Design of advanced encryption standard using Vedic Mathematics

International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2163
Volume 1 Issue 6 (July 2014) http://ijirae.com
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© 2014, IJIRAE- All Rights Reserved Page - 322
Design of advanced encryption standard using Vedic
Mathematics
Soumya Sadanandan Anjali.V
Dept. of ECE, MG University, Kerala Dept. of ECE, MG University, Kerala
Mangalam College of Engineering Mangalam College of Engineering
Abstract— This work describes about the designing of Advanced Encryption System suitable for areas requiring
maximal area minimization such as that for mobile phones. As the demand for secure transactions in banking and
such related areas is increasing, encryption and decryption using cryptography plays a very important role. Nowadays,
as majority of secure transactions occurs on smart phones and other handheld devices, an algorithm that consumes
less area and that without compromising with overall performance becomes a necessity. In order to meet this
requirement, several algorithms have been designed and implemented in the past, but each of these algorithms possess
their own shortcomings with respect to an ASIC or an FPGA implementation. The design is done using Verilog
hardware description language which provides an immediate hardware implementation possibility. The hardware
implementation of the system is faster when compared to the conventional designs. We utilize the techniques involved
in Vedic mathematics to realize the same. Comparisons are carried out with the conventional designs to state the
advantages of the proposed design.
Keywords— AES, Urdhwa Tiryakbhyam Sutra, Galois field multiplication.
I . INTRODUCTION
The encryption of data that is to be transmitted is always of major concern in wireless communication systems.
cryptographic algorithms have been proposed to encrypt and decrypt data to ensure security. It is useful to transmit and
store data through insecure networks in 2001,national institute of standard and technology (nist) replaced previous
encryption standards like des and triple des with advanced encryption standard because of its efficiency, implementation
and flexibility. The advanced encryption standard is a subset of much larger encryption algorithm known as rijndael. The
cryptographic algorithms involves encrypting the data to be transmitted or shared by means of unique keys, for
encryption and decryption, which are known only to the authorized parties, thereby ensuring data security. This serves as
a great boon to common man as well as for military applications. Several cryptographic algorithms have been discovered
and researched upon in the recent times ,giving importance to the problem of vulnerability of the algorithms especially in
applications which demand high security i.e. for smart cards, ATMs,WWW servers etc. Among these,the Advanced
Encryption Standard (AES) algorithm is one of the highly preferred algorithms as it has higher immunity towards attacks.
However, when considering the hardware implementation of the design, the AES is losing, since it involves several
complex operations implemented in the Galois Field .Also, these complex operations are iterative in nature which in turn
disturbs the speed of the encryption system and therefore increases the vulnerability. In this paper, an area efficient
architecture for performing the various operations involved in the Advanced Encryption Standard(AES) method of
cryptography is introduced. Here we make use of techniques used in ancient Vedic mathematics. Vedic mathematics is
an archaic style of mathematics which subsisted in India in 1500 B.C,and was later on brought to limelight by a famous
scholar Sri Bharathi Krishna Tirthaji between 1911 and 1918.He systemized it into 16 simple sutras, which are used by
most of the researchers and mathematicians due to its ease of use.Out of the 16 formulae available in Vedic Mathematics,
the Urdhwa Tiryakbhyam Sutra was utilized in order to address the flaws observed in the conventional mix columns
architecture utilized in AES.
II . OVERVIEW OF AES ALGORITHM
AES is a symmetric encryption block cipher which encrypts and decrypts 128 bits of electronic data in several rounds. It
was originally called Rijndael algorithm which was developed by Belgian cryptographers Joan Daemen and Vincent
Rijmen. It was later adopted by National Institute of Standards and Technology(NIST), USA in November 2001. The key
size required to encrypt the data is the sender’s choice i.e. the key size can either be 128 bits or 192 bits or 256 bits
keeping the data/plain-text fixed i.e. 128 bits. A copy of the 128 bit plain text is stored in a 4*4 matrix called the state
array, with each location supporting 1byte of the plain text. Example: if the plain text is a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a11 a12 a13 a14 a15 a16, it is arranged in a matrix format as shown in Fig. 1.

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Fig. 1. Arrangement of 128 bit plain text in a state array in a 4*4 fashion
Based on the key size, 10, 12, or 14 rounds of transformations are performed on the state matrix for key sizes of 128,
192 or 256 bits respectively, in order to encrypt it.These transformations, for a single round, include several steps such as
addition of round key, Sub-byte transformation, Shift Rows and Mix Columns. Decrypting the cipher involves inverse
transformations on the cipher but in the reverse order. The inverse transformations comprise of addition of round key,
inverse sub-byte transformation, inverse shift rows and inverse mix columns which once again constitute a single round.
The mix column operation is skipped in the last round i.e. 10th round during encryption as well as decryption. The
various steps involved , depicted in Fig. 2.
Fig. 2. Single round of AES encryption. Decryption is performed in the reverse order.
A. Add Round Key
The Add round key step involves logically XORing a round key with the state array which is in turn generated using a
main key. This is performed using a unique key expansion algorithm.
B. Subbyte transformation & inverse subbyte transformation
The data obtained after the add-round key operation is further transmogrified by means of a Subbyte
transformation.Traditionally, Subbyte transformation and Inverse Subbyte transformation is accomplished by computing
the Multiplicative Inverse of the input byte procured from the previous stage, followed by an affine transformation. In
order to reduce complexity, these steps are usually performed by simply looking up an “S-box” table which contain pre-
computed values for the multiplicative inverse and the transformation.
C. Shift rows and inverse shift rows
The Shift Rows and Inverse Shift Rows Stage mainly focus on cyclically shifting all the elements of a particular row of
the matrix obtained after Subbyte transformation and Inverse Subbyte transformation respectively, keeping the elements
of the 1st row of the state matrix unoperated.
D. Mix columns and inverse mix columns
Mix columns (MC) and inverse mix columns (IMC) are implemented by performing matrix multiplication over Galois
field i.e. GF(28) using the irreducible polynomial x^8 + x^4 + x^3+ x+ 1. The constant matrices used for mix columns
and inverse mixcolumns are unique and are defined by the FIPS.

