Design, Develop and Implement an Efficient Polynomial Divider

International Journal of Latest Technology in Engineering, Management & Applied Science (IJLTEMAS)
Volume VI, Issue III, March 2017 | ISSN 2278-2540
www.ijltemas.in Page 78
Design, Develop and Implement an Efficient
Polynomial Divider
Purbayan Deb1
, Anindya Sen2
1,2
Department of Electronics and Communication Engineering (VLSI), Heritage Institute of Technology, Kolkata, West Bengal
Abstract- Polynomial Division is a most common numerical
operation experienced in many filters and similar circuits next to
multiplication, addition and subtraction. Due to frequent use of
such components in mobile and other communication
applications, a fast polynomial division would improve overall
speed for many such applications. This project is to design,
develop and implement an efficient polynomial divider
algorithm, along with the circuit. Next its output performance
result is verified using Verilog simulation. A literature survey on
the normal division algorithms currently used by ALU’s to
perform division for large numbers, yielded Booth’s algorithm,
Restoring and Non-restoring algorithm. Verilog simulation of
these algorithms were used to derive efficiency in terms of the
timing characteristics, required chip area and power dissipation.
Initially, performance analysis of the existing algorithms was
done based on the simulated outputs. Later similar analysis with
the updated polynomial divider circuit is performed.
Keywords- Division, Polynomial, Booth’s algorithm, Restoring
algorithm, Non-restoring algorithm, Verilog.
I. INTRODUCTIONS
hroughout the years, mathematicians and engineers have
developed many algorithms to divide numbers. The
ALU which is primarily used for division has gone through
many changes in its design. One of these changes was in its
division algorithms. In a typical computer, an ALU is called
upon to do hundreds of division operations per second.
Divider circuits are used for various purposes like error
correcting codes. So to perform at its peak, the ALU„s
algorithms need to be as efficient as possible. However, some
of these algorithms work better when computing the result of
the operation by hand than using a computer and so these
algorithms are not efficient in every case [1].
The traditional pen and paper algorithm, when converted
to computer algorithm, resulted in the Booth‟s algorithm,
Restoring Division algorithm [5]. Smaller improvements
have been made to the restoring division algorithm, which
resulted in Non-Restoring Division algorithm and many high
radix algorithms, later combinational array divider circuits are
also implemented for the division purpose [6]. In many cases,
division is performed by taking the inverse of the divider and
then multiplying the two numbers.
Division methods are divided in five classes that include
iteration, digit recurrence, very high radix, table look up and
variable latency. Each of these classes of division is
implemented differently in hardware (using multiplication,
subtraction, table look up, etc.).
Some algorithms use multiple classes rather than just one
in particular. This report focuses on subtraction-based
methods, such as restoring and non-restoring division
algorithms, to obtain the final answer in a division
computation. Digit recurrence algorithm is another division
algorithm and they produce one digit of the final quotient per
iteration.SRT (Sweeney Robertson and Tocher) division
algorithm is very commonly use for digit recurrence purpose
[12].
Use of Verilog code, which is a Hardware Descriptive
language (HDL) will be done for the simulation purpose of
the project.
II. MOTIVATION
Despite the recent improvement in division algorithms,
division remains a complex operation and is therefore not
implemented in many low cost or low power ALUs. Division
can add more complexity to the computations since it can
have invalid inputs such as division by zero and can have
multiple machine cycles used in it. The time taken for its
operation is also high and the circuits get more complex if
more precision is needed and so it is avoided by most ALU‟s
[1].
A division operation is an indispensible tool for a high
performance system. A common perception of division is that
it is an infrequent operation whose implementation need not
receive high priority. However, it has been shown that
ignoring its implementation can result in significant system
performance degradation for many applications. Refining
the polynomial division algorithm helps in improving the
precision of the result and reduces the delay to obtain output.
So implementation of a proper divider circuit is very
important in an ALU.
III. OBJECTIVE
The main objective of the project work will be to focus
on using the most optimal division algorithm and to design an
T

