Report

CHAPTER I

INTRODUCTION

1.1 CRYPTOGRAPHY

Data security and cryptography are critical aspects of conventional computing. Here

we provide basic terminology used in cryptography. Cryptography is the most important

aspect of communications security. In data and telecommunication, cryptography is necessary

when communicating over any untrusted medium, which just about any network, particularly

the Internet. The goal is to transmit a message between a sender and receiver such that an

eavesdropper is unable to understand it. Plaintext refers to a sequence of characters drawn

from a finite alphabet, such as that of a natural language.

Encryption is the process of scrambling the plaintext using a known algorithm and a

secret key. The output is a sequence of characters known as the cipher text. Decryption is the

reverse process, which transforms the encrypted message back to the original form using a

key.

The goal of encryption is to prevent decryption by an adversary who does not know

the secret key. An unbreakable cryptosystem is one for which successful cryptanalysis is not

possible. Such a system is the one-time-pad cipher. It gets its name from the fact that the

sender and receiver each possess identical notepads filled with random data. Each piece of

data is used once to encrypt a message by the sender and to decrypt it by the receiver, after

which it is destroyed.

Fig 1: General diagram of cryptography

The terms are

Plaintext: original message is known as plaintext.

Ciphertext: coded message is called ciphertext.

Encryption: The process of converting from plaintext to ciphertext.

Decryption: The process of restoring from the ciphertext

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Plaintext Ciphertext PlaintextDecryptionEncryption

The two main types of cryptographic schemes are

Secret key cryptography or symmetric encryption

Public key cryptography or asymmetric encryption

Symmetric Encryption

It is also referred as conventional encryption. Symmetric encryption is a form of

cryptosystem in which encryption and decryption are performed using the same key.

Symmetric cryptography is susceptible to plain text attacks and linear cryptanalysis

meaning that they are hackable and at times simple to decode. With careful planning of the

coding and functions of the cryptographic process these threats can be greatly reduced.

Asymmetric cryptography uses different encryption keys for encryption and decryption. In

this case an end user on a network, public or private, has a pair of keys; one for encryption

and one for decryption. These keys are labelled or known as a public and a private key; in

this instance the private key cannot be derived from the public key.

Secret key Secret key

Transmitted

Cipher text

Fig 2: Simplified Model of Symmetric Encryption

Asymmetric Encryption

The asymmetrical cryptography method has been proven to be secure against

computationally limited intruders. The security is a mathematical definition based upon the

application of said encryption. Essentially, asymmetric encryption is as good as its applied

use; this is defined by the method in which the data is encrypted and for what use. The most

common form of asymmetrical encryption is in the application of sending messages where

the sender encodes and the receiving party decodes the message by using a random key

generated by the public key of the sender.

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Plaintext Input

Encryption algorithm

Decryption algorithm

Plaintext Output

Asymmetric key encryption uses different keys for encryption and decryption. These

two keys are mathematically related and they form a key pair. One of these two keys should

be kept private, called private-key, and the other can be made public (it can even be sent in

mail), called public-key. Hence this is also called Public Key Encryption.

A private key is typically used for encrypting the message-digest; in such an

application private-key algorithm is called message-digest encryption algorithm. A public

key is typically used for encrypting the secret-key; in such a application private-key

algorithm is called key encryption algorithm.

Popular private-key algorithms are RSA (invented by Rivest, Shamir and Adlemen)

and DSA (Digital Signature Algorithm). While for an ordinary use of RSA, a key size of 768

can be used, but for corporate use a key size of 1024 and for extremely valuable information

a key size of 2048 should be used. Asymmetric key encryption is much slower than

symmetric key encryption and hence they are only used for key exchanges and digital

signatures.

It is also referred as Public key encryption. Asymmetric encryption is a form of

cryptosystem in which encryption and decryption are performed using the different keys.

This encryption can be used for confidentiality, authentication or both.

Public key Private key

Transmitted

Cipher text

Fig 3: Simplified Model of Asymmetric Encryption

A asymmetric encryption scheme has six ingredients

Plaintext : This is the message or data that is fed into the algorithm as input.

Encryption algorithm : The encryption algorithm performs various

transformations on the plaintext.

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Plaintext Input

Encryption algorithm

Decryption algorithm

Plaintext Output

Public and private keys : This is a pair of keys that have been selected so that

if one is used for encryption, the other is used for decryption.

Cipher text : This is the scrambled message produced as output.

Decryption algorithm : This algorithm accepts the ciphertext and the

matching key and produces the output.

Public-key cryptography is a cryptographic approach which involves the use of

asymmetric key algorithms instead of or in addition to symmetric key algorithms. Unlike

symmetric key algorithms, it does not require a secure initial exchange of one or more secret

keys to both sender and receiver. The asymmetric key algorithms are used to create a

mathematically related key pair: a secret private key and a published public key. Use of these

keys allows protection of the authenticity of a message by creating a digital signature of a

message using the private key, which can be verified using the public key. It also allows

protection of the confidentiality and integrity of a message, by public key encryption,

encrypting the message using the public key, which can only be decrypted using the private

key.The most widely used public key cryptosystem is RSA. The RSA scheme is a block

cipher in which the plaintext and cipher text are integers between 0 and n-1 for some n.

1.2 PRIVATE KEY CRYPTOGRAPHY

In private-key cryptography, the sender and recipient agree beforehand on a secret

private key. The plaintext is somehow combined with the key to create the cipher text. The

method of combination is such that, it is hoped, an adversary could not determine the

meaning of the message without decrypting the message, for which he needs the key.

Fig 4 : Encryption process in Secret Key Cryptography

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Message to be encrypted or plain text

Encryption Algorithm

Encrypted message or Cipher text

Private Key known only to sender and receiver

To break a message encrypted with private-key cryptography, an adversary must

either exploit a weakness in the encryption algorithm itself, or else try an exhaustive search

of all possible keys (brute force method). If the key is large enough (e.g., 128 bits), such a

search would take a very long time (few years), even with very powerful computers.

Fig 5: Decryption Process in Secret Key Cryptography

Private-key methods are efficient and difficult to break. However, one major

drawback is that the key must be exchanged between the sender and recipient beforehand,

raising the issue of how to protect the secrecy of the key. When the President of the United

States exchanges launch codes with a nuclear weapons site under his command, the key is

accompanied by a team of armed couriers. Banks likewise use high security in transferring

their keys between branches. These types of key exchanges are not practical, however, for e-

commerce between, say, amazon.com and a casual web surfer.

The main problem with secret-key cryptosystems is getting the sender and receiver to

agree on the secret key without anyone else finding out. This requires a method by which the

two parties can communicate without fear of eavesdropping. However, the advantage of

secret-key cryptography is that it is generally faster than public-key cryptography.

1.3 PUBLIC KEY CRYPTOGRAPHY

Public Key cryptography uses two keys Private key (known only by the recipient) and

a Public key (known to everybody). The public key is used to encrypt the message and then it

is sent to the recipient who can decrypt the message using the private key.

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Message to be decrypted or cipher text

Decryption Algorithm

Decrypted message or Plain text

Private Key known only to sender and receiver

The message encrypted with the public key cannot be decrypted with any other key

except for its corresponding private key. The following Diagram illustrates the encryption

process in the public key cryptography.

Fig 6: Encryption Process in the public key cryptography

Fig 7: Decryption Process in the public key cryptography

The public-key algorithm uses a one-way function to translate plaintext to ciphertext.

Then, without the private key, it is very difficult for anyone (including the sender) to reverse

the process (i.e., translate the ciphertext back to plaintext). A one-way function is a function

that is easy to apply, but extremely difficult to invert. The most common one-way function

used in public-key cryptography involves factoring very large numbers. The idea is that it is

relatively easy to multiply numbers, even large ones, with a computer; however, it is very

difficult to factor large numbers. The only known algorithms basically have to do a sort of

exhaustive search (Does 2 go in to? Does 3? 4? 5? 6? and so on). With numbers 128 bits

long, such a search requires performing as many tests as there are particles in the universe.

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Message to be encrypted or plain text

Encryption Algorithm

Encrypted message or Cipher text

Public Key known to everyone

Message to be encrypted or plain text

Encryption Algorithm

Encrypted message or Cipher text

Private Key known only to receiver

For instance, someone wishing to receive encrypted messages can multiply two very

large numbers together. She keeps the two original numbers a secret, but sends the product to

anyone who wishes to send her a message. The encryption/decryption algorithm is based

upon combining the public number with the plaintext. Because it is a one-way function, the

only way to reverse the process is to use one of the two original numbers.

However, assuming the two original numbers are very large, their product is even

bigger; it would be impractical for an adversary to try every possibility to determine what the

two original numbers were.

In a public-key cryptosystem, the private key is always linked mathematically to the

public key. Therefore, it is always possible to attack a public-key system by deriving the

private key from the public key. Typically, the defense against this is to make the problem of

deriving the private key from the public key as difficult as possible. For instance, some

public-key cryptosystems are designed such that deriving the private key from the public key

requires the attacker to factor a large number, it this case it is computationally infeasible to

perform the derivation. This is the idea behind the RSA public-key cryptosystem.

