description pages 2 BACKGROUND OF THE ART N- FIELD OF THE INVENTION [00011 This invention pertains generally to cryptography and more precisely elliptical curve cryptography on wireless sensor network platform which is having limited resources. DESCRIPTION OF THE RELATED ART 100021 The rapid progress of wireless communications has become popular in our daily life, together with rapid growth in very large scale integrated (VLSI) technology, embedded systems and micro electro mechanical systems (MEMS) has enabled production of inexpensive sensor nodes which can communicate information over shorter distances with N. efficient use of power. 10003] Normally in the WSN systems, the sensor node will detect, record and transmit interested information designed by the special target, processes it with the help of an in-built microcontroller and communicates results to a sink or base station. The base station is a more powerful node, which can be linked to a central station via satellite or internet communication to form a network. There are many deployments for wireless sensor networks depending on various applications such as environmental monitoring, volcano detection , distributed control systems, agricultural and farm management, detection of radioactive sources, and computing platform for tomorrows' internet. [0004] Comparison with to conventional networks, a wireless sensor network normally has many resource constraints due to the limited size. For example, the MICA2 mote consists of an 8 bit ATMega 128L microcontroller working on 7.3 MHz. As a result nodes of WSN have limited computational power. The flash memory that is available on the MICA mote is only 512 Kbyte and radio transceiver of MICA motes can normally achieve maximum data rate of 250 Kbits/s, which restricts available communication resources. Apart from these limitations, the onboard battery is 3.3.V with 2A-1 Ir capacity. Therefore, the above restrictions with the current state of art protocols and algorithms are expensive for sensor networks due to their high communication overhead. 10005] Neal Koblitz and Victor Miller independently in the early eighties introduced elliptic curve cryptography (ECC). The advantage of ECC over other public key cryptography techniques such as RSA, Diffie-Hlellman is that the best known algorithm for solving elliptic curve discrete logarithm problem (ECDLP) which is the underlying hard mathematical problem in ECC which will take the fully exponential time. On the other hand the best algorithm for solving RSA and Diffie-Hellman takes sub exponential time . In summary, the ECC problem can only be solved in exponential time and, to date, there is a lack of sub exponential methods to attack ECC. 100061 An elliptic curve E over GF(p) can be defined by the format with equation y2 3+crb where a, b e GF(p) and 4a' +27b2 eo in the GF(p), wherep is larger than 3 prime integer. The point (x, y) on the curve satisfies the above equation and the point at infinity denoted by x is said to be on the curve. [0007] If there are two points on the curve namely, P (x1, yi), Q (x2, y2) and their sum is given by point R(x;, y3) as shown in Figure 1, the algebraic formulas for point addition and point Co doubling are given by following equations: We have: x, = A' - x2 y= =4x] -x) A= , iflPQ X2 -X1 .Z----,if P= 2y, [0008] Before we get into our innovation method, we need to have a closer look at the popular legacy scheme for WSN. The original Diffie-Hellman algorithm with RSA requires a key of 1024 bits to achieve sufficient security but Diffie Hel/man based on ECC can achieve the same security level with only 160 bit key size. Therefore, the ECC is more attractive in comparison with other public-key systems such as RSA. The classical Elliptic Curve Diffie Hellman scheme operates as shown in the Figure 2. [0009] It is well known that there are two heavily used operations are involved in ECC, namely scalar multiplication and modular reduction. Gura et. al. have showed that 85% of execution time is spent on scalar multiplication. Scalar Multiplication is the operation of multiplying point P on an elliptic curve E defined over a field GF(p) with positive integer k which involves point addition and point doubling. Operational efficiency of kP is affected by the type of coordinate system used for point P on the elliptic curve and the algorithm used for recoding of integer k in scalar multiplication.

O [00101 In this patent application , an innovative algorithm is to be proposed that is based on O one's complement for representation of integer k which accelerates the computation of scalar Cl multiplication in wireless sensor networks. 100111 Generally speaking, the number of point doubling and point addition operations for ECC in scalar multiplication depends on the recoding of integer k. Expressing integer k in binary format highlight this dependency. [0012] The number of zeros and number of ones in a binary form, their places and the total number of bits will affect the computational cost of scalar multiplications. The Hamming weight as represented by the number of non-zero elements, determines the number of point additions and bit length of integer K determines the number of point doublings operations in scalar multiplication. [0013] One point doubling when P = Q requires 11+ 4M as we can neglect the cost of field additions as well as the cost of multiplications by small constant 2 and 3 in the above fonnulae. One point addition when P # Q requires one field inversion and three field multiplications. Squaring is counted as regular multiplication. This cost is denoted by 11 + 3M, where I denote the cost of inversion and Mdenotes the cost of multiplication. [0014] Scalar multiplication is the computation of.the form Q = kP, where P and Q are the elliptic curve points and k is positive integer. This is obtained by repeated elliptic curve point addition and doubling operations. In binary method the integer k is represented in binary form: k=-K,2', K, {0.) The binary method scans the bits of K either from left-to-right or right-to-left. The binary method for the computation of kP is given in the following algorithm 1, as shown in "Algorithm I". 10015] The cost of multiplication when using binary method depends on the number of non zero elements and the length of the binary representation of k. If the representation has k, , 0 then binary method require (I - 1 ) point doublings and (W-I) where / is the length of the binary expansion of k, and W is the I slamming weight of k (i.e., the number of non-zero elements in expansion of k).

