A Novel Binary Division Algorithm Based on Vedic Mathematics and Applications to Polynomial Division

7/25/2019 A Novel Binary Division Algorithm Based on Vedic Mathematics and Applications to Polynomial Division

1/6

A Novel Binary Division Algorithm based on Vedic

Mathematics and Applications to Polynomial Division

Ashish Joglekar1, Shaunak Vaidya

1, Ajinkya Kale

2,

1T.Y.B.Tech, Department of E&TC,

2Final Year, B.Tech, Department of Computer & IT,College of Engineering, Pune 05, Maharashtra, INDIA.

Abstract - The Parvartya Stra from ancient Indian

Vedic Mathematics is extended to Radix 2 Binary

number system. An enhanced algorithm for binary number

division is presented and a corresponding hardware

module is proposed. Two algorithms are proposed to

perform polynomial division in binary system and may findapplications in various digital systems.

Keywords: Vedic Mathematics, Algorithm, Binary

Division, Synthetic Division, Polynomial Division.

1. Introduction

1.1. Need for Division Algorithm

Designers of processors with enhanced Arithmetic

and Logic Unit (ALU) are always on a look out for

algorithms for performing basic operations one of them

being Division. The algorithm should be such that itinvolves processes requiring compact hardware while

satisfying efficiency constraints.

1.2. Role of Vedic Mathematics

Ancient Indian Vedic Mathematics deals mainly

with sixteen Sutras and their applications for carrying out

tedious and cumbersome arithmetical operations, and to avery large extent, executing them mentally [2]. Parvartya

Stra is one of these sixteen Sutras used for division, in

Vedic Mathematics.Synthetic division [3], which is a shortcut method

for dividing a polynomial by a linear divisor of the form can be considered as a special case of theParvartya Stra [2].

1.3.

Vedic Mathematics Based Division

Algorithm

This paper presents an algorithm to perform binary

number division based on the Parvartya Stra. Anenhancement in this algorithm has led to process

optimization with respect to number of intermediateoperations involved.

Two algorithms for polynomial division viz. Split

Coefficient Method and Scaled Coefficient Method have

been proposed in binary format as required by any digital

system, which may find application in scientific

calculators.

2. The Parvartya Stra

The Polynomial Remainder Theorem states that ifpolynomial ( ) is dividedby ( then the remainder is

The Parvartya Stra however, provides us with both

Quotient and Remainder as illustrated in examples below.

2.1. Polynomial division

Figure 1: Example illustrating Parvartya Stra for

polynomial division

Unlike Synthetic division, the divisor inFigure 1

is non linear. This helps illustrate the scope of the

Parvartya Stra.

2.2. Application of the Stra to Decimal

Number Division

Following the representation similar to the case of

polynomial division, Figure 2 illustrates the Parvartya

Stras application to Arithmetical Operations.

Divisor Dividend

2 1 2 7 2 18 3 8

2 0 1

4

0

2

22 0 1140 0 202 11 20 20 8 12 2 1 1 2 0 2 0 8 1 2


2/6

Figure 2: Application to Decimal System

3. Extension to Binary Number

System

The Parvartya Stra as described in [2] is applied todecimal, i.e., Radix 10 system only. We have extended the

Sutra to perform Binary Number Division as illustrated in

Figure 3:

Figure 3: Extension to Binary Number System

4. Hardware Implementation

The binary form of Parvartya Stra is implemented

using Dedicated Register Bank Structure. Figure 4 shows

one such register bank.At each stage subtract contents of bit wise

vertically and add contents of bit wise vertically. Thisexplains hardware representation of -1.

