A novel block cipher involving keys in a key bunch

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 02 | Feb-2014, Available @ http://www.ijret.org 694
A NOVEL BLOCK CIPHER INVOLVING KEYS IN A KEY BUNCH
MATRIX AS POWERS OF THE PLAINTEXT ELEMENTS
K. Anup Kumar1
, V.U.K Sastry2
1
Associate Professor, 2
Director R & D, CSE Department, SNIST, A.P, India
Abstract
In this analysis, we have developed an asymmetric block cipher which is involving a key bunch matrix A ( = [aij]) in the process of
encryption, and B ( = [bij]) in the process of decryption. The keys aij are used as powers of the plaintext elements, and the keys bij are
used as the powers of the ciphertext elements. Here, we have made use of Euler’s totient function and Euler’s theorem in the
development of the cipher. The cryptanalysis clearly shows that the strength of the cipher is quite significant.
Keywords: Encryption, Decryption, Key, Plaintext, Cipher text and Cryptanalysis.
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1. INTRODUCTION
The literature of Cryptography [1] is replete with a number of
block ciphers, in which majority of ciphers are symmetric and
a few are asymmetric. The popular symmetric ciphers are Hill
cipher [2], Feistel cipher [3], Data Encryption Standard (DES)
[4], Advanced Encryption Standard (AES) [5] and several
variants of these ciphers. The well known asymmetric cipher
is RSA [6] developed by Ron Rivest et al.
In a recent investigation, Sastry and Sirisha [7] have
developed a block cipher in which the encryption is carried
out by a key bunch matrix E ( = [eij ]), and the decryption is
done by using another key bunch matrix D ( = [dij ]) which is
obtained by using the relation
(eijxdij) mod 256 = 1, (1.1)
in which both eij and dij are odd integers lying in the interval
[1,255]. Here it is to be noted that the encryption key and the
decryption key are different though they are related in a
particular way. In all the ciphers, developed basing upon the
cipher under consideration, the keys in the key bunch matrix
are used as multiplicands of the plaintext elements.
In RSA, we have two keys -- one key called public key and
another one called private key. The encryption is carried out
by using the public key of the receiver and the decryption (at
the receiver’s end) is done by using his own private key. Here
it is to be noted that, the public key {e, n} and the private key
{d, n}, in which, n is the product of two distinct prime
numbers p and q, and e and d are two positive integers
governed by the relation
ed mod Φ(n) = 1 (1.2)
Where,
Φ (n) = (p -1) (q - 1). (1.3)
In the RSA, both e and d are used as exponents, one in the
process of encryption and the other one in the process of
decryption.
In the present investigation our objective is to develop a block
cipher, wherein, we use the keys in a key bunch matrix, say A
= [aij], as powers of the plaintext elements (P = [pij]) in the
process of encryption, and the corresponding keys (B=[bij])
are used as powers of the ciphertext elements, C=[cij], in the
process of decryption.
The basic concepts utilized in the development of the cipher
under consideration can be summarized [1] as follows. When
n is a prime number, the number of integers less than n can be
obtained in the form of the Euler’s totient function [1] given
by
Φ(n) = n-1. (1.4)
On the other hand, when n is the product of two prime
numbers, say p and q then
Φ(n) = Φ(pq) = (p-1)(q-1). (1.5)
The Euler’s theorem well known in the literature of
Cryptography is given by
a Φ(n)
mod n = 1. (1.6)
On using this theorem, we readily prove that
p kΦ(n)+1
mod n = p mod n (1.7)