[0016] C c-i Algorithm 1: Left to right binary method for point multiplication Input: A point P E E (Fq), an I bits integer k K,2. K, 1 10,1) Output: Q = kP 1. Q+-cc 2. Forj=/M1to0do: 2.1 q <-2Q, 2.2 if k, = 1 the Q+- Q+ p. 3 Return Q. 10017] The subtraction has virtually the same cost as addition in the elliptic curve group. The negative of point (x, y) is (x, -y) for odd characters. This leads to scalar multiplication methods based on addition -subtraction chains, which help to reduce the number of curve operations. When integer k is represented with the following form, it is a hinary signed digit representation. k = IS, 2' ., S ,0,I 1-0 I0018J When a signed-digit representation has no adjacent non zero digits, i.e. S/Syi = 0 for all] 0 it is called a non-adjacent form (NAF).The following algorithm 2 computes the NAF of a positive integer given in binary representation. [0019] Algorithm 2: Conversion from Binary to NAF 1-I Input: An integer k = K, 2', K (0,1) Output: NAF k= S,2 , Sc {,0,-I} 1. C 0 o 2. ForI=Ototdo: 3. Ci+- 1(K,+ K.1 +C,)12] 4. S K + C 0 2C>, 5. Return (S,,.. Sc) o [00201 NAP usually has fewer non-zero digits than binary representations. Ihe average O hamming weight for NAF form is (n - )/3.0. So generally it requires (n - 1) point doublings and (n - 1) /3.0 point additions. The binary method can be revised accordingly and is given another algorithm for NAF, and this modified method is called the Addition Subtraction method. 10021J We are going to use the algorithm based on subtraction by utilization o the I's complement is most common in binary arithmetic. [he l's complement of any binary number may be found by the following equation: O C, 4(2" -1)- A O where C, = l's complement of the binary number, a = number of bits in N in terms of binary Cl form, N= binary number [00221 From a closer observation of the equation (1), it reveals the that any positive integer can be represented by using minimal non-zero bits in its l's complement form provided that it has a minimum of 50% Hamming weight. The minimal non-zero bits in positive integer scalar are very important to reduce the number of intermediate operations of multiplication, squaring and inverse calculations used in elliptical curve cryptography as we have seen in previous sections. [0023] The equation (1) can therefore be modified as per below: '=(2" -C, -1) (2) Let's take an example as a case study, we take N =1788 then it appears N= (11011111100)2 in its binary form

C

1 = 1's Complement of the numberof N= (00100000011)2 a is in binary form so we have a = 11 After putting all the above values in the equation (2) we have: 1788 = 00100000011 , this can be reduced as below: 1788 = 100000000000-00100000011-1 (3) So we have 1788= 2048 256 2 1-1 [0024] As is evident from equation (3), the Hamming weight of scalar N has reduced from 8 to 5 which will save 3 elliptic curve addition operations.

O [0025] Let us compute [763] P (in other words k = 763) as an example, with a sliding window O: algorithm with K recorded in binary form and window sizes ranging from 2 to 10. CA Window Size w = 2 763 = (1011111011)2 No of precomputations = 2' - 1= 22 - 1 [3] P 763= 10 11 11 10 11 The intermediate values of Q are P, 2P 4P, 8P, 11P, 22P, 44P, 47P, 94P, 95P, 190P, 380P, 760P, 763P Nq Computational cost = 9 doublings, 4 additions, and 1 pre-computation. 10026] Algorithm for sliding window scalar multiplication on elliptic curves. 1. Q <- Pandi +-I 2.while i > 0 do 3.if n, = 0 thenQ +-[2]Qand i <i- I 4 .else 5. s +- max(- k +1,0' 6.while n, = 0 do s <- s + I 7.forh =ltoi-s+IdoQ -[2]Q 8 u <- (n,....n,) 2 In, = I andi -s + I kJ 9.Q +- QT [u]P [u is odd so that[u]P is precomputd) I0 .i +- sI I 1.returnQ [00271 It is clear, from above description that there is a tradeoff between the computational cost and the window size. However, this tradeoff is underpinned by the balance between computing cost (or the RAM cost) and the pre-computing (or the ROM cost) of the node in the network. [00281 It is also clear that, from above description that the variety of wireless network working states will make this control complex and calculations could be relatively more expensive. Therefore, we propose a fuzzy dynamic control system, to provide dynamic control to ensure that the optimum window size is to be obtained by tradeoff between pre computation and computation cost.

