Chaos-based Information Hiding and

Security: an Emergent Technology

1Safwan El Assad

Safwan El Assad

IETR Laboratory, UMR CNRS 6164; Image team - site of

Nantes

Polytech Nantes, school of engineering of the university of

Nantes – France

International Workshop on Cryptography and its Applications,

IWCA’ 2016, Oran, Algeria, 26-27 April 2016

2Safwan El Assad

6 Ph.D studentsDaniel Caragata (16/10/2007 - 01/04/2011)

Title : Communication protocols secured with chaotic sequences.

Applications for: IP over DVB-S and the UMTS

Dalia Battik (Janvier 2012 – 18/05/2015)

Title : Information Security by steganography based on chaotic sequences.

Mousa Farajallah (28/11/2012 – 30/06/ 2015)

Title : Chaos-based crypto and joint crypto-compression for images and

videos

Ons Jallouli (08/10/2014 – September 2017)

Title: Chaos-based data security of the Internet of Things under real-time and

energy constraints

Mohammad Abu Taha (20/10/2014 – October 2017)

Title : Real-Time and Portable Chaos-based Crypto-Compression Systems for

Efficient Embedded Architectures

Nabil Abdoun (12/03/2015 - February 2018)

Title : Design and efficient implementation of one-way hash functions based

on chaotic maps and neural networks

Information Hiding

Steganography CryptographyWatermarking

InvisibleVisible

Robust Fragile

Used for tamper

detection Data

integrity

Used in copy

protection

applications

Process that embeds a

watermark (tag or label) into a

multimedia object such that

watermark can be detected or

extracted later to make an

assertion about the object

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Confidentiality

through

obscurity

Confidentiality

through

encryption

3

Cryptology

CryptanalysisCryptography

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Blocs Ciphers

Symmetric

Ciphers

Asymmetric

CiphersProtocols

Classical

Cryptanalysis

Implementation

Attacks

Social

Engineering

Stream Ciphers

Cryptanalytic Attacks Statistical Attacks

Safwan El Assad 5

Design of Robust and Fast Chaos based Cryptosystems

Chaos-Based

Decryption

Algorithm

Chaotic

GeneratorSecret key

Encrypted

and Perturbed

Information

Decrypted

Information

Errors impact on the

decryption information

Cryptographic modes

Bob

Key management

and Protocol Cryptanalyst

Useful

Information Encrypted

Information

Chaotic

Generator

Chaos-Based

Encryption

Algorithm

Channel

Noise

Secret key

Text, Audio,

Image, Video

Design of robust

and fast chaotic

generators

Design of robust

and fast encryption

algorithms

Alice

Eve

Principle

Outline

Why using chaos to secure information?

Some known chaotic maps used in chaos-based encryption

Effects of the finite precision N

How to avoid the effects of the finite precision N and to obtain

randomness.

Structure of proposed generators of discrete chaotic samples

Performances: Security analysis and time consuming

General structure of chaos-based cryptosystems: Encryption side

Chaos-based cryptosystems of 1er type: Separate layers of confusion

and diffusion: Example

6Safwan El Assad

Outline

Chaos-based cryptosystems of 2nd type: Combined layers of

confusion and diffusion: Example

Comparative performance : Time consuming and Security analysis

Joint Crypto-Compression & Selective Stream Encryption

Conclusion, current and future works.

7Safwan El Assad

Why using chaos to secure information?

Useful properties of chaos in secure information

Easy to generate: simple discrete-time dynamical system is

capable to generate a complex and random like behavior

sequences :

Chaotic signal is deterministic, not random (we can

regenerate it) and it has a broadband spectrum

Chaotic signal is extremely difficult to predict because of

the high sensitivity to the secret key

Very big number of orbits in finite region of phase space

Safwan El Assad8

( ) ( 1) X n F X n

Some known chaotic maps used in chaos-based

encryption

Chaotic maps used as PRNG:

1-D: Logistic, PWLCM, Skew Tent

2-D: Hénon map, Lozi map

3-D: Lorenz

Chaotic maps used as permutation layer :

2-D : Cat, Standard, and Baker maps

Chaotic map used as nonlinear substitution layer :

1-D : Skew Tent

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Discrete Skew Tent Map

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[ 1 , ]

