An Enhanced Nihilist Cipher Using Blum Blum Shub Algorithm

Abstract

The Nihilist cipher is a symmetric encryption cipher that works by substituting the characters of the plaintext and the keyword and form bigrams based on their character placement and coordinates within the Polybius square. The resulting bigrams from both plaintext and key are added to form the ciphertext. However, the pairing process is unsatisfactory due to its key distribution scheme making the entire process vulnerable to attacks. With this, the Blum Blum Shub (BBS) algorithm is added to randomize the key pairing process of the Nihilist cipher. Therefore, this paper discusses the processes of the proposed cipher methodology based on the combined Nihilist cipher and Blum Blum Shub algorithm. The hybrid methodology offers secure encryption and decryption process, as evident in the simulation results conducted.

Full text

Allemar Jhone P. Delima et al., International Journal of Advanced Trends in Computer Science and Engineering, 9(3), May – June 2020, 3270 – 3274
3270
ISSN 2278-3091
Volume 9, No.3, May - June 2020
International Journal of Advanced Trends in Computer Science and Engineering
Available Online at http://www.warse.org/IJATCSE/static/pdf/file/ijatcse122932020.pdf
https://doi.org/10.30534/ijatcse/2020/122932020

ABSTRACT
The Nihilist cipher is a symmetric encryption cipher that
works by substituting the characters of the plaintext and the
keyword and form bigrams based on their character
placement and coordinates within the Polybius square. The
resulting bigrams from both plaintext and key are added to
form the ciphertext. However, the pairing process is
unsatisfactory due to its key distribution scheme making the
entire process vulnerable to attacks. With this, the Blum
Blum Shub (BBS) algorithm is added to randomize the key
pairing process of the Nihilist cipher. Therefore, this paper
discusses the processes of the proposed cipher methodology
based on the combined Nihilist cipher and Blum Blum Shub
algorithm. The hybrid methodology offers secure encryption
and decryption process, as evident in the simulation results
conducted.
Key words: Blum Blum Shub, cryptography, encryption,
hybrid ciphers, Nihilist cipher
1. INTRODUCTION
Cryptography [1] has been part of human’s daily lives. Most
of the digital communications require authentication over
open channels such as the Internet as hackers, crackers,
eavesdroppers, and other adversaries are keeping an eye on
different data available almost everywhere to be used for
everything [2]. Governmental and non-governmental
agencies, commercial and non-profit enterprises, as well as
the general public all rely on security [3], of which the cipher
process of cryptography is an intrinsic part. A cipher is a
process that transforms data, in the form of plain text into an
incomprehensible format called the encryption with a
reversing decryption process that transforms ciphertext back
into its original plain text format [4].
Some of the commonly used ciphers in the literature are the
Nihilist cipher [5], Base64 cipher [6]–[8], Beaufort cipher [4],
Bifid cipher [9], Caesar cipher [10]–[12], Enigma Machine
cipher [4], Four-square cipher [4], Grille cipher [13], Hill
cipher [14], Homophonic substitution cipher [15], [16], and
Permutation cipher [4], among others. In this study, the
famous Nihilist cipher is modified and added with the Blum
Blum Shub algorithm. The Blum-Blum-Shub algorithm is a
secure pseudorandom number generator that produces a
sequence by reducing squares modulo the product of two
Blum primes [17]. The proposed hybridization of the two
algorithm aims to produce a more secure ciphertext by
enhancing the key generation process of the traditional
Nihilist cipher.
2. METHODOLOGY
2.1 Nihilist Cipher
The Nihilist cipher is a substitution cipher that makes use of a
matrix to produce ciphertext. In using the Nihilist cipher, the
plaintext and the keyword are translated into its numerical
form through character substitution using a Polybius square
to generate bigrams that represent the coordinates of the
character in the grid. The bigrams from the plaintext and
keyword are added together to generate the ciphertext [18].
The traditional Nihilist cipher uses a 5x5 grid matrix, as
presented in Table 1. The grid is filled with alphabets written
from left to right, then from top to bottom.
Table 1: Nihilist cipher table
1
2
3
4
5
1
A
B
C
D
E
2
F
G
H
I/J
K
3
L
M
N
O
P
4
Q
R
S
T
U
5
V
W
X
Y
Z
First, the plaintext is converted to a numeric value by
matching each character to the given matrix and retrieving its
row-column index known as a bigram. For instance, the
plaintext PASSAGE is translated into 35 11 43 43 11 22 15,
as presented in Table 2.
Table 2: Plaintext conversion
Plaintext
P
A
S
S
A
G
E
Position
1
2
3
4
5
6
7
Converted
Plaintext
35
11
43
43
11
22
15
Second, the keyword is also converted to its numeric
equivalent by matching each character to the given matrix
and retrieving its equivalent bigrams. For example, the
keyword KEY is converted to 25 15 45, as shown in Table 3.
An Enhanced Nihilist Cipher Using Blum Blum Shub
Algorithm
Allemar Jhone P. Delima1, Jan Carlo T. Arroyo2
1,2College of Computing Education, University of Mindanao, Davao City, Davao del Sur, Philippines
1College of Engineering, Technology and Management, Cebu Technological University-Barili Campus,
Cebu, Philippines
allemardelima@umindanao.edu.ph 1, jancarlo_arroyo@umindanao.edu.ph2

