Secure Linear Transformation Based Cryptosystem using Dynamic Byte Substitution

Adi Narayana Reddy K & Vishnuvardhan B
International Journal of Security (IJS), Volume (8) : Issue (3) : 2014 24
Secure Linear Transformation Based Cryptosystem using
Dynamic Byte Substitution
Adi Narayana Reddy K aadi.iitkgp@gmail.com
Dept. of CSE & HITAM
Gowdavelli, Medchal, Hyderabad,India
Vishnuvardhan B
Dept. of IT & JNTUH College of Engineering
Nachupally, Kondagattu, Karimnagar, India
Abstract
Many classical cryptosystems are developed based on simple substitution. Hybrid cryptosystem
using byte substitution and variable length sub key groups is a simple nonlinear algorithm but the
cryptanalyst can find the length of each sub key group because the byte substitution is static and
if the modulo prime number is small then byte substitution is limited to first few rows of S-box. In
this paper an attempt is made to introduce the nonlinearity to the linear transformation based
cryptosystem using dynamic byte substitution over GF (2
8
). The secret value is added to the
index value to shift the substitution to a new secret location dynamically. This adds extra security
in addition to non-linearity, confusion and diffusion. The performance evaluation of the method is
also studied and presented.
Keywords: Dynamic Byte Substitution; Hill Cipher; Pseudo-random Numbers; Sub Key Groups.
1. INTRODUCTION
The information must be secure while transmission. Cryptography is one of the methods to attain
security. The encryption and decryption process converts intelligible information into unintelligible
form and vice versa. There are various encryption algorithms to provide security for the
information. Traditional symmetric key ciphers use substitution, in which each character is
replaced by other character. The mathematician Lester S. Hill invented the Hill cipher in 1929. Hill
cipher is a classical substitution technique that has been developed based on linear
transformation. Many modifications improved the security of the Hill cipher but all modified
algorithms are linear. The nonlinearity is one of the measures to make the modified algorithms
more secure. The byte substitution using S-box provides nonlinearity. Consider the table of S-box
of size 16 x16, contains the permutation of all possible values of 8 bits with the leftmost 4 bits as
row value and the rightmost 4 bits as a column value. The row and column values serve as
indices into the S-box to select a unique 8-bit output. The dynamic byte substitution shifts the
substitution to a new secret location. The efficiency of the algorithm is one of the most important
aspects to be studied.
Hill cipher was developed based on simple linear transformation. It is easy to implement, the
processing speed is high and high throughput. But it is vulnerable to known plaintext attack and
the inverse of every shared key matrix may not exist all the time. It is a simple traditional
symmetric key cipher algorithm. In Hill cipher, the plaintext to be transmitted through the insecure
communication channel is partitioned into ‘m’ groups, each of size ‘n’ and these partitioned
groups are called blocks. Assume that both ‘n’ and ‘m’ are positive integers and Mi is the ith
partitioned block. Transform each of the block Mi, one at a time using secret key matrix. Map
each character with unique numeric value like A = 0, B = 1 ... to produce the ‘n’ characters in
each of the partitioned block. The ith cipher text block Ci can be obtained by encrypting the ith
plaintext block Mi using Equation (1) as:

