A Division Algorithm Using Bisection Method in Residue Number System

Transcription

1 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 59 A Division Algorith Using Bisection Method in Residue Nuber Syste * Chin-Chen Chang and Jen-Ho Yang Abstract. Introduction Residue Nuber Syste ) has coputational advantages for large integer arithetic. It provides the benefits of parallel, carry-free, and high-speed arithetic in addition, subtraction, and ultiplication. However, overflow detection, sign detection, agnitude detection, and division are tie-consuing operations in. The ost interesting one of the above operations is division, and any related researches have been proposed to ake it ore efficient. Hiasat and Abdel-Aty-Zohdy proposed a high-speed division algorith using the highest power coparison to speed up division in. Their algorith evaluates the quotient according to the highest powers of in the dividend and the divisor. Nevertheless, the evaluated quotient is underestiated such that there are redundant execution rounds in their algorith. Thus, Yang et al. proposed a division algorith in using parity checking technique in 004. In this algorith, the evaluated quotient is estiated precisely such that the actual quotient can be found quickly. However, coputing the highest powers of in the dividend and the divisor is tie-consuing in, and this coputation ust be perfored in each execution round of Yang et al. s algorith. Consequently, we propose the bisection ethod in to design a division algorith in the paper. Our new algorith uses the bisection ethod in to find the quotient in a possible interval efficiently. Copared with Yang et al. s algorith, our algorith has less execution rounds and greatly reduces the ties of the highest-power coputation in. Keywords: Residue Nuber Syste, division, bisection ethod *Corresponding Author: Chin-Chen Chang E-ail: alan3c@gail.co). Departent of Coputer Science and Inforation Engineering, Asia University, Taichung, 43, Taiwan; Departent of Inforation Engineering and Coputer Science, Feng Chia University, Taichung, 407, Taiwan Departent of Multiedia and Mobile Coerce, Kainan University, No., Kannan Rd., Luzhu, Taoyuan County, 33857, Taiwan Nowadays, nuerous nuber systes have been utilized to ake coputers ore and ore powerful. The ost popular one of these nuber systes is Residue Nuber Syste ) because it has any advantages of coputing large nubers in coputers. In, a nuber is represented by the residues of all oduli, and the arithetic can be perfored on each odulus independently. Thus, offers the properties of parallel, carry-free, and high-speed arithetic []. However, the overflow detection, sign detection, agnitude detection, and division are tie-consuing operations in. The ost interesting research topic of the is division because it has any applications such as odular operations. Thus, several algoriths have been proposed to solve division proble in [-9]. However, all proposed algoriths have the drawbacks of long execution tie and large hardware requireents because Mixed-Radix Conversion MRC) and Chinese Reainder Theore CRT) are used. Instead of eploying MRC and CRT, Hiasat and Abdel-Aty-Zohdy used the highest power coparison between the dividend and the divisor to design a division algorith in [0]. The execution tie and hardware requireents in their algorith are less than those in other division algoriths. Their algorith coputes the evaluated quotient according to the highest powers of the dividend and the divisor, and obtains the actual quotient by coputing the su of all evaluated quotients until the product of the divisor and the evaluated quotient is less than the dividend. However, the evaluated quotient is underestiated in their algorith such that the nuber of execution rounds increases. Thus, Yang et al. proposed a division algorith in using the parity checking technique in 004 []. In the algorith, the evaluated quotient is estiated precisely such that approxiating the actual quotient

