Markus A. Hitz and Erich Kaltofen. Adaption to symmetric RNS is not dicult, but destroys. some of the clarity in the descriptions.

Transcription

1 Integer Diviion in eidue Number Sytem arku A. Hitz and Erich Kaltofen Abtract Thi contribution to the ongoing dicuion of diviion algorithm for reidue number ytem (NS) i baed on Neton iteration for computing the reciprocal. An extended NS ith tice the number of moduli provide the range required for multiplication and caling. Separation of the algorithm decription from it NS implementation achieve a high level of modularity, and make the complexity analyi more tranparent. The number of iteration needed i logarithmic in the ize of the quotient for a xed tart value. With preconditioning it become the logarithm of the input bit ize. An implementation of the converion to mixed radix repreentation i outlined in the appendix. Index Term Integer diviion, reciprocal, Neton iteration, extended reidue number ytem, mixed radix converion, bae extenion. I. Introduction Already Szabo and Tanaka [9] devied to algorithm for general diviion in reidue number ytem (NS). But only after 980 the number of paper dedicated to NS diviion tarted to increae. Chren [3] ummarize ome of the previou eort, and dicue an improved verion of Banerji' et al. algorithm []. ot NS diviion algorithm ue ome form of binary expanion for the quotient or the reciprocal. They are uually cloely tied to their repective hardare implementation, making the complexity analyi dicult. An exception in thi category i the ork of Davida and Lito [4]. They give the complete analyi for an almot uniform logdepth diviion circuit. Another more recent reult by Lu and Chiang [8] i baed on comparion and binary earch. In our approach, e ue an extended NS hich provide roughly the quare of a normal NS range for intermediate reult. We try to avoid the term \redundant", becaue the additional reidue are ignicant for everal computational tep. We compute the (integer) reciprocal of the divior ith repect to the original range of the NS, and, after multiplication by the dividend, ue caling to get the quotient (and remainder). The algorithm for the reciprocal i in the pirit of Aho, Hopcroft and Ullman []. Alo Gamberger [5] ue an extended NS. Hi diviion algorithm trie to nd common divior of the nominator and denominator, removing them iteratively by caling. Davida and Lito have to increae the number of moduli in their algorithm to n + for intermediate value, hich probably make their reult impractical. anucript received onth nn,993. Thi material i baed on ork upported in part by the National Science Foundation under Grant No. CC and under Grant No. CDA The author are ith the Department of Computer Science, enelaer Polytechnic Intitute, Troy, N We only dicu implementation for unigned NS number. Adaption to ymmetric NS i not dicult, but detroy ome of the clarity in the decription. II. Definition Let m 0 ; ; m 0 n Z ; pairie relatively prime, uch that < m 0 < m 0 < < m 0 n? < m 0 n ; here n Z; n. We group thee n moduli into to vector of ize n and m m 0 ; m m 0 3 ; ; m n m 0 n? ; m n+ m 0 ; m n+ m 0 4 ; ; m n m 0 n We call the NS dened by thoe moduli an extended NS, the NS dened by the rt n moduli bae NS, and the NS dened by the moduli ith indice n + to n extenion NS. The repective range for the bae and the extenion NS i n m i ; n in+ m i ; here the range for the entire extended NS i. An integer X; 0 X < ith reidue x X mod m ; ; x n X mod m n ill be repreented in the extended NS by [x ; ; x n ; x n+ ; ; x n ] The repreentation in the aociated mixed radix ytem (S) ill be denoted by here hv ; ; v n ; v n+ ; ; v n i ; X v i P i? ; P 0 ; P i m m m i ; for i < n ; and 0 v i < m i ; for i n We note the folloing propertie. and are relatively prime and <, thu the mutliplicative invere? mod exit. Furthermore the bae part (left of the `;') of an extended NS number

