Implementation of Low-Complexity Redundant Multiplier Architecture for Finite Field

International Journal on Cybernetics & Informatics (IJCI) Vol. 5, No. 4, August 2016
DOI: 10.5121/ijci.2016.5436 333
IMPLEMENTATION OF LOW-COMPLEXITY
REDUNDANT MULTIPLIER ARCHITECTURE FOR
FINITE FIELD
JyothiLeonoreDake1
,Sudheer Kumar Terlapu2
and K. Lakshmi Divya3
1
M.Tech-VLSID,ECE Department, SVECW (Autonomous),Bhimavaram, India
2
Associate Professor, ECE Department, SVECW (Autonomous), Bhimavaram, India
3
Assistant Professor, ECE Department, SVECW (Autonomous),Bhimavaram, India
ABSTRACT
In the present work, a low-complexity Digit-Serial/parallel Multiplier over Finite Field is proposed. It is
employed in applications like cryptography for data encryption and decryptionto deal with discrete
mathematical andarithmetic structures. The proposedmultiplier utilizes a redundant representation because
of their free squaring and modular reduction. The proposed 10-bit multiplier is simulated and synthesized
using Xilinx VerilogHDL. It is evident from the simulation results that the multiplier has significantly low
area and power when compared to the previous structures using the same representation.
KEYWORDS
Digit-Serial, Finite Field multiplication, Redundant Basis.
1. INTRODUCTION
In cryptography and coding theory, there are many applications using arithmetic operations for
Finite Field [1]. In general, multiplication is extremely expensive in terms of time delay and
physical area. Therefore, more focus is concentrated on designing high speed multipliers and on
reduction of area [2]. The complexity mainly depends on representation of field elements. The
most commonly used basisincludes polynomial (PB), normal (NB), dual (DB) and redundant
(RB) [3]. Dual and normal basis multipliers require a conversion of basis, in which heavily rely
on the simplifiedpolynomial. There is no need of conversion of basis in case of standard basis
multipliers, the most commonly used multiplier is the polynomial basis multiplier due to their
simple design and which also provide scalability.
Redundant basis (RB) is attractive when performing exponentiations and squaring operations [5].
The major advantage of redundant basis is squaring operation, as normal basis and also involves
lower computational complexity. The multipliers of finite field are designed and classified into
full parallel multipliers and word level multipliers [6]. The hardware used and power required by
the bit-serial multipliers is less but it is slow.
Tounderstand the complication between area and speed, the Digit-Serial multipliers are reported
previously. These are scalable multipliers and classified into different forms .An effective
multipliers which utilizes RB is presented previously. Multipliers with systolic structures are
presented in [7].A comb architectures are also presented formerly. Word level multipliers over
finite field with high speed are also reported .And several other multipliers are also been
developed for reducing complexity.

334
In this paper, a low-complexity digit-serial/parallelis presented by utilizing a redundant basis over
finite field (GF (2m
)).The recursive decomposition scheme for digit-serial/parallel multipliers is
same as the previous, where the multiplier is modified. In his work, a low-complexity multiplier
is introduced which involves significantly low area and power complexities when verified with
the previous techniques.
Organization of the paper is as follows: Review of existing digit-serial RB multiplier is presented
in section 2. Proposed digit-serial RB multiplier mentioned in section 3. Implementation and
Comparison are shown in section 4. The paper ends with conclusion in section 5.
2. EXISTING DIGIT SERIAL RB MULTIPLIER
In Digit serial RB multiplier [4] the input operands A and B are divided into the number of
integers to attain Digit Serial Multiplication, to achieve the final product the partial products are
summed.
Assume x is an nth root of unity, components in Finite Field GF (2m
) are often described within
the form:
A=a0+a1x+a2x2
+ ⋯+an-1xn-1
(1)Which ai belongs to GF (2), for 0 ≤i ≤ n - 1, alike the set {1, x,
x2
,⋯ , xn-1
} is outlined as the RB for Finite Field components, wherevern could be a positive
number not below m.
And just then (m + 1) is prime and number 2 is a primitive root modulo (m + 1), for a finite field,
there being a type I Optimal Normal Basis (ONB) [8].X is component of GF (2m
), &n= m+1
Let A, Bbelongs toGF (2m
) can be demonstrated in the form of RB:
= (2)
= (3)
Thus ai ,bibelongs to GF(2). , Let A and B are input operands which obtain product C, is
demonstrated as follows
C= A. B = . (4)
= ∑ ∑ ( )
(5)
= ∑ ∑ ( ) (6)
= ∑ ∑ ( ) (7)
Where (i – j)n denotes modulo n reduction. Define C =∑ , where ci∈ GF (2), we have
[10].
ci= ∑ ( ) (8)

