In this paper, we propose a new One-Way Hash Algorithm, which is also obtains better efficiency and security, compared with a particular conventional hash algorithm, Hash algorithm can be used to determine if two values are equal, a hash function maps keys to small values[1]. DILH algorithm using linear combination of matrices to find non-invertible matrix, that takes advantage about of the compact representation of a set of numbers in a matrix, and fast calculations

Hamamreh, Rushdi A.

Jamoos, Mohammed A.

English

Al-Quds University Digital Repository

IJCSNS International Journal of Computer Science and Network Security, VOL.14 No.9, September 2014 
 
94 
Manuscript received September 5, 2014 
Manuscript revised September 20, 2014 
 
Abstract 
In this paper, we propose a new One-Way Hash Algorithm, which 
is also obtains better efficiency and security, compared with a 
particular conventional hash algorithm, Hash algorithm can be 
used to determine if two values are equal, a hash function maps 
keys to small values[1]. DILH algorithm using linear combination 
of matrices to find non-invertible matrix, that takes advantage 
about of the compact representation of a set of numbers in a matrix, 
and fast calculations.  
IndexTerms 
computational complexity, cryptography, hashing, collision, one- 
way algorithm, DILH, non-invertable matrix. 
1. Introduction 
There are three key aspects of data security: confidentiality, 
integrity and availability. In this paper, we introduced a 
concept of Data Integrity.  One-Way Hash function  have an 
important primitive cryptographic used for authentication, 
privacy and integrity[1][11]. 
Hash algorithm is a mathematical functions that converts a 
relatively large message into small strings of data and/or 
texts,  to check whether the message has been trampled by 
attackers which could compromise the information, these 
strings of texts, called hashes, can be used [3][4]. 
More precisely, a hash algorithm  maps bits strings of 
arbitrary finite length to string of fixed length called hashes 
 where  is the message block, given the original data, 
the encrypted data easy to calculate, for a given encrypted 
data to look for the original data is computationally 
infeasible, for different original data to get the same 
encrypted data is computationally infeasible [2][4][5]. 
Hash algorithm has played an important role in modern 
cryptography, it is generate because of the needs of 
message identify. So that it is widely used in the document 
verification and digital signature in information security[6]. 
According to its nature, you can’t restore the original 
plaintext ( ) from the encrypted cipher text ( ), so it can’t 
be used for conventional data encryption and it can only be 
used for identification[2]. 
MD5, SHA-1 are an excellent one-way hash algorithm. 
There are three main properties of such algorithms, in 
addition to one-way, it is collision-free and strongly 
collision-free [7]. 
Hash algorithm in practical use is mostly constituted of two 
parts, which could be called as compression function and 
iterative structure. So, we believe that the security of hash 
algorithm depends not only on the difficult problems, but 
also on the compression function structure and iterative of 
the hash algorithm [1][6]. 
There are many various hash algorithms including MD5, 
SHA-1, RIIPEMD and so on. However, MD5 has been 
broken by Wang Xiaoyun and she had found the effective 
method of attack the SHA-1 algorithm, so the difficulty of 
the break is reduced [8]. Thus, in order to meet practical 
application, it is desirable to study new hash generation 
algorithm . 
In this paper, we describe non invertible matrix which can 
be used as multiplication matrix in Hill Cipher technique 
for one way hash algorithm. Hill cipher algorithm was used 
for symmetric encryption, where the multiplication matrix 
is the key. The Hill cipher requires the inverse of the matrix 
to recover the plaintext from cipher text. We propose a 
linear combination of matrix  which is non invertible and 
easy to generate. We call this Data Integrity using Linear 
Combination for Hash Algorithm (DILH).  
The rest of the paper is organized as follows. In section II, 
the matrix hash algorithm is introduced. In section III, the 
proposed DILH algorithm is presented. The analysis and 
results are shown in section IV. Conclusion remarks are 
drawn in section V. 
The basic requirements for a cryptographic hash algorithm 
are: 
 The input can be in any length. 
 The output has a fixed length. 
  is relatively easy to compute for any given  . 
  is one-way. 
  is collision-free. 
A hash function  is said to be one-way if it is hard to invert, 
where "hard to invert" means that given a hash value  [9], 
Data Integrity Mechanism Using Hashing Verification 
Rushdi A. Hamamreh 
Computer Engineering Department 
Faculty of Engineering 
Al-Quds University 
 
Mohammed A. Jamoos 
Computer Science Department 
Faculty of Science and Technology 
Al-Quds University 
 
 
IJCSNS International Journal of Computer Science and Network Security, VOL.14 No.9, September 2014 95 
it is computationally infeasible to find some input  such 
that  
. 
For more secure transmitted data, padding using.  Padding 
(Pad) is an algorithm where the plaintext  is combined 
with  a random values " " that is as long as the plaintext 
[1]. 
In our work , we have been used Byte padding. Byte 
padding can be applied for messages that can be encoded as 
an integral number of bytes. 
 