2 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 60 by adding the evaluated quotients is two ties faster than that in Hisat and Abdel-Aty-Zohdy s algorith. However, coputing the highest power of of a nuber is tie-consuing in, and Yang et al. s algorith needs to perfor this coputation in each execution round. This extra load is the bottleneck of this algorith. As a result, a bisection ethod in is proposed to design a division algorith in this paper. A new algorith in this research uses the bisection ethod to find the quotient in a probable interval efficiently, and the algorith only needs to copute the highest-power coparison in in the first round. This iproveent greatly decreases the execution tie of each round in our algorith. Furtherore, Yang et al. s algorith needs to deterine the agnitudes of the dividend and the divisor by using the parity checking technique twice in each round. However, using the bisection ethod in, our algorith only perfors the parity checking technique once to deterine the agnitude of the residue in each execution round. Moreover, the proposed division algorith can be efficiently applied to any applications, such as iage processing [], cryptosystes [3], and odular exponentiation arithetic [4]. The reainder of this paper is organized as follows. A review of popular iage copression algoriths that support the ROI capability is given in Section II. In Section III, we shall address our proposed ultiple-roi iage copression schee. Then the siulation results and a further discussion are provided in Section IV. Finally, a brief conclusion is drawn in Section V.. Hiasat and Abdel-Aty-Zohdy's division algorith Given a set of relatively prie oduli B,,..., }, where gcd, ), i j, { n n M i i we copute i j. Here, B and M are called the base and the range of. In, any nuber X less than the range M can be denoted as a vector X x, x,..., x ), where i i n x X X od. According to Chinese i Reainder Theore, any X less than M has only one representation. We assue that two integers x, y [0, M) in are X x, x,..., x ) and Y y, y,..., y ), n n respectively, and copute Z X Y z, z,..., z ), where can be n addition, subtraction or ultiplication in. Then Z can be obtained by coputing X Y x n n n y, x y,..., x y ) z, z,..., z ). That is, the coputation can be perfored concurrently and independently on each residue in. Therefore, addition, subtraction, and ultiplication of two large nubers, individually converted into representations, can be efficiently coputed because of the properties of parallelis and no carries. Assue that we want to copute Q = [ X Y ] in, where X, Y, and Q are positive integers. In Hiasat and Abdel-Aty-Zohdy s algorith, h I) can be coputed the highest power of in the variable I represented by. According to [0], h I) is ipleented by a proper cobinational circuit like a priority encoder and an n-operand binary adder, where n is the nuber of the oduli in. Hiasat and Abdel-Aty-Zohdy s algorith is shown as follows. Step : Set the quotient Q 0. Step : Copute j hx ) and k hy ), where j and k are the highest powers of in X and Y, respectively. Step 3: If j k, then copute Q ' Q, X ' X * Y, Q Q', X X '. Go to Step. Step 4: If j k, then copute X' X Y Step 5: If j k and j' h X '). If j' j then Q Q. Otherwise, Q is unaltered. End the procedure. procedure., then Q is unaltered. End the n

3 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 6 Hiasat and Abdel-Aty-Zohdy s algorith utilizes addition, subtraction, and ultiplication according to the highest powers of in dividend and divisor to achieve division in. Besides, the algorith perfors division efficiently by avoiding the agnitude detection and overflow detection in. 3. Yang et al. s algorith Assue that we want to copute Q = [ X ] in, Y where X, Y, and Q are integers. Note that h I) denotes the highest power of in the variable I, where I is representation of an integer. For exaple, 0 = 0,, 0) and 7) =,, 3) with the oduli, 3, 5) in, then we have h 0,, 0)) 3 and h,, 3)). Besides, S X ) is defined as the sign of variable X. If S X ) S, it denotes that X and Y are with the sae sign. In addition, the parity checking technique [8] is used in this algorith to deterine the sign of a nuber in. Unlike other sign detection algoriths in using MRC or CRT, the parity checking technique can easily obtain the sign of a nuber in by checking its parity and looking up a table. This technique akes the sign detection and the agnitude coparison in ore efficient. The details of the parity checking technique are available in [8]. Yang et al. s algorith is shown in the following. Step : Set the quotient Q 0 and c 0. Step : Copute j hx ) and k hy ), where j and k are the highest power of in X and Y, respectively. Then, check the signs of X and Y by parity checking. According to the relationships between j and k, we perfor one of the following three steps. Step 3: If j k, we perfor the following operations: Set c. If S X ) S, then we copute Q Q X ' X *, jk jk ', Y Q Q', X X ' ; otherwise, we copute Q Q jk ', jk X ' X * Y, Q Q', and X X '. Go to Step. Step 4: If j k, we perfor the following operations. If S X ) S and c, then we copute X' X Y and j' h X '). If j' j then we copute Q Q. Otherwise, Q is unchanged. End procedure. If S X ) S and c, then we copute X' X Y and check the sign of X ' by parity checking. If X ' 0, then we copute Q Q. Otherwise, we set Q Q. End procedure. If S X ) S and c 0, then we set Q. End procedure. Step 5: If j k, we perfor the following operations. If c 0, then Q is unchanged. End procedure. Otherwise, we copute Q Q. End procedure. In this algorith, the evaluated quotient is jk which is larger than in Hiasat and Abdel-Aty-Zohdy s algorith such that the actual quotient can be obtained quickly in Yang et al. s algorith. Copared with Hiasat and Abdel-Aty-Zohdy s algorith, Yang et al. s algorith reduces the nuber of execution rounds by 50%.

