Abstract In this paper we have investigated the security of different variants the golden cryptography, and showed that all the variants of the cryptosystem are not secure against chosen-plaintext attack

A Jaafar

M Md Said

M Tahghighi

R Mahmod

S Turaev

CiteSeerX

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 7, 327 - 335
On the Security of Golden Cryptosystems
M. Tahghighi1, S. Turaev1, A. Jaafar1, R. Mahmod1 and M. Md. Said2
1Faculty of Computer Science and Information Technology
University Putra Malaysia
43400 UPM Serdang, Selangor, Malaysia
mtahghighi@yahoo.com, {sherzod, azmi, ramlan}@fsktm.upm.edu.my
2Institute for Mathematical Research
University Putra Malaysia
43400 UPM Serdang, Selangor, Malaysia
mrushdan@putra.upm.edu.my
Abstract
In this paper we have investigated the security of different variants
the golden cryptography, and showed that all the variants of the cryp-
tosystem are not secure against chosen-plaintext attack.
Mathematics Subject Classification: 11B39, 11T71, 11C20
Keywords: Fibonacci numbers and functions, Golden matrices, Cryptog-
raphy, Cryptanalysis
1 Introduction
Fibonacci numbers or Fibonacci sequences, closely connected to Golden Sec-
tion, appear in various fields such as nature, architecture, physics, etc., and
are used in modeling phenomena appearing in economy, computer science,
physics, etc. The recent intensive studies of Fibonacci numbers have resulted
different extensions of Fibonacci numbers. For instance, k-Fibonacci numbers,
hyperbolic Fibonacci functions, FibonacciQ-matrices, Golden matrices, and so
on. The simplicity and beauty of Fibonacci numbers allow developing matrix
cryptosystems, which are useful in digital communications, i.e., digital TV,
digital telephony, digital measurement, etc. One of such cryptosystems, called
golden cryptography based on golden matrices, which are a generalization of
Fibonacci Q-matrices for continuous domain, was introduced by Stakhov in [7].
Later, in his another paper, he improved golden cryptosystem by using golden
Gk-matrices based on the k-Fibonacci hyperbolic functions (see [8]). Another
328 M. Tahghighi, S. Turaev, A. Jaafar, R. Mahmod and M. Md. Said
interesting cryptosystem based on Hadamard product of golden matrices was
introduced by Nally in [4]. There are also other simple cryptographic methods
(for instance, see [9], [2]) based on extensions of golden matrices.
Any cryptosystem has a practical value if it is secure against different types
of cryptanalytic attacks such as ciphertext-only attack (cryptanalysis has to
based on only the sample of ciphertext), known-plaintext attack (the cryptan-
alyst knows in advance some pairs of plaintexts and the corresponding cipher-
texts), chosen-plaintext attack (the cryptanalyst can obtain the ciphertexts
corresponding to an arbitrary set of plaintexts of his own choosing).
Unfortunately, the above-mentioned cryptosystems are not resistable against
some cryptanalytic attacks. Rey and Sánchez (see [6]) showed that the cryp-
tosystem proposed in [7] is not secure, i.e., it is not robust against chosen-
plaintext attack. In this paper, we show that the improved version of golden
