chaos based-cryptogarphy report

7/25/2019 Chaos Based-cryptogarphy Report

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AbstractRecently, as a result of essential

progress in cryptanalysis of conventional

cryptosystems there has been tremendous

interest in chaotic cryptography. This brief

article, presents a new possibilities for a design of

chaotic cryptosystems on the basis of paradigms

of continuous and discrete chaotic maps. In this

talk, we give a description of basic paradigms to

design chaos-based cryptosystems which are

analog chaos-based cryptosystems and digital

chaos-based cryptosystems, according to design

chaotic cryptosystems on the basis of discrete-time or continuous-time dynamic chaotic

systems.

I ndex Terms: chaos, analog chaos-based cryp-

tosystems, digital chaos-based cryp-tosystems.

I.INTRODUCTION

In recent years , due to widespread computerization

and their interconnection via network, information

security has become the most fascinating andinteresting technology field in todays world. The

principles of any security mechanism areconfidentiality, authentication, integrity, non-

repudiation, access control and availability.

Cryptography, with the purpose of design of

technique to provide secret communication as itprotects the information transmission from the

influence of adversaries, is an essential aspect for

secure communications..Current cryptographic techniques are based on

number theoretic or algebraic concepts. Chaos is

another promising technique, which isdeterministic process, but its nature causes it looks

like a random one, especially owing to the strongsensitivity and the dependency on the initial

conditions and control parameters. This is the

reason why it seems to be relevant for the de- signof cryptographic algorithms. Determinism of chaos

creates the possibility for encryption, and itsrandomness makes chaotic cryptosystems resistant

against attacks.Chaos can be the basis formechanisms and techniques used in chaos-based

cryptography, also known as chaotic cryptography.In this paper, we give an overview of chaos

based cryptosystems. The rest of the paper is

organized as follows.Section II describes basic

paradigms to design chaos-based

cryptosystems In Section III, we present the

analog chaos-based cryptosystems. Section

IV describes the digital chaos-based cryptosystems,

before concluding.

II. PARADIGMS TO DESIGN CHAOS-BASED

CRYPTOSYSTEMS

Signals containing enciphered information can be

sent in an analog or a digital form. The carrier for

the first form is usually radio waves and digital

telecommunication links are used for the second.

Both forms of chaotic signals transmissiondescribed above define two different paradigms for

the design of chaotic cryptosystems. Using the

terminology introduced by Li [1], we call chaoticcryptosystems designed according to the first

paradigm (analog signal transmission) analogchaos-based cryptosystems and those designed

according to second one (digital signaltransmission) digital chaos-based cryptosystems.

Generally it is considered [1] that analog chaos-

based cryptosystems are designed mainly for the

purpose of the secure transmission of information innoisy channels, and that they cannot be used

directly for the design of digital chaos-basedcryptosystems. This type of system is designed

rather for implementation of steganographicsystems rather than for cryptographic ones [2].

III.ANALOG CHAOS BASED CRYPTOGRAPHY

The principle of enciphering in analog chaos-based

cryptosystems is to combine the message mk withthe chaotic signal generated by the chaotic system in

such a manner that even after the interception of that

signal by an attacker it is impossible to recover thatmessage or protected chaotic system parameters.

The transmitter chaotic system can be described by

the following general discrete time-dynamic system

[3].

xk+1 = f (x k, , [mk, . . .])

(1)

yk

= h(xk

, , [mk, . . .]) +v

k

where xk and f( ) = [ f1( ),... , fn( )] are the n-

dimensional discrete state vector and the n-

dimensional vector of chaotic maps, respectively ,=[1,... ,L] is the L-dimensional system parameter

vector, mk is the transmitted message, yk and h() =

Chaos-based Cryptography: An Overview

Adnan Adil Ebrahim HajOmerShanghai Jiao Tong university

Email([email protected])


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[h1 ( ),... , hm ( )] are the m-dimensional inputsignal sent to the receiver and the m-dimensional

output function vector for chosen or all components

of state vector x k , and v k are the transmissionchannel noises. .The

chaotic system of the receiver has to be synchronizedwith the system of the transmitter Therefore; the

model of the receiver should ensure one can recover

unknown components of the transmitter state vector.Its general form is given below:

xk+1 = f (x k , , yk , [. . .])