2 repreent the remainder modulo.. For uciently large n and moduli of imilar magnitude, the dierence beteen and ill be mall. Otherie the election of the m i from the m 0 i can be adapted in order to balance the relative ize ithout changing the rt property. 3. ultiplication of to bae NS number (in the range ) performed in the extended NS ill never overo becaue <. The reult can then be reduced to the bae range uing caling by. III. The Diviion Algorithm In thi ection e give a high level decription of our diviion algorithm; in the next ection e hall explain ho to ue baic NS operation to implement it. \DIVE" take to integer X ; 0 X < and ; X <, a input and return the quotient bx c, and the remainder X mod. Algorithm DIVE begin Input (X; ) Output (bx c; X mod ) Q bx ECIP ( ) c X? Q if < then return (Q; ) ele return (Q + ;? ) end. One might notice that other then being maller than, e do not imply any retriction on X and. DIVE make a call to ECIP (given belo), hich return the reciprocal b c of ith repect to. The folloing Lemma explain the neceity for the correction tep at the end of the algorithm. Lemma Algorithm DIVE i correct. Proof Let Z b c, i.e., Z + S, or Z (?S) ; here 0 S <. Then XZ X? X S. We note that X < and S < ; therefore X S <. Thu either XZ X or XZ X? In \ECIP" e ue Neton iteration to compute the reciprocal b c. Thi algorithm i imilar to the one in Aho, Hopcroft and Ullman []. But their divide and conquer approach, here the number of valid bit double in each recurive tep, i not uitable for NS, becaue here all reidue change imultaneouly. Still, the advantage of quadratic convergence i preerved in our algorithm. Applying the Neton iteration cheme Z i+ Z i? f(z i) f 0 (Z i ) ; to f(z i ) Z i? ith f 0 (Z i )? Z i recurion Z i+ Z i + Z i? Z i ; yield the Z i(? Z i ) In NS e have only integer diviion, o the nal verion become Zi (? Z i ) Z Z i+ Z i? i The algorithm for the reciprocal take an integer, ith <, a input and return b c Algorithm ECIP Input Output b c begin Z 0 Z hile Z 6 Z do Z Z Z bz (? Z ) c if? Z < then return Z ele return Z + end. The neceity for the correction tep at the end ill follo from Lemma 4. Lemma 3 to 5 ho the correctne of ECIP and indicate the number of iteration needed. The proof in the remainder of thi ection may be kipped ithout lo of undertanding. Lemma Subequent approximation atify the folloing inequalitie? Z i? Z i+ <? Z i+ + i? <? Z i + ()? Z i + i?? () Proof () follo from the baic inequalitie for the ceiling operator (X dxe < X + )? Z i+ Z? Z i + i <? Z i + Z i +? Z i + The ame argument i applied to prove the left ide of (). Alo the inductive proof for inequality () make ue of the

3 ame propertie of the ceiling operator? Z i+ + i?? <? Z i + i? +? (Z i + i?? ) Z i? l Z i m + i? Zi ( ) + + Z i ( )? (Z i + i?? ) + (Z i + i?? ) ( )? Z i + i?? Lemma 3 ECIP halt for any ith <. 3 Proof For Z and > 4 the iteration top ith Z Z 3 0; for < 34, it top at Z Z 3. In the cae of e can ho by induction on i that? Z i 0 for i ; herea? Z? 0 ;? Z i+? Z i 0 ; by the induction hypothei and the left ide of () in Lemma. Therefore Z i ; and Z i+ Z i? Z i Z i? Z i Z i Hence the equence i monotonically increaing, and the halting condition Z i+ Z i d Zi e ill eventually be reached. Lemma 4 Z i ill be ithin one quarter of after O(log( )) iteration. Proof For eae of argumentation, e chooe Z 33 and aume and uch that 33 <. For all other cae e need a contant number of iteration to compute b c. Expanding () of Lemma 3 iteratively don to Z yield? Z i + i?? No let i uch that <? Z? 33 i? i? (3) i? < i ; ie; i > log 33 + Then? 33 i? 4? Note that i? {z } < e?6 {z } < e?3 < i? 6 + ; and? x < x e for all x > 0, here e i Euler' bae of the natural logarithm. Finally, e multiply (3) by e 3 thu > Z i >? 33 i?!? e 3? 6 > {z } 0768 or >?? Z i + i?? ;? Z i < 4 >? Z i ; Let K uch that K < K+. Uing the right ide of () from Lemma, e get + <? Z i+ <? Z i 6 + < K? Lemma 5 If 0 <? Z i+ < K?3 +, it take O(loglog( )) tep to get ithin a dierence of to the exact quotient. Proof e prove by induction on j that? Z i+j < K?3j? +, here the given condition for Z i+ repreent the bae of the induction. By the induction hypothei and the right ide of () from Lemma, e get +? Z i+j+ <? Z i+j <?K K?3j? + + K?3j + 4?3j? + 4?K + < K?3j + 3

