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3 RNS to Binary Conversion Based on New CRT-I, New CRT-II, Mixed-Radix CRT and New CRT-III

3 RNS to Binary Conversion Based on New CRT-I, New CRT-II, Mixed-Radix CRT and New CRT-III

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96



5 RNS to Binary Conversion



Thus, Y can be first found and appending x2 as LSBs, we can obtain X.

Wang et al. [14] have suggested computation of Y in (5.24) as A ỵ 2n Bị 2n

2 1

where

"



x1 ỵ x10 ẩ x30 ị2n ỵ 2n 1 x3 ị ỵ 2n 1

Aẳ

2



#

5:25aị



and

"



x1 ỵ x10 ẩ x30 ị2n ỵ x3 ỵ 22n 1 x2 ị

Bẳ

2



#

5:25bị



where x10 and x30 are the LSBs of x1 and x3, respectively. The value A can be

computed using a 2-input adder to yield the sum and carry vectors A1 and A2.

Similarly, B can be estimated to yield the sum and carry vectors B1 and B2 and a

carry bit using a three-input n-bit adder. Next, Y can be obtained from A and B using

a 2n-bit adder (Converter I) or n-bit adders to reduce the propagation delay.

Two solutions for n-bit case have been suggested denoted as Converter II and

Converter III.

Bi and Gross [34] have described a Mixed-Radix Chinese Remainder Theorem

(Mixed-Radix CRT) for RNS to binary conversion. The result of RNS to binary

conversion can be computed in this approach for an RNS having moduli {m1, m2,

. . ., mn} with residues (x1, x2, . . ., xn) as

"

#

 1 x1 ỵ 2 x2 ỵ 3 x3 





X ẳ x1 ỵ m1 j 1 x1 ỵ 2 x2 jm2 ỵ m1 m2 



m2

m

"

#3

5:26aị

 1 x1 ỵ 2 x2 ỵ 3 x3 ỵ ỵ n xn 



ỵ þ m1 m2 . . . mnÀ1 



m2 m3 . . . mnÀ1

mn

where



M1

γ1 ¼





1

À1

M 1 m1

m1



and γ i ¼



 

M

1

:

m1 mi Mi mi



ð5:26bÞ



Note that the first two terms use MRC and other terms use CRT like expansion.

The advantage of this formulation is the possibility for parallel computation of

various MRC digits enabling fast comparison of two numbers at the expense of

hardware since many terms in the numerators for expressions for several Mixed

Radix digits and division by product of moduli and taking integer value are

cumbersome except for special moduli. The topic of comparison using this technique is discussed in Chapter 6. An example will be illustrative.



5.4 RNS to Binary Converters for Other Three Moduli Sets



97



Example 5.7 We wish to find the number corresponding to residues (1, 2, 3, 4) in

the RNS {3, 5, 7, 11}. Wecancompute asin CRT,

3 ¼ 165,

 M1 ¼

 385,M2 ¼ 231,

 M

1

1

1

1

as

¼ 1,

¼ 1,

¼ 2,

M4 ¼ 105 and various

Mi mi

385 3

231 5

165 7





1

¼ 2. Next, we compute γ 1 ¼ 128, γ 2 ¼ 77, γ 3 ¼ 110, γ 4 ¼ 70. Thus,

105 11

X can be computed as

"

X ẳ 1 ỵ 3 128 ỵ 154ị5 ỵ 15

"





128 ỵ 154 ỵ 330 ỵ 280

35



#



128 ỵ 154 þ 330

5



#

þ 105

7



11



¼ 1 þ 3 Â 2 þ 15 3 ỵ 105 3 ẳ 367



New CRT III [35, 36] can be used to perform RNS to binary conversion when the

moduli have common factors. Considering two moduli m1 and m2 with common

factors d, and considering m1 > m2, the decoded number corresponding to residues

x1 and x2 can be obtained as



X ẳ x1 ỵ m1







1

x 2 x 1 Þ

m1 =d

d

m2 =d



ð5:27Þ



As an illustration, consider the moduli set {15, 12} with d ¼ 3 as a common factor

and given residues (5, 2). The decoded number can be obtained from (5.27) as

X ẳ 5 ỵ 15



 



1 2 5ị

ẳ 50:

5

3

4



We will later consider application of this technique for Reverse conversion for an

eight moduli set.