(2)

yk= h(x k , , [. . .])

where xk^

and f ( ) = [ f1 ( ),... , fn ( )] arethe n- dimensional recovered discrete state vector

and the n-dimensional vector of chaotic maps (anapproximation of the transmitter behavior),

respectively, = [1, . ., L] is the L-dimensional

receivers systemparameter vector , y kand h ()

= [h1 (), . ., hm ()] are the m-dimensional

recovered input signal of the transmitter and the m-dimensional output function vector for chosen or all

components of state vector xk. .

In practice the transmitters parameter vector is

the secret enciphering key. Usually it is assumed

that = .The task of the receiver system is to reconstruct the

message transmitted by the transmitter, i.e., toachieve such a state of x k that m k = mk .

Usually this task is put in two steps.

The first step is the synchronization of the

transmitter and the receiver. The goal is to estimate

(at the receiver side) the transmitters state vectorxk on the basis of the output information ykobtained. For the purpose of estimation of the

system state (1) it is required to choose

synchronizing parameters of the transmitter (2) insuch a manner that the following criterion is met:

(3)

where E {} is the average value. It is a typicaltask of a nonlinear optimal filtering [4], and the

solution is the nonlinear optimal Kalman filter orsome extension of it.

The conclusion from (3) is that when there are

noises in communications lines, then it isimpossible to synchronize perfectly the receiver

and the transmitter. If it is assumed that thenoises are deterministic (or that there are no

noises at all), then the criterion (3) is simplified

to the form limk ||xk xk || = 0. Thesolution for this problem is the full-state or

reduced-order observer [5]. In the second step themessage value mk is estimated. The basic data for

this estimation are the recovered state x k and the

output signal y k.Some typical techniques for hiding the message

in interchanged signals are presented below [3].They are especially interesting, because after the

elimination of chaos synchronization mechanisms

they can be used in stream ciphers from the familyof digital chaos-based cryptosystems. Additionally,

for the sake of simplicity, transmission channel

noises are neglected .

A.ADDITIVE CHAOS MASKING

The scheme of an additive chaos masking is pre-

sented in Fig.1. It can be seen that the hiding of the

message mk is obtained simply by the addition ofthat message to the chaotic system output. The

observer built in the receivers chaotic system tries

to recover the corresponding state vector xk of thetransmitters system. The message mk plays the

role of an unknown transmission channel noise inthe system; therefore, it is hard to build an observer

that is able to recover the state properly. As a

consequence, m k mk.

Fig.1 Additive chaos masking

B. CHAOTIC SWITCHING

The principle of the cryptosystem based on chaotic

switching is as follows: at the transmitter side everymessage mk is assigned to another signal, and each

of them is generated by an appropriate set of chaoticmaps and output functions relevant for mk. The

scheme of the chaotic switching operation is

presented in Fig.2, where i(mk) means thedependency of the index i on the message mk.

Depending on the current value of mk, where k =

jK, the receiver is switched periodically (switching

is performed every K samples) and it is assumed

that the message mk is constant in the time interval

[ jK, ( j + 1)K 1]). .


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Fig.2 chaotic switching

III.DIGITAL CHAOS-BASED CRYPTOSYSTEMS

Digital chaos-based cryptosystems based on classes

of discrete chaotic systems are very interesting andpromising alternatives to conventional

cryptosystems based on number theory or algebraic

geometry ,due to lack of synchronization

mechanisms and eliminates security threats resultingfrom the need for reconstruction of the transmitterstate ,therefore ; two basic cipher types in

conventional cryptography: block ciphers and

stream ciphers can be design based on it . The blockcipher maps plaintext blocks into ciphertext blocks.

