1            Introduction

Public key cryptographic is the fundamental technology in secure communications. It was devised by Diffie and Hellman in 1976 to secret key distribution. The mathematical problems more used are the discrete logarithm problem (DLP). In 1985 the elliptic curve discrete logarithm problem (ECDLP) was proposed independently by Koblitz and Miller. In this paper, we present the Matrix discrete logarithm problem in a new cryptographic scheme. Consider a finite field L = F_q , where q is a power of p the characteristic

of L.[1, 2, 3] Throughout this work, we denote:

L^∗ multiplicative group of L. Let a,b ∈ L^∗ and let x,y ∈ L,

M_yx =  yba 1              xyb +1− 1 !  x −

                                                +1      a

G = {Myxdet(Myx) = 1} G_q = G mod p.

Myx11 4Myx22 = Myx33

where,

b



^{(1) :}  yx33 ⁼= x12yx21+x_a2abx+22ya12y1y2





               Mk =  k222+1k − 1 kk22−k1 − 1 ,k ∈ L∗.

                                k −1 +1               +1 

2k

m = |Gq |

The next theorem whose proof is evident.

Theorem 1 The set G_q with the operator 4 defined by (1) is a abelian group.

The identity element is M₀^a, that if M = M_y^x then N = M₋^x_y is the invertible element of M.

Remark 1 The MDLP consists of following for two elements M,N ∈ G_q, determine the scaler k ∈ _Zm such that M^4k = N. It is necessary that M be a generator of the group G_q.

Assumption 1 Given a group G_q and tow elements M and N ∈ G_q, there exists non polynomial time algorithm θ(logq) deciding the integer k such that M^4k = N if such a k exists.

Assumption 2 Given a group G_q and θ(logq) elements N_i on G_q, there exists non polynomial time algorithm

(θ(logq)) deciding the integers k_i, such that

N₁4k1 4N24k2 4…….4Nθ4(klogqθ(logq) ) = M

if such k_i exist, where M is a random element on G_q.

2            Matrix Cryptosystem

2.1         Key distribution protocols

Let M_y^x be a generator of the group G_q. Alice take a private key 1 < l < m, and computes M_y^x_l^l =

M_y^x4l, then she transmits M_y^x_l^l to Bob. Similar, Bob takes a private key 1 < t < m, and computes M_y^x_t^t = M_y^x4t and transmits M_y^x_t^t to Alice. In the same way Alice and Bob compute M_y^x_tl^tl = M_y^x_t^{t 4l} and Myxltlt = Myxll 4t respectively.

Theorem 2

                            xlt         ylt           xtl          ytl

+ = + mod p a b a b

The secret key is α = ^x_a^tl + ^y_b^tl mod p

2.2         Description of This Cryptosystem

Let L = F_q with q = pⁿ.

1)Space of lights: P = G_q.

2)Space of quantified: C = G_q .

3)Space of the keys: K = L^∗ .

4)Function of encryption:∀α ∈ K ,

                       eα:P          −→     C

                             Myx      7−→    eα (Mxy) = Myx 4Mα

5)Function of decryption:∀α ∈ K ,

                      dα:C          −→     P

                             Myx      7−→    dα(Mxy) = Myx 4M−α

Remark 2

dαoeα(Myx) = Myx 4Mα 4M−α = Myx

Remark 3 a) Secret key :α b) Public keys: 1) Space of lights; P

2) Space of quantified; C 3) Space of the keys; K

• Generator of the group P ; M_y^x
• Function of encryption; e_α
• Function of decryption; d_α

Remark 4 The M_y^x_l^l , M_y^x_t^t and m are public and can known by another person, but to obtain the private key α, it is necessary to solve the Matrix problem discrete logarithm in G_q, what returns the discovery of the difficult key α.

2.3         Numerical Example

Alice and Bob Choose the following public numbers; p = 41, a = 2, b = 5, and n = 1. They determine the group G₄₁ =< M₃₁²⁶ >, with the identity element M₀².