5.4



RNS to Binary Converters for Other Three Moduli Sets



Premkumar [37], Premkumar et al. [38], Wang et al. [39], and Globagade et al. [40]

have investigated the three moduli set {m1, m2, m3} ¼ {2n + 1, 2n, 2n À 1}. The

reverse converter for this moduli set based on CRT described by Premkumar [37]

uses the expressions



98



5 RNS to Binary Conversion



&

Xẳ



'

m m 

M m2 m3 

1 2



x1 ỵ

x3 m1 m3 x2 mod M

2

2

2



for x1 ỵ x3 ị odd

5:28aị



and

Xẳ



nm m 

o

m m 

2 3

1 2

x1 ỵ

x3 m1 m3 x2 mod M

2

2



for x1 ỵ x3 ị even 5:28bị



where M ¼ 2n(4n2 À 1).

Note that the output of the adder computing the value inside the brackets needs

to be tested and based on the sign, M has to be added or subtracted once.

The hardware implementation needs three 2k-bit  k-bit multipliers where

k ¼ log2(2n + 1) and a four-input 3k-bit adder. Premkumar et al. [38] suggested

simplification which needs one 2k-bit  k-bit multiplier and one k-bit  k-bit multiplier and 7 or 9 adders in Architecture A and B, respectively. They divide both

sides of CRT expression by m2 and find the integer part as

"



# 

X

x1 x3 



ẳ nx1 ỵ x3 2x2 ị ỵ



m2

2

m1 m3



both x1 , x3 odd or both even

5:29aị



and

"



# 

X

x1 x3 ỵ m1 m3 



x1 even, x3 odd or vice-versa:

ẳ nx1 ỵ x3 2x2 ị ỵ



m2

2

m1 m3

5:29bị



Note that in this

m1 ẳ 2n 1, m2 ¼ 2n and m3 ¼ 2n + 1. The final result is

j case,

k

X

given by X ẳ m2 m2 ỵ x2 . The authors suggest a high-speed version as well as a

cost-effective version.

Wang et al. [39] have given another technique for reverse conversion using the

formula based on new CRT II,

X ẳ x2 ỵ 2nfx2 x3 ị ỵ x1 2x2 ỵ x3 ịn2n ỵ 1ịgmod2n ỵ 1ị2n 1ịị

5:30ị

which needs one 2k-bit  k-bit multiplier and one k-bit  k-bit multiplier and few

adders. Note that in this case, m1 ¼ 2n À 1, m2 ¼ 2n and m3 ¼ 2n + 1. More recently,

Gbolagade et al. [40] have suggested computing X as



5.5 RNS to Binary Converters for Four and More Moduli Sets





x ỵ x





 1



3

x2 

X ẳ m2 x2 x3 ị ỵ x2 ỵ m3 m2 

2

m



99













5:31ị



1 M



Note that in this case, m1 ¼ 2n À 1, m2 ¼ 2n and m3 ¼ 2n + 1. This needs at most one

corrective addition of M. The critical path has been shown to be less than Wang

et al. converter [37] with reduced hardware complexity.

Premkumar [41, 42], Wang et al. [39] and Gbolagade [43] have considered

another moduli set {2n, 2n + 1, 2n + 2} which has 2 as a common factor and hence

half the dynamic range compared to the moduli set {2n + 1, 2n, 2n À 1}. It may be

remarked that the moduli sets {2n + 1, 2n, 2n À 1} and {2n, 2n + 1, 2n + 2} are not

attractive compared to powers of two related moduli sets since the hardware needed

has quadratic dependence on the bit size of the moduli.

Reverse converters for the moduli set {2k, 2k À 1, 2kÀ1 À 1} have also been

described [44–47]. The design due to Hiasat and Abdel-Aty-Zohdy [44] was

based on CRT. Denoting m1 ¼ 2k, m2 ¼ 2k À 1,j andk m3 ¼ 2kÀ1 À 1, the authors



X

M3 where M3 ¼ M/m3. Wang

et al. [45, 46] have used New CRT II and have shown that the conversion time

can be reduced whereas area is increased. Ananda Mohan [47] has suggested both

CRT and MRC-based converters. The CRT-based converter has reduced conversion time and uses ROM. On the other hand, the MRC-based converter has reduced

area but higher conversion time.