Bits always interpreted as negative

1 1 1 1 0 1 1 0 1

1 0 1 0 0 00 0 0

0 0 0

1 0 1

0 0 0

0 0 0

1 0 1

1 0 1

0 0 0

0 0 0

1 0 1

Note: 0, 1 and -1 are coded vertically in and as:

0: 1: 1:

0 0 1 0 1 0

Truth table:

Figure 4: Register Bank Structure

Divisor Dividend

1 1 3 1 3 9 0 5 1 3 1 3

2 6

4

12

1 2 4 10 71 2 3 06

1 2 4 1 1 2 3 11310706

Divisor Dividend1 1 0 1 1 1 1 1 0 1 1 0 11 0 1 1 0 10 0 01 0 11 0 11 0 1 1

0 1 1 0 1 1 1 1 11 1 10 101010101100101 1 1 0 0 1 0 1 01100

1 1 0 1

1 0 1

Result bits

1 0 1 1 1 1 1 1 0

Carry 1

1Sign bits 0 0 0 1 0 1 0 1

Interpretation0 x x 01 0 0 0

1 1 0 -1

1 1 1 1


3/6

The hardware can be reduced further by retaining only

two of the shown registers at the Result position, onebeing the current register and the other being the Result

Register which is continuously refreshed at every stage

of computation. The Result Register will show finalresult when computations for all bits have been done.

5. Process Enhancement

The number of operations performed, in binary version

of the Parvartya Stra are reduced significantly by

modifying its process algorithm. The new enhancedalgorithm, which remarkably simplifies the process, is

represented in Figure 5.

Figure 5: Process enhancement flowchart

Figure 6: Process enhancement Example

1. Here, 111101101 , 1101.2. Compare first 4 bits 1111 with. Since

, replace by, Get intermediate.3. Intermediate 1001101 isthe new.4. Compare first 4 bits 1001with.

Since , perform Parvartya Division asshown.

6. Polynomial Division

Polynomial Division, by itself is a cumbersomeprocess and not a frequently seen feature, even on

scientific calculators. Our enhanced algorithm is modifiedto suit generalized polynomial division in a binary format,

provided both polynomials have integer coefficients. This

enables applications in various digital systems, thereby

equipping them with another useful tool.

6.1. Scaled Coefficient Method

Performsgeneralizedpolynomialdivision

Dividend Divisor 6.1.1.

Need for Scaling of Coefficients

Scaling of coefficients by powers of enables us toeliminate fractional values in calculation stages.

The coefficients of the final result are conveniently

expressed in the form of fraction where numerator and

denominator are stored in separate locations.

Divisor Dividend Quotients1 1 0 1 1 1 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1000001 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 11 0 11

0

1

1 0 1 1 1 1 1 11 1 10 101 1010101101 110010 101100

100101

Yes

No

Yes

No Intermediate 0 ,

END

Compare first (starting fromMSB)

?

Proceed by our

Binary Parvartya

Stra,

Replace by ,Get intermediate ,Intermediate R is the new.

Merge all

intermediate Qs

START

Accept

dividend

?

Yes


4/6

Illustration :

Let m = 3 and n = 3

2 3 3 23

1 32 3 1 332

0 33 3 0 3233 Now, 3 3 33 22 1 0 23 323 2 13231332 033303233

33 22 1 0Thus

Figure 8: Scaled Coefficient Method Illustration

This can be extended further for the general case when thedividend is and divisor is

:

:

Start

Accept Dividend andDivisor where,

Get

and

where

0 and 110

Get Coefficients of Q by further dividing each of the first 1 coefficients by The remaining coefficients of result directly represent R

Apply Parvartya Stra with

Get

and

END

Figure 7: Scaled Coefficient Method Flowchart

END


5/6

32 , 52 1Figure 9: Scaled Coefficient Method Example

3 4 12 1

32 34 , 54 12Figure 10: Scaled Coefficient Method Example

4 13 23 1

43 359 , 479 5127

6.2. Split Coefficient Method

This method works for the special case when thedivisor has coefficients 0 or 1 and generalized dividend

with integer coefficients. The coefficients of the dividend

are split to form binary segments. This is a relativelysimple approach considering the operations involved,

compared to Scaled Coefficient Method.