1.3.1 Advantage and Disadvantage of Public-Key Cryptography Compared with

Secret-Key Cryptography

1. The primary advantage of public-key cryptography is increased security and

convenience. Private keys never need to transmitted or revealed to anyone. In a secret-key

system, by contrast, the secret keys must be transmitted (either manually or through a

communication channel), and there may be a chance that an enemy can discover the secret

keys during their transmission.

2. Another major advantage of public-key systems is that they can provide a method

for digital signatures. Authentication via secret-key systems requires the sharing of some

secret and sometimes requires trust of a third party as well. As a result, a sender can repudiate

a previously authenticated message by claiming that the shared secret was somehow

compromised by one of the parties sharing the secret. For example, the Kerberos secret-key

authentication system involves a central database that keeps copies of the secret keys of all

users; an attack on the database would allow widespread forgery.

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Public-key authentication, on the other hand, prevents this type of repudiation; each

user has sole responsibility for protecting his or her private key. This property of public-key

authentication is often called non-repudiation.

3. A disadvantage of using public-key cryptography for encryption is speed; there are

popular secret-key encryption methods that are significantly faster than any currently

available public-key encryption method. Nevertheless, public-key cryptography can be used

with secret-key cryptography to get the best of both worlds. For encryption, the best solution

is to combine public- and secret-key systems in order to get both the security advantages of

public-key systems and the speed advantages of secret-key systems. The public-key system

can be used to encrypt a secret key, which is used to encrypt the bulk of a file or message.

Such a protocol is called a digital envelope.

4. Public-key cryptography may be vulnerable to impersonation, however, even if

users' private keys are not available. A successful attack on a certification authority will allow

an adversary to impersonate whomever the adversary chooses to by using a public-key

certificate from the compromised authority to bind a key of the adversary's choice to the

name of another user.

5. In some situations, public-key cryptography is not necessary and secret-key

cryptography alone is sufficient. This includes environments where secure secret key

agreement can take place, for example by users meeting in private. It also includes

environments where a single authority knows and manages all the keys (e.g., a closed

banking system) Since the authority knows everyone's keys already, there is not much

advantage for some to be "public" and others "private" Also, public-key cryptography is

usually not necessary in a single-user environment. For example, if you want to keep your

personal files encrypted, you can do so with any secret-key encryption algorithm using, say,

your personal password as the secret key. In general, public-key cryptography is best suited

for an open multi-user environment.

6. Public-key cryptography is not meant to replace secret-key cryptography, but

rather to supplement it, to make it more secure. The first use of public-key techniques was for

secure key exchange in an otherwise secret-key system; this is still one of its primary

functions.

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1.4 SECURITY

Security often requires that data be kept safe from unauthorized access. And the best

line of defence is physical security (placing the machine to be protected behind physical

walls). However, physical security is not always an option, due to cost and/or efficiency

considerations. Instead, most computers are interconnected with each other openly, thereby

exposing them and the communication channels that they use.

With regards to confidentiality, cryptography is used to encrypt data residing on

storage devices or travelling through communication channels to ensure that any illegal

access is not successful. Also, cryptography is used to secure the process of authenticating

different parties attempting any function on the system. Since a party wishing be granted a

certain functionality on the system must present something that proves that they indeed who

they say they are. That something is sometimes known as credentials and additional measures

must be taken to ensure that these credentials are only used by their rightful owner. The most

classic and obvious credential are passwords. Passwords are encrypted to protect against

illegal usage.

Authorization is a layer built on top of authentication in the sense that the party is

authenticated by presenting the credentials required (passwords, smart cards ... etc.). After the

credentials are accepted the authorization process is started to ensure that the requesting party

has the permissions to perform the functions needed.

Security services

Authentication : Assure the recipient of a message the authenticity of the claimed

source.

Confidentiality : Protects against unauthorised release of message content.

Integrity : Gaurantees that a message is received as sent.

Non-Repudiation : Protects against sender/receiver denying sending/receiving a

message.

Availability : Guarantees that the system services are always available when needed.

Security audit : Keeps track of transactions for later use(diagnostic, alarm).

Key Management : Allows negotiating, setup and maintaining keys between

communication entities.

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1.5 DNA CRYPTOGAPHY

Deoxyribonucleic acid (DNA) is a long linear polymer found in the nucleus of a cell

and formed from nucleotides and shaped likes a double helix, associated with the

transmission of genetic information. DNA is the king of molecules. Each spiral strand,

composed of a sugar phosphate backbone and attached bases, is connected to a

complementary strand by hydrogen bonding (non- covalent) between paired bases Adenine,

thymine, guanine and cytosine. Adenine and thymine are connected by two hydrogen bonds

while guanine and cytosine are connected by three.

Fig 8: DNA Structure

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The DNA cryptography is a new and very promising direction in cryptography research.

DNA can be used in cryptography for storing and transmitting the information, as well as for

computation. Although in its primitive stage, DNA cryptography is shown to be very

effective.

Currently, several DNA computing algorithms are proposed for quite some

cryptography, cryptanalysis and steganography problems, and they are very powerful in these

areas. However, the use of the DNA as a means of cryptography has high tech lab

requirements and computational limitations, as well as the labour intensive extrapolation

means so far. These make the efficient use of DNA cryptography difficult in the security

world now.

As some of the modern cryptography algorithms (such as DES, and more recently,

MD5) are broken, the new directions of information security are being sought to protect the

data. The concept of using DNA computing in the fields of cryptography and steganography

is a possible technology that may bring forward a new hope for powerful, or even

unbreakable, algorithms.

It is Adelman, with his pioneering work [Adelman, 1994]; set the stage for the new field

of bio-computing research. His main idea was to use actual chemistry to solve problems that

are either unsolvable by conventional computers, or require an enormous amount of

computation. By the use of DNA computing, the Data Encryption Standard (DES)

cryptographic protocol can be broken [Boneh, et. al, 1995].However, researchers in DNA

cryptography are still looking at much more theory than practicality. The constraints of its

high tech lab requirements and computational limitations, combined with the labor intensive

extrapolation means. Thus prevent DNA computing from being of efficient use in today’s

security world.

1.6 RSA Algorithm

RSA is one of the most popular and successful public key cryptography algorithms.

The algorithm has been implemented in many commercial applications. It is named after its

inventor’s Ronald L. Rivest, Adi Shamir, and Leonard Adleman. They invented this

algorithm in the year 1977. They utilized the fact that when prime numbers are chosen as a

modulus, operations behave “conveniently”. They found that if we use a prime for the

modulus, then raising a number to the power (prime - 1) is 1.

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It is based on a very simple number-theoretical idea, and yet it has been able to resist

all cryptanalytic attacks. The idea is a clever use of the fact that, while it is easy to multiply

two large primes, it is extremely difficult to factorize their product.

Thus, the product can be publicized and used as the encryption key. The primes

themselves cannot be recovered from the product and are used for decryption. There is no

formal proof whatsoever that factorization is intractable or is intractable in the special case

needed for RSA, and that factorization is needed for the cryptanalysis of the RSA.

RSA algorithm simply capitalizes on the fact that there is no efficient way to factor

very large integers. The security of the whole algorithm relies on that fact. If someone comes

up with an easy way of factoring a large number, then that’s the end of the RSA algorithm.

Then any message encrypted with the RSA algorithm is no more secure.

The security of RSA algorithm depends on the ability of the hacker to factorise

numbers. Newer faster and better methods for factoring numbers are constantly being

devised. The current best for long numbers of the number field sieve. Prime number of a

length that was unimaginable a mere decade ago are now factored easily.Obiviously the

larger the number is, the harder it is to fact and so the better the security of RSA.As theory

and computers improve large and larger keys will have to be used.

The disadvantage in using extremely long keys is the computational overhead

involved in encryption/decryption.This will only become a problem if a new factoring

technique emerges that requires keys of such lengths to be used that necessary key length

increase much faster than the increasing average speed of computers utilising the RSA

algorithm.

1.7 ECC

Elliptic Curve Cryptography is a public key Cryptography. ECC is ideal for

environments such as pager, PDAs, cellular phones and smart cards. Moreover, because

of the apparent hardness of the underlying elliptic curve discrete logarithm problem

(ECDLP), ECC systems are also well suited for applications that need long-term security

requirements. Elliptic Curve Cryptography (ECC) is a public key technology that offers

performance advantages at higher security levels.

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Every user taking part in public key cryptography will take a pair of keys, a public

key and a private key. Only the particular user knows the private key whereas the public keys

are distributed to all users taking part in the communication. Some public key algorithm may

require a set of predefined constants to be known by all the devices taking part in the

communication. In ECC we call these predefined constants as ‘Domain Parameters.