[00291 The fuzzy decision problem introduced by Bellman and Zadeh has as a goal the maximization of the minimum value of the membership functions of the objectives to be optimized. Accordingly, the fuzzy optimization model can be represented as a multi objective programming problem as follows: -Max : min{ps (D)) & min{p,(U;)} Vs c S& V cEL such that A, < C VI C L, -1 I Vp C P &VS GuS, x,, =0 or I VrER&Vs es 100301 In above equation, the objective is to maximize the minimum membership function of C) all delays, denoted by D, and the difference between the recommend value and the measured Cl value, denoted by U. [0031] The Fuzzy control system is extended from and shown in Figure 3. We designed a three inputs fuzzy controller. The first input is storage room, which has three statuses, showing storage room in one of the three, namely (a) low, (b) average, and (c) high. [00321 The second input is pre-computing working load (PreComputing) in one of three states, namely (a) low, (b) average, and (c) high. [0033] The third input is Doubling, expressing how much working load for the calculation "doubling" which has three cases, namely (a) low, (b) average, and (c) high. The output is one, called WindowSzne to express the next window size should be moved in which way, which has three states for the window sizes, namely (a) down (the current window size should be decreased), (b) stay (the current window size should be no changing), and (c) up (the current window size should be increased). [0034] There are 26 Fuzzy Rules designed as follows (weights are unit): 1. If (SlorgeRoom is low) and (PreConmputing is low) and (Doubling is low) then (windoSize is tip) 2. [f(StorgeRoom is low) and (PreComputing is low) and (Doubling is average) then (windowSize is Up) 3If(StargeRoom is low) and (ProComputing is low) and (Doubling is high) hen (windowSize is stay) 4 fIf(StorgeRoom is low) and (PreComsaputing is average) and (Doubling is low) then (w\indowsie is Up) 5. If(SorgeRoom is los) and (PreCoaputing is average) and (Doubling is average) then (windowSiza is Up) 6. lf(StorgeRoom is low) and (PreComputing is average) and (Doubling is high) then (windowSize is slay) 7. If(Stoargekoom is low) and (PreComputing is high) and (Doubling is low) then (windosSize is Up) 8. I f(StorgeRoom is low) and (PreComputing is high) and (Dolbling is average) then (WindowSize is stay) 9. If(StorgeRoom is low) and (IPreConiputing is high) and (Doubling is high) the (WindowSize is slay) 10. lf(StorgeRoom is average) and (PreConpating is low) and (Doubling is low) then (WindowSize is Up) I. lf(StorgeRoonm is average) and (PreComputing is low) and (Doubling is average) hen (WindowSize is Up) 12. lf(StorgeRoom is average) and (PreComputing is low) and (Doubling is high) than (WindowSize is stay) 13 If(StorgeRoom is average) and (PreConpating is average) and (Doubling is low) then (windowSize is Up) 14 I (StorgeRooa is average) and (PreComputing is average) and (Doubling is average) then (WindowSize is slay) 15. If(Stlorgekoo is average) and (ProComputing is average) and (Doubling is high) then (Wiindowhize is Down) 16. If (StorgeRoom is average) and (PreConputing is high) and (Doubling is average) then (WindowSize isstay) 17 I (StorgeRoom is average) and (PreConputing is high) and (Doubling is high) then (WindowSize is stay) 18 lf(SorgeRsoom is higih) and (ProeComputing is low) and (Dosbling is low) than (Windowsize is stay) 19. If (StorgeRoom is high) and (PreConputing is low) and (Doubling is average) then (WindowSize is say) 20 I f(StorgeRoom is high) and (PreCanput ing is low) and (Doubling is high) then (Windosgize is Down) 21 If(StorgeRoom is high) and (PreCopuating is average) and (Doubling is low) then (windowSize is stay) 22 tf(StorgeRoom is high) and (PreCoiputing is average) and (Doubling is average) then (WindowSize is stay) 23 lf(Storgeoom is high) and (PreComputing is average) and (Doubling is high) then (WindowSize is Down) 24 If (Storgekoon is high) and (ProComputing is high) and (Doubling is low) then (windowSize is Down) 25 If (StorgeRoom is high) and (PreComputing is high) and (Doubling is average) hen (WinlowSize is Down) 26 tf(StorgeRoon is high) and (PreComputing is high) and (Doubling is high) then (WindowSize is Down) 100351 The number at each fuzzy condition with a bracket is the weight number, currently it is unit. Later we shall change it with different number according to the running situations as described in the next. [00361 The three inputs with 26 fuzzy rules in Mamdani model running fuzzy controller part is shown in Figure 4. The three inputs are StorageRoom, PreCompu/ing and Doubling. The output is WindowSize. 10037] The output with StorageRoom and PreComputing is shown in Figure 5. The surface StorageRoom vs. Doubling is shown in Figure 6. The surface SlorageRoom vs. PreCompuing is shown in Figure 7. 100381 From above figures, it is clearly observed that in the low window size side, if the storage room is low the dominated function of "doubling" will play role as Figure 5 shown but if the window size is at the high side, the storage room will be fairly stay at the middle either for PreComputing or Doubling, which is the doubling will sharply increased when window size a little bit larger that also can be shown from Table I. From Figure 6 it is clearly o to show when the storage room is getting big, it would be nice to have larger window size for O the "doubling". o 00391 Now if we change the weight for above fuzzy rules as such the rules 1,5 10, 13, 14, 15, 16, 18, 20 ,21, 22,23, 25, and 26 are set in 0.5 (the rest will keep the same) due to the major functions are controlled by the storage room, and doubling will rapidly increasing by the window size larger. The outputs will changed as the average storage room will increased 0.04% and the other two inputs are decreased by 0.02% the output become window staying a little wider side by 0.003%. [0040] It is clear that this fuzzy controller for the dynamic window is also involved a tradeoff between accuracy and control costs. For example the same system may go further for the second order parameters, not just check the changes about the input variables but also check the change tendencies of the variables, which will be discussed in our another paper. [0041] If we keep the storage constant and the situation shown by Figure 7 is how those two major factors shown in Table I to impact on the output. [00421 [he simulations of the example described in above were implemented. With equation (2), the computational cost has been reduced from 3 additions as in the binary method to only I addition in one's complement subtraction form. The number of pre-computations has remained the same. This can be proved for different window sizes. 100431 In our simulations, the proposed method together with a fuzzy window size controller makes the ECC calculation almost 15% more efficient than traditional methods in ECC wireless sensor network. [00441 It may be concluded that , positive integer in point multiplication may be recorded with one 's complement subtraction to reduce the computational cost involved in this heavy mathematical operation for wireless sensor network platforms. As the NAP method involves modular inversion operation to get the NAP of binary number, the one's complement subtraction can provide a very simple way of recoding the integer. Here is always decision between pre-computing and computing, the former is related to the storage and the latter is associated with computing capability and capacity. The window size may be the subject of trade-off between the available RAM and ROM at a particular instance on a sensor node, which can be controlled by fuzzy controller. The final simulation in a sensor wireless network shows that about 15% more efficient than transitional method can be obtained with ECC.

