IRJET- An Efficient Reverse Converter for the Three Non-Coprime Moduli Set {4,2n-1,2n}

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AN EFFICIENT REVERSE CONVERTER FOR THE THREE NON-COPRIME MODULI SET { Valentine, Aveyom1, Mohammed, .I., Daabo (PhD)2, Abdul-Barik, Alhassan (PhD)3 Notre Dame Seminary SHS, Box 10, Navrongo.

Computer Science Department, University for Development Studies, Navrongo.

3Senior Lecturer, Computer Science Department, University for Studies, Navrongo.

---------------------------------------------------------------------***---------------------------------------------------------------------2HOD,

1Tutor,

Abstract - In this paper, residue to binary conversion is discussed for the three moduli set { sharing a common factor. A new and efficient converter for the moduli set is proposed. Larger multipliers in (Premkumar, 1995) are replaced by smaller multipliers and adders reducing the hardware required by the proposed converter. The hardware implementation and comparison with other state-of- the- art schemes shows that the proposed scheme performed better.

2.1 Chinese Remainder Theorem (CRT) The Chinese Remainder Theorem (CRT) can be used to backward convert the residue digits ( ) of the moduli set } to its decimal number (X) as follows; For a moduli set with the dynamic range the residue number can be converted into the decimal number , according to the CRT as follows;

Key Words: Residue Number System(RNS), Chinese Remainder Theorem (CRT), Mixed Radix Conversion (MRC), Reverse Conversion, Non coprime Moduli set.

1. INTRODUCTION Residue Number Systems (RNS) are considered suitable for the implementation of high-speed digital signal processing devices due to their inherent parallelism, modularity, fault tolerance and localized carry propagation properties (Garner, 1959). Some arithmetic operations, such as addition and multiplication, can be carried out more efficiently in RNS than in conventional two’s complement systems.

Where,

The traditional moduli set { } has been one of the most popular studied in RNS. However, these moduli set does not offer some of the advantages that moduli sets with common factors offer such as consecutiveness and ability to allow for equal width adders and multipliers. The moduli set { which shares a common factor of 2 is therefore of study significance since it offers these advantages. It however, will have a limitation in high performance Digital Signal Processing (DSP) applications as it has a smaller dynamic range.

(Gbolagade et al, 2009).

2.2 Mixed Radix Conversion (MRC) The Mixed Radix Conversion (MRC) approach serves as an alternative method to the CRT as it does not involve the use of the large modulo-M computation. This method also used to perform `residue to binary conversion of based on the moduli set { } as follows;

2. FUNDAMENTALS OF RESIDUE NUMBER SYSTEM (RNS)

=

RNS is defined in terms of a set of relatively prime moduli set such that the for , where gcd means the greatest common divisor (gcd) of and , while , is the dynamic range. The residues of a decimal number can be obtained as , thus X can be represented in RNS as , . This representation is unique for any integer . is the modulo operation of with respect to (Gbolagade, 2011).

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Where be computed as;

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(2) (MRDs) which can

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(Gbolagade et al, 2008).

The theorem aims at reducing the magnitude of values involved in the computation.

2.3 Revised CRT for Moduli Set with Common Factors

Proof:

The revised CRT which is used for moduli set with factors is stated as;

The research uses lemmas presented by Wang, (1998) to achieve the proof as follows: Lemma 1:

Lemma 2: is the Least Common Multiple (LCM) of { Where the moduli set sharing a common factor.

where

Lemma 3:

,

Expanding equation (1) for

is

gives;

X is the decimal equivalent of {

is any integer such that

{

and

=1and

a set of integers such that and

(4) Putting Lemmas 2 and 3 into equation (4) results in;

divides

may not exist for some values of

Note however that,

3. PROPOSED CONVERSION TECHNIQUES (5)

Two conversion methods are discussed by the paper. These -Modulus Conversion Technique and computation are without modulo arithmetic.

3.1

Simplifying further yields;

-Modulus Conversion Technique (6)

This technique seeks to reduce the cost of computing by eliminating the computation of the dynamic Range (M) from the Chinese Remainder Theorem (CRT). The Technique first presents the modified CRT for general 3-moduli set { } which does not use the dynamic range (M) in computations.

Thus applying Lemma 1, gives;

(7)

Theorem 1:

Here

with common factors, the For any moduli set { decimal equivalent X of the residue number ( can be computed using; X= ( Where;

)+

form:

(3)

This equation uses only modfor computation instead of mod- M. The approach then further proceeds to eliminate from the computations.

is the multiplication inverse of

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the above equation reduces to the

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For moduli sets with common factors, not all residues are valid numbers. For a 3-moduli set sharing a common factor to represent a valid number, the following proposition must hold;

Theorem 2: For any moduli set sharing a common factor which is being mapped to a relatively prime moduli set { ( ) is computed as;

Proposition 1: For any RNS moduli set sharing a common factor, then ( will represent a valid number if and only if ( is even.