Allemar Jhone P. Delima et al., International Journal of Advanced Trends in Computer Science and Engineering, 9(3), May – June 2020, 3270 – 3274
3271
Table 3: Keyword conversion
Plaintext
K
E
Y
Position
1
2
3
Converted Keyword
25
15
45
Further, each character of the keyword is paired with each
character of the plaintext. Since the given keyword is
composed of only three letters, the characters are repeatedly
matched up to the length of the plaintext, as seen in Table 4.
Table 4: Plaintext-keyword pairing
Plaintext
P
A
S
S
A
G
E
Converted
Plaintext
35
11
43
43
11
22
15
Keyword
K
E
Y
K
E
Y
K
Converted
Keyword
25
15
45
25
15
45
25
Lastly, each pair of plaintext and keyword bigrams are
summed to generate the ciphertext. Based on the example, the
first character P is encrypted as 60 (35+25), A is encrypted as
26 (11+15), and so forth. Hence, the plaintext PASSAGE is
encrypted as 60 26 88 68 26 67 40 using the keyword KEY,
as presented in Table 5.
Table 5: Nihilist cipher encryption
Plaintext
P
A
S
S
A
G
E
Converted
Plaintext
35
11
43
43
11
22
15
Keyword
K
E
Y
K
E
Y
K
Converted
Keyword
25
15
45
25
15
45
25
Final
Ciphertext
60
26
88
68
26
67
40
To decrypt the ciphertext, the keyword is first identified. The
keyword is converted to its numerical equivalent. This is then
deducted from the ciphertext value to generate the bigrams
for the plaintext. Each resulting bigram is matched with the
Polybius square to retrieve the plaintext.
One advantage of using the Nihilist cipher over the Polybius
cipher is that the former may generate varied ciphertext
values for identical characters as opposed to the latter, which
produces the same values for identical characters. For
instance, the character S is encrypted as 88 or 68 and E as 30
or 40, as seen on the results above.
Cryptanalysis for Nihilist cipher is done through pattern
analysis and factoring. The keyword length can be guessed
by looking at low and high number patterns in the ciphertext.
This is more obvious if a lengthy plaintext is used. The
keyword may also be guessed by factoring, such that the
ciphertext 89 can only be made by the factors 45+44 based on
the limited combination of 1, 2, 3, 4, 5 from the matrix. This
is due to the simple addition approach of the repeating
keywords paired with the plaintext. For example, in a 5x5
matrix, if a ciphertext value is more than 100, then that would
easily mean that both plaintext and keyword values come
from the 5th row of the Polybius square.
2.2 Blum Blum Shub Algorithm
Blum Blum Shub (BBS) is a well-known probabilistically
secure pseudo-random number generator (PRNG) proposed
by Lenore Blum, Manuel Blum, and Michael Shub in 1986
[19]. The BBS algorithm produces random numbers by using
two primes p and q, such that p and q ≡ 3 (mod 4) and
gcd(p,q) = 1. These primes, together with a random seed, is
computed as:
(1)
where:
(i) b is a random seed value
(ii) M is the product of two large prime numbers p and q
(iii) p and q are congruent to 3 (mod 4)
The example below shows how BBS generates random
numbers:
 Let, p = 7 ≡ 3 (mod 4)
Let q = 19 ≡ 3 (mod 4)
Then, n = p * q = 7 * 19 = 133
 Choose a seed value b0 = 100
 b1 = 1002 (mod 133) = 25
 b2 = 252 (mod 133) = 93
 b3 = 932 (mod 133) = 4
 so forth...
2.3 Proposed Cipher Process
The proposed process introduces the use of BBS and the
XOR operation to further enhance the security of the cipher.
The addition of BBS and XOR increases the randomness of
the cipher to minimize patterns in the ciphertext values. The
proposed process also extends the Nihilist cipher into a 6x6
matrix to include digits in the range of allowable characters to
be encoded and decoded. The encryption process is presented
in Figure 1, while the decryption process is presented in
Figure 2.
Figure 1: Encryption process