Claims (5)

1. A method of sliding window scalar multiplication in elliptical curve cryptography for the wireless sensor networks (WSN) comprising the steps of, (a) Recoding scalar in the one's complement subtraction form with signed binary numbers, (b) Selecting dynamic window size w before doing scalar multiplication, (c) Performing pre-computation based on window size w and, (d) Doing scalar multiplication in the evaluation stage with help of pre-computed values.

2. The method of sliding window scalar multiplication as recited in claim 1 wherein the dynamic window size w will be selected by the fuzzy controller comprising the following steps, (a) The fuzzy controller will have three inputs namely storage room, pre computations and doublings. (b) All the three inputs will have three states namely low, average, and high depending on the available resources and loads. (c) The fuzzy controller will have single output called window size which will have three states namely up window size, down window size or stay as it is depending on the nature of inputs.

3. The method of sliding window scalar multiplication as stated in claim 1 wherein the fuzzy controller will take decision of window size w based on rules stored in its monitor programme.

4. The method of elliptical curve cryptography in which sliding window scalar multiplication will be done with dynamic window size w selected by the fuzzy controller based on available memory and number of pre-computations trade off.

5. A method of sliding window scalar multiplication on wireless sensor network platform with the fuzzy controller recommended window size w substantially as hereinbefore described with reference to the accompanying drawings and description of art.