i iC M K mod m= (1)
In which K is an n×n key matrix. The plain text can be obtained from the decrypted cipher text
using Equation (2) as:
1
i iM C K mod m−
= (2)
In which K-1 is the key inverse and it exist only if the GCD (det K (mod m), m) = 1. According to
Overbey [12] the key space of the Hill cipher is precisely GL (n, Zm) - the group of n x n matrices
that are invertible over Zm for a predetermined modulus m and the key space of a prime modulus
is larger than composite modulus.
Many researchers improved the security of linear transformation based cryptosystem. Yeh, Wu et
al. [21] presented an algorithm which thwarts the known-plaintext attack, but it is not efficient for
dealing bulk data, because too many mathematical calculations. Saeednia et al [16] improved
the original Hill cipher, which prevents the known-plaintext attack on encrypted data but it is
vulnerable to known-plaintext attack on permutated vector because the permutated vector is
encrypted with the original key matrix. Ismail [8] tried a new scheme HillMRIV (Hill Multiplying
Rows by Initial Vector). Rangel-Romeror et al. [13] proved it is vulnerable to known-plaintext
attack and also proved that if IV is not chosen carefully, some of the new keys to be generated
may not be invertible over Zm, this make encryption/decryption process useless. Lin C.H. et al.
[11] improved the security of Hill cipher by using several random numbers. It thwarts the known-
plaintext attack but the algorithm is not efficient and is vulnerable to the chosen-cipher text attack
and it was proved by Mohsen Toorani et al. [19, 20] and he improved the security with one-way
hash function but Keliher, L. et al [10] proved that it is still vulnerable to attacks . Ahmed Y
Mahmoud et al [3, 4, 5] improved the algorithm by using eigen values but it is not efficient
because the time complexity is more and too many seeds are exchanged. Reddy, K.A. et al. [14,
15] improved the security of the cryptosystem by using circulant matrices but the time complexity
is more and is linear. Dunkelman, O. et al. [6] presented that byte substitution step in AES is a
perfectly nonlinear function. Kaipa, A.N.R. et al. [9] improved the algorithm by adding
nonlinearity using byte substitution over GF (28) and substitution using variable length sub key
groups. The cryptanalyst can gain the length of each sub key group because the byte substitution
is static. In the study, an attempt is made to introduce dynamic byte substitution over GF (28)
and variable length sub key groups using pseudo random number sequence.
2. PROPOSED METHOD
In this paper the nonlinearity is introduced using byte substitution over GF (2
8
). The byte
substitution is static; to make it dynamic a secret value is added to the static index to shift the
substitution to a secret location. To add extra security variable length sub keys are generated
using sequence of pseudo random numbers.
2.1 Algorithm
Let M be the message to be transmitted securely. The message is divided into m blocks each of
size n where is positive integer (greater than 1) and pad the last block if necessary. Let Ci be the
ciphertext of the i
th
block corresponding to i
th
block of plaintext Mi. Choose a prime number p <
(2
8
) and base value b. The algorithm is explained in the following steps.
1. Select a vector of ‘n’ relatively prime numbers (k1, k2, … kn). Rotate each row vector
relatively right to the preceding row vector to generate a shared key matrix Knxn.
2. Let r = ∑=
n
i
ik
1
mod p. Generate a sequence of ‘p’ pseudo random numbers Si with initial
seed value as r. Now generate the sub-key group as i from pseudo-random number
sequence as, j = (i + Si) mod b, for all i ∈ SGj and b < p/2. (i.e. the sub-key groups are
formed with the pseudorandom number sequence.)

3. Encryption: The encryption process follows the following steps
I. Initially apply the transformation on the plaintext block as Y = KM mod p.
II. The index of very element of the vector Y is calculated as, Index = Y, where Index =
(Index1, Index2, …, Indexn)
III. Share a secret value to shift the substitution to a new secret location. The following
equations explains the dynamic byte substitution
NewIndex1 = (Index 1+ secret value) mod 256
NewIndexi = (NewIndexi-1 + Indexi) mod 256, for i = 2, 3, …, n (3)
The static Index has been shifted to new secret location. Now substitute each
element of the vector NewIndex by an element from the S-box table and the resultant
vector as Z
IV. Select ‘n’ elements in queue from the corresponding sub key group SG using z1 mod
b as index of the sub key group and add to Z over mod 28 to generate cipher text
block C.
4. Transfer the pair (C, index), where index = z1 mod b to other end
5. Decryption: The decryption process follows the following step. After receiving the pair,
select ‘n’ elements in queue from the received indexed sub key group and subtract these
elements from C and the resultant vector becomes Z. Substitute each element of vector Z
by an element from the inverse S-box table. The inverse S-box table produces the
NewIndex vector. Now apply the following inverse process on the NewIndex vector to get
Index vector as:
Index1= ( NewIndex1 – secret value) mod 256
Indexi = (NewIndexi – NewIndexi-1) mod 256, for i = 2, 3, …, n (4)
The vector Index becomes Y. The multiplication of the resultant vector Y with the inverse
key matrix produces the original plaintext as
M = K
-1
Y mod p (5)
The following example explains the algorithm
2.2 Example
Consider a prime number ‘p’ as 29 and the set of relatively prime numbers (5, 27, 13). Generate
shared key matrix K3x3. Assume the plaintext block M= [8, 13, 5]. The Table 1 and Table 2 shows
the S-box and inverse S-box using GF (2
8
) field with irreducible polynomial x
8
+ x
4
+ x
3
+ x +1 and
are presented in the Appendix. Generate a sequence of ‘p’ pseudo-random number with initial
seed value as 45 and secret value as 45. Assume b = 3 and generate the three sub-key groups
(SG) from the random number sequence.
SG [0] = {1, 3, 4, 6, 7, 8, 11, 14, 16, 19, 23, 24, 26, 27}
SG [1] = {0, 2, 5, 9, 13, 13}
SG [2] = {10, 15, 17, 18, 20, 21, 23, 25, 28}
The process of encryption outputs the ciphertext C = [160, 172, 69]. After communication of the
pair (C, 1) and the decryption process outputs the plaintext M = [8, 13, 5].