cryptography developed in [8] and the cryptosystem defined in [4] are also
insecure against chosen-plaintext attack.
The paper is organized as follows: in Section 2 we cite some necessary
notions and notations which are used in sequal. In Section 3 we show that
the cryptographic method based on Hadamard product of golden matrices (in
Subsection 3.1) and the generalized golden cryptography are not secure against
the chosen-plaintext attack (in Subsection 3.2).
2 Preliminaries
The reader is assumed to be familiar with basic notations of cryptography and
cryptanalysis, for details refer to [5].
2.1 Golden matrices
For some real number x, the golden matrices are defined as
Q2x =
(
cFs(2x+ 1) sFs(2x)
sFs(2x) cFs(2x− 1)
)
, (1)
Q2x+1 =
(
sFs(2x+ 2) cFs(2x+ 1)
cFs(2x+ 1) sFs(2x)
)
, (2)
where
sFs(x) =
τx − τ−x√
5
and cFs(x) =
τx + τ−x√
5
On the security of golden cryptosystems 329
are the symmetrical hyperbolic Fibonacci sine and the symmetrical hyperbolic
Fibonacci cosine with τ = 1+
√
5
√
2
(the Golden Proportion). For the detailed
information on golden matrices, we refer the reader to [7].
The Hadamard product of golden matrices is defined as
Q2x ◦Q−2x =
(
cFs(2x+ 1)cFs(2x− 1) −[sFs(2x)]2
−[sFs(2x)]2 cFs(2x+ 1)cFs(2x− 1)
)
, (3)
Q2x+1 ◦Q−2x−1 =
(
−sFs(2x+ 2)sFs(2x) [cFs(2x+ 1)]2
[cFs(2x+ 1)]2 −sFs(2x+ 2)sFs(2x)
)
. (4)
The inverses of the matrices (3) and (4) are defined as
(Q2x ◦Q−2x)−1 = 1
1 + 2q1
(
1 + q1 q1
q1 1 + q1
)
, (5)
(Q2x+1 ◦Q−2x−1)−1 = 1
1− 2q2
(
1− q2 −q2
−q2 1− q2
)
, (6)
where q1 = [sFs(2x)]
2 and q2 = [cFs(2x + 1)]
2. The detailed information on
the Hadamard product of golden matrices can be found in [3, 4].
For some positive integer k and real number x, the generalized golden ma-
trices of the order k is defined as
G2x
k
=
(
cFk(2x+ 1) sFk(2x)
sFk(2x) cFk(2x− 1)
)
, (7)
G2x+1
k
=
(
sFk(2x+ 2) cFk(2x+ 1)
cFk(2x+ 1) sFk(2x)
)
, (8)
where
sFk(x) =
σx
k
− σ−x
k
σk + σ
−1
k
and cFk(x) =
σx
k
+ σ−x
k
σk + σ
−1
k
, σk =
k +
√
k2 + 4
2
,
are k-Fibonacci hyperbolic sine and cosine, respectively.
The inverse matrices of (7) and (8) are defined as
G−2x
k
=
(
cFk(2x− 1) −sFk(2x)
−sFk(2x) cFk(2x+ 1)
)
, (9)
G−2x−1
k
=
(
−sFk(2x) cFk(2x+ 1)
cFk(2x+ 1) −sFk(2x+ 2)
)
. (10)
The detailed information on the k-Fibonacci hyperbolic functions and the
330 M. Tahghighi, S. Turaev, A. Jaafar, R. Mahmod and M. Md. Said
corresponding golden matrices is given in [1, 8].
2.2 Golden cryptographic methods
The basic idea of the cryptographic methods proposed in [4, 8] is as follows:
• a plaintext a1, a2, a3, a4, . . . is presented in the form of 2× 2 matrices
M =
(
a1 a2
a3 a4
)
, . . .
• For some i = 1, 2, . . . , 24, the permutation π(i) of a1, a2, a3, a4, i.e., aπ(1),
aπ(2), aπ(3), aπ(4) is designated.
• The matrix (3) or (4) (the matrix (7) or (8)), which called the enciphering
matrix, and its inverse matrix (5) or (6) (its inverse matrix (9) or (10)),
respectively, which is called the deciphering matrix, are chosen.
• The ciphertext E1(x) or E2(x) is obtained by using matrix multiplication
of the plaintext M and the matrix Q2x ◦ Q−2x or Q2x+1 ◦ Q−2x−1 (the