• Exchange of the key deprived between Alice and Bob:

Alice take a private key; l = 13 < 39, calculation; M₂₃¹³ = M₃₁²⁶⁴¹³ and send to Bob M₂₃¹³. In turn, Bob take a private key; t = 21 < 39, calculation; M₁₀¹⁵ = M₃₁²⁶⁴²¹ and send it to Alice. Alice and Bob calculate separately :

M1813 = M1015413 and M1813 = M2313421. They determine their secret key:

α =  +  = 6 mod 41

• Message to send:

Alice wants to send the following message

me = {M1828,M40,M2723,M3634 } It encrypts it using the encryption function

Mxy	e6(Myx)
M2818	39 1	40 0	!
M04	15 39	37 17	!
M2327	28 17	13 30	!
M3436	28 27	25 30	!

• Message received:

Bob receives message crypt send by Alice

28 mr = { 27

25 ! 28 , 30           17

13 ! 15 , 30           39

37 ! 39 , 17            1

40 ! } 0

It decrypts it using the decryption function

Mxy			d6(Myx)
39 1	40 0	!	M2818
15 39	37 17	!	M04
28 27	25 30	!	M3436
28 17	13 30	!	M2327

3          Example for cryptography

In this example we take p = 3, a = b = 1, n = 3, and α root of the polynomial X³ + 2X + 1. We have P = G₂₇, C = G₂₇, and K  and M_α^2α₂ ⁺² is a generator of the group P .

• Exchange of the key deprived:

Alice take a private key l = 12 < 25, send to Bob M_α^α₂⁺¹ = M_α^2α₂ ^+24l. Bob take a private key t = 20 < 25, send to Alice

M22αα22++2α+1 = Mα2α2 +24t. Their secret key is

β = 2α² +2+α² +2α +2 = 2α +1.

• Message Encryption:

It is known that the encryption functions and the decryption functions are defined by:

eβ(Myx) = Myx 4Mα2α2+22+2α+2 dβ(Myx) = Myx 4M22αα22++2α+1

Lets x = iα²+jα+k and y = lα²+mα+n, we denote M_y^x by ijklmn.

Each letter is represented by a ijklmn character. Often the simple The scheme a = 001000, b = 010112, …, z = 221102 is used, but this is Not an essential feature of encryption. To encrypt a message, each letter will be decrypted by the decryption function, for a message one obtains a block of n letters (considered as an n-component vector). Consider the message ’bonjour’ Which will be encrypted by the message: ”crvzrng”.

Table of the Symbol Encryption

Mxy	Symbol	eβ(Myx)	Encrypt Symbol
001000	a	202120	h
010112	b	010220	c
010220	c	012210	d
012210	d	211110	s
012122	e	010112	b
122110	f	021210	q
122222	g	112102	m
202120	h	011101	j
202212	i	001000	a
011101	j	221102	z
011201	k	202212	i
112200	l	122110	f
112102	m	022101	w
002001	n	101212	v
020112	o	021122	r
020220	p	020112	o
021210	q	020220	p
021122	r	122222	g
211110	s	221200	y
211222	t	012122	e
101120	u	002001	n
101212	v	022201	x
022101	w	101120	u
022201	x	112200	l
221200	y	011201	k
221102	z	211222	t

4          Conclusion

Although matrix multiplication can not provide security for the encryption of a message [4, 5, 6], we have been able to construct a law of internal composition other than the law of multiplication, which allows us to create a cryptography on the matrices and which is safer for a key of reasonable length.

1. A. Chillali, “Cryptography over elliptic curve of the ring Fq[e], e4 = 0 .”, World Academy of Science, Engineering and Technology., 78 , 848-850, 2011.
2. A. Tadmori, A. Chillali, M. Ziane, “The binary operations calculus in Ea;b;c.”, International Journal of Mathematical Models and Methods in Applied Sciences., 9 , 171-175, 2015.
3. A. Tadmori, A. Chillali, M. Ziane, “Elliptic Curve over Ring A4.”, Applied Mathematical Sciences., 35(9), 1721-1733, 2015.
4. Lester S. Hill, “Cryptography in an Algebraic Alphabet.”, The American Mathematical Monthly., 36, 1929.
5. Lester S. Hill, “Concerning Certain Linear Transformation Apparatus of Cryptography.”, The American Mathematical Monthly., 38(9), 1931.
6. Christos Koukouvinos and Dimitris E. Simos, “Encryption Schemes based on Hadamard Matrices with Circulant Cores.”, Journal of Applied Mathematics and Bioinformatics., 1(3), 17-41, 2013.

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