4 Hence eventually?z i+j < 3, and?z i+j+ <, by applying () one more time and oberving that < 33. From the condition K?3j + < 3, e can deduce the number of iteration needed at mot here K?3j < ; j > K3 ; or j > log (K3) ; K log ( ) < K + ; hence K blog ( )c; and j > log (blog ( )c3) IV. NS Implementation Here, e ho ho to implement DIVE and ECIP uing NS operation. Hoever, it i not the cope of thi paper to dicu hardare realization. Variou model of computation ere ued o far to derive the complexity of baic NS operation. ore recently, a thorough analyi for N C circuit a carried out by Davida and Lito [4]. Unfortunately converion ere left out. We ill ho in the Appendix that converion from NS to mixed radix repreentation can be implemented in depth O(log n) uing O(n ) NS proceor element; by \NS proceor element" e mean a circuit for arithmetic or boolean operation, uch a addition, multiplication or tet for equality, modulo any of the m i. For the folloing, e aume that all number are already in extended NS format. Otherie, e have to do \bae extenion" from bae NS to extended NS. Bae extenion The converion from bae NS to the extenion NS (or vice vera) amount to a converion to the aociated S repreentation hv ; ; v n i ; for X v i P i? given by [x ; ; x n ] ; ith ubequent evaluation of the um v i P i?! mod m j (v i mod m j ) (P i? mod m j )! mod m j ; for every modulu m j ith n + j n. The contant P i? mod m j can be kept in a lookup table, hile the product v i P i? are computed in parallel on n n NS proceor element. Finally the ummation mod m j i done in binary tree ith O(log n) depth. Tet for \" and \6" In ECIP, the halting condition for the Neton iteration i \Z Z ". To integer X and in NS repreentation are equal if and only if they agree in each component x i y i. The overall reult can be obtained from the individual tet by performing AND operation on a ingle bit in a binary tree ith depth O(log n). If neceary, the reult can be propagated back to the proceor element uing the tree in the invere direction. In the tet for \not equal", AND i replaced by O. Diviion by (Scaling) In both DIVE and ECIP the major operation i integer diviion by. Let U be the intermediate reult in extended NS, before diviion by (U X ECIP ( ) or U Z i (? Z i )), ith reidue [u ; ; u n ; u n+ ; ; u n ]. The left part of thi vector repreent the remainder U mod. By performing bae extenion it can be converted into extenion NS repreentation. After ubtracting U mod from U, e multiply by? mod to get the quotient Q in the extenion NS. Q ha to be extended to bae NS by another bae extenion. The reidue of? mod (denoted by m n+ ; ; m n ) can be precomputed. Thu, the equence of operation i a follo [u ; ; u n ; u n+ ; ; u n ] U mod! bae extenion ; u 0 n+; ; u 0 n ] ; t n+ ; ; t n ]? mod ; m n+ ; ; m n ] Q ; q n+ ; ; q n ] bae extenion [q ; ; q n ; q n+ ; ; q n ] Although U i greater than, the range of the extenion NS i ucient becaue the reult Q ill alay be maller than. An alternate method ork directly in S, avoiding the ubtraction and multiplication in the extenion NS. Hoever, becaue the mixed radix converion ue O(n ) proceor element, the hardare requirement increae almot by a factor of 4, hen n reidue have to be converted intead of n. Let U have the extended mixed radix repreentation hv ; ; v n ; v n+ ; ; v n i ; U v i P i? By denition, the product P i ith n i < n are multiple of P i S i ; here S n ; and S i m n+ m i for n < i < n. We no plit the um into U v i P i? + in+ v i S i? + Q ; 4