efficient polynomial divider circuit & simulate its
performance.
To find the most optimal algorithm timing characteristics
and area report generated by the Xilinx tool for various
division algorithms will be compared and analyzed.
IV. TOOL USED
For this project Verilog, HDL is used under Xilinx ISE
Design Suite 13.4 platform.
V. LITERATURE SURVEY
A. Booth’s Algorithm
Fig.1 Flowchart of booth‟s algorithm for division
Division is done by doing shifts and subtractions.
Dividing a number of 2n bits by a number of n bits results in a
quotient of up to 2n bits and a remainder of up to n bits.
At start, the n bits divisor is shifted to the left, while n
0‟s are added to its right. This way the dividend and the
divisor are 2n bits long.
At each step (repeating the following n+1 time), subtract
the divisor from the dividend. If the result is non-negative,
Shift the quotient left and place 1 in the new place.
Else, Shift the quotient left and place 0.
Restore the dividend by adding the divisor to it. Shift the
divisor to the right [2].
At start, the dividend occupies the right half of the
remainder register. The left half of the reminder register is full
with zeros. Shift reminder left 1 position.
At each step, the control subtracts the divisor from the
left half of the remainder register, putting there the result. If
the remainder is negative, it restores it. Then, instead of
shifting the divisor to the right, it shifts the remainder to the
left and inserts 0 or 1, according to the sign of the remainder.
0 if the sign bit is 1 and 1 if the sign bit is 0.
At the end, the remainder register contains the quotient in
its right half and the remainder in its left half [3].
The following example (7/2) will illustrate the division
process of booth‟s algorithm clearly.
TABLE I
Shows Division of 0111 / 0010 by Booth‟s Algorithm
Itera-
tion
Step Quotient Divisor
Remai
nder
0 Initial values 0000
0010
0000
0000
0111
1
1: Rem=Rem-Div 0000
0010
0000
1110
0111
2b: Rem<0=>+Div, sll
Q,Q0=0
0000
0010
0000
0000
0111
3: Shift Div right 0000
0001
0000
0000
0111
2
1: Rem=Rem-Div 0000
0001
0000
1111
0111
Q, Q0=0
0000
0001
0000
0000
0111
0000
1000
0000
0111
3
1: Rem=Rem-Div 0000
0000
1000
1111
1111
Q, Q0=0
0000
0000
1000
0000
0111

0000
0100
0000
0111
4
1: Rem=Rem-Div 0000
0000
0100
0000
0011
2a: Rem≥0=> sll Q,
Q0=1
0001
0000
0100
0000
0011
0000
0010
0000
0011
5
1: Rem=Rem-Div 0001
0000
0010
0000
0001
2a: Rem≥0=> sll Q,
Q0=1
0011
0000
0010
0000
0001
0000
0001
0000
0001
B. Restoring Division Algorithm
Digital Recurrence algorithms use subtractive methods to
calculate quotients one digit per iteration. Restoring division
algorithm is based on the digital recurrence algorithm [1].
Fig.2 Flowchart for restoring division algorithm
Restoring division follows the same method as the pen
and paper long division algorithm. In the long division
algorithm, the divisor is compared to the left digits of the
dividend. If the divisor is bigger than the dividend numbers
being compared, then a 0 in appended to the quotient and
divisor is shifted to the right to compare with bigger dividend
digits. If the divisor is smaller than the dividend, then the
divisor being compared is subtracted from the dividend and
the result is stored as remainder, while the number of times
the divisor can go into the dividend is appended to the
quotient [4].
During the next loop, the dividend and the remainder
need to be appended together to form the new dividend. This
process is repeated until the dividend cannot be divided
further by the divisor. The same process is applied in the
Restoring division algorithm.
To decide whether the divisor is bigger than the dividend
bits it is being compared to, it subtracts the divisor from the
dividend bits and stores the result in remainder field. If the
divisor is bigger than the dividend bits, then the result will be
negative.
If the result is negative, then the remainder is wrong and
it must be “restored” to the previous value and a 0 must be
appended to the quotient before the divisor is shifted to the
right (or dividend shifted to the left) and subtraction is tried
again.
If the result is positive, then divisor is bigger than the
dividend bits being compared to and the result is valid.
Therefore, a 1 is appended to the quotient.
Below the algorithm for restoring division method is
explained with an example by dividing 0011 1101 (61), by
01010 (10) and how it works.
Following the Restoring division algorithm discussed
above, the first step is to shift the quotient 1 bit to the left.
Second, subtract the divisor from the left half of the
dividend and update the left half of the quotient with the
answer. This new dividend value now has the remainder and
rest of the quotient appended together.
To avoid destroying the initial dividend value, the
dividend can be copied into the remainder register and use
remainder register as the dividend value.
If the new dividend is less than zero, then shift the
quotient left 1 bit and restore the previous value of the
dividend. Otherwise, shift the quotient left 1 bit and set the
least significant bit to 1. Repeat for procedure 4 times to
evaluate all bits of the dividend [5].
At the end of the procedure, the quotient value will be the