335
3. PROPOSED DIGIT-SERIAL RB MULTIPLIER
In Digit serial RB multiplier [4], to attain Digit Serial Multiplication both the inputs are divided
into a number of units and the partial products related to these units are summed to achievethe
desired product.
Considering equations (1) and (7) of Digit serial RB Multiplier
Where (i – j)n denotes modulo n reduction. Define C in the form of:
⋮
= !
⋯
⋮
⋯
⋮ ⋱
#
⋮
# ⋯
$
⋮
(9)
From (9), shifted form of the inputs bits B can be defined as follows
= = + + ⋯ + (10)
= = + + ⋯ + # (11)
⋯ ⋯ ⋯
= = + # + ⋯ + (12)
Where,
=
= , for 1 ≤j ≤ n – 2. (13)
The recursions on (13) can be extended further to have
&
='
& ,
&
) *+, 0 ≤ . ≤ / − 2
+2ℎ4, 5674
(14)
Where 1 ≤ 7 ≤ / − 1, Let P and Q are two integers alike n = QP + r, where 0≤ r ≤ :.for ease,
assume that r = 0and divide the input of A into Q units of vectors operands Au, where u= 0,1….Q-
1follows:
A0 = [a0 aQ ⋯ an-Q] (15)
A1 =[a1aQ +1 ⋯an – Q+1] (16)
⋯ ⋯ ⋯
AQ-1=[aQ-1a2Q-1 ⋯ an-1] (17)
Identically, we produce the Q units of shifted vector operandsBu , where u= 0, 1, ⋯ ,Q -1, follows:

336
B0 = [ B0
BQ
⋯Bn- Q
] (18)
B1= [B1
BQ+1
⋯ B n-Q+1
] (19)
⋯ ⋯ ⋯
BQ -1= [BQ-1
B2Q-1
⋯ Bn-1
]. (20)
The product C =AB which is obtained from (9) are broken down intoproducts Q of vectors Au
and Bu, where u = 0, 1, ⋯ , Q-1 as:
C = AB =B0
;
+B1
;
+ ⋯ + BQ-1 <
;
=∑ = >
;<
= =∑ ?<
= u(21)
Where ?= denotes
?== = >
;
(22)
Note that Au for u =0, 1, ⋯ , Q -1 is a P point bit – vector. Bu for u = 0, 1, ⋯ , Q – 1 is a P bit-
shifted forms of operand B.
The proposed structure shown in below Fig.1 is derived from the processor space flow graph in
[4], consists of S nodes, M nodes and A nodes which S node performs shifting operation and M
node, A node performs multiplication and addition operation.
The proposed Digit- serial RB multiplier consists of three block, bit-permutation block, partial
product generation block and finite field accumulator block. The BPB performs the rewriting of
inputs of B to consume the output according the shifting S node. The PPGM consists of AND cell
which performs the multiplication operation and XOR cell which performs addition operation.
And finite field accumulator blockconsistentwith n-bit parallel accumulation units. The recent
input which is received is added with past accumulated result, and the sum is retain in the register
cell and used in the next cycle. And successive output is obtained.Fig.2 shows the structure of
finite field accumulator which consists of XOR cell and register cells with n parallel input bits
and n parallel outputs bits.
In Fig.1 AND cell performs the multiplication of A input bits with the B input bits by bit-shifting
form, XOR cell performs an addition operation of the outputs obtained from the ANDcells, the
operation can be done concurrently and the partial products obtained at the XOR cell.