12 0 1

2 12 1 2

2

N

NN N

N

X n F X n P

X nif X n P

P

X nif P X n

P

0 2 1 NP Control parameter

Attractor Mapping

Discrete PWLCM Map

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1

1

1

1

2 ( 1)0 ( 1)

2 ( 1)( 1) 2

2

( ) ( 1),2 2 1 ( 1)

2 ( 1) 22

2 2 1 ( 1)2 ( 1) 2

N

NN

N

N N

N N

N

N N

N N

X nX n P

P

X n PP X n

P

X n F X n PX n P

X n PP

X nP X n

P

10 2 NP Control parameter

Attractor Mapping

Effects of the finite precision N

12Safwan El Assad

In finite precision N bits with 1-D chaotic map

1X

1 l cX

1lX

lX1lX0X

Transient branch of length l

Cycle of period cOrbit : o = l + c

Pseudo-orbit of a digital chaotic value

Maximum length of the orbit : max 2 1No

Average orbits : D

𝑋 𝑛 = 𝐹 𝑋 𝑛 − 1 , 𝑃 ∈ 1, 2𝑁−1 , n = 1, 2, …

𝑋𝑛 = 𝑋 𝑛

How to avoid the effects of the finite precision N

and to obtain randomness.

13Safwan El Assad

( )X n

1k

2k

mk

( )u n

1D

2D

mD

Chaotic

map

Recursive structure Cascading Technique (Li et. al., 2001)

Ultra-weak Coupling Technique and

Chaotic mixing (Lozi, 2007 & 2012)

Perturbation Technique (Tao, 2005, El

Assad 2008)

Recursive structure and Orbits

Multiplexing (El Assad et. al., 2008 &

2011)

Average length of the orbit : D

Average length of the orbit of

the recursive structure :max 1Dmo

Perturbation Technique

14

Implementation

Perturbation every delta iteration

min 2 1 D koLower length of the orbit :

Safwan El Assad

( 1)X n [ ( 1)]F X n

( )X n

( )Q n

Chaotic-map

LFSR

Linear Feedback

Shift Register

1 2 0( ) ( ) ( )... ( ).... ( ) ( ) 0,1N N i i bX n x n x n x n x n x n

0,1,2,...DIf n l l

[ ( 1)] 1( )

[ ( 1)] ( ) 0 1

i

i

i i

F x n k i Nx n

F x n Q n i k

: ( ) [ ( 1)]

Else

No perturbation X n F X n

Primitive polynomial

generator of degree k

D: orbit of the chaotic-map

without perturbation

Sequence of

disturbance

Chaotic generator_v1: PCNG

15Safwan El Assad

𝑋𝑔 𝑛 = 𝑋𝑠 𝑛 ⨁𝑋𝑠 𝑛

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Security Analysis of PCNG

Generic scheme

Long dynamic integer values X(n):

Large secret key size

Delay = 3: Size of the secret key is 555 bits :

8 I.Cs (8 x N bits); 8 parameters (7 x N+ N-1 bits); LFSR : 2 I.Cs (k1 + k2) bits:

= 15 x N +N - 1+ k1 + k2 = 15 x 32 + 31 + 23 + 21 = 555 bits

Delay = 2: Size of the secret key is 427 bits; Delay = 1: Size is 299 bits

For all delays the Brute-Force Attack is infeasible

Statistical analysis:

Passing statistical tests: Pseudo-random mapping, Nist, uniformity of

histograms, Chi2 test, delta-like auto-correlation, nearly zero cross correlation

1 2

min1 2 1 , 2 2 1 D D

k ko lcm

3 /2 48 71 140min32, 1 23, 2 21 2 2 2 2 D N

nomWith N k k and o

Security Analysis of PCNG

17

Mapping NIST

Histogram Chi2_th = 1073.64Chi2_ex = 992.37 Auto & cross correlation

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Security Analysis of PCNG and of UWCT-CM

18

Key sensitivity test

A good PCNG should be very sensitive to the secret key.

𝐾1 ====> 𝑆1

𝐾1 𝑤𝑖𝑡ℎ 1 𝑏𝑖𝑡 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 ====> 𝑆2 ≠ 𝑆1

𝐷𝐻𝑎𝑚𝑚𝑖𝑛𝑔 𝑆1, 𝑆2 =

𝑖=1

𝑙_𝑠𝑒𝑞

𝑆1 𝑖 ⨁𝑆2 𝑖 = 49.999%

The probability of bit changes is close to 0.5.