4 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 6 4. The proposed algorith After observing Yang et al. s algorith, we find h I) and the parity checking technique ust be perfored twice in each round of their algorith. If we decrease the nubers of these two coputations in each round, the execution tie can be reduced. Thus, we propose a bisection ethod in to accoplish this purpose. In the following, our algorith is shown. Input: X, Y : are expressed in ) Output: Q : is expressed in ) Step : Copute j hx ) and k hy ), where j and k are the highest powers of in X and Y, respectively. Step : If j k, then use the parity checking technique to copare X with Y. If X Y, then set Q. Otherwise, set Q 0. End the progra. If j k, then set Q 0. End the progra. j k Step 3: Copute Q u and Q l. Step 4: Copute Q' Q u Ql. If Q ' is odd, then set Q' Q',,...,). Step 5: In our algorith, we define,, and as addition, subtraction, and ultiplication in, respectively. Besides, Copute Q = Q and Z X Q Q u and Q l are the upper and the lower bound of the actual quotient in our algorith, respectively. In Step 4, we copute Q' Q to be the evaluated quotient in each round of our algorith. If all chosen oduli in are odd Y. Then, use the parity checking technique to deterine the sign and the agnitude of Z. Step 6: If Z Y 0, then set Q l Q. Go to Step 3. Step 7: If Z 0, then set Q u Q 3.. Go to Step Step 8: If Y Z 0, then Q is the quotient. End the progra integers, then coputing Q' can be directly ade in without converting it into the binary or the decial syste. For exaple, assue that the ultiplicative inverses of odulo all oduli,,..., n ) in are,,..., ), and the representation of Then, we have n Q ' is q, q,..., q ). n Q' q, q,..., qn ). n n Moreover, the ultiplicative inverses of odulo all oduli in can be precoputed to reduce the coputation tie. 5. Analyses We siulate Hiasat and Abdel-Aty-Zohdy s algorith, Yang et al. s algorith, and our algorith in Visual C++ 6.0, and run the siulation progras for the values of X and Y both ranging fro to Table shows the nubers of execution rounds in these three algoriths, and Table shows the nuber of coputing h I) and the ties of perforing the parity checking technique in these three algoriths. The analysis of the siulation results is shown as follows. In Hiasat and Abdel-Aty-Zohdy s algorith, the evaluated quotient is the power of i.e. = ), and the actual quotient is fored by these evaluated quotients. For exaple, 58 the actual quotient Q 7 33 in Table. After running Hiasat and Abdel-Aty-Zohdy s algorith, Q is fored by Note that even if 7 represented in the binary for is ) 0, Q is probably fored by or another for in Hiasat and

5 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 63 Abdel-Aty-Zohdy s algorith. This is because the evaluated quotient is underestiated in their algorith such that is fored by 0 0 in the above exaple. In this case, their algorith ust execute two rounds to obtain. This property causes that there are soe redundant rounds in their algorith. Thus, it can be easily observed that the nuber of execution rounds in Hiasat and Abdel-Aty-Zohdy s algorith depends on the coponents of the quotient Q represented in the binary syste. Assue that representing Q in the binary syste requires log Q bits, then the nuber of execution rounds in Hiasat and Abdel-Aty-Zohdy s algorith denoted as rounds ) ust fall in the range shown as Equation ). This is because the axiu nuber of the binary digits used to for Q is log ) Q. log Q rounds ) In our algorith, the range of the actual quotient Q is Q and we use soe techniques in to find Q in this range. With another point of view, our algorith can be regarded as perforing the bisection ethod in to find Q in the interval, ). Thus, the range of the nuber of execution rounds in our algorith denoted as rounds ) is shown as Equation ). rounds log ) ) ) To copare rounds ) with rounds ), we adjust Equations ) and ) in the following steps. Since Q obtained fro Equation )., Equation 3) can be rounds log ) 3) ) For the convenience of the analysis, Equation 3) is rewritten as Equation 4). rounds log ) 4) ) Because log ) log ) log ) log4 Equation 5) can be obtained fro Equation 4)., rounds log ) log4) 5) ) Fro Equation ), Equation 6) is obtained as follows. rounds log3 ) 6) ) Again, Equation 6) can be rewritten as Equation 7). j k rounds log 3 ) 7) Because log3 ) log3 log ) obtain Equation 8) fro Equation 7). ), we rounds log ) log3 8) ) Copared Equation 5) with Equation 8), rounds ) is the double of rounds ). The analysis confirs that Yang et al. s algorith reduces the nuber of execution rounds by 50% and also by coparing with Hiasat and Abdel-Aty-Zohdy s algorith. As the sae reason, the nuber of execution rounds in Yang et al. s algorith also depends on the coponents of the quotient Q represented in the binary syste. Each binary digit of Q in Yang et al. s algorith can be obtained by executing one round because the evaluated quotient is set larger and ore precise. Thus, assue that representing Q in the binary syste requires log Q bits, then the nuber of execution rounds in Yang et al. s algorith, denoted as rounds 3), is shown as Equation 9). 3) rounds logq 9) Using the sae thinking steps fro Equation 3) to Equation 5), Equation 0) can be obtained fro Equation 9). rounds log ) log 4 0) )3 Copared with Equation 8) and Equation 0) denoted as rounds ) is a little less than rounds 3). Table 4 also shows this fact. On the other hand, coputing h I) in is tie-consuing, but Hiasat and Abdel-Aty-Zohdy s algorith and Yang et al. s algorith both needs to copute h I) in each round. However, Our algorith only coputes h I) in the first round. This property greatly reduces the execution tie in