337
Figure 1.Proposed Digit-serial RB multiplier
Figure 2.Structure of finite field accumulator
The partial products generated are fed to the finite field accumulator and result is accumulated
and stored in register cell of the finite field accumulator and then finally the desired output is
obtained. The partial products generatedin this multiplier are lesser in numbers than those
previous multipliers, and also reduces the area complexities and reduce in power.
4. IMPLEMENTATION AND COMPARISON
The proposed digit-serial RB multiplier for 10-bit is coded using Verilog HDL in Xilinx ISE
12.2.The simulatedresults for the proposed structure are shown in Fig.3. In the above waveform
the inputs are a and b and the output is c. when the 10-bit inputs a=0000000100 and
b=0000000011 are given, by performing the shifting operation of the input operand b and
multiplying with the each bit of the operand a, then each value is accumulated and stored
inaccumulator to obtain the desired output c=00000000000000001100. And the remaining values
shown are the signals.
b0,b1, ,bn-1
finite field accumulator
C
aQ-1, ,a0
aQ , ,a2Q-1
S S
1 1n
n
n n
n
n
n
BPM
PPGM
R R R R
1
1 1 1
1 111
accumulatorn parallel input bits
n parallel output bits

Figure 3. Simulation waveform of 10
Figure 4. RTL Schematic
Figure 5. Detailed RTL Schematic
Simulation waveform of 10-Bit Proposed Digit-Serial RB multiplier
RTL Schematic of 10-Bit Proposed Digit Serial RB multiplier
Detailed RTL Schematic of 10-Bit Proposed Digit-Serial RB multiplier
338
erial RB multiplier

339
The RTL schematic of 10-bit Digit-Serial RB multiplier is shown in Fig.4,contain inputs a,b and
clock and output c. Here a and b are the 10-bit inputs and which obtain the 20-bit output. The
detailed view of 10-bit proposed digit-serial RB multiplier is in Fig.5, which gives the clear
explanation of logic required.
Table 1.Comparison table
Structures Area(Number of slices) Power(W)
Structure-I [4] 75 0.066
Structure-II [4] 105 0.067
Proposed Structure 73 0.065
The 10-bit Structure-I, 10-bit structure-II [4] and 10-bit proposed digit-serial multiplier is
implemented in Xilinx ISE .A table is formulated to show the results. The number of slices used
in each structure is estimated and tabulated in table 1.This comparison table indicates the
reduction in area .In similar manner; the power is also estimated and compared. The comparison
result for number of slices that is area and power are also shown in fig.6 as a graphical plot for
better comparison.
Figure 6. Area and Power comparison
5. CONCLUSION
The proposed 10-bit digit-serial RB multiplier is implemented. It is evident from comparison
table that the performance of the proposed architecture is good with respect to speed and the area.
Theproposed 10-bit digit-serial multiplier using redundant basis is used based on application
requirement and mostly in modern cryptographic applications. The proposed multiplier is derived
to obtain less complexity than the previous multipliers.
0.064
0.0645
0.065
0.0655
0.066
0.0665
0.067
0.0675
0
20
40
60
80
100
120
Structure-I Structure-II Proposed
Area
Power

340
REFERENCES
[1] Swamy.M.N, May (2007) “Cryptographic applications of bhaskara equations” IEEE Trans.Circ.Sys.I,
vol.54, no.7, pp. 927-928.
[2] M.Nikooghadam, March (2013) “ Low Power and High-Speed Design of a Versatile Bit- Serial
Multiplier in Finite Field, the VLSI journal, vol 46, Issue 2, pp.211-217.
[3] L.S.Hsu, H.M.Shao, (1987) “Comparison of VLSI Architecture of Finite Field Multipliers Using
Dual, Normal or Standard basis,” IEEE Tran. Comp, pp.63-75.
[4] Zhi-Hong Mao, J.Xie, Jan (2015) “High-throughput finite field multipliers using redundant basis for
FPGA and ASIC implementation,” IEEE Trans. Cirt. Sys-I, vol.62, no.1, pp.110-119.
[5] M.Uma.Maheswari, S.Bhaskar, November (2014) “High Speed Finite Field Multiplier GF(2m) for
Cryptographic Applications,” IJARECE, vol.3, Issue 11, pp.1705-1708.
[6] B.Sargunam, Dr.R.Dhanasekaran, April (2014)“Word Level Finite Field Multipliers Using Normal
Basis,” JTAIT, vol.62, no.3, pp.805-811.
[7] J-S.Pan, Mehar. P.K,December(2013) “Low latency digit serial digit parallel systolic multipliers for
large binary extension fields,” IEEE Trans. Circt.andSyst-I, pp.1-11.
[8] I.-C.Jou, C.-Y Lee, Sep. (2005) “Bit-Parallel Systolic Montgomery Multipliers for Special Classes of
GF(2m) ,” IEEE Trans. Compt., vol.54,no.9, pp.1061-1070.

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