Understanding ECC needs full mathematical background on elliptic curves. Elliptic

curves are not ellipses. The general cubic equation of elliptic curves is

y2+axy+by=x3+cx2+dx+e. But for our purpose it is sufficient to limit the equation to the

form y2 = x3 + ax + b. Say EP (a,b) consisting of all the points (x,y) that satisfy the above

equation together with element at infinity O. A group can be defined based on the set EP(a,b)

for specific values of a and b.

If P,Q R are points on EP(a,b) the relations commutativity, associativity, existence of

an identity element and existence of inverse hold good. The heart of ECC is discrete

logarithm problem that can be stated as “it should be very hard to find a value k such that

Q=kP where P and Q are known’. But ‘it should be relatively easy to find Q where k and P

are known’ P, Q are points on the elliptic curve.

1.8 ORGANISATION OF THE REPORT

The project report is organised as follows. Chapter II deals with literature review of

the recently published papers and related books on cryptography. Chapter III describes DNA

computing based RSA cryptography. Chapter IV describes DNA computing based ECC

cryptography. In chapter V simulation result is prescribed. Chapter VI concludes the report.

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CHAPTER II

LITERATURE REVIEW

A new way to show how cryptography works with DNA computing to transmit

message effectively and securely. The RSA algorithm belongs to asymmetric key

cryptography combined with DNA computing technique to encrypt message [1].

The DNA computing gives a new way to cryptography. Real DNA is not utilized to

perform the cryptography process, rather a new cryptography method based on central dogma

of molecular biology. Since this method simulates some critical processes in central dogma, it

is a pseudo DNA cryptography method[2].

The initial unencrypted data is referred as plaintext, it will be encrypted into cipher

text. Public key cryptography has been considered to be the most significant development in

cryptography recently years. It is also called as asymmetric key cryptography[3].

Different weakness of the RSA algorithm could be observed and many attacks against

it are developed successfully. Improving this algorithm was performed in order to ensure a

higher data security and an increased compact process speed[4].  

ECC Encryption and Decryption methods can only encrypt and decrypt a point on the

curve not messages. A systematic way of finding points on Ep (a,b) relating somehow to the

plaintext message use probabilistic algorithms to do this, where the chance of failure is

acceptably small. Thus Encoding (message to a point) and Decoding (point to a message)

methods are important while Encryption and Decryption[5].

A text based elliptic curve cryptosystem is implemented each character in the message

is represented by its ASCII values. Each of these ASCII value is transformed into an affine

point on the EC, by using a starting point called pm. this transformed character of the message

is encrypted by the ECC technique. Decryption of ECC encrypted message is itself quite a

formidable task, unless we have knowledge about the private key, the secret integer the affine

point[6]. 

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CHAPTER III

EXISTING SYSTEM

3.1 DNA COMPUTING RSA CRYPTOGRAPHY

In the existing method, they proposed a encryption design by the com DNA based

computing deals with combination of DNA computing theory with RSA algorithm. The private

and public keys in RSA are based on extremely larger prime numbers. This algorithm is

quite easy, whereas the real challenge for RSA is the selection and generation of the public

keys and private keys or else the attacker can track it.

In this, we are combining DNA based computing with RSA to get a better security.

The DNA based computing deals with the encoding and decoding of the given input text

message. The encryption and decryption is done with the help of RSA algorithm.

3.2 DEOXYRIBONUCLEIC ACID

DNA is a double helical structure consists of four nucleotides namely adenine

(A),thymine(T),guanine(G),cytosine(C).

Fig 9: Structure of DNA molecule

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DNA is a polymer which is strung together from monomers called deoxyribo

nucleotide or simply nucleotide. Each nucleotide consists of three distinct chemical subunits:

a five-carbon sugar, acidic phosphate and a nitrogen-rich base(B).

According to Watson-Crick complement rule

A only pairs with T and

G only pairs with C.

Nucleotides can link in two different ways

1) Through a strong covalent phosphodiester bond which joins a 5’-phosphate group of one

nucleotide to a 3’-hydroxyl group of another nucleotide .

2) Through a weak hydrogen bond between two base from two nucleotides. But adenine(A)

can pair with only thymine(T) and cytosine(C) can pair with only guanine(G). This paring

principle is called Watson-Crick complementarity or WC complementarity.

Using only phosphodiester bond we get single strand DNA molecule . The 5’-

phosphate group and 3’-hydroxyl group induces an orientation in the DNA molecule, which

is called a 5’-3’ molecule if we start from the 5’-phosphate group and go towards 3’-hydroxyl

group or a 3’-5’ molecule if we start from 3’-hydroxyl and go towards 5’-phosphate group. If

two single strands satisfy WC complementarity and they are opposite in orientation (i.e., one

in 5’-3’ orientation and the other in 3’-5’ orientation) then the two strands can give a double

strand DNA molecule. But this molecule is not linear in structure; the two single strands are

wound around each other to form the famous double helix.

3.2.1 DNA computing

DNA computing is a novel method for solving intractable computational problems. In

1994, Adleman first demonstrated that a directed Hamiltonian path problem (HPP) could be

encoded in DNA and evaluated. After that, many researchers around the world have been

attracted to the new field of DNA computing. In 1995, Lipton extended Adleman’s idea to

solve the satisfiability problem. In 1997, Quyang presented a molecular biology-based

experimental solution to the maximal clique problem. In 2000, Liu designed a DNA

computing model system, which is called surface-based DNA computing, and solved the

satisfiability problem .In 2001, Wu analyzed and improved their surface-based method .

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In the same year Benenson designed a programmable and autonomous computing

machine made of biomolecules, on which a finite automaton can run . Independently of the

future technological success of DNA computing, this area has led already to interesting new

computing paradigms which are certainly enriched our understanding of the nature of

computation .

In order to introduce the principles of DNA computing we briefly review the model

which Prof. Adleman had used to solve a directed Hamiltonian path problem. The

Hamiltonian path problem is to find a path that begins at vin, ends at vout and enters every

other vertex exactly once on a directed graph. For each vertex i in the graph, a random

20-mer oligonulecotide DNA sequence was generated.

The process to solve the directed Hamiltonian path problem is list as follows:

(1) Generate random paths through the graph.

(2) Keep only those paths which begin with vin and end with vout.

(3) If the graph has n vertices, then keep only those paths which enter exactly

n vertices.

(4) Keep only those paths which enter all of the vertices of the graph at least once.

(5) If any paths remain, say “yes”, otherwise say “no”.

Generally speaking, Adleman used the DNA sequence encoding of all possible

answers to the problems, removing the solutions that do not meet the requirements through a

series of restrictive conditions. Finally, Adleman found the solution. From the algorithm

solving the directed Hamiltonian path problem, we can see the difference between DNA

computing and traditional computing that DNA computing has massive parallelism and high

density information of biomolecules.

DNA computing pioneer Adleman reviewed DNA computing as follows:“For

thousands of years, humans have tried to enhance their inherent computational abilities using

manufactured devices. Mechanical devices such as the abacus, the adding machine, and the

tabulating machine were important advances. But it was only with the advent of electronic

devices and, in particular, the electronic computer some 60 years ago that a qualitative

threshold seems to have been passed and problems of considerable difficulty could be solved.

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It appears that a molecular device has now been used to pass this qualitative threshold

for a second time. The development of DNA cryptography benefits from the progress of

DNA computing. On the one hand, cryptography always has some relationship with the

corresponding computing model more or less. On the other hand, some biological

technologies used in DNA computation are also used in DNA cryptography.

SOME POINTS IN FAVOUR OF DNA COMPUTING

DNA strand has the capability of processing information due to its chemical

properties.

DNA strand can store an incredible amount of data in a very small volume.

It is massively parallel. This gives DNA computing very high speed.

Various complex structure like living organism is the result of applying few simple

operation to an initial DNA sequence. This makes us believe that DNA can be a

potential tool for computation.

3.2.2 DNA encryption techniques

It is known that, one-time-pads security is completely decided by the random of

cipher key, this kind of algorithm is absolutely secure in theory. The algorithm required:

firstly, the data of cipher key is completely random; secondly, the cipher key can’t be

recycled. But physically, if one-time-pads provides a complete security exists two basic

difficulties: on the one hand, it is very difficult in producing a large-scale random cipher key

as well as have to in accordance with the plain code length; on the other hand, facing the

problem of cipher key saving and distributing. These problems result in the one-time-pads

algorithm physically infeasible. However, DNA as information carrier has high memory

density, which is one hundred billion to one thousand billion times compared to the

commonly used disk memory. It will be better solved the huge cipher key producing and

saving problem and providing a even road for the one-time-pads algorithm.

Prof. Gehani utilized this thought to present one-time-pads mechanism based on DNA

to design two encryption methods of one-time-pads of DNA sequence. One method is to

translate the fixed length DNA plain code sequence cell to DNA cryptograph sequence

according to the defined mapping graph, we call it mapping substitute.