From the point of view of the nonlinear systemdynamics, the block cipher can be considered as the

static linear mapping [6]. Next, the stream cipher

processes the plaintext data sequence into theassociated ciphertext sequence; for that purpose,

dynamic systems are used. Both approaches to the

cipher design can be used in digital chaos-basedcryptography. In the chaotic block cipher, the

plaintext can be an initial condition for chaoticmaps, their control parameter, or the number of

mapping iterations required to create the In thechaotic stream cipher, chaotic maps are used for the

pseudorandom keystream generator; that keystream

masks the plaintext. Many of existing chaotic

ciphers have been cryptanalyzed successfully. The

cryptanalysis demonstrated substantial flaws inthe security of those ciphers.

A.

CHAOS-BASED STREAM CIPHERS

Stream ciphers based on chaos theory are usually

used for the purpose of an unpredictablepseudorandom sequence generation. Relevant

enciphering algorithms using operate in a floating-point arithmetic domain. This invokes many

problems [9]: (1) a proper selection of the

representation of floating-point numbers, (2) round-

up errors and finite-precision computation errors,

and (3) an equivalence of many keys.

AMODEL OF CHAOS-BASED STREAM CIPHER

The scheme of a chaos- based stream cipher can be

presented as the extension of the stream cipher

scheme given in [8]. The additional components are

the feedback function and mapping transformations.

The chaotic system plays the role of the next-statefunction. The feedback function is used in some

enciphering algorithms to modify the ciphers

internal state. Mapping transformations are used fortransformation of plaintext symbols to the values

relevant for the cipher in question (e.g., to define thepart of an attractor assigned to given plaintext

symbol or the relevant value of a chaotic orbit ). The

functions of chaos-based stream ciphers can be de-fined as follows:

(4)

(5)

(6)

Fig.3 Model of a chaos-based stream cipher

where: k is the key, m is the plaintext, c is the

ciphertext, z is the keystream, i is the cipher

internal state, h is the output function,g is the

keystream generation function (the filter function),t1, t2, and t3 are the mapping transformations,j isthe feedback function, and f is the chaotic system

(the next-state function).

THE KEY

Key components significantly depend on the detailsof the chaos-based stream cipher design. Usually the

following parameters are used:

1.Initial condition of chaotic systems

2. Dynamic systems controlparameters

3. Mappings (i.e., bindings) between plaintextsymbols and values used in chaotic system

iterations.

NEXT-STATE FUNCTION AND KEYSTREAMFUNCTION(FILTER)

The next-state function in chaos-based stream

ciphers is a chaotic map. The dynamics of thechaotic system depends merely on the chaotic map


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chosen. Good chaotic and statistical properties areessential to make it resistant against any

cryptanalysis. The main task of the filter function

is to process the inner state to make the keystreamindistinguishable from a random sequence.

OUTPUT FUNCTION AND FEEDBACK FUNCTION

An output function combines a plaintext witha keystream. The function has to be reversible to

make the deciphering process possible. The idea of

stream ciphers is an extension of Vernams cipher,when a random key is mixed with a plaintext by

means of the xor operator. The total security of

this cipher depends on the security of the key it hasto be used once and be random. .

A feedback function is used in self-synchronizing

stream ciphers. When it is used, then the keystreamdepends on a specified number of bits from pre-

viously enciphered plaintext symbols. The maindesign problem for that type of cipher is to design

the keystream considering the feedback function ina proper way [9]. The standard for self-

synchronizing stream cipher design is to use one-bitCFB mode in a block cipher. Then the next-state

function depends not only on the key and the

previous state, but on the feedback as well. That

property causes the security of the cipher to depend

significantly on the proper design of the feedbackfunction.

B.

CHAOS -BASED BLOCK CIPHERS

In block ciphers a plaintext m is partitioned to m iblocks and then enciphered. Therefore, every

plaintext can be considered as an ordered sequence

of blocks m = m1 , m2, . . . , mN, where N is thenumber of blocks the message consists of. If the

block is shorter , then this is padded withappropriate bits (e.g., with ones) to the full length

of the block. Blocks mi of the plaintext message m

belong to some set of plaintext blocks M. Eachplaintext block mi M consists of elements

(symbols) from the alphabet AM. The set M forms

the space of all plaintext blocks. An encryptionfunction Ee transforms plaintext blocks to

ciphertext blocks belonging to the ciphertext space .