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Let us now take the basic equations of a block cipher in the
form
C = Pe
mod n (1.8)
and
P = Cd
mod n. (1.9)
From (1.8) and (1.9), we get
P = Ped
mod n. (1.10)
On writing ed = kΦ(n)+1, (1.11)
we find that (1.8) and (1.9) are valid relations of the cipher in
view of (1.7). From (1.11), we find that
ed mod Φ(n) = 1 (1.12).
In the light of the above discussion, the basic equations
governing the cipher are given by (1.8), (1.9) and (1.12).
In the present paper, our interest is to develop a block cipher
wherein n is a prime number. Thus in view of the relation
(1.4), equation (1.12) assumes the form,
ed mod (n – 1) = 1 (1.13)
In this analysis, we take n=257 as 257 is the nearest prime
number which encompasses all the EBCIDIC codes, which are
lying in [0-255].
In what follows we present the details of the organization of
this paper. In Section 2, we deal with the development of the
cipher. In this we display the flowcharts and the algorithms
describing the cipher. In Section 3, we present an illustration
of the cipher and examine the avalanche effect. Section 4 is
devoted to the study of the cryptanalysis. Finally in Section 5,
we mention the computations carried out in this analysis and
draw conclusions.
2. DEVELOPMENT OF THE CIPHER
Consider a plaintext. On using the EBCIDIC code, this can be
written in the form of a matrix P given by
P = [pij], i= 1 to m, j = 1 to m (2.1)
in which each pij is an integer lying in [1-255].
Let
A = [aij], i= 1 to m, j = 1 to m (2.2)
be the encryption key bunch matrix,
and
B = [bij], i= 1 to m, j = 1 to m (2.3)
be the decryption key bunch matrix.
Here aij and bij are governed by the relation
(aijxbij) mod 256 = 1 (2.4)
as we have taken n=257.
The basic equations governing the cipher can be written in the
form,
C = [cij] = [pij
aij
] mod 257 (2.5)
and
P = [pij] = [cij
bij
] mod 257. (2.6)
The flowcharts and the algorithms concerned to the encryption
and the decryption can be presented as shown below.
Fig 1 The process of Encryption
for k = 1 to r
for i = 1 to m
pij = (pij
aij
) mod 257
P=Mix (P)
C = P
for j = 1 to m
P= [pij]
Write (C)
Read P, A, r and m

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Fig 2 The process of Decryption
2.1 Algorithm for Encryption
1. Read P, n, A, r and m
2. for ( k = 1 to r )
begin
for ( i = 1 to m )
begin
for ( j = 1 to m )
begin
pij = (pij
aij
) mod 257
end
end
P = [pij]
P = Mix(P)
end
3. C = P
4. Write (C)
2.2 Algorithm for Decryption
1. Read C, n, B, r and m
2. for ( k = 1 to r )
begin
C = IMix (C)
for ( i = 1 to m )
begin
for ( j = 1 to m )
begin
cij = (cij
bij
) mod 257
end
end
C = [cij]
end
3. P = C
4. Write (P)
In this analysis, r represents the number of rounds in the
iteration process. Here we have taken r=16. The function Mix
( ) which is used in each round of the iteration process can be
explained as follows.
Let P = [pij], i = 1 to m, and j = 1 to m be the plaintext at a
particular round of the iteration process. Let us suppose that
there are ‘s’ elements in the matrix P which are having their
values as 256. Then the (m2
– s) elements of P, which are not
equal to 256, can be written in the form of a matrix having 8
rows and (m2
-s) columns. On writing the binary bits in each
column as a decimal number we get (m2
-s) decimal numbers.
These numbers can be placed, in a row wise manner, in all the
other places, one after another, excluding the places wherein
256 is present, we get, a new matrix P of size mxm. Thus
mixing is done in a perfect manner.
3. ILLUSTRATION OF THE CIPHER
Consider the plaintext given below
Respected uncle,
I am very much delighted to inform you that the Andhra
Pradesh is now getting partitioned into two parts. The
Hyderabad which was ruled earlier by our own people, now
clubbed with another nine districts, is going to be carved as
Telangana, and the rest of the thirteen districts are going to be
branded as Seemandhra. This bifurcation is similar to the
disintegration as Pakistan and India which we had just before
Independence. We cannot forget the instance in which
Pakistan was divided into two segments, and the role played
by India in that bifurcation. This division of Andhra Pradesh is
certainly going to help our community. With regards, Your
Mohammad. (3.1)
Read C, B, r and m
for k = 1 to r
for i = 1 to m
cij = (cij
bij
) mod 257
C= IMix (C)
Write (P)
for j = 1 to m
C = [cij]
P = C

