Efficient Large Integers Arithmetic by Adopting Squaring and Complement Recoding Techniques

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1 The th Worshop on Combnatoral Mathematcs and Computaton Theory Effcent Large Integers Arthmetc by Adoptng Squarng and Complement Recodng Technques Cha-Long Wu*, Der-Chyuan Lou, and Te-Jen Chang *Department of Avaton and Communcaton Electroncs Chnese Ar Force Insttute of Technology Department of Electrcal Engneerng Chung Cheng Insttute of Technology, Natonal Defense Unversty Abstract The effcent squarng algorthm plays an mportant role n large number arthmetc. All multplcaton algorthms can be used for squarng large ntegers, but ther performance can be greatly mproved by usng the standard squarng algorthm. Complement recodng method can be dvded nto two groups: s complement recodng method and s complement recodng method. In ths paper, we propose an effcent squarng algorthm by combng s complement recodng method wth standard squarng algorthm to fast evaluate the squarng. The computatonal complexty of the proposed squarng method s approxmately multplcatons, whch s faster than that of the standard squarng method multplcatons, where s the bt-length of X. We can therefore effcently speed up the overall performance of squarng arthmetc. Keywords: Squarng arthmetc, nformaton securty, multplcaton, s complement recodng, algorthm desgn.. Introducton Squarng s an easer operaton than multplcaton snce half of the sngle-precson multplcatons can be spped. Ths s due to the fact that t j =a *a j =t j, as shown n Fgure. Thus, we can modfy the standard multplcaton procedure to tae advantage of ths property of squarng operatons. So, the effcent squarng s an mportant role n large arthmetc. In ths paper, a new method s proposed for speedng up the performance of squarng. The proposed method combnes the standard squarng method and the s complement recodng method. The computatonal complexty of the proposed algorthm s, where s the bt-length of the exponent. The rest of ths paper s organzed as follows. In Secton, we revew the standard squarng method and the complement recodng method. The proposed method s descrbed n Secton 3. Fnally, we put our conclusons for the proposed method and state some future wors.. Mathematcal Prelmnares. Squarng Method Squarng s used to substtute for multplcaton because squarng s much more effcent than multplcaton [3]. Ths s due to the fact that s j =x *x j = s j. The standard squarng procedures are descrbed n Table and n Algorthm []. Thus, we can modfy the standard multplcaton -60-

2 The th Worshop on Combnatoral Mathematcs and Computaton Theory procedures to tae advantage of ths property of the squarng operaton. Table The standard squarng procedures. x x 3 x x * x x 3 x x S S 3 S S S S 3 S S S 3 S 33 S 3 S 3 + S S 3 S S S S 3 S S S S 3 S + S S 3 S 33 S S 7 S 6 S 5 S S 3 S S Algorthm (The standard squarng algorthm) Input: Integer X=(x n x n- x x ) b Output: Integer S=X =(s n s n- s s ) b begn s =0 for = to n for = to n (u, v) b =s +- +x *x s +- =v for j = + to n (u, v) b =s +- +*x j *x +u s +j- =v endfor s +n =u endfor Return S=(s n s n- s s ) b end. Example : Let X = 3,.e., n = 3 and b = 0. The squarng procedures of X usng Algorthm are shown n Table.. Complement Recodng Method In 003, Chang, Kuo and Ln proposed an effcent method to compute the general multplcaton operaton and for the common-multplcand multplcaton by performng complements. Assume the bnary representaton of multpler B s b b... bb 0. Performng complements s advantageous n the speed up of multplcaton computatons. The equaton s shown as follows. B b 0, (0...0) B, () ( )bts where B b b... b0, b { b 0 For example, f f b, b 0, for = 0,,,. B=(9) 0 =(00) =(0 0) (+)bts - B =(0 0) (+)bts -(000000) = Clearly, the Hammng weght of B can be reduced. Therefore, the multplcaton computatons wll be speeded up. Table. The procedures of 3 usng by Algorthm. j s +- +x *x s +j- +*x j *x +u u v S s s 3 s s 5 s 6-0+* *3* ** * ** *

3 The th Worshop on Combnatoral Mathematcs and Computaton Theory 3. The Proposed Method In Secton, we descrbed a squarng method and a complement recodng method respectvely. Now the proposed method s descrbed as follows. We assume there are bts n X. So X should be calculated by combnng a squarng method and a complement recodng method. Therefore, two cases should be dscussed as Case : Ham(X)> and Case : Ham(X). Then the overall computatonal complexty of X should be represented *the computatonal complexty of Case () or *the computatonal complexty of Case. (3) The frst and the second tems n Equaton () and Equaton (3) mean the probabltes of Ham(X)> and Ham(X). Now, we ntroduce the computatonal complextes of Case and Case respectvely. At frst, we defne X a s complement representaton for X. Case : Ham(X)> : When Ham(X)> n Case, we tae s complement for X. In other words, t can be represented by X. It means that Ham( X ). We can replace the pattern for X wth [(0 0) (+)bts - X -], where s the bt-length of X. For computatonal complexty, we wll dscuss t n detal n Secton. Case : Ham(X) : When Ham(X) n Case, we square X drectly. For computatonal complexty, we wll dscuss t n detal n Secton.. Computatonal Complexty In Secton 3, we dscuss the s complement applcatons for X between Case and Case. The computatonal complextes for Case and Case are dscussed respectvely as follows. Case : Ham(X)> : The pattern for X can be changed nto [(0 0) (+)bts - X -]. It can be expanded as follows. (0 0) (+)bts -* X *(0 0) (+)bts -*(0 0) (+)bts -( X ) +( X )+ () There are sx tems n Equatons (). We only consder ( X ) tem and other tems only consdered shft poston problem [0, ]. That s, other fve tems are not consdered for ther computatonal complexty. Then the number of the multplcatons for X s, where s the bt-length of X. Case : Ham(X) : Snce the Hammng weght of X s smaller than or equal to, the number of the multplcatons for X s also, where s the bt-length of X. From the above Case and Case depcted n the proposed method, we can therefore get the computatonal complexty of X as followng by usng Equaton () or Equaton (3) to be * =. Case or Case can not exst at the same tme. -6-

