Overflow Detection Scheme in RNS Multiplication Before Forward Conversion

7/29/2019 Overflow Detection Scheme in RNS Multiplication Before Forward Conversion

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OVERFLOW DETECTION SCHEME IN RNS MULTIPLICATION BEFORE

FORWARD CONVERSION

M.I. DAABO1,2

, K. A. GBOLAGADE1

1 Department of Computer Science, Faculty of Mathematical Sciences, University for Development

Studies, Navrongo, Ghana.

2 Department of Mathematical Applications, Wisconsin International

University College, Accra, Ghana.

AbstractOverflow detection is one of the major issues that preclude Residue Number System (RNS) usage in

general purpose computing. Contemporary authors have presented various schemes that rely on either the Chinese

Remainder Theorem (CRT) or the Mixed Radix Conversion (MRC). This paper presents an overflow detection

scheme in RNS multiplication before forward conversion. Overflow in RNS multiplication of integers X and Y

occurs when the product of the calculated quotients, ab , where M = is the system dynamic range. Our

proposal is a multiplicative overflow detector, which does not require computations involving the use of the time

consuming CRT or MRC. The newly proposed scheme utilizes lesser modulo computations and hence has the

advantage of having smaller hardware architecture with lesser delay.

Keywords: Residue Number System, Multiplicative overflow detector, Dynamic Range. CRT. and MRC

-------------------------------------------------------- -----------------------------------------------------------

1.0IntroductionThe advantages of Residue Number System

(RNS) over the conventional binary numbersystem include parallelism, fault tolerance

low power and high speed computations and

are well documented in [2], [8], [10], [12].However, the inability of RNS to efficiently

manage dynamic range overflow is one of the

major disadvantages [1], [7], [11].

Overflow in RNS is a condition where a

calculated number falls outside the validdynamic range of a particular RNS [4].

Contemporary authors over the past decades

have developed overflow detection

algorithms, which rely on the ChineseRemainder Theorem (CRT) or the Mixed

Radix Conversion (MRC) using one or more

of relatively prime moduli as scale factors [9],[10]. Siewobr and Gbolagade [5] proposed a

scheme that uses the large modulo (M+1)

computation. Siewobr and Gbolagade againdeveloped an additive overflow detection

scheme that reduces the large modulo M to Mi

by scaling M and the integers X and Y with

the modulus mi = 2n

[4]. However, theproposed scheme in [4] does not consider

overflow in multiplication of two numbers.

In this paper, we present an overflow

detection scheme in RNS multiplication

before forward conversion. The algorithmsare based on first, the computation of the

quotients, a = and b = for integers X and

Y. Secondly, we make the proposition that

overflow occurs in RNS multiplication of X

and Y when ab , where M = is the

system dynamic range. This new proposal

detects multiplicative overflow. It also

eliminates the time consuming reverseconversion approach characterized by the use

of CRT and MRC with large modulo M. Theoverall achievement is reduced area cost and

improvement in delay.

JOURNAL OF COMPUTING, VOLUME 4, ISSUE 12, DECEMBER 2012, ISSN (Online) 2151-9617

https://sites.google.com/site/journalofcomputing

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2012 Journal of Computing Press, NY, USA, ISSN 2151-9617


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2.0The Proposed SchemeIn this work, we propose a simple overflow

detection scheme in RNS multiplication

before forward conversion. Given the

binary/decimal numbers X and Y and themoduli set {m1, m2 m3,, mn}, with the

dynamic range M = . The modulus

m = 2 is chosen such that:

a = (1)

b = (2)

= (3)

ab (4)

Proposition 1: An overflow only occurs in

RNS multiplication of the binary/decimal

numbers X and Y ifab

Proof:

If we let

a = (5)

b = (6)

= (7)

Then we proposed that an overflow will occur

in RNS multiplication of X and Y if ab

(8)

From equations (5), (6), and (7) we see that

X = a m (9)

Y = b m (10)

M = m (11)

But overflow only occurs if XY M

That is mm (ab) m

Thus ab = as in equation (8)

But we then choose m = 2

Therefore a =

b =

=

and ab as in equations (1)-(4)

2.1 proposed overflow detection algorithm

1. Given the binary/decimal numbers Xand Y and any moduli set {m1, m2m3,,mn}, with the dynamic range M

= .