Ciphertext attack is infeasible :

It is computationally infeasible to retrieve the secret key from the generated sequences

The chaotic system is CSPCNG (Cryptographically secure pseudo-

chaotic number generator): a CSPCNG is PCNG which is unpredictable

Given n output samples of the keystream (sequence) si, si+1,…,si+n, it is

computationally infeasible to compute the next subsequent samples si+n+1,

si+n+2, …or the preceding subsequent samples si-1, si-2, ….

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Bit rate (Mbit/s) / NCpB

Sequential

implementation

Parallel

Implementation (4 cores)

PCNG

Delay 1 930.1 / 22.3 1450.19 / 14.3

Delay 2 890.56 / 23.3 1368.29 / 15.2

Delay 3 750.57 / 27.7 1276.95 / 16.3

UWCT-CM 888.4 / 24 1372.4 / 16

AES/CTR

AES/OFB

1107 / 21.2

787.48 / 29.7

Chaotic generators : Bit rate and NCpB

Pseudo chaotic generator NCpb

François et al., 2012 97254

Akhshani et al., 2014 23

Jallouli et al., 2015 151

UWCT-CM 16

PCNG (with delay = 3) 16.3

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eStream: Bit rate and NCpB

Cipher Test Encryption

throughput

(Mbps)

NCpB

Rabbit 500 packets of 40 B 648.75 34.45

100 packets of 576 B 1898.35 11.77

42 packets of 1500 B 2077.41 10.76

HC-128 11 packets of 40 B 15.32 1458.6

11 packets of 576 B 213.16 104.86

10 packets of 1500 B 524.18 42.64

Salsa20/12 500 packets of 40 B 701.55 31.86

100 packets of 576 B 2039.98 10.96

40 packets of 1500 B 1987.12 11.25

General structure of chaos-based cryptosystems

21Safwan El Assad

Cipher

blockPlain

block

rc rd

r

Kc Kd

Confusion

layer

Diffusion

layer

Chaotic generatorSecret key

X(n)

Shannon [1949]

Confusion : measures how a change in the secret key affects the ciphered massage

Diffusion : assesses how a change in the plain message affects the ciphered one

Fridrich [1998]:

Most popular structure adopted in many chaos-based cryptosystems

Chaos-based cryptosystem-1: 1er type

22Safwan El Assad

[El Assad et. Farajallah, 2016] in Signal Processing: Image Communication

Cipher

block

Plain

block

rd

rd

r

Kp

Confusion

Bit-permutation

Cat map

Chaotic generatorSecret

key

Diffusion

Binary

matrix

Int2

Bin

Bin2

Int

rp

Int2BIN: Nonlinear converter

2D cat : Efficient formulation for C implementation

When a bit permutation layer is applied on a block ,

it performs, on one scan, a substitution and a

diffusion operations on the bytes

Chaos-based cryptosystems: 1er type

23Safwan El Assad

1,

1

n

n

i u i ri rj MMod

j v uv j rj M0 , , , 1 2 1 qu v ri rj M

2-D Cat map as permutation layer

1 2

, , , 1, ,

r

l l l l l

Kp kp kp kp

kp u v ri rj l rp

Where i, j and in, jn are the original and permuted pixel positions

of the M X M square matrix, with M = 2q.

The Cat map is bijective, so each point in the square matrix

is transformed to another point uniquely.

Structure of the dynamic key Kp

Chaos-based cryptosystems: 2nd type

24Safwan El Assad

2nd type : Combined layers of confusion and diffusion

The confusion and diffusion processes are performed simultaneously

in a single scan of plain-image pixels. More Speed

Cipher

block

Plain

block

rp

r

Kp Kd

Confusion layer

2D-Pixel

Permutation

Chaotic generator_sSecret key

Diffusion layer

Sequential Pixel

Value Modification

[Wong et al., 2009], [Wang et al., 2011], [Zhang et al., 2013],

[Farajallah et al., 2016] in IJBC Journal

Chaos-based cryptosystems: 2nd type

25Safwan El Assad

2nd type : The diffusion process at the pixel level is governed

by the confusion one

, ( , , , , , ),

( , ) ( , ) ( ), (1)

( , )

n n

n n

n n

i j Cat i j u v ri rj M

c i j p i j q f z L

z c i j

1

( ) (1 ) (2)

, 2

L

z Kd

f z z z

q b L b

p(i, j)

p(i, j+1)