Allemar Jhone P. Delima et al., International Journal of Advanced Trends in Computer Science and Engineering, 9(3), May – June 2020, 3270 – 3274
3272
Figure 2: Decryption process
Encryption using the proposed method involves a plaintext,
keyword, and randomly generated seed. For instance, the
given plaintext is PASSAGE is used where the keyword is
KEY, and the seed is 7 0 7 3 1 0 2. The plaintext and the
keyword are translated into bigrams using the 6x6 matrix.
Further, each character from the keyword is paired with each
character from the plaintext. This is done repeatedly until the
length of the plaintext. Each seed value is paired as well. A
sample 6x6 matrix is presented in Table 6, while the
converted values are presented in Table 7.
Table 6: A 6x6 nihilist cipher table
1
2
3
4
5
6
1
A
B
C
D
E
F
2
G
H
I
J
K
L
3
M
N
O
P
Q
R
4
S
T
U
V
W
X
5
Y
Z
0
1
2
3
6
4
5
6
7
8
9
Table 7: Plaintext and keyword conversion
Plaintext
P
A
S
S
A
G
E
Converted
Plaintext
35
11
43
43
11
22
15
Keyword
K
E
Y
K
E
Y
K
Converted
Keyword
25
15
45
25
15
45
25
Seed
7
0
7
3
1
0
2
The next step involves performing addition and XOR
operations. Each seed value is XORed with the
corresponding keyword bigram and then added to the
plaintext bigram using the equation P + (K
⊕
S). Based on
the given example, the first character M is encrypted as 64 =
(35 + (25 ⊕ 7)), E is encrypted as 26 = (11 + (15 ⊕ 0), and
so forth. The plaintext PASSAGE is now encrypted as 64 26
93 67 25 72 42 using the keyword KEY, as seen in Table 8.
Table 8: Nihilist cipher encryption
Plaintext
P
A
S
S
A
G
E
Converted
Plaintext
35
11
43
43
11
22
15
Keyword
K
E
Y
K
E
Y
K
Converted
Keyword
25
15
45
25
15
45
25
Final Ciphertext
64
26
93
67
25
72
42
The decryption process requires access to the ciphertext,
keyword, and seed values. The seed is XORed with the
keyword bigram and deducted from the plaintext bigram.
Each resulting bigram is now matched with the matrix to
retrieve the plaintext.
3. RESULTS AND DISCUSSION
In order to assess the viability of the proposed method, two
test cases were conducted using a variety of plaintext and
keys. In this study, a 6x6 matrix was used to test both the
standard and modified Nihilist ciphers.
In test case 1, a 43-byte plaintext is encrypted using the
keyword DOG, as shown in Table 9. Each encrypted
ciphertext for the standard and enhanced Nihilist cipher is
checked for its low and high patterns. Based on the results of
the standard Nihilist cipher, a cryptanalyst may easily
recognize the keyword pattern LOW HIGH LOW (d-o-g) in
the ciphertext because of how frequent it appears (6 times).
Cryptanalysts may also use factoring to easily identify the
keyword bigram in the standard Nihilist cipher, such that the
first ciphertext bigram 27 can only be produced using the
digits 11+16, 12+15, or 14+13 when using the matrix. The
enhanced process performs better as the identified LOW
HIGH LOW pattern is not as apparent since it only appears
three times. On the other hand, attackers may not use
factoring to determine the keyword bigram because the
matrix could not generate bigrams lower than 11 and higher
than 66.
Table 9: Test case 1
Plaintext
cryptographyisthescienceofhidinginformation
Size
43 bytes
Keyword
Dog
Seed
5 4 3 3 5 0 7 5 6 1 1 3 0 6 4 7 7 7 2 4 7 2 2 4 2 6 2 6 4 5 3
3 7 3 2 0 6 6 6 3 3 7 2
Nihilist
Ciphertext
27 69 72 48 75 54 35 69 32 48 55 72 37 74 63 36 48 62
27 56 36 46 46 36 47 49 43 37 47 44 46 54 44 46 49 54
50 64 32 56 56 54 46
Pattern
Low High High Low High Low Low High Low High
High High Low High Low Low High High Low High
Low High Low Low High High Low Low High Low
High High Low High High High Low High Low High
Low Low Low
Modified
Nihilist
Ciphertext
243 57 200 48 115 186 35 245 96 208 199 214 225 202
63 26 176 60 211 182 90 46 50 98 175 69 155 93 175 74
208 184 194 176 181 184 172 64 32 166 184 54 206
Pattern
Low Low High Low High High Low High Low High
Low High High Low Low Low High Low High Low
Low Low High High High Low High Low High Low
High Low High Low High High Low Low Low High
High Low High
In test case 2, a 77-byte plaintext is encrypted using the
keyword INFO, as shown in Table 10. Each encrypted
ciphertext generated by the standard and enhanced Nihilist
cipher is checked for its low and high patterns. Based on the
results generated by the standard Nihilist cipher, a
cryptanalyst may easily recognize the keyword pattern LOW
HIGH LOW HIGH (i-n-f-o) in the ciphertext because of how
frequent it appears (8 times). Cryptanalysts may also use
factoring to easily identify the keyword bigram in the
standard Nihilist cipher, such that the ciphertext bigram 31