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The other is called exclusiveor method, which uses biological molecular techniques to

carry through exlusiveor operation of DNA plain code and cipher key sequence. It is

absolutely secure to use these two methods of one-time-pads encryption mechanism. But in

this case, it is crucial that how to set down the encryption mapping graph or cipher key carrier

(called DNA material) between the two communicators and ensure this material can’t be

filched and replicated. In the meantime, how to carry on the error-correction disposal and

long period conservation of DNA cipher key is also a problem that has to be resolved.

Gehani also lead the DNA computing into dissymmetric encryption mechanism. They

utilize the super parallel computing ability and incomparable information saving capacity of

DNA to improve the strength of code system with the more complex algorithm, which is a

brave speculation.

Essentially, Gahani’s way is in virtue of DNA as the information carrier, which is a

coding and decoding techniques based on traditional encryption algorithm. It will increase the

amount of information to realize more great and complicated data structure if adding precise

coding information.

3.2.3 DNA Steganography

With the development of biotechnology, the way of transmitting DNA are more and

more abundant and brief. The advanced transmitting way does not only reduce the cost but

increase the information security [20, 21]. DNA steganography has more a layer of protection

than the simplex code encryption techniques, which provides a novel thought for information

security and a new orientation for its research.

The principle of DNA steganography is conceal the information which needs

encryption in the large numbers of irrelevant DNA sequence chains. This way of decoding

like looking for a needle in a bottle of hay which make attackers difficult to ascertain the

correct DNA fragment. Only the proper receiver can find the correct DNA fragment based on

the conventional information in advance between the two parties as well as require the

information which conceal in it. One can argue that steganography is not actually encryption,

since plaintext is not encrypted but only disguised within other media.

19

The experimentation about DNA steganography have firstly accomplished by

Bancroft, who apply the DNA steganography in a piece of celebrated information of the

Second World War, which withdraw successfully. By using an alphabet of exoteric short

nucleic acid sequence, they encode the plain code information in DNA chains and add a

special section marked information on the bottom of DNA chains. This kind of DNA will

blend with the same length DNA chains. These amounts of DNA chains were divided into

lots of microdots, However, just a microdot contains DNA molecules which take count of a

hundred million. In this way, to attackers, though they can ascertain the information exist in a

microdot among numerous microdots, it like looking for a needle in a bottle of hay to select

one which contains in a hundred million DNA chains.

As a matter of fact, the key of deciphering information lies in looking for a special

section of bottom mark which is able to utilize the method of DNA computing to search.

Once the DNA chain have confirmed through the mark, the receivers will adopt PCR to

replicate this DNA chain as well as acquiring information by deciphering. By introducing

these methods, Bancroft and his colleagues successfully code and decode a piece of

information: “June 6 Invasion: Normandy”.

3.2.4 DNA Certification

Strictly speaking, DNA certification doesn’t deal with much DNA computing

techniques, but mainly employ the biological characteristics of DNA. Currently, the DNA

certification is broadly applied in the field of justice, finance, and so on, which will certificate

biological individuals accurately.

3.2.5 The problem and prospect that the DNA technique faces

Although the DNA computing is a fire-new computing mode, it can’t get away from

the influence of Turing in the corresponding theoretical computing model. The DNA

computing is still placed in a theoretical stage, its computing model was mostly just using

molecular technique to resolve a certain problems, and put on an experiment to resolve a

certain problem, the varieties of problems lead to the discrepancy of computing schemes,

there still haven’t an uniform computing and coding model currently.

20

Under the existing DNA computing mode, the time complexity of DNA computing

compared to the space complexity doesn’t increase with the computational complexity

remarkably. That is, DNA computing only converts the time complexity into space

complexity. Then, once the complication of problems break the physical limit of DNA

segment which operated by the bio-chemical technique, DNA computing is still too far away

to reach. Boneh spend nearly 4 months to construct DES-1(E) solution, however, the

quantities of cipher key of AES algorithm utilized by the US federal government is 21 times

compared to DES algorithm.

Therefore, according to the Boneh’s way, it will cost several years if we construct

AES-1(E) solution. So we can say that Boneh’s method can only break the symmetric system

under 64 bits. Mathematical cryptography can be easily increasing the length of the cipher,

thereby it’ll prevent the cryptography from powerful attack using DNA computer. Therefore,

in terms of existing DNA computing mode, though DNA computer greatly improve the

ability of the cipher break of people, it is disable to construct genuine intimidation to the

security of cryptography. DNA cipher is the beneficial supplement to the existing

mathematical cipher, it is a good prior choice especially to the lower demand real-time

encryption system.

DNA has a bright development potential in steganography and DNA certification,

which has a more layer protection than a single encryption. With the rapid development of

modern biotechnology, the costly biological experiment has been a formal one. The further

development of biotechnology and the appearance of a better DNA cipher design alternative

will provide a new orientation for the research of the information security. However, the

security, feasibility, stability of DNA cryptography is still need a further research.

3.2.6 ASCII codes

Before ASCII was developed, the encodings in use included 26 alphabetic characters,

10 numerical digits, and from 11 to 25 special graphic symbols. More than 64 codes were

required in ASCII. ASCII codes (Fig-1) represent as text in computers, communications

equipment, and other devices that work with text.

21

Most modern character encodings which support many more characters than did the

original have a historical basis in ASCII. ASCIIdeveloped from telegraphic codes and its first

commercial use was as a seven-bit tele printer code promoted by Bell data services. Work on

ASCII formally began October 6, 1960 with the first meeting of the ASA X3.2

subcommittee. The first edition of the standard was published in 1963, a major revision in

1967, and the most recent update in 1986. Compared to earlier telegraph codes, the proposed

Bell code and ASCII were both ordered for more convenient sorting (i.e., alphabetization) of

lists, and added features for devices other than teleprinters.

32 - Space

33 - !

34 - “

35 - #

36 - $

37 - %

38 - &

39 - ,

48 – 0

49 - 1

50 - 2

51 - 3

52 - 4

53 - 5

54 - 6

55 - 7

64 - @

65 - A

66 - B

67 - C

68 - D

69 - E

70 - F

71 - G

80 - P

81 - Q

82 – R

83 - S

84 - T

85 - U

86 - V

87 – W

96 - `

97 -a

98 - b

99 - c

100 - d

101 – e

102 - f

103 - g

112 - p

113 - q

114 - r

115 - s

116 - t

117 - u

118 - v

119 - w

40 - (

41 - )

42 - *

43 - +

44 – ‘

45 - -

46 - .

47 - /

56 - 8

57 - 9

58 - :

59 - ;

60 - <

61 - =

62 - >

63 - ?

72 -H

73 - I

74 - J

75 - K

76 - L

77 - M

78 - N

79 - O

88 - X

89- Y

90 - Z

91 - [

92 - \

93 - ]

94 - ^

95 - _

104 - h

105 - i

106 - j

107 - k

108 - l

109 - m

110 - n

111 – o

120 - x

121 – y

122 - z

123 - {

124 - |

125 - }

126 - ~

127 – DEL

Table 1: ASCII Tabel

22

Some ASCII features, including the "ESCape sequence", were due to Robert Bemer.

ASCII includes definitions for 128 characters: 33 are non-printing, mostly obsolete control

characters that affect how text is processed; 94 are printable characters (the space is not

printable). The ASCII character encoding or a compatible extension is used on nearly all

common computers, especially personal computers and workstations. The representation of

each and every character is with the seven bits (b7 to b1),e.g. the representation of A is

(1000001)=65,similarly other characters are coded like this.

3.2.7 Block diagram and description

Fig 10: Flow diagram of DNA computing based cryptography

The original message is the plaintext. The plaintext is mapped with the nucleotides.

After the mapping process is done it is converted to numbers. Then the public key

cryptography algorithm is followed to encrypt and decrypt the message. Here the public key

cryptography algorithm used is RSA. Then the demapping process is done to recover the

original plaintext[1].

3.2.8 Mapping the plaintext

The plaintext is to be mapped with the nucleotides. The mapping formation is given in

the table. Use of mapping is to encode the original message. For example if the plaintext is

‘think’ for each character it will encode with corresponding nucleotides given in Table 3.2.1.

23

Setup plaintext

Matching with nucleotides

Nucleotides to numbers

Encrypted message

Recovery plaintext

By the use of DNA nucleotides, the given text message is converted into a encoded

text by arranging nucleotides in a difficult manner. Perform the mapping process which

converts the nucleotides to numbers. Here, each character in the encoded text will be

converted into number which are already defined. The encoded messages in the form of

numbers are given as an input plain text to RSA encryption.

A-CCA B-GTT

C-TTG D-GGT

E-TTT F-TCG

G-CGC H-ATG

I-AGT J-CGA

K-GAA L-CGT

M-CCT N-TCT

0-CGG P-ACA

Q-CAA R-ACT

U-GTC V-TCC

Y-AAA Z-TCA

S-GCA T-CTT

W-GCC X-ACT

Table 2: Encode Plaintext

A-01 C-03 G-07 T-20

Table 3: Mapping process from base nucleotides to numbers

3.3 RSA ALGORITHM

RSA is a public key cryptography which have the following important characteristic:

It is computationally infeasible to determine the decryption key given only

knowledge of the cryptographic algorithm and the encryption key[2].

