Hybrid RNS-to-Binary Converter for the Moduli Set {2 2n, 2 n -1, 2 n +1, 2 n+1-1}

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1 Research Joural of Appled Sceces, Egeerg ad echology 6(): 07-0, 0 ISSN: ; e-issn: Mawell Scetfc Orgazato, 0 Submtted: November, 0 Accepted: Jauary 9, 0 ublshed: July 5, 0 Hybrd RNS-to-Bary Coverter for the Modul Set {, -, +, -} Somayyeh Jafaral Jassb ad Amr Sabbagh Molahosse Faculty Member of Computer Egeerg Departmet, Scece ad Research Brach, Islamc Azad Uversty, ehra, Ira Departmet of Computer Egeerg, Kerma Brach, Islamc Azad Uversty, Kerma, Ira Abstract: he four-modul Resdue Number System (RNS) sets such as { -,, +, -} have attracted a lot of researches durg recet years. However, owadays applcatos requre hgher dyamc rage. hs study troduces the RNS four-modul set {, -, +, -} whch s obtaed by ehacg the modul set { -,, +, -}. hs ehacemet dd t crease the total speed of RNS arthmetc ut sce the crtcal modulo both of the modul sets { -,, +, -} ad {, -, +, -} s +. Besdes, a effcet RNS-tobary coverter for the proposed modul set s desged usg a two-level archtecture where a prevous coverter desg for subset {, -, +} s used the frst level ad the a two-chael Med-Rad Coverso (MRC) algorthm s cosdered to acheve the fal result. Comparso wth a recetly troduced RNS-to-bary coverter for a four-modul set wth the same dyamc rage show that the proposed desg results hgher speed. Keywords: Med-Rad Coverso (MRC), Resdue Number System (RNS), reverse coverter INRODUCION Nowadays, the Resdue Number System (RNS) have foud cosderable applcatos such as Dgtal Sgal rocessg (DS) ad cryptography systems where we eed hgh-speed ad low-power hardware mplemetatos of addto, subtracto ad multplcatos (Soderstrad et al., 986; Cardarll et al., 007; Coway ad Nelso, 00). he RNS have capablty to perform addto, subtracto ad multplcato a fully parallel maer sce there s ot ay carry-propagato betwee resdue dgts RNS. However, other operatos such as dvso, magtude comparso ad sg detecto are omodular processes that are cosdered as hard RNS operatos (Moha, 00). I order to costruct a RNS, frst some par-wse relatvely prme umbers should be selected to forms the modul set of the system. he product of these selected umbers determes the rage of umbers whch ca be represeted that s called as Dyamc Rage (DR). o adapt RNS wth other dgtal systems two coverters are eeded. Frst, bary-to-rns coverter trasforms the weghted bary umber to RNS represetato by computg the resdue of dvso of that umber to the each modulo of modul set. Also, RNS-to-bary coverter decodes the RNS represeted umber to ts equvalet weghted represetato. I cotrast to bary-to-rns coverter, the RNS-to-bary coverter s so comple. wo famous algorthms to perform RNS-to-bary coverso are Chese Remader heorem (CR) ad Med-Rad Coverso (MRC) (Omod ad remkumar, 007). he kd of the modul set has a mportat mpact o other parts of the RNS system. Due to ths the problem of selecto of approprate RNS modul set attracts may researches durg the prevous years ad as a result some modul sets have bee proposed for dfferet RNS applcatos. Frst, the three-modul sets such as { -,, +} (Wag et al., 00), { -,, -} (Moha, 00), { -,, - -} (Wag et al., 000) ad {, -, +} (Hasat ad Sweda, 00) have bee cosdered sce the less umber of modul results low-complety coverters. However, the eed for more parallelsm lead to troducg fourmodul sets such as { -,, +, -}, { -,, +, - -} (Cao et al., 005) ad { -,, +, +} (Moha ad remkumar, 007). Nowadays, hgh-performace applcatos requre larger dyamc rage. Hece, hgh DR modul sets such as { -,, +, -}, { -, +,, +} (Molahosse et al., 00), { -,, +, +} (Cao et al., 00) ad { -,, +, -, - -} (Cao et al., 007) were proposed. I ths study, we are gog to desg effcet reverse coverter for the large DR four-modul set {, -, +, -} to acheve fast RNS system sce ths modul set ca provde hgh dyamc rage wth relatvely the same total RNS arthmetc ut speed tha the modul set { -,, +, -}. Because the crtcal modulo the proposed set s + ad so the Correspodg Author: Somayyeh Jafaral Jassb, Faculty Member of Computer Egeerg Departmet, Scece ad Research Brach, Islamc Azad Uversty, ehra, Ira 07