1 2 3

Diffusion

Process

c(in, jn)

c[i*, (j+1)*]

1

2

3

Plain-image Ciphered-image

Safwan El Assad

Diffusion

𝐿𝑆𝐵8(𝑦0(𝑘))

𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 2𝐷 𝑐𝑎𝑡 𝑀𝑎𝑝(𝒊, 𝒋)

⊕ ⊕

𝐶ℎ𝑎𝑜𝑡𝑖𝑐 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟Secret Key

Kdm

iv(k)

p0(k)

Kpm

f(y0(k-1))

(𝒊𝒏, 𝒋𝒏)

C0(kn)

C0(0) C0(1) C0(2) … C0(M-1) … C0(bs-1)

Diffusion

𝐿𝑆𝐵8(𝑦1(𝑘))

𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 2𝐷 𝑐𝑎𝑡 𝑀𝑎𝑝(𝒊, 𝒋)

⊕ ⊕

𝐶ℎ𝑎𝑜𝑡𝑖𝑐 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟Secret Key

p1(k)

Kpm

f(y1(k-1))

(𝒊𝒏, 𝒋𝒏)

C1(kn)

C1(0) C1(1) C1(2) … C1(M-1) … C1(bs-1)

y0(bs-1)

C0(k)

C1(k)

Chaos-based cryptosystem-2: 2nd type

Diffusion process :

V1: Logistic map

with N = 32 bits

V2: Skew tent map

with N = 32 bits

V3 : Look up table

with N = 8 bits

of the Skew tent

[ Farajallah et al., 2016] in IJBC Journal

𝒚𝒍(𝒌) = 𝒑𝒍(𝒌)⊕ 𝑺𝒍−𝟏 𝒌 ⊕ 𝒇(𝒚𝒍(𝒌 − 𝟏))

𝑪𝒍 𝒌𝒏 = 𝑳𝑺𝑩𝟖[𝒚𝒍(𝒌)]

𝑺𝒍−𝟏 𝒌 = 𝒊𝒗(𝒌) 𝒊𝒇 𝒍 = 𝟎𝑪𝒍−𝟏(𝒌) 𝒊𝒇 𝒍 > 𝟎

𝒌𝒏 = 𝒊𝒏 ×𝑴+ 𝒋𝒏𝒌 = 𝒊 ×𝑴+ 𝒋

Performance in terms of time consuming

27Safwan El Assad

Average Encryption / Decryption time

Encryption Throughput

Number of needed Cycles per Bytes

C language, PC: 3.1 GHz processor Intel Core TM i3-2100 CPU, 4GB RAM

Windows 7, 32-bit operating system.

Average is done by encrypting the test image 1000 times

with different secret keys each time

Performance in terms of time consuming

28Safwan El Assad

Cryptosystem Enc / Dec times

(ms)

ET (MBps) NCpB

Crypto 1 8.38 / 8.48 22.3 132

Crypto 2-V1 2.1 / 2.6 93.9 32

Crypto 2-V2 4.15 / 4.79 45.3 65

Crypto 2-V3 1.3 / 1.4 140.7 21

Zhang et al 7.5 / 8.25 25 122

Wang et al 7.79 / 8.39 24.1 208

Wong et al 15.59 / 16.77 7.2 417

AES 1.75 / 1.8 122 24

Lena image of size 256 X 256 X 3

Crypto2-V1 : Discrete Logistic map-32 bit (as diffusion)

Crypto2-V2: Discrete Skew tent map-32 bit (as diffusion)

Crypto2-V3: Look up table-8 bit of the Skew tent map (as diffusion)

Safwan El Assad 29

Statistical analysis: Histogram and correlation (Confusion property)

N=8000 pairs (x, y) of two adjacent pixels randomly

selected in vertical, horizontal, and diagonal

directions from the original and encrypted images.

Cameraman Ciphered Histograms :Plan Ciphered

Theoretical Chi-square is 293 in case of

alpha=0.05 and number of intervals = 256.

Correlation of adjacent horizontal

pixels of plain and ciphered images

r = 0.898492 r = 0.010523

Performance in terms of security analysis

Chi-square Exp value = 255.12

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Performance in terms of security analysis

Plaintext sensitivity attack: Diffusion property

To resist the chosen plaintext attack and the differential attack, the

cryptosystem should be highly sensitive to one bit change in the plaintext.