Figure.1  One-way hash algorithm 
As it is shown in figure.1, if a given message , it is 
computationally infeasible to find a message  not equal 
to  such that  then  is said to be a 
weakly collision-free hash algorithm. On the other hand,  
is a strongly collision-free hash algorithm if and only if it is 
computationally infeasible to find any two messages  and 
 such that . 
2. Matrix Hash Algorithm 
Matrix  is a square matrix, the inverse is written . 
When K is multiplied by  the result is the identity 
matrix . Non-square matrices do not have inverses. 
Not all square matrices have inverses.  square matrix 
which has an inverse is called invertible or nonsingular, and 
a square matrix without an inverse is called non-invertible 
or singular [12]. 
     
We call  here identity matrix. As we previously mentioned, 
not all matrices have Inverses but most of them have [12], 
so here we proposed an algorithm that converts any 
invertible matrix non-invertible one. 
Hill Cipher 
The core of Hill-cipher is matrix manipulations. It is a 
multi-letter cipher, is a type of monoalphabetic polygraphic 
substitution cipher. Hill cipher requires inverse of the key 
matrix while decryption. In fact that not all the matrices 
have an inverse and, therefore they will not be eligible as 
key matrices in the Hill cipher scheme [6][9]. Moreover, 
Hill cipher has several advantages such as disguising letter 
frequencies of the plaintext ( ), its simplicity because of 
using matrix multiplication and inversion for enciphering 
and deciphering, its high speed, and high throughput . 
A. Hill Cipher (Encryption , Decryption) 
To Encrypt Plaintext Block size of  [9],  we need key 
matrix ( ) with entries are between  
included, but the determinant must be relatively prime to , 
each entry in the plaintext block is between , 
included each block of plaintext is then an n-dimensional 
vector  .We encrypt vector  simply to produce the cipher 
text vector  using the following linear algebra equation 
 
To Decrypt cipher text  vector  [9], we need first to find the 
inverse matrix  to   , where that matrix must be 
invertible over  , then can calculate  from the 
mathematical model 
 
In [5][9] they proposed new technique to convert any 
non-invertible matrix’s to invertible ones. As a result, Hill 
cipher being a efficient algorithm because any encrypted 
text will  decrypted using the key matrix [5][10]. 
B. Hill Cipher (Hashing algorithm) 
The main point of one-way hash algorithm is that any 
encrypted text cannot be decrypted [11]. From this 
point ,we need to choose the non-invertible matrix from the 
hill cipher to use it inside the practical one- way hash 
algorithm.  
 
 
Where  is the non-invertible matrix. In [11] author works 
on an algorithm that generate non-invertible matrix and 
multiply it by plaintext as column vector with modular 
value  to generate the hash value .  
3. Proposed Algorithm Models 
Proposed algorithm  DILH uses non-invertible matrix to 
produce hash value . This algorithm selects and generates 
non-invertible matrixes using Linear combination[1] of 
rows or columns of a matrix to ensure that the hash values 
are collision-free and one-way properties. 
In this section, we plot the diagram for our proposed 
algorithm which showing each step inside it and how it 
works , It is also showing the mathematical proof of our 
work. Besides that, we have the DILH algorithm analysis 
and  result  and its comparison with other hashing 
algorithms including SHA-1, MD5. 
 
 
IJCSNS International Journal of Computer Science and Network Security, VOL.14 No.9, September 2014 96 
A. DILH algorithm 
 
Figure.2  DILH diagram 
According to figure 2, the step of DILH algorithm 
structured as the followings: 
Step1 (Input): input . 
Step2 (Padding): Pad  .   
Step3 (Splitting):  is split into  . 
          blocks     each of length  
suitable     
          for the hashing block. 
Step4 (Key generation): Key matrix generation  :  
 4.1 Generate a random matrix  with 
size        
       and value in a interval (0,1). 
 4.2 Casting (R). 
 4.3 Introduce a new row in the matrix  so as 
       the size of the resulting matrix  is . 
       The added row is obtained by linear  combination                      
       of row  and row  of matrix  according to: 
 = 1  +  2   
where ,  are arbitrary integer constants. This 
ensure that the inverse of  denoted by  does not 
exist. 
Step5(Generation ) : Hash value  generation using this    
        formula : 
 
Step6(Digest) : Digest . 
Padding is the function where the plaintext is combined 
with a random data or "pad” to produce fixed length of the 
plaintext, and message digests are used to produce arbitrary 
values which is ensure data integrity [1]. 
B.Mathematical Model Proof 
The first step is to define linear independence. Given a set 
of vectors[12] 
 
 we look at their linear combinations 
 
Where  are arbitrary constant weights. The 
trivial combination, with all weights , obviously 
produces the zero vector  . The 
question is whether this is the only way to produce zero. If 
so, the vectors are independent. If any other combination of 
the vectors gives zero, they are dependent [12]. 
It is well known that a matrix  is invertible if and only if its 
rows and columns are linearly independent [12]. This mean 
that if rows and columns of a matrix are linearly 
independent then the matrix is invertible, otherwise it’s not 
invertible. 
4. Analysis and Results 
In this subsection we will analyze the time delay in the 
proposed DILH algorithm. Given a range of data files with 
sizes (8KB – 256KB), we first convert each file into matrix 
with size between (2x2 – 12x12). Then, we calculate the 
needed time in second to generate hash values. Table 1 
shows the performance of the proposed DILH algorithm in 
terms of the required time to generate hashes for different 
file sizes and with different matrix sizes. According to 
Table 1, increasing the file size will increase the required 
time for all matrix size considered. 
Table 1: Required time in seconds for the proposed DILH algorithm with 
different file size and matrix size m=127 
 