2 Res. J. Appl. Sc. Eg. echol., 6(): 07-0, 0 ehaced modulo s ot result creasg the total delay of RNS arthmetc ut. Besdes, a two-level RNS-to-bary coverter for ths modul set s proposed. he preseted coverter uses a estg coverter for the subset {, -, +} followed by a two-chael MRC coverso crcut to compute the fal weghted umber based o the composte set { ( -)( +), -}. LIERAURE REVIEW he resdue umber system s costructed based o the modul set {,,, } where each two modul ad j are par-wse relatvely prme (Moha, 00; Omod ad remkumar, 007). Now, a weghted umber X s showed as (,,, ) where: X mod X () I order to covert the RNS umber (,,, ) to ts equvalet weghted bary umber X, the CR has bee used usually. O the other had, the MRC algorthm ca also do the coverso. Chese Remader heorem (Omod ad remkumar, 007): hs algorthm descrbes a method to acheve X by some modulo multplcato ad summato as follows: X N M M () Note that M s DR, N M - s the multplcatve verse of M modulo ad M M/.. Med-Rad Coverso (Cao et al., 005): I cotrast to CR whch s a parallel algorthm, the MRC s a sequetal algorthm based o the followg formulas: I ths study, MRC s used oly for two modul coverso. Hece, the Eq. (6) ca be smplfed to: X a + a + ( (7) HE RNS-O-BINARY CONVERSION ALGORIHM he umbers of the modul set {, -, +, -} are par-wse relatvely prme oly for eve values of. Hece, the RNS-to-bary coverter for ths modul set works oly for eve values of. Besdes, Due to the propertes of ths modul set a two-level coverso process s selected to acheve a effcet coverter. Fgure shows the geeral block dagram of the coverter. he frst level reles o a estg RNS-to-bary coverter for the subset {, -, +} (Hasat ad Sweda, 00). As descrbed Hasat ad Sweda (00), the formulas for coverso of resduerepreseted umber (,, ) to the weghted umber Z are as below: Z + Y (8) Y (9) k,,,0 bts,0,,,0,, bts bts (0) (),0,,,0,, (), bts bts X a a + a + a () k, 0 00, () bts bts he coeffcets a s should be calculated usg hs three-modulo coverter cossts of two -bt these equatos: carry-save adders (CSAs) wth ed-aroud carres (EACs) followed by a modulo ( a () -) adder whch ca be mplemeted by a -bt carry-propagate adder (CA) wth EAC (estrak, 99). a ( a (5) he secod level uses a two-chael MRC algorthm to combe the result of frst level wth the fourth resdue. Frst, the requred multplcatve verse s computed usg the followg lemmas. a ((( a a a (6) Lemma : the multplcatve verse of ( -)( +) modulo - s as follows: 08