remainder.
Dividend z: 0011 1101 (61),
Divisor d: 01010 (10)
Quotient : 0110 (6),
Remainder : 0001 (1)
TABLE II
Division of 00111101 / 01010 by Restoring Method
C. Non-Restoring Division Algorithm
The non-restoring divide does not “restore” the
remainder to the correct value but leaves it incorrect until the
next cycle [1]. In the restoring divide algorithm, if we had
restored the partial remainder to its correct value, we would
proceed with the next shift and trial subtraction getting the
result. Instead, because we used the incorrect partial
remainder, a shift and trial subtraction would yields result
which is not the intended. However, an addition would do the
trick.
The non-restoring algorithm can result in a negative
remainder, which is incorrect. Therefore, a correction step is
needed to obtain the correct remainder. The algorithm to
perform non-restoring division is as follows:
Non-restoring divide algorithm [6]:
i. Shift remainder left 1 bit.
ii. If remainder is negative, add divisor to the left half
of
iii. the remainder. Shift quotient left 1 bit.
iv. If remainder is positive, subtract divisor from the left
v. half of the remainder. Shift quotient left 1 bit and
add 1.
vi. Repeat for number of bits in divisor.
vii. Correction step: If remainder is negative, add divisor
to the remainder to obtain the correct value.
Fig.3 Flowchart for non-restoring division algorithm
Here, the dividend is not restored after each unsuccessful
subtraction operation as was in restoring algorithm.
Instead, the following logic is followed:
If the current remainder is positive,
Then Q0 = 1.
Next operation will be shift and subtract
Else (remainder negative),
Then Q0 = 0.
Next operation will be shift and add.
Example: 10 / 4
Quotient = 2, Remainder = 2, A = 0
Q = Dividend = 10 = 1010
B = Divisor = 4 =0100 ,
Iteration P Q
1 Shift z left once: 0111101
Subtract d from left half of z: 11111101
Result is negative. Restore to previous
value: 0111101
0
Subtracting d from left half of z: 010101
Result is positive. New z is 010101
01
3 Shift z left once: 1010 1
Result is positive. New z is 00001
011
Result is negative: Restore to previous
value: 0001
0110

- B means 2‟sComplement of B = 4
B =00100
11 011 1‟s Complement of B
+ 1 addition of 1 to 1‟s Complement of B
11100 2‟s Complement of B
If Q (Divisor) register contains 4 bits then Contents of A
register is 4+1=5 bits
TABLE III
Division of 1010 / 0100 by Non-Restoring Method
D. LFSR (Linear Feedback Shift Registers)
Linear Feedback Shift Registers (LFSR) can be used for
polynomial division, among its other uses.
Two example LFSRs are shown in Fig4 & Fig5. Note that
both these structures use D-type flip-flops and linear logic
elements (EOR gates) to realize LFSRs [8]. The basic
difference in these two structures being the circuit of Fig4
uses linear elements interspersed between the flip-flops
whereas the circuit of Fig5 has no linear element appearing
between the flip-flops instead the linear elements appear only
in the feedback path. It is for this reason the realization of
Fig4 is called internal-EOR LFSR and the realization of Fig5
is called external EOR LFSR. A equivalence exists between
the two structures in the sense that knowing the properties of
the first structure one can deduce the properties of the second
structure.
The circuit in Fig4 is better for its high-speed operation
since they don't have multiple adder combination propagation
delays [9].
Fig.4 Internal EOR type LFSR
Fig.5 External EOR type LFSR
Polynomial arithmetic can be performed using this
LFSR circuit. The LFSR here does the shifting operation at
each clock cycle and keeps on generating new patterns based
on the input values. One thing to be noted here during
polynomial division is that all these polynomials are not
usually written with minus signs, but they could be, therefore
a coefficient of –1 is equivalent to coefficient of 1 and
polynomials with co-efficient other than 1 cannot be used as
in digital domain we have only 0 & 1[10].
Fig.6 A basic polynomial division register
VI. METHODOLOGY
A. Preliminary Work
In this project firstly comparison of the restoring and non
Represents EOR gate