5 here U mod and Q buc. Q i the deired reult for the quotient, and ha to be converted back to extended NS by evaluating the um for each modulu m j ; ( j n), in the ame ay a decribed for bae extenion. Comparion For the correction tep at the end of both algorithm, one comparion i neceary. Davida and Lito [4] decribe an N C circuit for comparion of to NS integer ith ize O(b ) and depth O(log b), here b i the input ize in bit (dlog e). Here, e ant to ho that ome of the functional unit for diviion by can alo be ued for comparion (and overo detection). Firt e note that the number e ant to compare are trictly maller than. If e ubtract to uch (nonnegative) integer X and the reult Z ill \undero" henever Z? X < 0 (X > ), hich i equivalent to Z > in extended NS repreentation. After performing a bae extenion from bae NS to extenion NS, e tet the reult for equality ith the reidue z n+ ; ; z n. If all component agree, e return \X ", otherie \X > ". While uing the econd method for diviion by, e convert Z to (extended) mixed radix repreentation, and tet ether every component v i 0 for n < i n. From Lemma 4, Lemma 5, and the dicuion of thi ection e conclude that Theorem The diviion algorithm can be implemented in depth O(log n log( )) ith O(n ) NS proceor element. Speed up eplacing z in ECIP by a tart value hich i \cloer" to the reult, the number of iteration can be reduced to O(loglog( )). In order to achieve thi, e chooe the nearet poer of to by comparing in parallel againt b K c for K ; ; N, here N < N+. K i obtained by adding up the reult of thee comparion (0 or ) in a binary tree of height log N log blog c. The contant b K c can be precomputed. Lemma 6 With tart value Z K, here b K+ c < b K c ; ECIP terminate after O(loglog( )) iteration. Proof If bc < then b c, and e are done. For bc, e rt note that (for the choen K ) K+ K+ < + K+ K K ; thu Z K < K+ Hence? Z < ; and by Lemma? Z < 4 + Finally? Z 3 < < K?3 + (uing ). According to Lemma 4, e obtain the reult in O(loglog( )) tep from here. With thi modication, e can tate the overall complexity a Theorem The better tart value reduce the depth of the diviion algorithm to O(log n loglog( )+loglog ), increaing the number of proceor element to O(n log ). Proof Lemma 6 and the caling operation contribute log n loglog( ) to the depth. The comparion tep at the beginning can be implemented in S ith n N proceor element and depth O(log n + log N). The converion from NS to S ue O(n ) proceor and ha depth O(log n). Becaue N blog c > n, log ill dominate n. V. Concluding emark Our NS diviion algorithm i imple, and robut for a ide range of tart value. Although not being the optimum regarding circuit depth, it oer a balance beteen pace and time complexity. High level decription and analyi of the algorithm give the advantage of modular implementation. Any improvement in baic NS operation can eaily be incorporated. Future invetigation hould concentrate on reducing the depth of the caling operation (diviion by ). eference [] Aho, A. V., Hopcroft, J. E., and Ullman, J. D., The Deign and Analyi of Computer Algorithm; Addion and Weley, eading,.a., 974. [] Banerji, D. K., Cheung, T.., and Ganean, V., \A highpeed diviion method in reidue arithmetic," Proc. 5th. Symp. on Computer Arithmetic, pp. 58{64, 98. [3] Chren, W. A., Jr., \A ne reidue number ytem diviion algorithm," Computer ath. Appl., vol. 9, pp. 3{9, 990. [4] Davida, G. I. and Lito, B., \Fat parallel arithmetic via modular repreentation," SIA J. Comput., vol.0, pp. 756{765, 99. [5] Gamberger, D., \Ne approach to integer diviion in 5

6 µ µ µ,,,n * * *n µ, µ,n * *n µ n,n n *n v Adder Tree Adder Tree Adder Tree -lookahead correction tep v v n Overall Figure the multiplication tep reidue number ytem," Proc. 0th. Symp. on Computer Arithmetic, pp. 84{9, 99. [6] Huang, C. H., \A fully parallel mixed-radix converion algorithm for reidue number application," IEEE Tran. Comput., vol. 3, pp. 398{40, 983. [7] Leighton, F. T., Introduction to Parallel Algorithm and Architecture Array, Tree & Hypercube; organ Kaufmann Publ., San ateo, California, 99. [8] Lu, i and Chiang, Jen-Shiun, \A novel diviion algorithm for the reidue number ytem," IEEE Tran. Comput., vol. 4, pp , 99. [9] Szabo, N. S. and Tanaka,. I., eidue Arithmetic and it Application to Computer Technology, cgra-hill, Ne ork, N.., 967. Appendix ixed adix Converion One of the frequently cited reult for converion from NS to mixed radix repreentation i Huang [6]. Unfortunately, ith the kind of carry-pipelining outlined there, it i not poible to reach the claimed depth of O(log n) in term of NS proceor element. In thi ection e ho that, ith an additional carry lookahead correction tep in the mixed radix ytem (S), thi complexity can actually be achieved. Alo, e ho ho to incorporate multiplier intead of lookup table (hich ill be rather huge for large moduli). We retrict ourelve here to converion from either bae or extenion NS to the correponding S. We ill dicu the neceary adjutment for extended NS at the end of the ection. Given X in the bae NS (or extenion NS repectively) by it reidue [x ; ; x n ], e ant to nd it repreentation in the aociated S hv ; ; v n i. By virtue of the Chinee remainder theorem, e have X? xi? i mod m i i! mod v i P i? here i m i and P 0 ; P i m m i ; for 6