We evaluate the plaintext sensitivity as follows:

For each of the 1000 random secret keys, we compute the Hamming

distance, versus the number of rounds r, between two cipher-text images

C1 and C2, resulted from two chosen plaintext images I1 and I2, with:

I1 = [0, 0, …,0] and I2 = [0, 0, …1i,…,0], differ only by one bit (chosen

randomly).8

( 1, 2) 1( ) 2( )min

1

L C PD C C C k C k

Ham gk

If the Hamming distance is close to 50% (probability of bit changes

close to 1/2), then the previous attacks would become ineffective.

This test gives also the minimum number of rounds r, needed to

overcome the plaintext sensitivity attack.

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Performance in terms of security analysis

Average Hamming distance (over 1000

keys) versus the number of rounds r.

With r =1, the effect avalanche is reached.

Plaintext sensitivity attack: Diffusion property

1 1 1

( , , )

100%

P L C

p i j

D i j p

NPCRL C P

0 1( , , ) 2( , , )( , , )

1 1( , , ) 2( , , )

if C i j p C i j pD i j p

if C i j p C i j p

1 1 1

1( , , ) 2( , , )1100%

255

P L C

p i j

C i j p C i j pUACI

M N P

Number of pixel change rate (NPCR)

Unified average changing intensity (UACI)

NPCR and UACI criteria

For two random images the expected

values of NPCR and UACI are:

E(NPCR) = 99.609 %

E(UACI) = 33.463 %

Safwan El Assad 32

Key sensitivity test

A good encryption scheme should be sensitive to the secret key in process

of both encryption and decryption.

1

1 1

1 2 1

Key

Key with bit changes

I C

I C C1

1 1

1 2 1

Key

Key with bit changes

C I

C I I

To quantify the effectiveness of any algorithm, researchers use the NPCR

and UACI criteria

Performance in terms of security analysis

ProposedCrypto

Image Size HD NPCR UACI

Crypto 1 Lena 512x512 0.500173 99.607 33.463

Crypto 2-V1 Lena 512x512 0.499587 99.521 33.437

Crypto 2-V2 Lena 512x512 0.499987 99.609 33.459

Crypto 2-V3 Lena 512x512 0.499975 99.611 33.463

33

Coded Bit streamInput Video P T Q EC

1

2

3 5 7 9

4 6 8

1 9

P = Prediction

T = Transformation

Q = Quantization

EC = Entropy Coding

Joint Crypto-Compression

34

SE B

Context

Modeling

C-B SE

Encryptable

Bins

Not Encryptable

Bins

Arithmetic

Coding

Context

Coding

Bypass

Coding

Bitstream

Context update

Chaos-Based

Selective Encryption

Selective Encryption in HEVC at CABAC level

Syntax

element

Binarization

GA

GB

Not the same probability

Same probability

CABAC and Selective Encryption

35

Chaos-based Selective Stream Encryption

𝒄𝒊 = 𝒔𝒊 ⊕ 𝒙𝒊 + 𝒄𝒊−𝟏

Encryption algorithm

Chaotic generator

Interface

Syntax

element

𝒔𝒊

𝒍𝒆𝒏𝒈𝒕𝒉(𝒔𝒊)

𝒄𝒊−𝟏

𝒄𝒊

Secret key

& IVg

𝒙𝒊

𝒔𝒊 = 𝒄𝒊 ⊕ 𝒙𝒊 + 𝒄𝒊−𝟏

Stream encryption: Stream decryption:

Safwan El Assad 36

Conclusion, current and future works

Conclusion

We showed how to design efficient chaotic generators from

chaotic maps to overcome the effect of the finite precision and to

obtain randomness.

We gave an example of two chaos-based cryptosystems types:

- 1er type with separate confusion-diffusion layers. In general, such

cryptosystems are not very robust against chosen plain-text attack.

- 2nd type used dependent diffusion structure, such cryptosystems

offer high security levels and low computational complexity.

Current and future works

Efficient hardware implementation of chaotic systems

Design and Efficient implementation of Hash functions based on

Chaotic Neural Network

Joint crypto-compression for videos : HEVC and SHVC

Safwan El Assad 37

Thanks for your Attention

Questions ?

You are welcome to

Chaos – Information Hiding and Security

Workshop – C-HIS-2016

The 11th International Conference for Internet

Technology and Secured Transactions

(ICITST-2016), http://www.icitst.org

December 5-7, 2016, Barcelona, Spain

38Safwan El Assad

References

[Fridrich , 1998], “Symmetric ciphers based on two-dimensional chaotic maps”. Int. J.