6 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 64 each round of our algorith. Although our algorith needs to copute the evaluated quotient by Q l ' u Q Q, coputing Q ' by looking up a table is faster than coputing h I) in. As the densities and speed of RAMs increase, the use of looking up tables is beneficial. Besides, our algorith decreases the ties of perforing the parity checking technique in each round. This iproveent also reduces the execution tie in each round of our algorith. Table 5 shows the nubers of coputing h I) and the ties of perforing the parity checking techniques in Hiasat and Abdel-Aty-Zohdy s algorith, Yang et al. s algorith, and our algorith for coputing and Y are all ranging fro to X, where X Y Table : The nubers of execution rounds for the siulations Hiasat and Abdel-Aty-Zohdy s algorith Yang et al. s algorith The nuber of rounds of our algorith The percentage of the nubers of rounds X= ~0000, Y= ~0000 According to Table, our algorith greatly reduces the nuber of coputing h I) and the ties of perforing the parity checking techniques by 46% and 50%, respectively. Consequently, our algorith is uch faster than Yang et al. s algorith even if the execution rounds in our algorith are a little less than those in Yang et al. s algorith. X= ~0000, Y= ~0000 X= ~30000, Y= ~ % 49.3% 49.87% Table : The ties of coputing hi) and the parity checking techniques Hiasat and Abdel-Aty-Zo Yang et al. s hdy s algorith algorith Parity checking technique h I) Conclusions Our algorith In this paper, we propose a division algorith using the bisection ethod in. Assue that the highest powers of in the dividend and divisor are j and k, respectively, then the quotient would fall into the interval, ). Our algorith skillfully finds the quotient in the interval without converting the nubers fro into the binary or the decial syste. Copared with Yang et al. s algorith, our algorith reduces the nuber of the highest power coputation h I) and the parity checking techniques in by 46% and 50%, respectively. According to the siulation results and the analysis in Section 5, it is obvious that our algorith is ore efficient than Yang et al. s algorith. References [] N. Szabo and R. Tanaka, Residue Arithetic and Its Applications to Coputer Technology, McGraw Hill, New York, 967. [] E. Kinoshita, H. Kosako, and Y. Kojia, General Division in the Syetric Residue Nuber Syste, IEEE Transactions on Coputers, Vol., 973, pp [3] D. Banerji, T. Cheung, and V. Ganesan, A High-speed Division Method in Residue Arithetic, Proceedings of 5th IEEE Syposiu on Coputer Arithetic, Michigan, USA, 98, pp [4] W. Chren Jr., A New Residue Nuber Division Algorith, Coputer and Matheatics with Applications, Vol. 9, No. 7, 990, pp [] D. Gaberger, New Approach to Integer