(6)

where K is the key space .Any element c

(a binary string of length lm) is called a ciphertext

(a cryptogram) and consists of elements (symbols)from the alphabet AC. Particularly, where AM=AC,

the cipher is called an endomorphic cipher. A

transformation inverse to the encryption function Ee

is called a decryption function and it is denoted as

De. The function has to be a bijection from C to M:

(7)

Hence, the decryption functionDe is the inverse

function of the encryption functionEe.

(8)

Block ciphers defined by a mapping pair (Ee, De)

can be considered as static nonlinear

transformations. This means that invertible chaoticmaps are required for the design of chaos-based

block ciphers. .

A general inverse chaotic system approach isapplied for the selection of maps [6.20]. In most

cases, chaotic maps are noninvertible; therefore, it isnecessary to use discretization methods to ensure

such a type of inevitability.

AMODEL OF A CHAOS-BASED BLOCK CIPHER

general principles guiding the design of block

ciphers is (a) confusion and diffusion and (b)

completeness and avalanche effect. When a

cipher algorithm has the confusion property,

then plaintext bits are randomly and uniformly

distributed over the ciphertext. On the other hand,

the diffusion property guarantees that each

plaintext and key bit has an influence on many

ciphertext bits The diffusion property should result

in completeness and an avalanche effect. The

measure of the avalanche property is a number of

changed ciphertext bits after the change of a

single input bit. The completeness ensures that

each output bit is a complex function of all input

bits. They are some of the most important design

criteria to be considered in the case of block

ciphers; therefore, they are presented below [6].To obtain the properties of a block cipher stated

above an approach proposed by Shannon [10] is

used. It consists in the usage of simple elements

(components) performing substitutions, permuta-

tions, and modular arithmetic operations in an ap-

propriate order. Those functions are combined and

performed in so-called rounds; there can be a few or

several dozen such rounds...

A general scheme of such a type of a block ci-

pher (a so-called iterated block cipher) ispresented

in Fig. 4. It consists of two basic components: around function f anda round key generation blockKRG . The round function is based on

substitutionpermutation (SP) networks, which

are the com- bination of two basic cryptographic

primitives: S-boxes and P-boxes. The S-box ensures

a substitution of an input binary string by another


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binary string. The P-box reorders input bits

(performs their permutation).Each round consists

of one P-box and a layer of S-boxes. Many rounds

form an SP network, and an example is presented

in Fig 5 .Owing to the fact that S-boxes ensure the

confusion property and P-boxes ensure thediffusion property.

Fig. 4 General scheme of a block cipher

Considering that ergodicity and mixing properties of

chaotic maps ensure confusion and diffusion,

respectively [11], it is obvious for many researchers

to use them to design a round function f (this

concerns also chaotic SP networks). Conventional

round functions are defined on finite data sets and

depend on an enciphering key k. The design of asimilar round function on the basis of chaotic maps

requires them to be made discrete. It is necessary to

replace continuous variables of the map (elements of

the set of real numbers) and appropriate operations

by a finite set of integer numbers and respective

operations [12].

Fig.5 four-round substitutionpermutation network.

S substitution, P permutation

CONCLUSIONS

In spite of the significant achievements already

accomplished, there are still too many problems to

be solved in the field. The research of chaos-basedcryptography is far from exhaustive. At theoretical

level, it seems that chaotic systems are ideal

candidates for cryptographic primitives, but

at the practical level, chaotic ciphers are still less

efficient than the corresponding conventional ones.

As a consequence, chaotic cryptography has been an

active research field but with marginal impact in

classical cryptography. So further investigation is

needed to evaluate the efficient algorithms for real

applications

References

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chaos based-cryptogarphy report

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