(9) Where; ML=LCM

is the

multiplication inverse of

The Prove to this proposition can be seen in (Ahmad et al, 1999). Substituting the proposed moduli set into Theorem 1, gives;

with respect to

Proof: This is proved by relating equation (9) to equation (2) where all the conditions are present except for being an integer such that = 0 and = 1.

Corollary 1: For the moduli set sharing a common factor 2, the decimal equivalent X of a residue number ( for even and odd as stated in proposition 1 are computed as shown:

Assume that;

.It implies that

=1, which

1. If

is even, then;

implies that

. Therefore it can be written that

X=(

+

as is in equation (9) It can then be shown that

=0. ,

Since

(10)

which

Where; implies

that

and

, hence equation (9) is a more 2. If

formal way of representing equation (2).

X= (

To perform reverse conversion using equation (3) however, requires a method of computing the relatively prime { of the moduli set with common factor .

{

={ 2.

{

(11)

and Table -1: For even

n

Multiplicative Inverses

Equivalent Values

={

1

2

, when n is even and

2

1

={

3

={

, when n is odd and

Note that, the conditions ( ) and ( important as it is based on them that ( exists.

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)

Where

According to Ahmad et. al (1999), the moduli set { with a common factor of 2 can be mapped to a set of relatively prime moduli set,{ given by; 1.

is odd:

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4 ) are very ) and

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Table -2: For odd Multiplicative Inverses

3.2 Computation Without Modulo Operation Given the Residue Number System (RNS) number for the moduli set which shares a common factor of 2 between and the proposed algorithm calculates the decimal equivalent of an RNS number using a simplified version of the CRT stated in equation (2) as shown. Sets of relatively prime moduli sets are selected for the moduli set for being even and odd. As given by (Ahmad et al, 1999) the moduli set with a common factor of 2 can be mapped to a set of relatively prime moduli set,{ given by;

Equivalent Values

3

1

5

1

7 9 11

1.

{

2.

={ {

13

={ , when n is even , ={

{

Note that, the conditions ( ) and ( important as it is based on it that ( ) and

Table -3: For even Multiplicative Inverses

Case 1: For

Equivalent Values

4

{4, 7, 8}

{2, 7, 8}

6

{4, 11, 12}

{2, 11, 12}

8

{4, 15, 16}

{2, 15, 16}

10

{4, 19, 20}

{2, 19, 20}

12

{4, 23, 24}

{2, 23, 24}

14

{4, 27, 28}

{2, 27, 28}

{ {

}. ,

}= { ,

,

}

odd, will have a relatively prime moduli set

,

}.

Thus { ,

,

}= {

Given the moduli set {

Relatively Prime New Set

Thus

{4, 5, 6}

{4, 5, 3}

2

5

{8, 9, 10}

{4, 9, 5}

1

7

{12, 13, 14}

{4, 13, 7}

3

9

{16, 17, 18}

{4, 17, 9}

5

11

{20, 21, 22}

{4, 21, 11}

7

13

{24, 25, 26}

{4, 25, 13}

9

|

,

Theorem 3:

3

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will have a relatively prime moduli set ,

Case 2: For { {

) are very exists.

even,

Thus; { ,

Table 4: For odd Given Set

, when n is odd,

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,

,

}=

being even,

, implying . There exists a compact form of multiplicative inverses for any even integer n > 2 as follows:

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{

for {

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Proof: If we can show that is the multiplicative inverse of ( to

Thus equation (14) holds true.

, then

Theorem 4:

) with respect

.

Given the moduli set { Thus;

{

,

,

for

being odd,

}={

, it implies . There exists a compact form of multiplicative inverses for any even integer n > 2 as follows:

Thus equation (12) holds true.

Similarly, if we can show that then respect to

is the multiplicative inverse of (

,

Proof: If we can show that

) with

.

, then

is the multiplicative inverse of ( to

Thus equation (13) holds true.

Thus equation (15) holds true.

Again, If it can be shown that then

is the multiplicative inverse of (

respect to

.

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,

Similarly, if we can show that

) with

then respect to

|

) with respect

.

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,

is the multiplicative inverse of (

) with

.

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Thus equation (16) holds true.

Again, If it can be shown that then respect to

is the multiplicative inverse of

, with

Proposition 2

.