7 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 65 Division in Residue Nuber Syste, Proceedings of 0th Syposiu on Coputer Arithetic, Grenoble, France, 99, pp [] Y. Kier, P. Cheney, and M. Tannenbau, Division and Overflow Detection in Residue Nuber Systes, IRE Transactions on Electronic Coputers, Vol., 96, pp [3] L. Lin, E. Leiss, and B. Mcinnis, Division and Sign Detection Algorith for Residue Nuber Systes, Coputer and Matheatics with Applications, Vol. 0, 984, pp [4] M. Lu and J. S. Chiang, A Novel Division Algorith for the Residue Nuber Syste, IEEE Transactions on Coputers, Vol. 4, No. 8, 99, pp [5] A. A. Hiasat, and H. S. Abdel-Aty-Zohdy, A High-speed Division Algorith for Residue Nuber Syste, Proceedings of IEEE International Syposiu on Circuits and Systes, Vol. 3, Washington, USA, 995, pp [6] A. A. Hiasat, and H. S. Abdel-Aty-Zohdy, Design and Ipleentation of an Division Algorith, Proceedings of 3th IEEE Syposiu on Coputer Arithetic, California, USA, 997, pp [] J. H. Yang, C. C. Chang, and C. Y. Chen, A High-Speed Division Algorith in Residue Nuber Syste Using Parity-Checking Technique, International Journal of Coputer Matheatics, Vol.8, No. 6, pp , 004. [] D.K.Taleshekaeil and A. Mousavi, The Use of Residue Nuber Syste for Iproving the Digital Iage Processing, Proceedings of 0th IEEE International Conference on Signal Processing, Beijing, China, pp , 00. [3] J. Adikari, V. S. Diitrov, and L. Ibert, Hybrid Binary-Ternary Nuber Syste for Elliptic Curve Cryptosystes, IEEE Transactions on Coputers, Vol. 60, No., pp.54-65, 0. [4] F. Gandino, F. Laberti, P. Montuschi, and J. Bajard, A General Approach for Iproving Montgoery Exponentiation Using Pre-processing, Proceedings of 0th IEEE Syposiu on Coputer Arithetic, Tübingen, Gerany, pp , 0. Chin-Chen Chang received his Ph.D. degree in coputer engineering fro National Chiao Tung University. His first degree is Bachelor of Science in Applied Matheatics and aster degree is Master of Science in coputer and decision sciences. Both were awarded in National Tsing Hua University. Dr. Chang served in National Chung Cheng University fro 989 to 005. His current title is Chair Professor in Departent of Inforation Engineering and Coputer Science, Feng Chia University, fro Feb Prior to joining Feng Chia University, Professor Chang was an associate professor in Chiao Tung University, professor in National Chung Hsing University, chair professor in National Chung Cheng University. He had also been Visiting Researcher and Visiting Scientist to Tokyo University and Kyoto University, Japan. During his service in Chung Cheng, Professor Chang served as Chairan of the Institute of Coputer Science and Inforation Engineering, Dean of College of Engineering, Provost and then Acting President of Chung Cheng University and Director of Advisory Office in Ministry of Education, Taiwan. Professor Chang has won any research awards and honorary positions by and in prestigious organizations both nationally and internationally. He is currently a Fellow of IEEE and a Fellow of IEE, UK. And since his early years of career developent, he consecutively won Outstanding Talent in Inforation Sciences of the R. O. C., AceR Dragon Award of the Ten Most Outstanding Talents, Outstanding Scholar Award of the R. O. C., Outstanding Engineering Professor Award of the R. O. C., Distinguished Research Awards of National Science Council of the R. O. C., Top Fifteen Scholars in Systes and Software Engineering of the Journal of Systes and Software, and so on. On nuerous occasions, he was invited to serve as Visiting Professor, Chair Professor, Honorary Professor, Honorary Director, Honorary Chairan, Distinguished Alunus, Distinguished Researcher, Research Fellow by universities and research institutes. His current research interests include database design, coputer cryptography, iage copression and data structures. Jen-Ho Yang received the B.S. degree in coputer science and inforation engineering fro I-Shou University, Kaoshiung in 00, and the Ph.D. degree in coputer science and inforation

8 International Journal of Coputer, Consuer and Control IJ3C), Vol., No. 03) 66 engineering fro National Chung Cheng University, Chiayi County in 009. Since 009, he has been an assistant professor with the Departent of Multiedia and Mobile Coerce in Kainan University, Taoyuan. His current research interests include electronic coerce, inforation security, cryptography, authentication for wireless environents, digital right anageent, and fast odular ultiplication algorith.