ciphertext E1(x, k) or E2(x, k) is obtained by using matrix multiplication
of the plaintext M and the matrix G2x
k
or G2x+1
k
), i.e.,
E1(x) = M ×Q2x ◦Q−2x or E2(x) = M ×Q2x+1 ◦Q−2x−1
(E1(x, k) = M ×G2xk or E2(x, k) = M ×G2x+1k ).
• In order to recover the original plaintext M , the corresponding inverse
matrix (Q2x ◦Q−2x)−1 or (Q2x+1 ◦Q−2x−1)−1 (the corresponding inverse
matrix G−2x
k
or G−2x−1
k
) is used, i.e.,
M = E1(x)× (Q2x ◦Q−2x)−1 or M = E2(x)× (Q2x+1 ◦Q−2x−1)−1
(M = E1(x, k)×G−2xk or M = E2(x, k)×G−2x−1k ).
3 Results
In the following subsections we show that the above-mentioned cryptographic
methods do not resist against the chosen-plaintext attack.
On the security of golden cryptosystems 331
3.1 The chosen-plaintext attack against the cryptogra-
phic method with Hadamard product of golden ma-
trices
We choose the pairs of plaintexts and ciphertexts {Mi, Ei(x)}, i = 1, 2, 3, 4,
where
M1 =
(
−1 1
1 1
)
, M2 =
(
1 −1
1 1
)
, M3 =
(
1 1
−1 1
)
, M4 =
(
1 1
1 −1
)
. (11)
Without loss of generality, we can assume that the enciphering matrix is
Q2x ◦ Q−2x. Then 24 possible permutations of the initial matrix are reduced
to the following 4 matrices:
E1(x) = M1 ×Q2x ◦Q−2x =
(
−e2 e2
e1 e1
)
,
E2(x) = M2 ×Q2x ◦Q−2x =
(
e2 −e2
e1 e1
)
,
E3(x) = M3 ×Q2x ◦Q−2x =
(
e1 e1
−e2 e2
)
,
E4(x) = M4 ×Q2x ◦Q−2x =
(
e1 e1
e2 −e2
)
,
where
e1 = cFs(2x+ 1)cF (2x− 1)− [sFs(2x)]2,
e2 = cFs(2x+ 1)cF (2x− 1) + [sFs(2x)]2.
Thus, for the known real variables t1 and t2, we obtain the system
{
Fs(2x+ 1)cF (2x− 1)− [sFs(2x)]2 = t1
cFs(2x+ 1)cF (2x− 1) + [sFs(2x)]2 = t2.
(12)
of non-linear equations. By the definition of sF (x) and cFs(x),
{
(τ2x+1+τ−2x−1)
5
− (τ2x−τ−2x)2
5
= t1
(τ2x+1+τ−2x−1)
5
+ (τ
2x
−τ
−2x)2
5
= t2.
(13)
The substraction of the first equation from the second equation in (13)
leads to the equation
332 M. Tahghighi, S. Turaev, A. Jaafar, R. Mahmod and M. Md. Said
2(τ 2x − τ−2x)2 = 5(t2 − t1). (14)
Further, by some calculations, we obtain the equation
y2 + (2.5(t1 − t2)− 2)y + 1 = 0, (15)
where y = τ 4x. The solutions of the equation (15) are
y =
2− 2.5(t1 − t2)±
√
6.25(t1 − t2)2 + 5(t1 − t2)
2
.
Thus, by the simple calculations one can easily find the secret key
x =
1
4
logτ
(
2− 2.5(t1 − t2)±
√
6.25(t1 − t2)2 + 5(t1 − t2)
2
)
of the cryptosystem, which means that the cryptographic method based on
the Hadamard product of golden matrices is not resistable against the chosen-
plaintext attack.
3.2 The chosen-plaintext attack against the generalized
golden cryptographic method
We choose the following plaintext matrices
M1 =
(
1 0
0 0
)
, M2 =
(
0 1
0 0
)
, M3 =
(
0 0
1 0
)
, M4 =
(
0 0
0 1
)
, (16)
and the enciphering matrix G2x
k
(without loss of generality). The selection
of the plaintexts as the matrices in (16) reduces possible 24 variants of the
permuted initial matrix the following 4 ciphertexts
E1(x, k) = M1 ×G2xk =
(
cFk(2x+ 1) sFk(2x)
0 0
)
,
E2(x, k) = M2 ×G2xk =
(
sFk(2x) cFk(2x+ 1)
0 0
)
,
E3(x, k) = M3 ×G2xk =
(
0 0
cFk(2x+ 1) sFk(2x)
)
,
E4(x, k) = M4 ×G2xk =
(
0 0
sFk(2x) cFk(2x+ 1)
)
.
On the security of golden cryptosystems 333
Thus, the key of the cryptosystem can be found by the solution of the
following system