Allemar Jhone P. Delima et al., International Journal of Advanced Trends in Computer Science and Engineering, 9(3), May – June 2020, 3270 – 3274
3273
can only be produced using the digits 16+15 when using the
matrix. The enhanced process performs better as the
identified LOW HIGH LOW pattern is not as apparent since
it only appears two times. Also, attackers may not use
factoring to determine the keyword bigram because the
matrix could not generate bigrams lower than 11 and higher
than 66.
Table 10: Test Case 2
Plaintext
TheNihilistcipherisamonoalphabeticclassicalcipherusedb
ytheRussiansagainstCzar
Size
77 bytes
Keyword
Info
Seed
77 105 9 50 120 27 38 9 60 53 113 73 9 51 16 69
93 27 121 2 49 16 47 105 79 15 37 54 104 44 78 90
114 115 94 39 52 111 127 65 6 45 43 12 26 119 72
61 1 2 58 99 94 66 21 92 85 85 31 25 111 46 47
111 83 95 32 120 7 45 57 32 35 108 121 15 89
Nihilist
Ciphertext
65 54 31 65 46 54 39 59 46 73 58 46 46 66 38 48 59 55 57
44 54 65 48 66 34 58 50 55 34 44 31 75 46 45 29 59 34 73
57 56 36 43 42 46 46 66 38 48 59 75 57 48 37 44 67 75 45
47 52 76 64 73 39 44 55 73 27 54 34 55 48 74 65 45 68 44
59
Pattern
Low Low Low High Low High Low High Low High Low
Low Low High Low High High Low High Low High
High Low High Low High Low High Low High Low
High Low Low Low High Low High Low Low Low High
Low High Low High Low High High High Low Low
Low High High High Low High High High Low High
Low High High High Low High Low High Low High
Low Low High Low High
Modified
Nihilist
Ciphertext
132 95 40 51 134 81 77 66 66 62 139 117 53 53 22 115
110 82 146 46 69 81 95 105 99 73 87 45 138 24 109 165
124 96 91 32 46 120 152 119 30 24 85 58 36 121 110 43
58 77 83 81 87 110 56 167 88 132 51 99 161 55 86 89 100
168 59 110 27 36 73 42 94 89 157 57 114
Pattern
Low Low Low High High Low Low Low High Low High
Low Low High Low High Low Low High Low High
High High High Low Low High Low High Low High
High Low Low Low Low High High High Low Low Low
High Low Low High Low Low High High High Low
High High Low High Low High Low High High Low
High High High High Low High Low High High Low
High Low High Low High
4. CONCLUSION
In this paper, the use of the extended Nihilist cipher is
introduced to increase the cipher capability of the Nihilist
cipher as recommended in the study of [20]. Further, the use
of the Blum Blum Shub algorithm, together with the XOR
process prior to the generation of ciphertext, increases the
cipher’s randomness, concealing the low and high number
patterns and as well as preventing attackers on guessing the
keyword used through factoring. The proposed method
shows ciphertext variability and pattern obscureness, making
the encrypted data difficult to break.
REFERENCES
[1] W. Stallings, Cryptography and Network Security
Principles and Practices. Prentice Hall, 2015.
[2] S. Dey, J. Nath, and A. Nath, “An Integrated
Symmetric Key Cryptographic Method –
Amalgamation of TTJSA Algorithm, Advanced
Caesar Cipher Algorithm, Bit Rotation and Reversal
method: SJA Algorithm,” Int. J. Mod. Educ. Comput.
Sci., vol. 4, no. 5, pp. 1–9, 2012.
https://doi.org/10.5815/ijmecs.2012.05.01
[3] A. Rayarapu, A. Saxena, N. V. Krishna, and D.
Mundhra, “Securing Files Using AES Algorithm,”
Int. J. Comput. Sci. Inf. Technol., vol. 4, no. 3, pp.
433–435, 2013.
[4] M. S. Hossain Biswas et al., “A systematic study on