restoring algorithms for division are done and the results are
obtained.
The comparison is on the basis of area analysis, timing,
delay and power consumed. The algorithms are implemented
using Verilog code and done the comparison.
I have done the coding in Xilinx ISE Design Suite 13.4
tool.
B. Proposed Implementation in Polynomial Divider Circuit
Steps to be followed for implementation of polynomial
division-
i. Firstly the circuit is built on basis of the denominator
given.
ii. The input is given serially in the circuit.
iii. The output of each stage is calculated at each clock
cycle.
iv. The process continues until all the bits of input are
given.
v. The quotient is calculated from the outputs of the
final register.
vi. Finally, the remainder is calculated from the output
of all the registers in the final clock cycle.
Below two examples are given to explain the process of
polynomial division-
i) (X3
+X2
+1)/(X2
+1)
Fig.7 Circuit for (X3
+X2
+1)/(X2
+1)
Fig.8 Division using long division method
TABLE IV
LFSR Stages Corresponding to the Input Applied
INPUT D0 D1 OUTPUT STREAM
1 0 0 0
1 1 0 00
0 1 1 001
1 1 1 0011
0 1
Here the final output stream is 0011.
So the quotient is 0*X3
+ 0*X2
+ 1*X1
+ 1*X0
= X+1
The value of D0 and D1 in final cycle is 0 & 1 respectively.
So, remainder is 0*X0
+ 1*X1
=X
Here any numerator can be applied and the output result will
be obtained using the same denominator.
ii) (X7
+X6
+X5
+X4
+X2
+1)/(X5
+X4
+X2
+1)
Fig.9 Circuit for (X7
+X6
+X5
+X4
+X2
+1)/(X5
+X4
+X2
+1)
TABLE V
LFSR Stages Corresponding to the Input Applied
INPUT D0 D1 D2 D3 D4 OUTPUT
STREAM
1 0 0 0 0 0 0
1 1 0 0 0 0 00
1 1 1 0 0 0 000
1 1 1 1 0 0 0000
0 1 1 1 1 0 00000
1 0 1 1 1 1 000001
0 0 0 0 1 0 000010
1 0 0 0 0 1 0000101
0 0 1 0 1

Here the final output stream is 0000101.
So the quotient is
0*X6
+ 0*X5
+ 0*X4
+ 0*X3
+ 1*X2
+ 0*X1
+ 1*X0
=X2
+1
The value of D0,D1,D2,D3,D4 in final cycle is 0,0,1,0,1
respectively.
So, remainder is
0*X0
+ 0*X1
+ 1*X2
+ 0*X3
+ 1*X4
= X2
+X4
.
Fig.10 Division using long division method
These are simulated using the Xilinx tool and the results
are compared.
The max power of the denominator can be 9312 as this is
the number of Flip Flops supported by the Xilinx tool.
This division operation can also be performed using
multiplication of LFSR. But that will be a much more time
consuming process.
Example- X2
/X= X
This will be the process if it is done normally using
polynomial division method.
But if multiplication is used it will be, X2
* X-1
= X
So the number of operations needed to be performed in
this method will be more and hence more time will be
consumed [11].
VII. RESULT
Fig.11 Division of 13 by 5 using restoring division
Fig.12 Area analysis for restoring division
Fig.13 Timing analysis for restoring division
Fig.14 Power analysis for restoring division
Fig 11, Fig 12, Fig 13, Fig 14 represents the waveform of
output of the division, area analysis, timing analysis and
power analysis respectively.
Fig.15 Division of 13 by 5 using non-restoring division

Fig.16 Area analysis for non-restoring division
Fig.17 Timing analysis for non-restoring division
Fig.18 Power analysis for non-restoring division
Fig.19 (X3
+X2
+1)/(X2
+1) using LFSR
Fig.20 Area analysis for polynomial division of (X3
+X2
+1)/(X2
+1)
Fig.21 Timing analysis for polynomial division of (X3
+X2
+1)/(X2
+1)
Fig.22 Power analysis for polynomial division of (X3
+X2
+1)/(X2
+1)
Fig23 (X7
+X6
+X5
+X4
+X2
+1)/(X5
+X4
+X2
+1) using LFSR
Fig.24 Area analysis for polynomial division of
(X7
+X6
+X5
+X4
+X2
+1)/(X5
+X4
+X2
+1)
Fig.25 Timing analysis for polynomial division of
(X7
+X6
+X5
+X4
+X2
+1)/(X5
+X4
+X2
+1)