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E. Mix column using vedic mathematics
One of the crucial mathematical operation performed during the mix column step in AES,is the Galois field
multiplication.Multiplication,being a tedious and a power hungry operation,causes the computation of mix columns and
its inverse to be an even more difficult task.This is due to the fact that,it involves matrix multiplication.Therefore,there
arises a necessity to ease the entire process of mix columns.In order to achieve the same,the Urdhwa Tiryakbhyam Sutra
of Vedic Mathematics,is utilized in our proposed architecture for mix columns and its inverse due to its excellence in
terms of speed and area.
III. MULTIPLER ARCHITECTURE
The Multiplier Architecture is based on the Vertical and Crosswise algorithm of ancient Indian Vedic Mathematics. In
the overlay architecture, grouping of 4 bits at a time is done for both multiplier and multiplicand and thereafter vertical
and crosswise algorithm is applied to decompose the whole of the multiplication operation into 4x4 multiply modules.
The algorithm is explained in Table-1 for 16x16 bit number. After getting the sub-product bits in parallel from the
4x4 multiply modules, we can employ an efficient method of addition to generate the final 32 bit product. This method
can be generalized for NXN bit multiplication where N is a multiple of 4 such as 8,12,16,20,24,....4n. Thus instead of
implementing the entire multiplication through a single NXN bit multiplier, we can get the same product efficiently by
using the proposed overlay architecture. The advantage of this is that the multiply operation of large number of bits can
now be performed by using smaller and efficient 4x4 multiplier.
TABLE 1- 8 x 8 bit Vedic multiplier Using Urdhva Tiryakbhyam.
CP- Cross Product (Vertically and Crosswise)
A= A7 A6 A5 A4 A3 A2 A1 A0
X1 X0
B= B7 B6 B5 B4 B3 B2 B1 B0
Y1 Y0
X1 X0 Multiplicand [8 bits]
Y1 Y0 Multiplier [8 bits]
------------------------------------------------------------------
F E D C
P3 P2 P1 P0 Product [16 bits]
Where X1, X0,Y1 and Y0 are each of 4 bits.
Note: Each Multiplication operation is an embedded parallel 4x4 multiply module

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IV. SIMULATION RESULTS
The algorithm for the Advanced Encryption Standard using Vedic Mathematics technique was designed and simulated in
Verilog HDL using Xilinx ISE.The design of the conventional implementation of the AES was also carried out in order
to compare with the proposed design.The designing was carried out so as to be implemented and synthesized for a
Spartan 3e series XC3s1600e Xilinx FPGA.The table below shows the results of the simulation proposed design,after
comparison with the conventional counterpart.
We can find that the proposed design provides savings in overall area required for implementation when compared with
the conventional design,with only a small increase in the timing requirement.
V. CONCLUSION
In this paper,an area efficient design of 128 bit advanced encryption standard that is suitable for carrying out
cryptographic applications is done.The architecture of design performs well when compared with the conventional
designs.The design provided good savings in the overall area with only neglectable increment in the timing requirement
that proves its applicability in mobile devices.The optimizations can be extended towards the design of Sub byte
operation in the future.
VI. ACKNOWLEDGEMENT
My sincere thanks to Peiyi Zhao,Member IEEE.
VII. REFERENCE
[1] Huang, Xu, Shirantha Wijesekera, and Dharmendra Sharma. "Quantum cryptography for wireless network
communications." Wireless Pervasive Computing, 2009. ISWPC 2009. 4th International Symposium on. IEEE,2009.
[2] Himanshu Thapliyal and M.B Srinivas,VLSI Implementation of RSA Encryption System Using Ancient Indian Vedic
Mathematics,Center for VLSI and Embedded System Technologies,International Institute of Information Technology
[3] H Thapliyal,and H R Arabnia,A time area-power efficient multiplier and square architecture based on Ancient Indian
Vedic Mathematics, Proceedings of the 2004 International Conference on VLSI,June 2004,pp. 434-9.
[4] M C Hanumantharaju,H Jayalaxmi,R K Renuka,and M Ravishankar,A high speed block convolution using Anciet
Indian Vedic Mathematics,International Conference on Computational Intelligence and Multimedia.
Algorithm Area occupancy(%) No.of gates Timing(ns)
Conventional AES 7.35 1210 11.816
Proposed AES using Vedic Technique 5.11 731 12.33

Design of advanced encryption standard using Vedic Mathematics

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Design of advanced encryption standard using Vedic Mathematics