The moduli set {22n À 1, 22n, 22n + 1} has been suggested by Ananda Mohan

[48, 49] for which using CRT, cost-effective as well as high-speed converters have

been described. Note that the moduli have word lengths of n bits, 2n bits and 2n + 1

bits. The dynamic range is 5n + 1 bits. Another moduli set with (3n + 1)-bit dynamic

range has also been explored {2n, 2n À 1, 2n+1 À 1} [50] using CRT as well as MRC

techniques. The multiplicative inverses needed in the case of MRC technique are

very simple. The CRT-based converter needs modulo (2n À 1)(2n+1 À 1) reduction

after a CPA which has been suggested to be realized by using ROMs by looking at

the MSBs and subtracting the appropriate residue. Thus, one converter using ROM

and two converters not using ROM have been suggested. This moduli set has the

advantage that due to absence of modulus (2n + 1), the multiplication and addition

operations for all moduli channels can be simpler.



start with CRT and estimate X mod M3 and



5.5



RNS to Binary Converters for Four and More

Moduli Sets



Some reverse converters of four moduli sets [51–54] are extensions of the converters for the three moduli sets. These use the optimum converters for the three

moduli set M1 {2n À 1, 2n, 2n + 1} and use MRC to get the final result to include the

fourth modulus 2n+1 À 1, 2nÀ1 + 1, 2nÀ1 À 1, 2n+1 + 1, etc.



100



5 RNS to Binary Conversion



The reverse converter due to Vinod and Premkumar [51] for the moduli set

n

n

n n+1

{m1, m2, m3, m4} ¼ {2

j Àk1, 2 À+ 1, 2 , 2 Á À 1} uses CRT but computes the higher

Mixed Radix Digit MX mod 2nỵ1 1 where X is the desired decoded number

4



using the three moduli

and Mi ¼ M/mi. On the other hand, X mod M4 is computed

j k

X

RNS to binary converter. Next, X is computed as M M4 ỵ x4 .

4



The reverse converter due to Bhardwaj et al. [52] for the moduli set

j {m

k 1, m2, m3,



m4} ¼ {2n À 1, 2n + 1, 2n, 2n+1 + 1} uses CRT but computes first E ¼



X

. Note that

2n

E can be obtained by using CRT on the four moduli set and subtracting the residue

r3 and dividing by m3. However, the multiplicative inverses needed in CRT are

quite complex and hence, E1 and E2 are estimated from the expression for E. Next,

from E1 and E2 using CRT, E can be obtained:



Â

Ã

E1 ¼ jEj 2n ẳ 2n1 2n ỵ 1ịr 1 2n r 2 À 2nÀ1 ð2n À 1Þr 3 2n

2 À1

2 À1

E2 ẳ j E j



2nỵ1 ỵ1



ẳ ẵ2r 2 2r 4



5:32aị

5:32bị



2nỵ1 ỵ1



Ananda Mohan and Premkumar [53] have suggested using MRC for obtaining

E from E1 and E2.

Ananda Mohan and Premkumar [53] have given an unified architecture for RNS

to binary conversion for the moduli sets {2n À 1, 2n + 1, 2n, 2n+1 À 1} and {2n À 1, 2n

+ 1, 2n, 2n+1 + 1} which uses a front-end RNS to binary converter for the moduli set

{2n À 1, 2n + 1, 2n} and then uses MRC to include the fourth modulus. Both

ROM-based and non-ROM-based solutions have been given.

Hosseinzadeh et al. [55] have suggested an improvement for the converter of

Ananda Mohan and Premkumar [53] for the moduli set {2n À 1, 2n + 1, 2n, 2n+1 À 1}

for reducing the conversion delay at the expense of area. They suggest using (n + 1)bit adders in place of (3n + 1)-bit CPA to compute the three parts of the final result.