Public key cryptography is a fundamental and widely used technology around the world.

It is the approach which is employed by many cryptographic algorithms and cryptosystems.

24

Needed to work

Public key encryption algorithm is used for encryption and decryption with a pair

of keys, one for encryption and one for decryption.

The sender and receiver must each have one of the pair of keys (not the same

one).

Needed for security

One of the two keys must be kept secret.

It must be infeasible or at least impractical to decipher a message if no other

information is available.

Knowledge of the algorithm plus one of the keys plus samples of ciphertext must

be insufficient to determine the other key.

In the RSA scheme makes use of an expression with exponentials. Plaintext is encrypted

in blocks, with each block having a binary value less than some number n, that is, the block

size must be less than or equal to log2(n). Encryption and decryption are of the following

form, for some plaintext block M and cipher text block C.

C = Me mod n

M = Cd mod n = (Me)d mod n

Both sender and receiver must know the value of n. the sender knows the value of e,

and only the receiver knows the value of e, and only the receiver knows the value of d. thus,

this is a public-key encryption algorithm with a public key of PU = {e, n} and a private key

of PR = {d, n} for this algorithm to be satisfactory for public-key encryption.

The following requirements must be

1. It is possible to find values of e, d, n such that Med mod n=M for all M < n.

2. It is relatively easy to calculate Me mod n and Cd mod n for all

values of M<n.

3. It is infeasible to determine d given e and n.

The relationship between e and d can be expressed as

edmod φ(n) = 1

e and d are multiplicative inverses of mod φ(n).

25

3.3.1 Efficient Operation Using The Public Key

To speed up the operation of the RSA algorithm using the public key, a specific

choice of e is usually made. The most common choice is 65537 (216 -1).

Each of these choices has only two 1 bits and so the number of multiplications required to

perform exponentiation is minimized. However, with a very small public key, such as e = 3,

RSA becomes vulnerable to a simple attack. It is required that during key generation the user

selects a value of e that is relatively prime to φ (n). Thus, for example, if a user has

preselected e = 65537 and then generated primes p and q, it may turn out that gcd(φ(n),e) is

not equal to 1, Thus, the user must reject any value of p or q that is not congruent to 1 (mod

65537).

The RSA algorithm involves three steps:

Key generation,

Encryption and

Decryption

RSA involves a public key and a private key. The public key can be known to

everyone and is used for encrypting messages. Messages encrypted with the public key can

only be decrypted using the private key.

3.3.2 RSA Software

RSA software is prototype software to demonstrate key generation using RSA

cryptography algorithm. Software is capable of generating 32-bit long keys.

It has the following features.

1. RSA software can generate two prime numbers randomly. Also, user can specify

one or both prime numbers.

2. Software calculates and displays the value of ‘n’.

3. It calculates and displays the value of ‘e’.

4. It generates and displays the value of ‘d’ using ‘e’ and ‘n’.

5. Displays both public and private keys for analysis.

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6. User can encrypt a text file using the public key.

7. User can decrypt the encrypted file using the private key.

8. User can compare the original text file and decrypted text file.

An example is shown below to describe the key generation process

1. Selecting two prime numbers, p=13 and q=19

2. Calculate n = p x q = 13 x 19 = 247

3. Calculate Φ (n) = (p-1) x (q-1) = 216

4. Select ‘e’ such that e is relatively prime to Φ (n)=216 and less than Φ (n); in this

case, e = 5

5. Determine d such that

d x e=1 mod 216 and d<216.

Where, d x e = k Φ (n)+1

The correct value of d = 173

6. Thus, the resulting public key KU = {5, 247} and private key KR = {173, 247}

3.3.3 Main Key Generation Algorithm

Algorithm: MAIN_KEY_GENERATION

1. Generate prime number (p and q) if not specified by the user (refer algorithm

PRIME_NO_GENERATE).

2. Calculate n = p x q

3. Calculate Φ (n) = (p-1) x (q-1)

4. Select the value of ‘e’ (refer algorithm SELECT_e).

5. Calculate the value of ‘d’ (refer algorithm CALCULATE_d).

6. Display public key KU={e,n} and private key KV={d,n}

7. End

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3.3.4 Prime Number Generation Algorithm

Algorithm: PRIME_NO_GENERATE

1. Generate a random number (p) using pseudorandom number generator

2. If the random number is not odd then add one to the number to make it odd.

If ‘p’ no odd then p =p+1;

3. Let I = 2

4. Compute remainder = p%I.

5. If remainder = 0 then goto step 2;//this odd number is not prime

Else I =I+1;

6. If I = p/2+1 then goto 7;

Else Goto 4;

7. Return the odd number as prime number

8. End

3.3.5 Algorithm for Selecting the Value of ‘e’

This algorithm is to determine the value of integer ‘e’ using Φ (n). As stated earlier

Φ(n) and e should be relatively prime.

Relatively Prime: To explain this let us consider two numbers a and b. If a and b are called

as relatively prime then

Mod(a,b) = 1

Algorithm: SELECT_e

Input: a, b (where X>Y)

1. X = b; Y = a (where X>Y)

2. If Y=0 return X=gcd(X,Y)

3. R = X mod Y

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4. X = Y

5. Y = R

6. goto 2

3.3.6 Algorithm for Calculating the Value of ‘d’

Following equation is used for calculating the value of ‘d’

ed = k Φ (n) + 1

Where, e and _(n) are known

Also, k is varied from 1 to 100000

Algorithm: Calculate_d

1. k =1

2. (Float calculation) d = (Φ (n) x k +1)/e

3. (Integer calculation) d_dash = (Φ (n) x k +1)/e

4. If d = d_dash then Return d as the calculated value

5. k = k+1

6. If k=100001 goto 8

7. Goto 2

8. Couldn’t find d

3.3.7 Algorithm for Encrypting Data

The algorithm presented in this section describes how data is encrypted in the software

using the public key. Basically for any encryption the formula used is given below

Plaintext M, where M<N

Ciphertext C = M^e (mod n)

Algorithm: Encrypt_Data

Input: data_in

Output: data_out

1. Convert ‘e’ into binary and store in ‘bits’

2. Get length of the converted binary number in ‘length’

3. c = 0, d =1, i =length-1

4. c= c*2

5. d =mod(d*d, n)

29

6. If bits_I = 1 then

{

c=c+1;

d=fmod(data_in*d,n);

}

7. i =i-1

8. If I != -1 goto 4

9. data_out = d

10. Return data_out

3.3.8 Algorithm for Decrypting Data

The algorithm presented in this section describes how data is decrypted in the

software using the private key. Basically for any decryption the formula used is given below

Plaintext C

Ciphertext M = C^d (mod n)

Algorithm: Decrypt_Data

Input: data_in

Output: data_out

1. Convert ‘d’ into binary and store in ‘bits’

2. Get length of the converted binary number in ‘length’

3. c =0, data_out = 1, i= length-1

4. c= c*2

5. data_out =mod(data_out * data_out, n)

6. If bits_I = 1 then

{

c=c+1;

data_out =fmod(data_in* data_out,n);

}

7. i _=i-1

8. If i != -1 goto 4

9. Return data_out

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Example of RSA encryption and decryption

This section presents an example of RSA encryption and decryption process with key

generation.

P = 61, first prime number (destroy this after computing E and D)

Q = 53,second prime number (destroy this after computing E and D)

PQ = 3233,modulus (give this to others)

E = 7 ,public exponent (give this to others)

D = 1783, private exponent (keep this secret!)

Your public key is (E, PQ).

Your private key is (D, PQ).

Public Key

e=7, n=3233 thus,

KU = {7, 3233}

Private Key

d=1783, n=3233 thus

KR = {1783, 3233}

The encryption function is:

encrypt(T) = (T^E) mod PQ

= (T^7) mod 3233

The decryption function is:

decrypt(C) = (C^D) mod PQ

= (C^1783) mod 3233

3.4 Key generation

Generate two large random primes, p and q, of approximately equal size such

that their product n = pq is of the required bit length, e.g. 1024 bits.

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Fig 11: RSA algorithm

Compute n = pq and (φ) phi = (p-1)(q-1).

Choose an integer e, 1 < e < phi, such that gcd(e, phi) = 1.

Compute the secret exponent d, 1 < d < phi, such that ed ≡ 1 (mod phi).

The public key is (n, e) and the private key is (n, d). Keep all the values d, p, q and

phi secret.In the process of RSA algorithm

n is known as the modulus. e is known as the public exponent or encryption exponent or just the exponent. d is known as the secret exponent or decryption exponent.

3.4.1 Encryption with RSA

Public key

Ciphertext

Fig 12 : RSA Encryption

32

Plaintext Encryption algorithm

P&Q (Two large prime numbers are choiced)

N=P*Q

Public key: E & private key: D

Encryption CT=PTe mod n.