Fig.26 Power analysis for polynomial division of
(X7
+X6
+X5
+X4
+X2
+1)/(X5
+X4
+X2
+1)
TABLE VI
Comparison of Timing, Area and Power of Restoring, Non- Restoring and Polynomial Division
Restoring Division Non-Restoring Division Polynomial Division
1101 / 101 1101 / 101 (X3
+X2
+1)/(X2
+1)
(X7
+X6
+X5
+X4
+X2
+1)/
(X5
+X4
+X2
+1)
Timing
Analysis
Maximum input arrival time
after clock-18.650ns
Maximum output required
time after clock-4.283ns
Maximum input arrival
Maximum output
required time after clock-
4.310ns
Area
Analysis
No. of Slice-18
No. of 4 input LUT- 30
No. of Slice-18
No. of 4 input LUT- 33
No. of Slice-1 No. of Slice-3
No. of Slice Flip-Flop-2 No. of Slice Flip-Flop-5
No. of 4 input LUT-2 No. of 4 input LUT-4
Power Analysis Total power- 0.083W Total power- 0.083W Total power- 0.081W Total power- 0.081W
VIII. CONCLUSION
In this paper analysis of various division algorithms are
done. Their timing, area & power comparison are obtained.
From the analysis I have seen that though the number of
LUT utilization in non-restoring algorithm is more but the
timing performance of the non-restoring algorithm is much
better. So from these to algorithm analysis it can be concluded
that the non restoring algorithm is by far better than restoring
algorithm.
Later in the next part a polynomial divider is
implemented using LFSR and the simulation is done using
Verilog. The results obtained are then compared and we see
that the delay is much less in this method. We also see that
more the maximum power of denominator more is the delay.
The power consumed by the polynomial divider circuit is also
bit less. Overall it can be said that by implementing the
polynomial division the delay and area can be minimized a
lot.
As the complexity in designing a polynomial divider
circuit is high, need for more efficient design is very high in
market. Thus seeing the growing needs my project can be
highly beneficial.
IX. FUTURE WORK TO BE DONE
Up to this point research on divider circuits like Booth‟s
algorithm, Restoring algorithm & Non-restoring algorithm are
done and a polynomial divider using LFSR‟s serially is
implemented.
So, the next work will be to implement the LFSRs in
parallel to reduce the time required for execution of the
division method.
ACKNOWLEDGEMENT
The road of knowledge is long and full of obstacles,
but going over those obstacles is what ultimately enlightens
the individual and strengthens ones spirit. I have been very
fortunate to walk through the road of knowledge, not alone,
but accompanied by the brightest and warm hearted individual
I have ever met.
I take this opportunity to thank my project guide Sri
Anindya Sen (Phd) for guiding me in this paper.

He has been a great source of motivation for me,
which inspired me a lot to take up the project and deliver it in
a nicest possible way.
Finally, I would also like to thank all the faculty
members and staff of Electronics and Communication
Engineering Department for their time to time support and co-
operation, which has helped me a lot.
PURBAYAN DEB
REFERENCES
[1]. Performance analysis of various multiplication and division
algorithms for large numbers Harpreet Kaur, 2010
[2]. Patterson, David and Hennessy, John. Computer Organization and
Design - The Hardware / Software Interface. San Francisco:
Morgan Kaufmann Publishers, 1998.
[3]. Lecture 5 Multiplication and Division. ECE 0142 Computer
Organization
[4]. Division Algorithms and Hardware Implementations, Sherif Galal
Dung Pham EE 213A: Advanced DSP Circuit Design
[5]. Computer Principles And Design In Verilog HDL Yamin Li Hosei
University, Japan
[6]. https://asnaikblog.wordpress.com/video-tutorial-for-non-restoring-
method-of-division-operation/
[7]. Linear Feedback Shift Registers Theory and. Applications. Kewal
K. Saluja. Department of Electrical and Computer Engineering.
University of Wisconsin-madison
[8]. Implementation and evaluation of a polynomial-based division
algorithm Examensarbete utfört vid Elektroniksystem, Linköpings
Tekniska Högskola Av Stefan Pettersson
[9]. Polynomial Multipliers and Dividers, Shift Register Generators
and Scramblers Phil Lucht Rimrock Digital Technology, Salt
Lake City, Utah 84103
[10]. Mr. Hiren G. Patel, Dr. D.M.Patel, Mr. Milan A. Chaudhari, Mr.
Mahavirsinh A. Zala “An Automated CRC Engine” International
Journal of Engineering Research And Management (IJERM) ISSN
: 2349- 2058, Volume-1, Issue-3, June 2014, PP. 11-15
[11]. https://www.freemathhelp.com/forum/archive/index.php/t-
56823.html
[12]. Oberman, Stuart F. and Flynn, Michael J. "Division Algorithms
and Implementations." IEEE Transcation on Computers (1997):
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