classical cryptographic cypher in order to design a
smallest cipher,” Int. J. Sci. Res. Publ., vol. 9, no. 12,
pp. 507–11, 2019.
https://doi.org/10.29322/IJSRP.9.12.2019.p9662
[5] E. V. Haryannto, M. Zulfadly, Daifiria, M. B. Akbar,
and I. Lazuly, “Implementation of Nihilist Cipher
Algorithm in Securing Text Data with Md5
Verification,” J. Phys. Conf. Ser., vol. 1361, no.
012020, 2019.
[6] F. Anwar, E. H. Rachmawanto, C. A. Sari, and de
Rosal Ignatius Moses Setiadi, “StegoCrypt Scheme
using LSB-AES Base64,” in International
Conference on Information and Communications
Technology, ICOIACT 2019, 2019, pp. 85–90.
[7] A. R. Pathak, S. Deshpande, and M. Panchal, “A
Secure Framework for File Encryption Using Base64
Encoding,” in Computing and Network
Sustainability, vol. 75, Springer Singapore, 2019, pp.
359–366.
https://doi.org/10.1007/978-981-13-7150-9_38
[8] R. Rahim, S. Sumarno, M. T. Multazam, S. Thamrin,
and S. H. Sumantri, “Combination Base64 and
GOST algorithm for security process,” J. Phys. Conf.
Ser., vol. 1402, 2019.
[9] A. Borodzhieva, “MATLAB-based software tool for
implementation of Bifid Ciphers,” in International
Conference on Computer Systems and Technologies,
2017, pp. 326–333.
https://doi.org/10.1145/3134302.3134333
[10] A. Singh and S. Sharma, “Enhancing Data Security
in Cloud Using Split Algorithm, Caesar Cipher, and
Vigenere Cipher, Homomorphism Encryption
Scheme,” in Emerging Trends in Expert Applications
and Security, 2019, vol. 841, pp. 157–166.
[11] I. Gunawan, Sumarno, H. S. Tambunan, E. Irawan,
H. Qurniawan, and D. Hartama, “Combination of
Caesar Cipher Algorithm and Rivest Shamir
Adleman Algorithm for Securing Document Files
and Text Messages,” J. Phys. Conf. Ser., vol. 1255,
2019.
https://doi.org/10.1088/1742-6596/1255/1/012077
[12] D. Gautam, C. Agrawal, P. Sharma, M. Mehta, and P.
Saini, “An Enhanced Cipher Technique Using
Vigenere and Modified Caesar Cipher,” in 2nd
International Conference on Trends in Electronics
and Informatics, ICOEI 2018, 2018.
[13] J. Liu et al., “The Reincarnation of Grille Cipher: A
Generative Approach,” Cryptogr. Secur., pp. 1–27,
2018.
[14] P. E. Coggins and T. Glatzer, “An Algorithm for a
Matrix-Based Enigma Encoder from a Variation of
the Hill Cipher as an Application of 2 × 2 Matrices,”
Primus, vol. 30, no. 1, 2020.
https://doi.org/10.1080/10511970.2018.1493010
[15] M. Shumay and G. Srivastava, “PixSel: Images as
book cipher keys an efficient implementation using
partial homophonic substitution ciphers,” Int. J.
Electron. Telecommun., vol. 64, no. 2, pp. 151–158,

Allemar Jhone P. Delima et al., International Journal of Advanced Trends in Computer Science and Engineering, 9(3), May – June 2020, 3270 – 3274
3274
2018.
[16] G. Zhong, “Cryptanalysis of Homophonic
Substitution Cipher Using Hidden Markov Models,”
2016.
[17] B. Shrestha, “Multiprime Blum-Blum-Shub
pseudorandom number generator,” 2016.
[18] D. Kahn, The Codebreakers: The Comprehensive
History of Secret Communication from Ancient
Times to the Internet. Scribner, 1996.
[19] L. Blum, M. Shub, and M. Blum, “Simple
Unpredictable Pseudo-Random Number Generator,”
SIAM J. Comput., vol. 15, no. 2, pp. 364–383, 1986.
https://doi.org/10.1137/0215025
[20] T. S. Kondo and L. J. Mselle, “An Extended Version
of the Polybius Cipher,” Int. J. Comput. Appl., vol.
79, no. 13, pp. 30–33, 2013.
https://doi.org/10.5120/13803-1836