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Consider the string of the first 16 characters of the plaintext
(3.1) given by
Respected uncle,
On using the EBCIDIC code we get
082 101 115 112
101 099 116 101
100 032 117 110
099 108 101 044
Let us now choose the encryption key bunch matrix A in the
form
021 057 171 039
101 067 089 223
067 157 171 001
037 203 233 017
On using the relation (2.4), the decryption key bunch matrix B
can be obtained in the form
061 009 003 051
109 107 233 031
107 181 003 001
173 227 089 241
On applying the encryption algorithm the ciphertext C can be
obtained as
011 049 137 225
237 042 153 007
115 036 177 086
059 123 231 136
On using the decryption algorithm, given in Section 2, we get
back the original plaintext P.
Now let us study the avalanche effect.
On replacing 82, the first row first column element of the
matrix P by 83 we get a one bit change in the plaintext. On
using this modified plaintext, the key given by (3.4), and the
encryption algorithm, given in Section 2, we get the ciphertext
C in the form
085 068 132 236
239 093 149 020
115 142 133 146
005 068 244 116
On comparing (3.6) and (3.7), after converting them into their
binary form, we notice that these two ciphertexts differ by 58
bits out of 128 bits. This shows that the cipher is expected to
be a strong one.
Now let us consider a one bit change in the key A. This is
achieved by replacing the first row second column element of
A by 56. On using this modified key, the original plaintext and
the encryption algorithm, we get the corresponding ciphertext
C given by
073 157 180 120
006 255 023 208
040 153 162 075
114 219 124 094
On comparing (3.8) and (3.6), brought into their binary form,
we find that these two ciphertexts differ by 69 bits out of 128
bits. From this also, we conclude that this cipher is a potential
one.
4. CRYPTANALYSIS
In the development of every cipher cryptanalysis plays a
predominant role in deciding whether a cipher is having
sufficient strength or not. The basic attacks that are available
in the literature of the Cryptography are
1. Ciphertext only attack (Brute force attack)
2. Known Plaintext attack
3. Chosen plaintext attack
4. Chosen ciphertext attack
Usually, every cipher is designed so that it cannot be broken,
atleast, by the first two attacks [1]. The first two attacks are
thoroughly studied by offering analytical proofs. However, the
strength of the last two attacks is also to be decided by
applying all possible intuitive approaches.
Let us now consider, the ciphertext only attack (Brute force
attack). In the case of this attack the ciphertext is known.
Keeping this ciphertext in view we have to determine the key
which leads to a sensible plaintext.
In the key bunch matrix A=[aij], each aij is an odd integer
lying in the interval [1-255] as its value. Thus there are 128
possible values for aij. In the matrix used in this analysis, as
we have m2
elements the size of the key space is
m2
7m2
0.7m2
2.1 m2
128 = 2 = 210 ≈
10.
On assuming that the time required for the computation of this
cipher with one value of the key in the key space is 10-7
seconds, the time required for the computation with all
possible keys in the key space is approximately equal to
2.1 m2
-7 2.1m2
-15
10 x 10 = 3.12x 10 years.
365x24x60x60
P =
A = (3.4)
B = (3.5)
C =
(3.6)
C =
(3.7)
C = (3.8)(3.2)