CT to receiver

Decryption PT= CTd mod n.

Sender A does the following:-

Represents the plaintext message as a positive integer m.

Computes the ciphertext c = me mod n.

Sends the ciphertext c to receiver.

3.4.2 Decryption with RSA

Private key

plaintext

Fig 13: RSA Decryption

Uses private key (n, d) to compute m = cd mod n.

Extracts the plaintext from the message representative m.

After the completion of the decryption process is done the demapping process

is followed. The source code is attached at appendix.

3.4.3 Demapping

The demapping is the reverse process of mapping with nucleotides.

Once the demapping process is completed the original plaintext is retrived.

3.5 EXISTING SYSTEM

Steps to be followed

Get the input as plain text.

The plain text is encoded by using the DNA based computing.

Each character of the plain text is matched with the nucleotides.

Then mapping the nucleotides to corresponding numbers. The encoded value

is obtained.

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ciphertext Decryption algorithm

Fig 14: A Hybrid Cryptographic scheme using DNA and RSA

Select two prime numbers. Get the encoded as the input for encryption.

The obtained number is separated to calculate the cipher text.

The encrypted message is transmitted.

At the receiver, decrypt the cipher text into plain text in the form of numbers.

The obtained numbers are matched with the base nucleotides to get the plain

text.

3.6 ADVANTAGES OF RSA

The primary advantage of public-key cryptography is increased security and

convenience, private keys never need to be transmitted or revealed to anyone.

3.7 DISADVANTAGES OF RSA

RSA is a highly secure algorithm, provided the keys are generated properly, the only

way to attack is to perform a brute- force attack on the modulus. This attack can be simply

defeated by increasing the key size.

34

Setup plain text

Matching with Nucleotides

Nucleotides to Numbers

Numbers to NucleotidesP&Q (Two large prime numbers are choiced

Public key: E & private key:D

Encryption CT=PTe mod n.

CT to receiver

N=P*Q

Decryption PT= CTd mod n.

De Matching with Nucleotides

Recovery plain text

However this approach can lead to a number of problems:

Increased processing time – Decryption time increases approximately eight

fold times as key sizes double.

Increased key storage requirement – Key storage requires significant amount

of memory for storage.

Furthermore, key generation is complex and time consuming. Time increases considerably as

the key size increases.

Three possible approaches to attracking the RSA algorithm are as follows:

Brute force : This invoves trying all possible private keys.

Mathematical attacks : There are several approaches,all equivalent in effect to

factoring the product of two primes.

Timing attacks : These depends on the running time of the decryption algorithm.

35

CHAPTER IV

PROPOSED SYSTEM

4.1 DNA COMPUTING ECC CRYPTOGRAPHY

In RSA, its security is based on the difficulty of factoring large integers. An important

advantage of ECC is the shorter key lengths. ECC-160 provides comparable security to RSA-

1024 and ECC-224 provides comparable security to RSA-2048.For these bit-lengths, signing

is about five times faster with elliptic curves, but verifying a signature is seven times faster

with RSA.

ECC is a public-key primitive that is increasingly important as alternative to RSA. So

in this paper we proposed the new encryption design by the combination of DNA computing

with ECC algorithm. The DNA based computing deals with the encoding and decoding of the

given input text message. The encryption and decryption is done with the help of ECC

algorithm.

4.2 ELLIPTIC CURVE CRYPTOGRAPHY

Elliptic curve cryptography (ECC) is a public key cryptography. In public key

cryptography each user or the device taking part in the communication, generally have a pair

of keys, a public key and a private key, and a set of operations associated with the keys to do

the cryptographic operations. Only the particular user knows the private key whereas the

public key is distributed to all users taking part in the communication. Some public key

algorithm may require a set of predefined constants to be known by all the devices taking part

in the communication. ‘Domain parameters’ in ECC is an example of such constants. public

key cryptography, unlike private key cryptography, does not require any shared secret

between the communicating parties but it is much slower than the private key cryptography.

The mathematical operations of ECC is defined over the elliptic curve y2 = x3 + ax +

b, where 4a3 + 27b2 ≠ 0. Each value of the ‘a’ and ‘b’ gives a different elliptic curve. All

points (x, y) which satisfies the above equation plus a point at infinity lies on the elliptic

curve. The public key is a point in the curve and the private key is a random number. The

public key is obtained by multiplying the private key with the generator point G in the curve.

36

The generator point G, the curve parameters ‘a’ and ‘b’, together with few more

constants constitutes the domain parameter of ECC.

y2=x3+x+1 y2=x3-x

Fig 15: Elliptic Curve

One main advantage of ECC is its small key size. A 160-bit key in ECC is considered to be as secured as 1024-bit key in RSA.

4.3 ECC PUBLIC KEY CRYPTOSYSTEM

Elliptic curves are used for cryptography because of the difficulty of the elliptic curve

DLP. While generic algorithms apply, the index calculus algorithm has no adaptation,

effectively eliminating one of our most powerful tools. Before we can look more in depth at

the cryptosystems, we need to modify the addition on elliptic curves for p Z .

Let P =(x1,y1) and Q =( x2 , y2 ) if x1 = x2 and y1= −y2 then P +Q = O otherwise P +Q = (x3 , y3) , where

X3= λ2-x1-x2,

Y= λ(x1-x3)-y,

λ=[(y2-y1)(x2-x1)-1]mod p, if P≠Q

λ=[(3x12+a)(2y1)-1]mod p, if P=Q

and P+O=O+P=P for all P∈E .

37

If E is small enough, it is possible to calculate all of the elements of E using a brute

force method. After we determine the elliptic curve, we need to calculate the order.

The brute force method is extremely slow when p is a large prime, and there are

several theorems which allow us to approximate the order, but our most powerful tool for

doing this is Schoof’s algorithm. Schoof’s algorithm computes E with a running time of

O(log p)8 , and is efficient for primes p up to several hundred digits. After we have

determined E we can easily see if the elliptic curve is cyclic, because if the order of a group is

prime, then the group is cyclic.

Now that we have established that p Z is a group, and we can determine relatively

easily if it is a cyclic group, we can look at the El Gamal cryptosystem, which translates

nicely to the elliptic curve. The first thing we need to do is to change the encryption and

decryption from multiplicative to additive notation.

In the public key elliptic curve cryptosystems, we assume that entity A wants to send

a message m to entity B securely. Order of a point on the curve can be defined as a value n

such that nP = P+P+..+P.. n times = O (infinity)

4.3.1 Discrete Logarithm Problem

The security of ECC depends on the difficulty of Elliptic Curve Discrete Logarithm

Problem. Let P and Q be two points on an elliptic curve such that kP = Q, where k is a scalar.

Given P and Q, it is computationally infeasible to obtain k, if k is sufficiently large. k is the

discrete logarithm of Q to the base P.Hence the main operation involved in ECC is point

multiplication. i.e. multiplication of a scalar k with any point P on the curve to obtain another

point Q on the curve.

4.3.2 Point multiplication

In point multiplication a point P on the elliptic curve is multiplied with a scalar k

using elliptic curve equation to obtain another point Q on the same elliptic curve.

i.e. kP=Q

Point multiplication is achieved by two basic elliptic curve operations

• Point addition, adding two points J and K to obtain another point L i.e., L = J + K.

• Point doubling, adding a point J to itself to obtain another point L i.e. L = 2J.

38

Here is a simple example of point multiplication.

Let P be a point on an elliptic curve. Let k be a scalar that is multiplied with the point

P to obtain another point Q on the curve. i.e. to find Q = kP.

If k = 23 then kP = 23.P = 2(2(2(2P) + P) + P) + P.

Thus point multiplication uses point addition and point doubling repeatedly to find the

result. The above method is called ‘double and add’ method for point multiplication. There

are other efficient methods for point multiplication such as NAF (Non – Adjacent Form)

and wNAF (windowed NAF) method for point multiplication.

4.3.3 Finite Fields

The elliptic curve operations defined above are on real numbers. Operations over the

real numbers are slow and inaccurate due to round-off error. Cryptographic operations need

to be faster and accurate. To make operations on elliptic curve accurate and more efficient,

the curve cryptography is defined over two finite fields.

• Prime field Fp and

• Binary field F2m

The field is chosen with finitely large number of points suited for cryptographic

operations.Section 7 and 8 explains the EC operations on finite fields. The operations in

these sections are defined on affine coordinate system. Affine coordinate system is the

normal coordinate system that we are familiar with in which each point in the coordinate

system is represented by the vector (x, y).

4.3.4 EC on Prime field Fp

The equation of the elliptic curve on a prime field Fp is y2 mod p= x3 + ax + b mod

p,where 4a3 + 27b2 mod p ≠ 0. Here the elements of the finite field are integers between

0 and p – 1. All the operations such as addition, substation, division, multiplication involves

integers between 0 and p – 1. This is modular arithmetic and is defined in session.