4 The th Worshop on Combnatoral Mathematcs and Computaton Theory 5. Conclusons and Future Wors In ths paper, we propose a new method to fast evaluate squarng, whch uses a complement recodng method. The computatonal complexty of the proposed method s multplcatons, whch s faster than that of the orgnal method multplcatons, where s the bt-length of X. We can therefore effcently speed up the overall performance of squarng. In the future wors, we wll do our best to well nvestgate and properly adopt ellptc curve encrypt methods [3, 7], fast mproved Montgomery s method [5, ], effcent cryptographc hardware desgn [, 6] and unfed-multplers method [9, ]. By explorng these methods and mprovng some technques, we wll desgn fast arthmetc algorthms to further decrease the overall computatonal complexty for nformaton securty usages (ncludng ey exchange schemes and smart card cryptographc dentfcaton applcatons). References [] Alar, A. Z., Sonmez, R., A Hardware Verson of the RSA Usng the Montgomery's Algorthm wth Systolc Arrays, VLSI Journal Integraton, vol. 3, no., pp , Dec. 00. [] Chang, C.-C., Kuo, Y.-T., Ln, C.-H., Fast Algorthms for Common Multplcand Multplcaton and Exponentaton by Performng Complements, Proceedng of 7 th Internatonal Conference on Advanced Informaton Networng and Applcatons, pp. 07-, March 003. [3] Chen, T.-S., A Threshold Sgnature Scheme Based on the Ellptc Curve Cryptosystem, Appled Mathematcs and Computaton, vol. 6, no. 3, pp. 9-3, March 5, 005. [] Eslam, Y., Sheholeslam, A., Gula, P.G., Masu, S., Muada, K., An Area-Effcent Unversal Cryptography Processor for Smart Cards, IEEE Transactons on Very Large Scale Integraton (VLSI) Systems, vol., no., pp. 3-56, Jan [5] Katja, S.-S., Semay, O., Taag, T., Analyss of Fractonal Wndow Recodng Methods and Ther Applcaton to Ellptc Curve Cryptosystems, IEEE Transactons on Computers, vol. 55, no., pp. -57, Jan [6] Koc, C. K., Hgh-Speed RSA Implementaton, Techncal Report TR 0, RSA Laboratores, 99. [7] Nada, N., Luza, M. M., A Revew of Modular Multplcaton Methods and Respectve Hardware Implementatons, INFORMATICA, vol. 30, no., pp.-9, Aprl 006. [] Nedjah, N., Mourelle, L. M., Four Hardware Implementatons for the M-ary Modular Exponentaton, Internatonal Conference on Informaton Technology: New Generatons, pp. 0-5, Aprl 006. [9] Nedjah, N., Mourelle, L. M., Three Hardware Archtectures for the Bnary Modular Exponentaton: Sequental, Parallel, and Systolc Crcuts, IEEE Transactons on Crcuts and Systems I: Fundamental Theory and Applcatons, vol. 53, no. 3, pp , March 006. [0] Nbouche, O., Nbouche, M., Ahmed, B., New Iteratve Algorthms for Modular Multplcaton, Sgnal Processng, vol., no. 0, Oct. 00, pp [] Sun, D.-Z., Hua, J.-P., Sun, J.-Z., Zhang, J.-W., Computatonal Effcency Analyss of Wu et al. s Fast Modular Mult-Exponentaton Algorthm, Appled Mathematcs and Computaton, In Press, Accepted Manuscrpt, Avalable onlne 7 Feb [] Tenca, A. F., Koc, C. K., A Scalable Archtecture for Modular Multplcaton Based on Montgomery's -63-

5 The th Worshop on Combnatoral Mathematcs and Computaton Theory Algorthm, IEEE Transactons on Computers, vol. 5, no. 9, Sep. 003, pp. 5-. [3] Wu, C.-L., Lou, D.-C., Chang, T.-J., An Effcent Modular Exponentaton Bnary Algorthm Usng Shortest Addton Chan, Mathematcal Meetng and Annual Meetng of the Mathematcal Socety, page, Dec [] Zya, A. A., Remzye, S., A Hardware Verson of the RSA Usng the Montgomery's Algorthm wth Systolc Arrays, Integraton, the VLSI Journal, vol. 3, no., Dec. 00, pp

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