3 Res. J. Appl. Sc. Eg. echol., 6(): 07-0, 0. (0) () Y Fg. : he block dagram of the coverter k ( )( + ) k () roof: It s clear that: k ( )( + ) ( ) + + (. ).( ) Lemma : he value of / - ca be calculated usg the followg formula where s eve (Cao et al., 005): / 0 (5) roof: he proof descrbed Cao et al. (005). Now, cosder the modul set { ( -)( +), -} ad X(Z, ). Usg the MRC techque (7), X ca be computed by ths equato: X Z + ( ) k( Z ) + (6) Cosderg (8), the above equato ca be rewrtte as: X + Y + ( ) + ( Y + ) (7),..., (),0 bts 00,... (), +, bts,..., () +,0 YY 0Y... YY bts ( Y... YY 0 ) bts Y Y... Y Y ( 00Y... Y Y ) bts bts bts ( Y... YY 0 ) bts ( Y... Y Y ) bts (5) (6) Note that two well-kow modulo k - arthmetc propertes whch descrbed Cao et al. (005) are used dervg ()-(6). Net, by substtutg (5) (0) we have: 0 6 ( ). ( ). (7) hs equato ca be wrtte as follows: () (6) ( ) (8) where () deotes -bt crcular left shftg of (Cao et al., 005). ( ) (8) k Z Substtutg the value of k ad Z from () ad (8), respectvely (8) results : ( + Y ) ( Y ) (9) Sce, ad Y are -bt umbers, they have to be dvded to ()-bt parts. Hece (9) ca be wrtte as: HE HARDWARE ARCHIECURE OF CONVERER Hardware realzato of the proposed RNS-tobary coverter for the modul set {, -, +, -} cossts of two parts. he frst part s the crcuts for combg the resdues, ad based o the modul set {, -, +}. Here, the method of Hasat ad Sweda (00) s used to mplemet ths part. Net, (7) should be mplemeted to acheve the fal result. But, before t, the formulas () ad (8) have to realze. he Eq.() requres a (7, -) 09

4 Res. J. Appl. Sc. Eg. echol., 6(): 07-0, 0 able : Detals of coverter hardware compoets arts FA NO XOR/AND XNOR/OR Delay hree Modul Cov. - - ()t FA OU tno MOMA 8 (5)t FA OU MOMA ((-)() /) (l)t FA OU tno CA ()t FA otal ((-)/)+ (6+) - (8+l)t FA +t NO able : erformace comparso arts Hardware requremets Delay (Moha ad remkumar, 007) (95+k)A FA +(6)A NO +()A XNOR/OR (.56)t FA (Cao et al., 005) (/+)A FA +()A NO (8+l)t FA (Molahosse et al., 00) (8)A FA + ( )A XOR/AND +()A XNOR /OR + (7)A NO +()A MUX (5)t FA roposed (( -)/)+(6+))A FA +()A NO +(-)A XOR/AND +( )A XNOR/OR (8+l)t FA Fg. : Hardware detals of the proposed coverter mult-operad modular adder (MOMA) (estrak, 995) whch ca be realzed by three ()-bt CSAs wth EACs followed by a ()-bt CA wth EAC. Besdes, (8) also eeds a (/, -) MOMA that cossts of ()-bt CASs wth EACs. Fally, mplemetato of (7) ca be easly doe by cosderg two cocateatos. Frst, sce Y s a -bt umber, computato of Y+ eeds o hardware. he the results of ths cocateato should be added wth two s complemet of plus oe to 00 perform the eeded subtracto. herefore, the result of ths addto should be cocateated wth to form the weghted umber X. Fgure shows the detals of the proposed coverter. Besdes, able descrbes detals of each compoet. Note that full adders (FAs) of the last CA ca be replaced by XNOR/OR gates due to the costat bts of oe of the operads. Furthermore, t FA s the delay of oe FA ad l s the umber of levels of a CSA tree wth / puts. COMLEXIY COMUAION hree related studes are selected for comparso. Frst, two RNS-to-bary coverters for the ()-bt DR four-modul set { -,, +, -} (Cao et al., 005; Moha ad remkumar, 007) are cosdered. Secod, the recet RNS-to-bary for the (5)-bt DR modul set (Molahosse et al., 00) s also vestgated. able compares the hardware requremets ad coverso delays of these coverters. Note that the delay of NO gates gored ths able. It ca be see from ths able that the proposed coverter s faster tha Molahosse et al. (00) ad also t has relatvely the same delay tha other two coverters. However, the coverter desg of Molahosse et al. (00) eeds less hardware. It should be oted that the proposed modul set has (5)-bt DR whle the modul set { -,, +, -} has ()-bt DR. So, the proposed coverter ca provde hgher DR wth relatvely the same delay tha Cao et al. (005) ad Moha ad remkumar (007). CONCLUSION hs study presets a effcet RNS-to-bary coverter for the ew RNS modul set {, -, +, -}. he coverter for ths set s acheved by twolevel coverter archtecture wth better delay compared to a recetly troduced RNS-to-bary coverter for a 5-bt DR modul set. Also, wth the ew proposed modul set the teral RNS arthmetc crcuts ca be

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