2. Choose the modulus m = 23. Compute the following; a = , b = ,

= , and ab .

4. If ab then overflow occurs inRNS multiplication.

2.2Numerical illustrationsFor purposes of illustrations, we show how

the scheme works with some examples.

1. Product of 30 and 4 using the moduli set

{7, 5, 3}.

Let X = 30, Y = 4 and M = 105. Then a =

= 15, b = = 2 and = 26.25.

If ab then overflow occurs in RNS

multiplication. That is 30 26.25 is true and

overflow does occur in RNS multiplication of

30 and 4 for the moduli set {7, 5, 3}.

2. Product of 20 and 4 using the moduli set

{7, 5, 3}.






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Let X = 20, Y = 4 and M = 105. Then a = =

10, b = = 2, = 26.25.

If ab then overflow occurs in RNS

multiplication. But 20 26.25 is false andoverflow does not occur in RNS

multiplication of 20 and 4 for the moduli set

{7, 5, 3}.

Table1

Summary of illustrated examples

X Y

a

=

b

= ab

Is

ab Is there

overflow

30 4 15 2 30 26.25 YES YES

20 4 10 2 20 26.25 NO O

8 10 4 5 20 26.25 NO O

6 30 3 15 45 26.25 YES YES

0 0 0 0 0 26.25 NO O

32 20 16 10 160 26.25 YES YES

40 4 20 2 40 26.25 YES YES

M = 105

3 ImplementationThe algorithms are implemented in stages.

First, since a = , b = and are made up of

integer divisions and remainders, we scale X

and Y by 2 and scale M by 4. The product of

a and b is implemented in three steps. Theresults in the previous two steps are multiplied

to obtain ab. A simple logic circuit is used to

implement ab and then output whether

there is overflow or not. We note here that

binary scaling and division by 2, that is,

powers of two requires no hardware since it isjust a right logical shift operations. Also,

computing the residue of a number with

respect to 2 type moduli results in the oneleast significant bit of the number in question

and also requires no hardware. Therefore the

computations of a and b can be hardwired.

The implementation is therefore reduced tosimple multiplication and magnitude

comparison as shown in Fig. 2.

4 Performance AnalysisTable 2

Table of comparison

ITEMS [5] [6, 3] TM OP

Operation

size

Mod

M+1

M M Mi

DOBFC No No yes yes

Redundant

moduli

yes yes No No

Sign

Detection

No No No No

CRT/MRC No yes No No

DOBFC = Detection of Overflow Before

Forward Conversion. TM = Traditional

Method. OP = Our Proposal

In Table 2, it can be seen that it is only the

new proposal, the TM and [4] that can detectoverflow before forward conversion. In

addition, the proposed scheme detects

overflow in multiplication operation and

requires lesser modulo operation whencompared with the TM model and others. This

results in the use of less hardware resources in

design and minimum operation time. Theoverall achievement is reduced area cost and

improvement in delay.

Fi . 2. Dia ram of ro osed scheme

No OverflowOverflow

(ab)

T F

Logic gate 1

ab

Multiplier

X Y






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5 ConclusionIn this paper, we presented a dynamic range

multiplicative overflow detection scheme in

RNS before forward conversion. Firstly, the

algorithms seek to compute the quotients a

= and b= . Secondly we propose that

overflow in RNS multiplication of integers Xand Y occurs when the product of the

calculated quotients, ab , where M =

is the system dynamic range. By

design, our proposal is a multiplicative

overflow detector and does not require

computations involving the use of the timeconsuming CRT and MRC techniques with

large modulo M. It is also found that the newscheme uses lesser modulo computations and

hence has the advantage of moderatehardware architecture with lesser delay

operations.

6 Reference[1] F.J Taylor and C. Huang, An Auto scale

Multiplier. IEEE Transactions on computers

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International Journal of Com putational

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Efficient RNS Overflow Detection Algorithm,

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[9] K.H Rosen, Discrete Mathematics and ItsApplications, Fourth Edition, McGraw-Hill,

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Overflow Detection Scheme in RNS Multiplication Before Forward Conversion