Bifurcat Chaos, vol. 8, no. 6, 1998, pp. 1259-1284.

[Chen et al., 2004], “A symmetric image encryption schemes based on 3D chaotic cat

maps”. Chaos Solitons and Fractals vol. 21, 2004, pp. 749-761.

[Lian et al., 2005a], “A bloc cipher based on a suitable use of the chaotic standard

map“. Chaos Solitons and Fractals, vol. 26, 2005, pp. 117-129.

[Lian et al., 2005b], “Security analysis of a chaos-based image encryption algorithm”.

Physica A vol. 351, 2005, pp. 645-661.

[Wong et al., 2008], “A fast image encryption scheme based on chaotic standard map“.

Physics Letters A, vol. 372, no. 15, 2008, pp. 2645-2652.

[Masuda et al., 2006], “Chaotic block ciphers: from theory to practical algorithms”.

IEEE Trans on Circuits and Systems-I, vol. 53, no. 6, 2006, pp. 1341–1352.

[Caragata et al.,], “On the security of a new image encryption scheme based on a

chaotic function”. Signal, Image and Video Processing, vol. 8, Issue 4, pp. 641-646.

39Safwan El Assad

References

[El Assad et al., 2008], “Design and analyses of efficient chaotic generators for

cryptosystems" . WCECS, pp. 3-12, 2008, Advances in Electrical and Electronics

Engineering - IAENG Special Edition of the World Congress on Engineering and

Computer Science, 2008.

[El Assad et al., 2014],“Chaos-based Block Ciphers: An Overview”, IEEE, 10th

International Conference on Communications, COMM-2014, Bucharest, Romania,

May 2014, pp. 23-26. Invited talk

[El Assad et Farajallah, 2016] “A new Chaos-Based Image Encryption System”.

Signal Processing: Image Communication, 41 (2016), pp. 144-157.

[Wang et al., 2009], “A chaos-based image encryption algorithm with variable

control parameters”. Chaos Solitons and Fractals vol. 41, 2009, pp. 1773-1783.

[Wong et al., 2009], “An efficient diffusion approach for chaos-based image

encryption”. Chaos Solitons and Fractals vol. 41, 2009, pp. 2652-2663.

[Wang et al., 2011], “A new chaos-based fast image encryption algorithm”. Applied

Soft Computing, vol. 11, 2011, pp. 514-522.

40Safwan El Assad

References

[Zhang et al., 2013], “An image encryption scheme using reverse 2-dimentional

chaotic map and dependent diffusion”. Commun Nonlinear Simulat, vo. 18, 2013,

pp. 2066-2080.

[Lozi, 2007] “Giga-period orbits for weakly coupled tent and logistic discretized

maps”, Proc. Conf. Intern. On Industrial and Appl. Math., New Delhi, India, Invited

conference.

[Lozi, 2012] “Emergence of randomness from chaos, International Journal of

Bifurcation and Chaos, vol. 22, n°. 2 (2012), pp. 1250021-1-1250021-15.

[François et al., 2012] “A novel pseudo random number generator based on two

plasmonic maps, Applied Mathematics vol. 3, n° 11, (2012), pp. 1664-1673.

[Akhshani et al., 2014] “Pseudo random number generator based on quantum

chaotic map, Communications in Nonlinear Science and Numerical Simulation vol.

19, n°. 1, (2014), pp. 101-111.

[Farajallah et al., 2016], “Fast and secure chaos-based cryptosystem for images”,

International Journal of Bifurcation and Chaos, vol. 26, n°. 2 (2016), pp. 1650021-1-

1650021-21.

41Safwan El Assad

References

[Jallouli, et al., 2015] "A Novel Chaotic Generator Based On Weakly-coupled

Discrete Skewtent Maps", IEEE, 10th International Conference for Internet

Technology and Secured Transactions, ICITST-2015, London, UK, December 2015,

pp. 38-43.

Invited talk

[El Assad et al., 2011], “Generator of chaotic Sequences and corresponding

generating system” WO Patent WO/2011/121,218,2011.

PCT Extension:

Europe : EP-2553567 A1, February 2013.

China : CN-103124955 A, May 2013.

United States: US-20130170641, July 2013.