Theydo not perform the final addition of the output of the multiplier evaluating



1

ðx4 Xa ị nỵ1 where Xa is the decoded output corresponding the

2 1

Xa 2nỵ1 1

moduli set {2n 1, 2n + 1, 2n} but preserve as two carry and sum output vectors and

compute the final output.

Sousa et al. [56] have described an RNS to binary converter for the moduli set

{2n + 1, 2n À 1, 2n, 2n+1 + 1}. They have used two-level MRC. In the first

level, reverse conversion using MRC for moduli sets {x1, x2} ¼ {2n + 1, 2n À 1}

and {x4, x3} ¼ {2n+1 + 1, 2n} is performed and the decoded words

 X12, X34 are

obtained. Note that the various multiplicative inverses are



nÀ1

1

x1 modx2 ¼ 2 ,



nÀ3

nÀ1

 





2

X

X

2iỵ1

1

1



1

and



m





2



22iỵ2 . Since the archimodx

mod

m

3

1 2

x4

m3 m4



iẳ0



iẳn1

2



tecture uses MRC, it can be pipelined. The multiplication with multiplicative

inverses mod (2n À 1), mod 2n, and mod (22n À 1) can be easily performed.



5.5 RNS to Binary Converters for Four and More Moduli Sets



101



The resulting area is more than that of Ananda Mohan and Premkumar converter

[53], whereas the conversion time is less.

Cao et al. [54] have described reverse converters for the two four moduli sets

{2n + 1, 2n À 1, 2n, 2n+1 À 1} and {2n + 1, 2n À 1, 2n, 2nÀ1 À 1} both for n even.

They use a front-end RNS to binary converter due to Wang et al. [14] for the three

moduli set to obtain the decoded word X1 and use MRC later to include the fourth

modulus m4 (i.e. (2n+1 À 1) or (2nÀ1 À 1)). The authors suggest three stage and

four stage converters which differ in the way the MRC in second level is

performed. In the three-stage converter considering the first moduli set, the

second stage computes

!!





and



the



third

!



1

À

Á

2n 22n À 1



stage

ẳ2



x4 X1 ị



computes

nỵ2



1







2n 22n 1



2nỵ1 1





X ẳ X1 þ 2n 22n À 1 Z.



where S ¼



Noting



that



À10



, the authors realize Z as

2 1



 nỵ2



1

2

10

x4 X1 ị

ẳ SQị nỵ1

Zẳ

2 1

3

3

2nỵ1 1

nỵ1



5:33aị



3



5:33bị



 



 nỵ2



1

2

10

, Q ẳ x 4 X 1 ị

. Note that S can be

3 2nỵ1 1

3

2nỵ1 1



realized as

Sẳ



 

1

ẳ 20 ỵ 22 ỵ 24 ỵ ỵ 2n :

3 2nỵ1 1



Thus, Z can be computed as sum of shifted and rotated versions of Q available in

carry save form using a tree of CSA with end-around-carry. In the four-stage

converter, the sum and carry vectors realizing Q are first added in a mod (2n+1 À 1)

adder and then multiplied with S realized by summing shifted and rotated terms.

Same technique has been used for the other moduli set as well.

The reverse converters for the four moduli set {2n À 1, 2n + 1, 2n À 3, 2n + 3}

have also been described which use ROMs and combinational logic

[48, 57–59]. The designs in [48, 57, 58] consider in the first level, two 2-moduli

sets {2n À 3, 2n + 1} and {2n + 3, 2n À 1} to compute the decoded numbers Xa and Xb

respectively using MRC. Sheu et al. [57] use a ROM-based approach. In the design

in [58], Montgomery algorithm is used to perform the multiplication with

 multi

plicative inverse needed in MRC. This takes advantage of the fact that m12 mod





À1Á

ðx 1 À x 2 Þm 1

n

n

m1 ¼ 4 modm1 where m1 ¼ 2 À 3 and m2 ¼ 2 + 1. Thus,

modm1

4



implies adding a multiple of m1 to ðx1 À x2 Þm1 to make the two LSBs zero so that



102



5 RNS to Binary Conversion



 

division by 4 implies ignoring the two LSBs. In the case of computation of Xb, m13

ÀÁ

modm4 ¼ 14 modm4 ¼ 2nÀ2 where m3 ¼ 2n + 3 and m4 ¼ 2n À 1. The multiplication with 2nÀ2 mod (2n À 1) can be carried out in a simple manner by

 bit rotation





of ðx3 À x4 Þm4 . In the case of MRC in the second level, note that m31m4 mod





1

ðm1 m2 Þ ẳ nỵ2

modm1 m2 ị enabling Montgomery technique to be used easily.