39

The prime number p is chosen such that there is finitely large number of points on the

elliptic curve to make the cryptosystem secure. SEC specifies curves with p ranging between

112-521 .

The graph for this elliptic curve equation is not a smooth curve. Hence the

geometrical explanation of point addition and doubling as in real numbers will not work here.

However, the algebraic rules for point addition and point doubling can be adapted for elliptic

curves over Fp.

Point Addition

Consider two distinct points J and K such that J = (xJ, yJ) and K = (xK, yK)

Let L = J + K where L = (xL, yL), then

xL = s2 - xJ – xK mod p

yL = -yJ + s (xJ – xL) mod p

s = (yJ – yK)/(xJ – xK) mod p, s is the slope of the line through J and K.

If K = -J i.e. K = (xJ, -yJ mod p) then J + K = O. where O is the point at infinity.

If K = J then J + K = 2J then point doubling equations are used.

Also J + K = K + J

Point Subtraction

Consider two distinct points J and K such that J = (xJ, yJ) and K = (xK, yK)

Then J - K = J + (-K) where -K = (xk, -yk mod p)

Point subtraction is used in certain implementation of point multiplication such as NAF .

Point Doubling

Consider a point J such that J = (xJ, yJ), where yJ ≠ 0

Let L = 2J where L = (xL, yL), Then

xL = s2 – 2xJ mod p

yL = -yJ + s(xJ - xL) mod p

s = (3xJ2 + a) / (2yJ) mod p, s is the tangent at point J and a is one of the parameters

chosen with the elliptic curve

If yJ = 0 then 2J = O, where O is the point at infinity.

40

4.3.5 Elliptic Curve Domain parameters

Apart from the curve parameters a and b, there are other parameters that must be

agreed by both parties involved in secured and trusted communication using ECC. These

are domain parameters. The domain parameters for prime fields and binary fields

aredescribed below. The generation of domain parameters is out of scope of this

paper.Generally the protocols implementing the ECC specify the domain parameters to be

used.

Domain parameters for EC over field Fp

The domain parameters for Elliptic curve over Fp are p, a, b, G, n and h.

p is the prime number defined for finite field Fp . a and b are the parameters defining

the curve y2 mod p= x3 + ax + b mod p. G is the generator point (xG, yG), a point on the

elliptic curve chosen for cryptographic operations. n is the order of the elliptic curve. The

scalar for point multiplication is chosen as a number between 0 and n – 1. h is thecofactor

where h = #E(Fp)/n. #E(Fp) is the number of points on an elliptic curve.

4.3.6 Modular Arithmetic

Modular arithmetic over a number p involves arithmetic between numbers 0 and

p – 1. If the number happens to be out of this range in any of the operation the result is

wrapped around in to the range 0 and p – 1.

Addition

Let p = 23, a = 15, b = 20

a + b (mod p) = 15 + 20 (mod 23) = 35 mod 23 = 12

Since the result of a + b = 35 which is out of the range [0 22], The result is wrapped

around in to the range [0 22] by subtracting 35 with 23 till the result is in range [0 22].

a mod b is thus explained as remainder of division a/b.

Subtraction

Let p = 23, a = 15, b = 20

41

a - b (mod p) = 15 - 20 (mod 23) = -5 mod 23 = 18

Since the result of a - b = -5 which is negative and out of the range [0 22], The result

is wrapped around in to the range [0 22] by adding -5 with 23 till the result is in range

[0 22].

Multiplication

Let p = 23, a = 15, b = 20

a * b (mod p) = 15 * 20 (mod 23) = 300 mod 23 = 1

Since the result of a * b = 300 which is out of the range [0 22], The result is wrapped

around in to the range [0 22] by subtracting 300 with 23 till the result is in range [0 22].

Division

The division a/b (mod p) is defined as a * b-1 (mod p). b-1 is the multiplicative

inverse of b over p.

Multiplicative Inverse

Multiplicative inverse of number b with respect to mod p is defined as a number b-1

such that b*b-1 (mod p) = 1. Multiplicative inverse exists only if b and n are relatively prime.

The algorithm such as extended Euclidean algorithm can be used to find the multiplicative

inverse of a number efficiently. Finding multiplicative inverse is a costly operation.

Finding x mod y

x mod y is the remainder of the division x/y. Finding x mod y by repeatedly

subtracting y with x till the result is in range [0 y-1] is a costly operation. Methods such as

Barrett Reduction can be used to find modulus of a number in efficient manner.

4.3.7 To make it more efficient

As we discussed earlier the point multiplication is the main operation in elliptic curve

cryptography. Point multiplication involves plenty of point addition and point doubling.

Each point addition and doubling involves a multiplicative inverse operation . Finding

multiplicative inverse is a costly operation in both finite fields,Fp and F2m.

42

Representing the points in projective coordinate systems can eliminate the need of

multiplicative inverse operation in point addition and point doubling and there by increasing

the efficiency of point multiplication operation. For using the projective coordinate in elliptic

curve one has to convert the given point in affine coordinate to projective coordinate before

point multiplication then convert it back to affine coordinate after point multiplication. The

entire process requires only one multiplicative inverse operation. The operation in projective

coordinate involves more scalar multiplication than in affine coordinate. ECC on projective

coordinate will be efficient only when the implementation of scalar multiplication is much

faster than multiplicative inverse operation.

4.4 Key generation

Fig 16: ECC Algorithm

43

Represent character to the ASCII value

Encode this value to a point on the curve

Convert ASCII value to the original character

Decode the point to its ASCII value

Decrypt the cipher text points to original plaintext

Encrypt the point to two cipher text points

Both the entities in the cryptosystem agree upon a,b,p,G,n which are called ‘Domain

Parameters’ of ECC.G is called generator point and n is the order of G. Now A generates a

random number nA < n as his private Key and calculates his public key Set PA = G+G+G…

+nA times. B generates a random number nB < n as his private Key and calculates his public

key, set PB = G+G+G…+nB times.

4.4.1 Key Exchange

Entity A computes his Shared Key by Computing K = PA +PA +..+… nB times

Entity A computes his Shared Key by Computing K = PB +PB +..+… nA times

The two above keys have same value because:

nA*PB = nA* (nB*G)= nB*(nA *G)= nB * PA

4.4.2 Encryption

Here, each character is converted into a point based on kolbitz method. Since we are

using ECC it is just to improve the security and to get a less processing time for the whole

process. The elliptic curve has number of points. The character which are converted into

points are listed. The ECC encryption is done with the help of its generated keys.

With the generated key and the obtained points each will be encrypted. The

encrypted points form the elliptic curve and which will be transmitted.

A sends Cm = 2 cipher text points those are { kG, Pm + k PB)}.

Where

G - generator Point

Pm - plaintext point on the curve

k - a random number chosen by A

PB - public key of B

44

4.4.3 Decryption

Pm + kPB - nB(kG) = Pm + k(nB)G - nB(kG) = Pm

The transmitted message will be received at the receiver. This will be decrypted and

the obtained points are converted into numbers with the help of kolbitz method. These

numbers will be decoded with the help of DNA nucleotides and matching with these

nucleotides, the required plain text will be obtained.

4.4.4 Encoding and Decoding a message in the implementation of ECC

ECC Encryption and Decryption methods can only encrypt and decrypt a point on the

curve not messages. Unfortunately, there are no known polynomial time algorithms for

finding a large number of points on an arbitrary curve. We are not simply looking for random

points on E, here.

Message Encoding and Decoding

Fig 17: ECC Encoding and Decoding

45

Encode this value to a point on the curve

Convert ASCII value to the original character

Decode the point to its ASCII value

Decrypt the cipher text points to original plaintext

Encrypt the point to two cipher text points

Represent character to the ASCII value

Start

End

We want a systematic way of finding points on Ep(a,b) relating somehow to the

plaintext message.

Therefore, we are forced to use probabilistic algorithms to do this, where the chance

of failure is acceptably small. Thus Encoding(message to a point) and Decoding (point to a

message) methods are important while Encryption and Decryption.

Let us suppose a text file has to be encrypted, a user can encrypt the ASCII code of

each and every printable character on the keyboard , let us say he has to encrypt an 8- bit

number , can represent 128 characters on the keyboard. the sequence of steps to be followed

when a message to be encrypted and decrypted using elliptic Curve Cryptography.

All the points on the elliptic curve can be directly mapped to an ASCII value, select a

curve on which we will get a minimum of 128 points, so that we fix each point on the curve

to an ASCII value. For example, ‘ENCRYPT’ can be written as sequence of ASCII

characters that is ‘ 69’ ‘78’ ‘67’ ‘82’ ‘89’ ‘80’ ‘84’ we can map these values to fixed points

on the curve. This is easiest method for embedding a message but less efficient in terms of

security. The steps to be followed during encoding and decoding are given the following

flowchart.

4.4.5 KOBLITZ’S METHOD FOR ENCODING PLAINTEXT

Step1: Pick an elliptic curve Ep(a,b).