3. PERFORMANCE ANALYSIS OF THE PROPOSED METHOD
The performance of the proposed method is evaluated by considering running time (i.e. time
complexity), result analysis and security analysis.
3.1 Time Complexity
The time complexity measures the running time of the algorithm. The time complexity of the
proposed algorithm to encrypt the ‘m’ plaintext blocks are O (mn
2
) and to decrypt ‘m’ ciphertext
blocks also O (mn
2
) which is shown in the equation (6),where ‘n’ is size of each block, which is
same as that of original Hill cipher. In this process TEnc and TDec denote the running time for
encryption and decryption of ‘m’ block of plaintext respectively. From step 3.1 the number of
multiplications is n
2
and the number of additions is n
2
-n because K is an n x n non-singular key
matrix and the size of plaintext/ciphertext is ‘n’.
SVISAddMulDec
SVSAddMulEnc
mnTmnTTnnmTnmmT
mnTmnTTnnmTnmmT
++−+≅
++−+≅
)()()(
)()()(
22
22
(6)
Where AddT , MulT , ST , IST and TSV are the time complexities for scalar modular addition,
multiplication, byte substitution, inverse byte substitution and secret vector addition respectively.
452
2
1
2
432
2
1
2
)()()(
)()()(
mncmnccnnmcnmmT
mncmnccnnmcnmmT
Dec
Enc
++−+≅
++−+≅
(7)
In which c1, c2, c3, c4 and c5 are the time for scalar multiplication, scalar addition, byte
substitution, inverse byte substitution and secret vector addition respectively. These times taken
for encryption and decryption are same as original Hill cipher and less than other modified Hill
ciphers.
3.2 Result Analysis
The result analysis of the method is carried with by considering a large encrypted data. The runs
test and correlation coefficients are carried on the data.
3.2.1 Runs Test
The runs test is carried on to check the randomness of the encrypted data. The data falls into two
separate categories such as above and below to a median. The test result shows that the
encrypted data is random since runs test gives -0.525 as a result. This value is better than the
hybrid cryptosystem proposed by Kaipa, A.N.R. et al [8].
3.2.2 Correlation Coefficient
Correlation coefficient is a number between -1 and 1 which quantifies the relation between two
variables. The correlation coefficient is 1 in case an increasing linear relation, -1 in case of
decreasing linear relation and some value in between in all other cases indicating the linear
dependence between the variables. The figure 2 shows that the correlation distribution of
plaintext and ciphertext for the proposed algorithm. The correlation coefficient has calculated for
the dynamic byte substitution and hybrid cryptosystem based on static byte substitution proposed
by Kaipa, A.N.R. et al [8] and the values are (-0.0545) and (-0.06005) respectively. The dynamic
byte substitution is better than static byte substitution. The results are presented in scatter plot in
figure 1.

FIGURE 1: Relation between Plaintext and Ciphertext.
3.3 Comparative Study
3.3.1 Time Complexity
The Fig 2 shows that the running time of encryption process in sec over various block sizes. The
5 million characters as input. The results show that our method takes same time as Hill cipher
and Toorani’s improved algorithm.
FIGURE 2: Encryption Time in Sec for various sizes of blocks.
3.3.2 Avalanche Effect
The avalanche effect is a characteristic of an encryption algorithm in which a small change in
plaintext or key should produce large change in the corresponding ciphertext. The avalanche
effect of our proposed method is same as AES algorithm. The avalanche effect has been
calculated on 5 million characters by considering random keys the results shows that it was at
least 71% because our proposed method uses dynamic byte substitution.
3.4 Security Analysis
The proposed method overcomes all the security drawbacks of the linear transformation based
cryptosystem. The byte substitution introduces non-linearity and dynamic byte substitution shifts