2

In [58], MRC using ROMs and CRT using ROMs also have been explored. In

MRC techniques, modulo subtractions are realized using logic, whereas multiplication with multiplicativeinverse

 is carried out using ROMs. In the CRT-based

1

values are stored in ROM. Carry-save-adder

method, the various Mi

M i mi

followed by CPA and modulo reduction stage are used to compute the decoded

result.

Jaberipur and Ahmadifar [59] have described an ROM less adder-only reverse

converter for this moduli set. They consider a two-stage converter. The first stage

performs mixed radix conversion corresponding to the two pairs of moduli {2n À 1,

2n + 1} and {2n À 3, 2n + 3} to obtain residues corresponding to the pair of composite moduli {22n À 1, 22n À 9}. The multiplicative inverses needed are as follows:















1

1

n1



2

,

ẳ 2n3 ỵ 2n5 ỵ þ 23 þ 2 for n even and

n

n

n

n

2 þ 3 2 3

 2 1 2 ỵ1





n3



1

1

n5

2

0

for

n

odd,



2



2











2



2

ẳ 22n3 :

2n þ 3 2n À3

22n À 9 22n À1



The decoded words in the first and second stages can be easily obtained using

multi-operand addition of circularly shifted words.

Patronik and Piestrak [60] have considered residue to binary conversion for a

new moduli set {m1, m2, m3, m4} ¼ {2n + 1, 2n, 2n À 1, 2nÀ1 + 1} for n odd. They

have described two converters. The first converter is based on MRC of a two moduli

set {m1m2m3, m4}. This uses Wang et al. converter [12] for the three moduli set to

obtain the number X1 in the moduli set {m1, m2, m3}. The multiplicative inverse

needed in MRC is

0n3

1

2 1

X

ẳ k1 ẳ @

22iỵ1 ỵ 1A



!

1





2n 22n 1



2



n1



ỵ1



5:34ị



iẳ0



Note that since the lengths of residues corresponding to the moduli m1m2m3 and

m4 are different, the operation (x4 À X1) mod (2nÀ1 + 1) needs to be carried out using

periodic properties of residues. The multiplication with the multiplicative inverse in

(5.34) needs circular left shifts, one’s complementing of bits arriving in LSBs due to

circular shift and addition of all these modified partial products with a correction

term using several CSA stages. Note that mod (2nÀ1 + 1) addition needs correction



5.5 RNS to Binary Converters for Four and More Moduli Sets



103



to cater for inverting the carry and

À addingÁ in the LSB position. The number of

partial products can be seen to be nÀ3

2 þ 2 . The final computation of X 1 þ m1 m2

m3 ðÀk1 ðx4 À X1 ÞÞm4 can be rearranged to take advantage of the fact that LSBs of

the decoded word are already available as x3.

The second converter uses two-stage conversion comprising of moduli sets

{m1m2, m3m4} using MRC. The numbers corresponding to moduli sets m1m2 and

m3m4 are obtained using CRT and MRC respectively in the first stage. The various

multiplicative inverse used in CRT and MRC in this stage are as follows:

!

1



n

2 ỵ1



!

1



n

2 1





2



n



1



!

ẳ2

2



n



ỵ1



n1



1



, n1

2

ỵ1



ẳ 2n1 ỵ 1 5:35aị

2



n



The multiplicative inverse needed in MRC in the second stage is

!

1





2n 2n1 ỵ 1



22n 1



0 0 n3

11

2

X





1

ẳ @ n@

22iỵ2 ỵ 22iỵnỵ2 ỵ 2AA

2 iẳ0



5:35bị

2



2n



1



The multiplication with this multiplicative inverse mod (22n 1) can be obtained

by using a multi-operand carry-save-adder mod (22n À 1) which can yield sum and

carry vectors RC and RS. Two versions of the second converter have been presented

which differ in the second stage.