Step 2: Let us say that E has N points on it.

Step 3: Let us say that our alphabet consists of the digits 0,1,2,3,4,5,6,7,8,9 and the

letters A,B,C,. . . , X,Y,Z coded as 10,11,. . . , 35.

Step 4: This converts our message into a series of numbers between 0 and 35.

Step 5: Now choose an auxiliary base parameter, for example k = 20. ( both parties

should agree upon this)

Step 6: For each number mk (say), take x=mk + 1 and try to solve for y.

Step 7: If you can't do it, then try x = mk +2 and then x = mk +3 until you can solve

for y.

46

Step 8: In practice, you will find such a y before you hit x = mk + k - 1. Then take the

point (x,y). This now converts the number m into a point on the elliptic curve.

In this way, the entire message becomes a sequence of points.

4.4.6 Decoding

Consider each point (x,y) and set m to be the greatest integer less than (x-1)/k.

Then the point (x,y) decodes as the symbol m.

Example

Say the parameters of curve are:

p(751),a(-1),b(188),n(727).

1. Say we have to send character ‘b’.

2. ‘B’ is first encoded as number 11.

3. x=mk+1 ie 11*20+1=221cannot solve it for a y such that y2= x3 + ax+ b mod p .

4. So go for x=mk+2 , x=222 , no y exists. x=mk+3, x=223,no y exists.

5. x=mk+4 so x=224 can solve it for y and y=248.

6. Now the point (224,248) is point is encrypted and decrypted as a message.

7. To decode just compute (x-1)/k ie (224-1)/20=223/20 ie 11.15.

8. Return 11 as original plaintext(greatest integer less than (x-1)/k ,that is 11.

9. The number 11 is now decoded to character ‘B’.

10. The probability that we fail to find a square (and hence fail to associate m to a

point) is about 1/2k[10].

How to select Curve Parameters for Koblitz’s Method

In Koblitz’s method the maximum possible value for m is 128, if an 8-bit number is

encrypted. Say value of k=10. Now the minimum value of x is mk+1 ie 128*10+1=1280 to

represent a character.To get a point on the curve whose x-coordinate is above 1280, we need

to select an elliptic curve with p value not less than 1280. So depending on the value of

k(>=10) we need to select the curve parameters.

47

4.5 PROPOSED SYSTEM

Fig18: A Hybrid Cryptographic scheme using DNA and ECC

Steps to be followed:

Get the input as plain text.

The plain text is encoded by using the DNA based computing.

Each character of the plain text is matched with the nucleotides.

Then mapping the nucleotides to corresponding numbers. The encoded value

is obtained.

The obtained value is to be converted into a point on the curve.

Encrypt the point to two cipher text points on the generated curve, it can be

transmitted.

For decryption, the cipher text points are converted into original points

Then the points are converted into the stream of numbers or values.

These values are de-mapped and obtained the output in nucleotides form.

These nucleotides are then converted into corresponding characters, which is

the plain text.

48

Encode this number to a point on the curve

Encrypt the point to two cipher text points

Decrypt the cipher text points to original plaintext

Decode the point to the number

Setup plain text

Matching with Nucleotides

Nucleotides to NumbersNumbers to Nucleotides

De Matching with Nucleotides

Recovery plain text

4.6 COMPARISON OF ECC WITH RSA

ECC is a public-key primitive that is increasingly important as alternative to RSA.An

important advantage of ECC are the shorter key lengths. ECC-160 provides comparable

security to RSA-1024 and ECC-224 provides comparable security to RSA-2048.For these bit-

lengths, signing is about five times faster with elliptic curves, but verifying a signature is

seven times faster with RSA. Moreover, very compact hardware implementations of ECC

have been developed.

At the 163-bit ECC/1024-bit RSA security level, an elliptic curve exponentiation for

general curves over arbitrary prime fields is roughly 5 to 15 times as fast as an RSA private

key operation, depending on the platform and optimizations. At the 256-bit ECC/3072-bit

RSA security level the ratio has already increased to between 20 and 60, depending on

optimizations. To secure a 256-bit AES key, ECC-521 can be expected to be on average 400

times faster than 15,360-bit RSA.

Symmetric ECC DSA/RSA

80 163 1024

128 283 3072

192 409 7680

256 571 15360

Table 4 : Key Sizes In Bits For Equivalent Levels

49

CHAPTER V

SIMULATION RESULT

The software used for simulation part is MATLAB. MATLAB is a commercial

"Matrix Laboratory" package which operates as an interactive programming environment.

5.1 COMPARISON OF DNA WITH RSA AND ECC

Fig:19: Comparison of DNA with RSA and ECC

The above fig(19) shows that DNA with RSA takes more processing time than the

DNA with ECC. For the key size 200,DNA with RSA takes the processing time of 6.5second

whereas DNA with ECC takes only 1.13second.Therefore by increasing the key size, the

processing time for DNA with RSA increases whereas the processing time for DNA with

ECC takes a minimum value as much as possible.

50

5.2 COMPARISON OF DNA WITH RSA AND ECC(STRINGS)

Fig 20: comparison of DNA with RSA and ECC Algorithm

The above fig(20) shows that RSA takes more processing time than the ECC. For the

string length 3, RSA takes the processing time of 0.16seconds whereas ECC takes only

0.12seconds.Therefore by increasing the string length, the processing time for RSA increases

whereas the processing time for ECC takes a minimum value as much as possible.

51

5.3 COMPARISON OF ECC WITH RSA

Fig 21: Comparison of ECC with RSA

The above fig (21) shows that RSA takes more processing time than the ECC. For the

key size 80bits, RSA takes the processing time of 1.8seconds whereas ECC takes only

0.09seconds. Therefore by increasing the key size, the processing time for RSA increases

whereas the processing time for ECC takes a minimum value as much as possible.

52

5.4 COMPARISON OF RSA ENCRYPTION WITH DECRYPTION

0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5

3

3.5

4comparison of RSA Encryption with Decryption

Key size in bits

proc

essi

ng ti

me

in s

ec

EncryptionDecryption

Fig 22: Comparison of RSA encryption with decryption

The above fig(22) shows that RSA encryption takes more processing time than the

RSA decryption . For the key size 200, RSA encryption takes the processing time of

2.5seconds whereas RSA decryption takes only 0.2seconds.Therefore by increasing the key

size, the processing time for RSA encryption increases whereas the processing time for RSA

decryption takes a minimum value as much as possible.

53

5.5 COMPARISON OF RSA WITH DNA ENCRYPTION AND DECRYPTION

0 200 400 600 800 1000 12000

1

2

3

4

5

6

7comparison of RSA with DNA Encryption and Decryption

Key size in bits

proc

essi

ng ti

me

in s

ec

EncryptionDecryption

Fig 23: Comparison of RSA with DNA encryption and decryption

The above fig(23) shows that RSA with DNA encryption takes more processing time

than the RSA with DNA decryption . For the key size 200, RSA with DNA encryption takes

the processing time of 4.6seconds whereas RSA with DNA decryption takes only

0.2seconds.Therefore by increasing the key size, the processing time for RSA with DNA

encryption increases whereas the processing time for RSA with DNA decryption takes a

minimum value as much as possible.

54

5.6 COMPARISON TABELS

Comparison of DNA with RSA and ECC

Key Size in bitsProcessing Time in Seconds

DNA With RSA DNA With ECC

80 6.8387 1.326

160 7.6255 1.345

493 8.3565 1.352

1023 8.7773 1.370

Table 5: Comparison of DNA with RSA and ECC

Comparison of DNA with RSA and ECC

String lengthProcessing Time in Seconds

DNA With RSA DNA With ECC

3 0.1683 0.1262

4 0.1731 0.1311

11 0.2214 0.1370

28 0.2334 0.1433

Table 6: Comparison of DNA with RSA and ECC(Strings)

55

Comparison of RSA and ECC

Key Size in bitsProcessing Time in Seconds

ECC RSA

80 0.0932 1.8326

160 0.1032 2.1568

493 0.1754 2.4412

1023 0.2053 2.537

Table 7: Comparison of RSA and ECC

56

CHAPTER VI

CONCLUSION

This paper proposes a new encryption design by the combination of DNA computing

theory with ECC algorithm. The elliptic curve cryptography has evolved from a fringe

activity to a major challenger to the popular RSA. Elliptic curves offer major advantages over

traditional systems such as increased speed, less memory and smaller key size. In addition,

less storage, less power and less memory than other systems make it possible to implement

cryptography in many special platforms such as wireless devices, laptop computers and smart

cards. So do the situations where efficiency is important.

SCOPE FOR FUTURE WORK

HECC can attain a faster encryption compared to ECC due to their rich algebraic

structure. Recently some efficient addition formula of HECC have been proposed and

implementation in system shows the performance of HECC to be competitive to that of

ECC.HECC has the advantage of shorter operand length than ECC.Since HECC has 80bits as

its minimum key size and offer more security, whereas ECC has minimum key size of

163bits.

57

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