the static location to the new secret location. The dynamic byte substitution provides security
against linear, differential and algebraic attacks and Dunkelman, O. et al. [5] presented that the
byte substitution is perfect nonlinear function. The variable length sub key groups add another
degree of security to the method. It thwarts the ciphertext-only attack if the modulo a prime
number is large and dimension of key matrix is at least 6 x 6. The proposed method thwarts
known-plaintext attack because ‘n’ plaintext, ciphertext pairs cannot be used to solve ‘n’ unknown
elements of key matrix K and n
2
unknowns of secret shift values and n
2
unknowns of sub key
groups. It also thwarts chosen-plaintext and chosen –ciphertext attacks because the dynamic
byte substitution using S-box over GF (2
8
) adds nonlinearity and confusion to the cryptosystem
and also variable length sub key groups adds extra security to diffuse the relation between key
and ciphertext. From the analysis the adversary needs
4. CONCLUSION
The proposed cryptosystem is similar to linear transformation based Hill cipher but the nonlinear
is included by using byte substitution over GF (2
8
). The dynamic byte substitution adds extra
security by shifting the substitution to a secret location. This overcomes the problems of hybrid
cryptosystem based on static byte substitution. This method encrypts the same plaintext blocks to
different ciphertext blocks and the cryptanalyst cannot know the length of sub key groups. It is
free from linear and differential cryptanalysis attack along with known-plaintext and ciphertext
only attack. It provides more security than the existing algorithms with less key size because set
of ‘n’ elements is enough to form the n x n key matrix. It required memory is less compared to
other improvements and the processing speed is high so we can use this algorithm in mobile
environment.
5. REFERENCES
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random eigen values HCM-PRE. Applied Mathematics and Information Sciences (SCI-E)
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4&sid=21102878411191
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Appendix
Table 1 S-box generated from the polynomial x
8
+ x
4
+ x
3
+ x +1
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 63 7C 77 7B F2 6B 6F C5 30 01 67 2B FE D7 AB 76
1 CA 82 C9 7D FA 59 47 F0 AD D4 A2 AF 9C A4 72 C0
2 B7 FD 93 26 36 3F F7 CC 34 A5 E5 F1 71 D8 31 15
3 04 C7 23 C3 18 96 05 9A 07 12 80 E2 EB 27 B2 75
4 09 83 2C 1A 1B 6E 5A A0 52 3B D6 B3 29 E3 2F 84
5 53 D1 00 ED 20 FC B1 5B 6A CB BE 39 4A 4C 58 CF
6 D0 EF AA FB 43 4D 33 85 45 F9 02 7F 50 3C 9F A8
7 51 A3 40 8F 92 9D 38 F5 BC B6 DA 21 10 FF F3 D2
8 CD 0C 13 EC 5F 97 44 17 C4 A7 7E 3D 64 5D 19 73
9 60 81 4F DC 22 2A 90 88 46 EE B8 14 DE 5E 0B DB
A E0 32 3A 0A 49 06 24 5C C2 D3 AC 62 91 95 E4 79
B E7 CB 37 6D 8D D5 4E A9 6C 56 F4 EA 65 7A AE 08
C BA 78 25 2E 1C A6 B4 C6 E8 DD 74 1F 4B BD 8B 8A
D 70 3E B5 66 48 03 F6 0E 61 35 57 B9 86 C1 1D 9E
E E1 F8 98 11 69 D9 8E 94 9B 1E 87 E9 CE 55 28 DF
F 8C A1 89 0D BF E6 42 68 41 99 2D 0F B0 54 BB 16
Table 2 Inverse S-box generated from the polynomial x
8
+ x
4
+ x
3
+ x +1
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 52 09 6A D5 30 36 A5 38 BF 40 A3 9E 81 F3 D7 FB

1 7C E3 39 82 9B 2F FF 87 34 8E 43 44 C4 DE E9 CB
2 54 7B 94 32 A6 C2 23 3D EE 4C 95 0B 42 FA C3 4E
3 08 2E A1 66 28 D9 24 B2 76 5B A2 49 6D 8B D1 25
4 72 F8 E6 64 86 68 98 16 D4 A4 5C CC 5D 65 B6 92
5 6C 70 48 50 FD ED B9 DA 5E 15 46 57 A7 8D 9D 84
6 90 D8 AB 00 8C BC D3 0A F7 E4 58 05 B8 B4 45 06
7 D0 2C 1E 8F CA 3F 0F 02 C1 AF BD 03 01 13 8A 6B
8 3A 91 11 41 4F 67 DC EA 97 F2 CF CE F0 B4 E6 73
9 96 AC 74 22 E7 AD 35 85 E2 F9 37 E8 1C 75 DF 6E
A 47 F1 1A 71 1D 29 C5 89 6F B7 62 0E AA 18 BE 1B
B FC 56 3E 4B C6 D2 79 20 9A DB C0 FE 7C CD 5A F4
C 1F DD A8 33 88 07 C7 31 B1 12 10 59 27 80 EC 5F
D 60 51 7F A9 19 B5 4A 0D 2D E5 7A 9F 93 C9 9C EF
E A0 E0 3B 4D AE 2A F5 B0 C8 EB BB 3C 83 53 99 61
F 17 2B 04 7E BA 77 D6 26 E1 69 14 63 55 21 0C 7D

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