sFk(2x) = t1
cFk(2x+ 1) = t2
cFk(2x− 1) = t3
(17)
of non-linear equations, where t1, t2 and t3 are known real numbers. Further,
we prove that the solutions of the system (17) can be found by using simple
calculations.
By the definitions of sFk(x) and cFk(x), the system (17) can be written as







σ2x
k
− σ−2x
k
= t1(σk + σ
−1
k
)
σ2x+1
k
+ σ−2x−1
k
= t2(σk + σ
−1
k
)
σ2x−1
k
+ σ−2x+1
k
= t3(σk + σ
−1
k
).
(18)
If we multiply the first equation of the system (18) by σ−1
k
, and add it to
the second equation, the equation
σ2x−1
k
+ σ2x+1
k
= (σk + σ
−1
k
)(t1σ
−1
k
+ t2)
is resulted. Further
σ2x
k
(σk + σ
−1
k
) = (σk + σ
−1
k
)(t1σ
−1
k
+ t2),
and finally
σ2x
k
= (t1σ
−1
k
+ t2). (19)
On the other hand, if we multiply the first equation of (18) by σk, and add
the obtained equation and the third equation, then
σ2x−1
k
+ σ2x+1
k
= (σk + σ
−1
k
)(t1σk + t3).
By the similar calculations above, we obtain
σ2x
k
= (t1σk + t3). (20)
From the equations (19) and (20),
t1σ
−1
k
+ t2 = t1σk + t3,
and
t1σ
2
k
+ (t3 − t2)σk − t1 = 0.
334 M. Tahghighi, S. Turaev, A. Jaafar, R. Mahmod and M. Md. Said
Since D = (t3 − t2)2 + 4t21 ≥ 0,
σk =
t2 − t3 ±
√
(t3 − t2)2 + 4t21
2t1
. (21)
Thus the value of σk can be easily calculated from the known variables t1, t2
and t3.
As
σk =
k +
√
k2 + 4
2
,
it is easy to see that
k =
σ2
k
− 1
σk
. (22)
From (20) (or from 19), we can calculate the value of x, i.e.,
x =
1
2
log
σk
(t1σk + t3) (or x =
1
2
log
σk
(t1σ
−1
k
+ t2)). (23)
Thus the secret key {k, x} is obtained, which shows that the generalized
golden cryptosystem is also vulnerable to the chosen-plaintext attack.
4 Conclusions
We have showed that the generalized golden cryptography and the golden
cryptography with Hadamard products are not resistable against the chosen-
plaintext attack. We have not mentioned the security of the cryptographic
methods proposed in [2] and [9] intentionally, as they are very simple and it
is very easy to show their insecurity against the chosen-plaintext attack. We
leave the proofs of this statement to an eager reader.
Acknowledgements
This work was partially supported by University Putra Malaysia via RUGS
05-01-10-0896RU.
References
[1] S. Falcón and Á. Plaza, The k-Fibonacci hyperbolic functions, Chaos,
Solutions and Fractals, Vol. 38 (2008), 409–420.
On the security of golden cryptosystems 335
[2] Ernatuti, R. Salim, Sulistyo, The applications of “ELC numbers” to
golden cryptography, The Fifth International Conference on Information
and Communication technology and Systems, 2009, 329–334.
[3] A. Nally, On the Hadamard product of FibonacciQn matrix and Fibonacci
Q−n matrix, Int. J. Contemp. Math Sciences, Vol. 1, No. 16 (2006), 753
– 761.
[4] A. Nally, On the Hadamard product of golden matrices, Int. J. Contemp.
Math Sciences, Vol. 2, No. 11 (2007), 537 – 544.
[5] A. Menezes, P. van Oorschot and S. Vanstone, Handbook of Applied Cryp-
tography, CRC Press, Boca Raton, 1996.
[6] A. Rey and G. Sánchez, On the security of the “golden cryptography,
International Jounal of Network Security, Vol. 7, No. 3 (2008), 448–450.
[7] A. Stakhov, The “golden matrices and a new kind of cryptography, Chaos,
Solutions and Fractals, Vol. 32 (2007), 1138–1146.
[8] A. Stakhov, Gazale formulas, a new class of the hyperbolic Fibonacci and
Lucas functions, and the improved method of the ”golden” cryptography,
Moscow: Academy of Trinitarism, 77-6567, publication 14098 (2006),
1138–1146.
[9] K.R. Sudha, A. Chandra Sekhar, Prasad Reddy PVGD, Cryptography
protection of digital signals using some recurrence relations, Int. J. of
Comp. Sci. and Network Security, Vol. 7, No. 5 (2007), 203–207.
Received: April, 2011


On the Security of Golden Cryptosystems

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