7 + + i i c c + + i i c c + + i i c c + i Sum S i Accumulated carry C i Figure an adder tree i n. The contant i? mod m i can be precomputed and tored directly in the NS? proceor element, herea the computation of i x i i? mod mi require one NS multiplication. (By uing the econd form of the Chinee remainder theorem X P n x i ^ i mod, ith contant ^i ( i? mod m i ) i, thi multiplication could be aved. Hoever, the overall carry ould become coniderably larger in thi cae). Finally e have to evaluate the hole um in S. Firt e look at the S repreentation of the contant i. It can be eaily veried that all S coecient ith index j < i are zero becaue i i a product of moduli. We denote the nonzero coecient for i j n by i;j. Therefore the S repreentation become ^ h ; ; ; ; ; ;n? ; ;n i ^ h 0; ; ; ; ;n? ; ;n i 3 ^ h 0; 0; ; 3;n? ; 3;n i n ^ h 0; 0; ; 0; n;n i No e ue n(n+) modular multiplier to compute the product i i;j in parallel. In Figure, modulo m i multiplier are repreented by \ i ". Each proceor element generate a remainder i i;j mod m i a ell a a carry b i i;j m i c. Both value are paed in regiter () to the next tage. Thee intermediate value have to be added up, rt in each column modulo m i, and then all reult acro in one carry-lookahead S addition. The modulo m i adder are arranged in binary tree of height dlog ie (ee Figure ). Each proceor element (denoted by \+ i ") compute the um () modulo m i of it input -line and update the accumulated carry (c). Finally, e get the um S i and the accumulated carry C i. The lat column of Figure i only needed if e are to compute the overall carry (e.g., for converion from extended NS to S); normal integer addition i ued in thi cae. We aume that the moduli are in order m < m < < m n. In the ith column, e have to add up i? number modulo m i. Each of the i? addition can reult in a carry of at mot. It can eaily be veried that m i+ i alay greater than i? (in the \ort" cae, e have m and m 3 >? 0). Therefore, the accumulated carry from each adder tree i alay maller than the modulu of the next column, o e need only one carry-lookahead S addition to get the reult j C C 3 C n? j C n + S j S 3 S 4 S n j S n+ v j v 3 v 4 v n j carry S n+ i the um of carrie from the nth column. v a already computed in the multiplication tep, and v i actually equal to S. -lookahead addition in S ue the ame method a in the binary ytem. We follo the notation of Leighton [7]. Addition in S pan over i 3; ; n (indicated by the `j' delimiter). For the lat column it i normal integer addition corrected by the carry from the S addition. For i 3; ; n e compute the um C i? + S i rt, and compare it againt m i? 8 < if C i? + S i 8 < a carry i 9 < ; m i? then > topped propagated generated () (p) (g) Parallel prex applied to the, p and g value give u the correction for each component, a ell a the carry forarded to the lat column. C i? +S i and the correction value (0 or ) are added together modulo m i. The parallel prex operation ha depth O(log n). Converion from extended NS to it aociated S need pecial attention. Becaue the mallet modulu m n+ of the extenion NS i in general maller than the larget modulu m n of the bae NS, the requirement m i > i? (in particular m n+ > n +,, 7

8 m n > 4n? ) i more retrictive. If ome of the moduli are too mall to atify all inequalitie, the ummation of the product and the carrie modulo m i, ith n + i n, can be done eparately, requiring additional carry-lookahead correction tep() at the end. 8