Didier and Rivaille [61] have described a two-stage RNS to binary converter for

moduli specially chosen to simplify the converter using ROMs. They suggest

choosing pairs of moduli with a difference of power of two and difference between

products of pairs of moduli being powers of two. Specifically, the set is of the type

È

É

fm1 ; m2 ; m3 ; m4 g ẳ m1 , m1 ỵ 2p1 , m3 , m3 ỵ 2p2 such that m1m2 m3m4 ẳ 2pp

where pp is an integer. In the first stage, the decoded numbers corresponding to

residues of {m1, m2} and {m3, m4} can be found and in the second stage, the

decoded number corresponding to the moduli set {m1m2, m3m4} can be found. The

basic converter for the two moduli set {m1, m2} can be realized using one addition

without needing any modular reduction. Denoting the residues as (r1, r2), the

decoded number B1 can be written as B1 ẳ r 2 ỵ r 1 r 2 , 0Þ where the second

term corresponds to the binary number corresponding to (r1 À r2, 0). Since r1 À r2

can be negative, it can be written as a α-bit two’s complement number with a sign

bit S and (α À 1) remaining bits. The authors suggest that the decoded number be

obtained using a look-up table T addressed by sign bit and p LSBs where

m2 À m1 ¼ 2p and using addition operation as follows:





ỵ T signr 1 r 2 Þ, LSBðr 1 À r 2 ÞpÀ1

B1 ¼ r 2 þ m2  MSBðr 1 À r 2 ÞαÀ1

ð5:36Þ

0

p

Some of the representative moduli sets are {7, 9, 5, 13}, {23, 39, 25, 41}, {127,

129, 113, 145} and {511, 513, 481, 545}. As an illustration, the implementation for

the RNS {511, 513, 481, 545} needs 170AFA, 2640 bits of ROM and needs a



104



5 RNS to Binary Conversion



conversion time of 78ΔFA + 2ΔROM where ΔFA is the delay of a full adder and

ΔROM is ROM access time.

We next consider four moduli sets with dynamic range (DR) of the order of 5n

and 6n bits. The four moduli set {2n, 2n À 1, 2n + 1, 22n + 1} [62] is attractive since

New CRT-I-based reduction can be easily carried out. However, the bit length of

one modulus is double that of the other three moduli. Note that this moduli set can

be considered to be derived from {22n À 1, 22n, 22n + 1} [48, 49].

The reverse converters for the moduli set {2n À 1, 2n + 1, 22n+1 À 1, 2n} with DR

of about (5n + 1) bits and {2n À 1, 2n + 1, 22n, 22n + 1} with a DR of about 6n bits

based on New CRT II and New CRT I respectively have been described in [63]. In

the first case, MRC is used for the two two moduli sets {m1, m2} ¼ {2n, 22n+1 À 1}

and {m3, m4} ¼ {2n + 1, 2n À 1} to compute Z and Y. A second MRC stage computes

X from Y and Z:





Z ẳ x1 ỵ 2n 2nỵ1 x2 x1 ị 2nỵ1

1

2





Y ẳ x3 ỵ 2n ỵ 1ị 2n1 x4 x3 ị 2n 1





X ẳ Z ỵ 2n 22nỵ1 1 2n Y Z ÞÞ 2n

2 À1



ð5:37aÞ

ð5:37bÞ

ð5:37cÞ



Due to the modulo reductions which are convenient, the hardware can be simpler.