An

RNS

number for the moduli set { } represented by ( the integer will represent a valid number if and only if is even and can be computed by the following formula;

Proof

Proposed Converter

The prove is done by showing that if ( represents a valid number, then is even. There exist some such that , Therefore , which must be even.

Using the Chinese Remainder Theorem (CRT) on the selected moduli set { } ={ is defined as follows;

If is not even, then ( does not represent a valid RNS number and should not be decoded.

Thus equation (17) holds true.

=

Consider

being even. It is easy to see that The CRT cannot directly be used since the selected moduli set shares a common factor. We therefore adopt the following relations from (Premkumar, 1995) using the weight concept.

Since

Given that;

Therefore;

For

is even, then

,

therefore

Which implies that, there exists a number and

not being even, then

such that

(20) Where

Proposition 1

must be in the interval

where a and b are integers

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Therefore;

From the diagram, the two circles represent multipliers of ranges respectively whiles are adder/subtractor units to generate the values of 1 and 2.

The unit is a modulo adder and is a normal adder unit.

which is same as formula (21) .

4. HARDWARE IMPLEMENTATION In this section, the paper implements the proposed hardware design for the converter by mainly using carry save adders, multipliers and a modular adder.

Table -1: Comparison with some Proposed Converter Proposed Converter

The formula;

Converter (Premkumar , 1995)

Converter (Gbolagade.

2009)

No. of Adder units

can be represented by;

No. of Multiplie r units The hardware sum of the Adders and the multiplier of range is less than 2 multipliers of range .

That is if takes bits, then will take bits.

, Let

,

and

Therefore, in terms of hardware size, the new converter is much smaller than the one proposed in (Premkumar, 1995) because of the fewer multipliers used since adders are much smaller that multipliers.

can then written as shown; for

5. CONCLUSION This paper proposes a new converter for the special moduli set { sharing a common factor of 2. The proposed converter is more efficient and smaller in hardware size than the converter presented in (Premkumar, 1995). The converter reduces cost of implementation and improves on the speed gain of earlier converters.

REFERENCES [1] Abdelfattah, (2011) “Data Conversion in Residue Number System”. McGill University. balanced moduli sets and enhanced modular arithmetic structures". Computers &Digital Techniques.

[2] Chaves, R., and Sousa, L. (2007) "Improving residue number system multiplication with more Circuits & Communication.

[3] Gbolagade, K. A. and Cotofana, S.D. (2008). MRC Techniques for RNS to Decimal Conversion Using the Moduli set Proceedings of the 16th Annual Workshop on Circuits, Systems and Signal processing, Veldhoven, the Netherlands.

Fig -1: Architecture for the proposed Scheme

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[4] Gbolagade, K. A. and Cotofana, S.D. (2009a). Residue-toDecimal Converters for Moduli Sets with Common Factors.

IEEE, Pp.624-627, 2009.

[5] Gbolagade, K. A. and Cotofana, S.D. (2009b). A Reverse Converter for the new 4 –Moduli set . Submitted to IEEE Newcastaisa Toulouse, France. July, 2009. IET, Volume 1, Issue 5, pages 472-480.

[6] Premkumar, A.B., (1992). “An RNS to Binary converter in 2n+l, 2n, 2n-1 moduli set”, IEEE Transactions on Circuits and Systems - I1 Vol. 39, No. 7, pp. 480-482.

[7] Premkumar, A.B., (1995). “An RNS to Binary converter in a three moduli set with common factors”, IEEE Transactions on Circuits and Systems -11, Vol. 42, No. 4, pp. 298-301.

[8] Garner, H.L. (1959). The residue Number System, IRE Trans. On Electronic Computers, pp. 140-147.

BIOGRAPHIES Valentine Aveyom is an MSc. Computational Mathematics Student at the Faculty of Mathematical Sciences, UDS, Ghana and an ICT Tutor at Notre Dame Sem. SHS, Navrongo. He has research interest in the area of Residue Number System (RNS).

Dr. Mohammed I. Daabo is a senior lecturer in Computer Science at the Department of Computer Science, UDS, Ghana. He has over (15) years teaching experience in Tertiary Education and has conducted academic research in the fields of Overflow Detection, Reverse conversion, Error Detection, Sensor Networks and Epidemiological Modeling.

Dr. Abdul-Barik Alhassan is a senior Lecturer and Exams Officer of the Computer Science Department of UDS Ghana. He has research interest in the areas of Computer Hardware and Architecture, Residue Number System (RNS), Information Security/ Cryptography, Algorithm Design and Analysis and Statistical Computing.

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