__________________________________________________________________________________________
In this analysis as we have taken m=4, the time required for
this cipher is
3.12 x 1018.6
years.
As this number is very large, we cannot break the cipher by
the ciphertext only attack.
Now consider the known plaintext attack. To proceed with this
we know the pairs of plaintext and ciphertext, as many as we
require. If we assume that we are carrying out only one round
of the iteration process, that is if take r = 1, the basic equations
involved in the cipher are
pij = (pij
aij
) mod 257, (4.1)
P = [pij], (4.2)
P= Mix(P), (4.3)
C = P. (4.4)
Here we know the pij, occurring in the right hand side of (4.1),
and the C present in the left hand side of (4.4). As this C is
known, we know the P occurring on the left hand side of (4.3).
On using the IMix ( ), we readily obtain the pij which is
occurring on the right hand side of (4.2). Thus we know the pij
which is on the left hand side of (4.1). As we know the pij
which is on the right hand side of (4.1) the key A (= [aij] ) can
be determined by making attempts with all the possible values
of aij. Thus this cipher can be broken by the known plaintext
attack when r=1.
Let us consider the cipher when r = 2. In this case the basic
equations governing the cipher are given by
pij = (pij
aij
) mod 257, (4.5)
P= [pij], (4.6)
P= Mix(P), (4.7)
pij = (pij
aij
) mod 257, (4.8)
P = [pij], (4.9)
P = Mix(P), (4.10)
C = P. (4.11)
Here also we know the initial plaintext (pij occurring in the
right hand side of (4.5)), and the ciphertext C in (4.11). On
using this C and the IMix ( ) we can find the pij occurring in
(4.9), this gives the pij which is on the left hand side of (4.8).
From here, we cannot proceed further in the upward direction.
Though pij, occurring on the right hand side of (4.5), is known
to us, we are stuck up at this stage as the other quantities
involved here are not known. In the light of the above facts the
key A ( = [aij] ) cannot be determined by any means. Hence
the cipher is unbreakable by the known plaintext attack when r
is equal to 2. This implies that the strength of the cipher is
quite significant as we have taken r=16.
Now let us explore what will happen in the case of the chosen
plaintext attack and in the case of the chosen ciphertext attack.
On inspecting the equations (4.5) to (4.11), and using our
intuition in an effective manner, we readily notice that this
cipher cannot be broken either by the chosen plaintext attack
or by the choosen ciphertext attack.
From the above analysis, we firmly conclude that this cipher is
a strong one, and it cannot be broken by any cryptanalytic
attack.
5. COMPUTATIONS AND CONCLUSIONS
In this paper, we have developed an asymmetric block cipher
in which we are having a key bunch matrix A ( = [aij]) for
encryption, and another key bunch matrix B ( = [bij]) for
decryption, wherein, these two matrices are connected by the
relation
(aij x bij) mod 256 =1 . (5.1)
In this analysis, we have used the keys in the key bunch matrix
A as powers of the plaintext elements in the process of
encryption, and the keys in the key bunch matrix B as powers
of the ciphertext elements in the process of decryption.
Basing upon the algorithms, given in Section 2, we have
written the programs for the encryption and the decryption in
C language.
From the cryptanalysis carried out in this investigation, we
have seen that the strength of the cipher is remarkable as the
powers are playing a vital role, and the Mix ( ) function is
supporting further very thoroughly.
REFERENENCES
[1]. William Stallings, Cryptography and Network Security,
Principles and Practice, Third Edition, Pearson, 2003.
[2]. Jack Levine and Richard Chandler, “The Hill
Cryptographic System with Unknown Cipher Alphabet, But
Known Plaintext”, Cryptologia 13:1, pp. 1 – 28, Jan 1989.
[3]. Feistel H, “Cryptography and Computer Privacy”,
Scientific American, Vol. 228, No. 5, pp. 15 – 23, 1973.
[4]. National Bureau of Standards NBS FIPS PUB 46, “Data
Encryption Standard (DES)”, National Bureau Standards, US
Department of Commerce, Jan 1977.
[5]. Daemen J and Rijmen V, “Rijndael, the Advanced
Encryption Standard (AES)”, Dr. Dobbs’ Journal, Vol. 26, No.
3, pp. 137 – 139, Mar 2001.

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[6]. Diffie,W. and Hellman, M. New directions in
Cryptography, IEEE Transactions Information Theory IT – 22,
(Nov, 1976) 644-654. people.csail.mit.edu/rivest/Rsapaper.pdf
[7]. V.U.K Sastry and K. Shirisha “ A Novel Block Cipher
Involving a Key Bunch Matrix” IJCA International Journal of
computer Applications (0095-8887), Vol 55 No.16, October
2012.
BIOGRAPHIES:
Dr. V. U. K. Sastry is presently working
as Professor in the Dept. of Computer
Science and Engineering (CSE), Director
(SCSI), Dean (R & D), SreeNidhi
Institute of Science and Technology
(SNIST), Hyderabad, India. He was
Formerly Professor in IIT, Kharagpur,
India and worked in IIT, Kharagpur
during 1963 – 1998. He guided 12 PhDs, and published more
than 90 research papers in various international journals. His
research interests are Network Security & Cryptography,
Image Processing, Data Mining and Genetic Algorithms.
Mr. K. Anup Kumar is presently
working as an Associate Professor in the
Department of Computer Science and
Engineering, SNIST, Hyderabad India. He
obtained his B.Tech (CSE) degree from
JNTU Hyderabad and his M.Tech (CSE)
from Osmania university, Hyderabad. He
is now pursuing his PhD from JNTU,
Hyderabad, India, under the supervision
of Dr. V.U.K. Sastry in the area of Information Security and
Cryptography. He has 12 years of teaching experience and his
interest in research area includes Cryptography,
Steganography and Parallel Processing Systems.

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