In the case of the moduli set {m1, m2, m3, m4} ¼ {2n À 1, 2n + 1, 22n, 22n + 1},

New CRT-I has been used. The decoded number in this case is given by

À

À

Á

À

Á

Á

X ¼ x1 ỵ 22n 22n x2 x1 ị ỵ 22n1 22n ỵ 1 x3 x2 ị ỵ 2n2 22n þ 1 ð2n þ 1Þðx4 À x3 Þ



24n À1



ð5:38Þ

Zhang and Siy [64] have described an RNS to binary converter for the moduli set

{2n À 1, 2n + 1, 22n À 2, 22n+1 À 3} with a DR of about (6n + 1) bits. They

consider two-level MRC using the two moduli sets {m1 ¼ 2n À 1, m2 ¼ 2n + 1}

and {m3 ¼ 22n À 2, m4 ¼ 22n+1 À 3}. The multiplicative inverses are very simple:







 





1

1

1

nÀ1

¼2 ,

¼ 1,

¼1

m2 m1

m4 m3

m3 m4 m1 m2



ð5:39Þ



Sousa and Antao [65] have described MRC-based RNS to binary converters for

the moduli sets {2n + 1, 2n À 1, 2n, 22n+1 À 1} and {2n À 1, 2n + 1, 22n, 22n+1 À 1}.

They consider in the first level {x1, x2} ¼ {2n À 1, 2n + 1} and {x3, x4} ¼ {2n(1+α),

22n+1 À 1} where α ¼ 0,1 correspond to the two moduli sets to compute X12 and

X34 respectively.

The  multiplicative

inverses in the first level are







1

1

ẳ 2n1 ,

ẳ 21ỵịn 1, and in the second

2nỵ1

n1ỵị

2n ỵ 1 2n 1



1

2

2









1

1

n



2

for





0

and

ẳ 1 for ẳ 1.

level are 3nỵ1

2n 22n 1

2

24nỵ1 2n 22n 1



5.5 RNS to Binary Converters for Four and More Moduli Sets



105



Note that all modulo operations are mod (2n À 1), 2(1+α)n and 22n À 1 which are

convenient to realize. The authors use X12 and X34 in carry save form for computing

ðX12 À X34 Þ 2n thus reducing the critical path.

2 À1

Stamenkovic and Jovanovic [66] have described a reverse converter for the four

moduli set {2n À 1, 2n, 2n + 1, 22n+1 À 1}. They have suggested exploring the

24 possible orderings of the moduli for being used in MRC so that the multiplicative

inverses are Ỉ1 and 2 nÀ1 . The recommended ordering is {2 2n+1 À 1, 2n, 2 n + 1,

2n À 1}. This leads to MRC using only subtractors and not needing modulo

multiplications. They have not, however, presented the details of hardware requirement and conversion delay.

The reverse converter for the five moduli set [67] {2n À 1, 2n, 2n + 1, 2n+1 À 1,

nÀ1

2

À 1} for n even uses in first level the converter for four moduli set {2n À 1,

n

n

2 , 2 + 1, 2n+1 À 1} due to [54] and then uses MRC to include the fifth modulus

(2nÀ1 À 1).

Hiasat

n [68] has described reverse converters for two

o five moduli sets based on

nỵ1



nỵ1



CRT 2n , 2n 1, 2n ỵ 1, 2n 2 2 ỵ 1, 2n ỵ 2 2 ỵ 1 when n is odd and n ! 5 and

n

o

nỵ1

nỵ1

2nỵ1 , 2n 1, 2n ỵ 1, 2n 2 2 ỵ 1, 2n ỵ 2 2 ỵ 1 when n is odd and n ! 7. Note



that this moduli set uses factored form of the two moduli (22n À 1) and (22n + 1) in

the moduli set {2n, 22n À 1, 22n + 1}. The reverse conversion procedure is similar to

Andraros and Ahmad technique [4] of evaluating the 4n MSBs since n LSBs of the

decoded result are already available. The architecture needs addition of eight

4n-bit words using 4n-bit CSA with EAC followed by 4n bit CPA with EAC or

modulo (24n À 1) adder using parallel prefix architectures.