Table 2: Required time in seconds for the proposed DILH algorithm with 
different file size and matrix size, m=256. 
 
 
In addition, by Increasing the matrix size will usually 
decreasing the require time. Nevertheless, one can notice 
that there is an optimal matrix size that results in the lowest 
required time. For example, the optimal matrix size that 
IJCSNS International Journal of Computer Science and Network Security, VOL.14 No.9, September 2014 97 
results in the lowest time when the file size 8KBand m=127 
is 10x10, and when m=256 is 11x11. Table 2 summarizes 
the optimal matrix size for considered file sizes . 
Table 3: Optimal time in seconds required for the proposed DILH 
algorithm. 
 
As shown in the previous table, the best time for calculating 
hash value at proposed model in 256KB is 8x8 when 
m=127 and 12x12 when m=256. 
0 50 100 150 200 250 300
0
20
40
60
80
100
120
140
160
180
File Size [Kbytes]
Ti
m
e 
[s
ec
]
DLIH:Optimal time in seconds
 
 
m=127
m=256
10x10
11x11
11x11
11x11
11x11
12x12
8x8
11x11
11x11
11x11
11x11
11x11
 
Figure 3: Optimal matrix size in terms of required time for different file 
sizes where m=127. 
0 50 100 150 200 250 300
0
100
200
300
400
500
600
700
File Size [Kbytes]
Ti
m
e 
[s
ec
]
 
 
2x2
4x4
6x6
8x8
10x10
12x12
 
Figure 4: The time delay versus file size of the proposed algorithm for 
different matrix sizes and m=127. 
0 50 100 150 200 250 300
0
100
200
300
400
500
600
700
800
900
File Size [Kbytes]
Ti
m
e 
[s
ec
]
 
 
2x2
4x4
6x6
8x8
10x10
12x12
 
Figure 5: The time delay versus file size of the proposed algorithm for 
different matrix sizes and m=256. 
According to the experimental data obtained by test, the 
efficiency of our hash algorithm is about 2.808 times slower 
than the efficiency of MD5 and about 5.813 times slower 
than SHA-1efficiency in 8KB file size. This ratio is 
increased in an positive relationship with file size, so you 
can find that our proposed algorithm is efficient 9.794 times 
that SHA-1 when the file size is 256KB. Table 3 
summarizes the experimental delay time for considered 
hash algorithms . 
Table  4: The time delay in seconds versus file size of the proposed 
algorithm compared to SHA-1 and md5 algorithm. 
Data Size 
KB 
Hash Algorithms 
LDIH 
SHA-1 MD5 
m=127 m=256 
8 4.7815 4.7090 27.7990 13.2750 
16 10.0950 10.7546 56.9980 25.3340 
32 21.8399 20.3674 117.1002 51.3510 
64 44.8996 38.6646 281.8830 135.4560 
128 88.1278 80.8281 669.4310 432.0470 
256 177.9706 174.0730 1743.1 1140.4 
From table 3, we can observe that our hash algorithm still 
has an advantage in the efficiency. 
 
IJCSNS International Journal of Computer Science and Network Security, VOL.14 No.9, September 2014 98 
Figure 6: Hash algorithms comparison ( SHA-1,MD5 and DILH) in term 
of  time delay versus file size where m=127. 
5. Conclusion and Future Work 
The algorithm can protect user key, because one-way hash 
function is not reversible. We describe a new one-way data 
integrity hash algorithm based on special kinds of  
non-invertible metrics. Particularly, we proposed to 
generate non-invertible key matrices by  linear 
combinations of rows and columns of a matrix. We have 
analyzed time delay of the proposed algorithm, compare it 
with other hash algorithms. In our experiment 
and padding standard is ISO 10126. 
We implement our hash scheme in a Dell vostro 1015 
laptop, 2.10GHz 2 core(s), 2GB RAM/ Microsoft 
windows7/MATLAB 7.9.0 simulator. 
From the result, our algorithm is faster than compared 
algorithm. Currently we are investigating  collisions, 
hamming distance and vulnerable time for this algorithm. In 
addition, we will compare the proposed algorithm with 
other hashing algorithms including RIPEMD ,MD5 and 
SHA-1.  
 
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Data Integrity Mechanism Using Hashing Verification

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Data Integrity Mechanism Using Hashing Verification

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