Figure 11: Scaled Coefficient Method ExampleFigure 12: Split Coefficient Method Flowchart

-00 -01*10 011 000*10 100*10*10 001*10*10*10

00 -11*10

00 00

11 00 1010 1000Divide by:

(representative)

10 10*10 10*10*10

Final

Division by:

(for Q only)10

-01 -01*10 011 000*10 100*10*10 001*10*10*10

-11 -11*10

11 11*10

11 -11 101 100

10 10*10 10*10*10

10

-01 -01*11 100 1101*11 0000*11*11 0010*11*11*11

-100 -100*11

-100011 -100011*11

100 100011 -101111 -11001111 11*11 11*11*11

11

No

Get Partial 2in binarywith sign of current dividend

segment MSB.

Subtract from current dividendsegment to get partial, where is

the binary number obtained by

replacing first bits of currentdividend segment by divisor.

END

Add 1s in partial R to next largestdividend segments available where

corresponding bit is 0. If corresponding

bit in all dividend segments is 1,createnew segment

Yes

Split the Coefficients of

dividend to form binary

dividend segments

Start from largest dividend

segment remaining

Is no. of bits in Divisor

> no. of bits inDividend Segment ?

START

AcceptgeneralizedDividend andDivisor where 0 1


6/6

Divisor DividendSegment

Partial

Quotients

Partial

Remainder

101 111011 1000 10011111000

100000

101 111011 1000 10011110000

101 110011 1000 1-1011

10000

101 10000 100 -100

1-1011

101 1-1-111 100 -1-111-100

101 -1-111 -10 -111

-100 1

101 -111 -1 11

-110 1

101 -111 -1 11

11 11

1

3 2 2 2 3

Figure 12: Split Coefficient Method Example

1. The dividend is initially split into three binary

segments.

2. The partial quotient is obtained from the first

dividend segment at each stage.3. The partial remainder is merged with the other

segments and the new segments thus formed

are shown in the next stage.

4. Representation of -1 can be done at hardware

level by assigning a sign bit.

7. Conclusion

The Parvartya Stra is successfully extended to

perform binary division. The Scaled CoefficientMethod and Split Coefficient Method enable us to

perform polynomial division in binary format. Thisenables applications in various digital systems, like

scientific calculators, thereby equipping them with

another useful tool.

Acknowledgements

The Authors are grateful to Prof. V.V. Joshi, Dept.

Of Mathematics, COEP and Prof. A.A. Sawant, Head,

Dept. of Computer Engineering & IT, COEP and Prof.(Mrs.) M.A. Joshi, Dept. of E&TC, COEP for their

valuable suggestions.

References

[1] Himanshu Thapliyal and Hamid R. Arabania ,

"High Speed Efficient N Bit by N Bit Division

Algorithm And Architecture Based On Ancient

Indian Vedic Mathematics", Proceedings of

VLSI04, Las Vegas, U.S.A, June 2004, pp. 413-

419(CSREA Press).

[2] Jagadguru Swami Sri Bharati Krsna TirthatjiMaharaja, Vedic Mathematics, Motilal

Banarsidass Publication,

1992.

[3] Wolfram MathWorld

http://mathworld.wolfram.com/SyntheticDivision.html

[4] Shaunak Vaidya, Ashish Joglekar, and Vinayak

Joshi, An Efficient Binary MultiplicationAlgorithm based on Vedic Mathematics,

Proceedings National Conference on Trends in

Computing Technologies 2008, Pg 37-43.[5]

Himanshu Thapliyal and M.B Srinivas,"VLSI

Implementation of RSA Encryption Systemusing Ancient Indian Vedic Mathematics ",

Proceedings of SPIE -- Volume 5837 VLSI

Circuits and Systems II, Jose F. Lopez, FranciscoV. Fernandez, Jose Maria Lopez-Villegas, Jose

M. de la Rosa, Editors, June 2005, pp. 888-892[6]

Chidgupkar Purushottam D., Karad Mangesh T.,

The Implementation of Vedic Algorithms inDigital Signal Processing, Global J. of Engng.

Educ., Vol.8, No.2 (2004), 153 157.

A Novel Binary Division Algorithm Based on Vedic Mathematics and Applications to Polynomial Division

Documents

Transcript of A Novel Binary Division Algorithm Based on Vedic Mathematics and Applications to Polynomial Division