Skavantzos and Stouraitis [69] and Skavantzos and Abdallah [70] have

suggested general converters for moduli products of the form 2a(2b À 1) where 2b

À 1 is made up of several conjugate moduli pairs such as (2n À 1), (2n + 1) or

À n





nỵ1

nỵ1

2 ỵ 2 2 ỵ 1 , 2n 2 2 ỵ 1 . The reverse converter for conjugate moduli is

quite simple which needs rotation of bits and one’s complementing and addition

using modulo (24n À 1) adders or modulo (22n À 1) adders. The authors suggest

two-level converters which will find the final binary number using MRC

corresponding to the intermediate residues. The first level converter uses CRT,

whereas the second level uses MRC. The four moduli sets {2n+1, 2n À 1, 2n+1 À 1,

2n+1 + 1} for n odd, {2n, 2n À 1, 2nÀ1 À 1, 2nÀ1 + 1}for n odd, the five moduli

sets {2n+1, 2n À 1, 2n + 1, 2n+1 À 1, 2n+1 + 1}, {2n, 2n À 1, 2n + 1, 2n + 2(n+1)/2 + 1,

2n À 2(n+1)/2 + 1} and the RNS with seven moduli {2n+3, 2n À 1, 2n + 1, 2n+2 À 1,

2n+2 + 1, 2n+2 + 2(n+3)/2 + 1, 2n+2 À 2(n+3)/2 + 1} have been suggested. Other RNS

with only pairs of conjugate moduli up to 8 moduli also have been suggested.

Note that care must be taken to see that the moduli are relatively prime. Note

that in case of one common factor existing among the two sets of moduli, this

should be taken into account in the application of CRT in the second level

converter.

Pettenghi et al. [71] have described general RNS to binary converters for the

moduli sets {2n+β, 2n À 1, 2n + 1, 2n + k1, 2n À k1} and {2n+β, 2n Ỉ 1, 2n Ỉ k1, 2n Ỉ k2,



106



5 RNS to Binary Conversion



j k

. . ., 2n Ỉ kf} using CRT. In the case of first moduli set, they compute mX1 where

 

5

j k X

Mi 1

m1 ¼ 2n+β as mX1 ¼

V i xi where V i ¼ m

xi for i ¼ 2, . . ., 5 which are

1 Mi

mi

i¼1

integers since m1 divides Mi exactly. On the other hand, in case of V1, we have

!



3n







1

n 2

2

2

k 1 ỵ 1 ỵ x1

M 1 m1





V1 ẳ

where is defined as





1

k2 ẳ m1 ỵ 1

M1 m1 1



ð5:40aÞ







ð5:40bÞ



"



#

X

can be removed using

m1 m1

this technique. As an illustration for m1 ẳ 2nỵ , k1 ẳ 3,

 ẳ n ¼ 3, m1 ¼ 64, m2 ¼ 15,

1

¼ 57 and V1 ¼ 14,024,

m3 ¼ 17, m4 ¼ 13, m5 ¼ 19, we have ψ ¼ 2,

M 1 m1

V2 ¼ 58,786, V3 ¼ 59,280, V4 ¼ 43,605 and V5 ¼ 13,260. Note that the technique

can be extended to the case of additional moduli pairs with different k1, k2, etc.

Skavantzos et al. [72] have suggested in case of the balanced eight moduli RNS

using the moduli set {m1, m2, m3, m4, m5, m6, m7, m8} ¼ {2nÀ5 À 1, 2nÀ3 À 1, 2nÀ3

+ 1, 2nÀ2 + 1,2nÀ1 À 1, 2nÀ1 + 1, 2n, 2n + 1}, four first level converters comprising of

moduli {2nÀ3 À 1, 2nÀ3 + 1}, {2nÀ5 À 1, 2nÀ2 + 1}, {2nÀ1 À 1, 2nÀ1 + 1}, {2n, 2n + 1}

to obtain the results B, D, C and E respectively. The computation of

Note that the fractional part in the computation of



D ẳ x4 ỵ m4 X01



5:41aị



where



X01 ẳ



1



2n2 ỵ 1







x 1 À x 4 Þ



2nÀ5 À1



ð5:41bÞ



needs a multi-operand modulo (2nÀ5 À 1) CSA tree followed by a modulo (2nÀ5 À 1)

CPA. The computation E is simpler where



E ẳ x8 ỵ m 8





1

x 7 x8 ị

m8

m7



5:42ị



where m8 ẳ 2n ỵ 1 and m7 ¼ 2n .

The second level converter takes the pairs {B, D} and {C, E} and evaluates the

corresponding numbers F and G respectively which also uses MRC which can

also be realized by multi-operand modulo (22nÀ6 À 1) CSA tree followed by a



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