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PUBLIC KEY CRYPTOGRAPHY-RSA ENCRYPTION 
ALGORITHM 
Meenakshi Shetti 
GENESYS Receipt No:79 
Department Of Computer Science And Engineering 
K.L.S.Gogte Institute Of Technology 
Belgaum, Karnataka, India. 
meenakshishetti_11@yahoo.com 
Muthu Gomathy V 
GENESYS Receipt No:130 
Department Of Computer Science And Engineering 
K.L.S.Gogte Institute Of Technology 
Belgaum, Karnataka, India. 
muthugomathy1003@gmail.com
Abstract—Practice and study of techniques for secure 
communication in the presence of third parties is the 
cryptography. In this paper we are explaining about 
public key cryptography also called as asymmetric key 
cryptography where two different keys are used. No 
other key can decrypt the message – not even the 
original (i.e. the first) key used for encryption. The 
beauty of this scheme is that every communicating 
party needs just a key pair for communicating with any 
number of other communicating parties. Once some one 
obtains a key pair, he /she can communicate with any 
one else. Here we explain about the wide used 
encryption algorithm the RSA algorithm developed in 
1977. Which is developed on the basis of Diffie Hellman 
key exchange algorithm due to its shortcoming in one 
sender and many receiver. In this paper we discuss 
about working of RSA algotithm , its application in 
various sectors and its weekness and limitations. 
I. INTRODUCTION 
Cryptography (from Greek means "hidden, secret" 
and ,graphein, "writing",respectively) is the practice 
and study of techniques for secure communication in 
the presence of third parties (called adversaries). 
Cryptography is heavily based on mathematical 
theory and computer science practice; cryptographic 
algorithms are designed around computational 
hardness assumptions , making such algorithms hard 
to break in practice by any adversary. It is 
theoretically possible to break such a system but it is 
infeasible to do so by any known practical means. 
These schemes are therefore termed computationally 
secure; theoretical advances, e.g., improvements 
in integer factorization algorithms, and faster 
computing technology require these solutions to be 
continually adapted. There exist information-theoretically 
secure schemes that provably cannot be 
broken even with unlimited computing power—an 
example is the one-time pad—but these schemes are 
more difficult to implement than the best 
theoretically breakable but computationally secure 
mechanisms. More generally, it is about constructing 
and analyzing protocols that overcome the influence 
of adversaries[3] and which are related to various 
aspects in information security such as 
data confidentiality, data integrity, authentication, 
and non-repudiation.[4] Modern cryptography 
intersects the disciplines of mathematics, computer 
science, and electrical engineering. Applications of 
cryptography include ATM cards, computer 
passwords, and electronic commerce. 
fig 1. cryptography 
II. PUBLIC KEY CRYPTOGRAPHY(PKC) 
Public-key cryptography is used where each user has 
a pair of keys, one called the public key and the other 
private key. Each user’s public key is published while 
the private key is kept secret and thereby the need for 
the sender and the receiver to share secret 
information (key) is eliminated. The only 
requirement is that public keys are associated with 
the users in a trusted (authenticated) manner using a 
public key infrastructure (PKI) . The public key 
cryptosystems are the most popular, due to both 
confidentiality and authentication facilities. PKC 
depends upon the existence of one-way functions, or 
mathematical functions that are easy to compute 
whereas their inverse function is relatively difficult to 
compute. Generic PKC employs two keys that are 
mathematically related although knowledge of one 
key does not allow someone to easily determine the 
other key. One key is used to encrypt the plaintext 
and the other key is used to decrypt the ciphertext. 
The important point here is that it does not matter 
which key is applied first, but that both keys are 
required for the process to work . Because pair of 
keys are required, this approach is also called 
asymmetric cryptography 
fig 2. publickey cryptography
III. RSA ALGORITHM 
The Rivest-Shamir-Adleman (RSA) cryptosystem is 
one of the best known publickey cryptosystems for 
key exchange or digital signatures or encryption of 
blocks of data. RSA uses a variable size encryption 
block and a variable size key. The key-pair is derived 
from a very large number, n, that is the product of 
two prime numbers chosen according to special rules; 
these primes may be 100 or more digits in length 
each, yielding an n with roughly twice as many digits 
as the prime factors. The public key information 
includes n and a derivative of one of the factors of n; 
an attacker cannot determine the prime factors of n 
(and, therefore, the private key) from this information 
alone and that is what makes the RSA algorithm so 
secure. RSA's safety is due to the difficulty in 
factoring large prime numbers. The main arithmetic 
operation in the RSA Cryptosystem is modular 
exponentiation defined as 
C = Me mod n for encryption and 
M = Cd mod n for decryption, 
where C is the cipher, 
M is the message, 
e is the public key, 
d is the private key and 
n is the modulus. 
RSA algorithm has some important parameters 
affecting its level of security and speed. By 
increasing the modulus length plays an important role 
in increasing the complexity of decomposing it into 
its factors.This will increase the length of private key 
and hence difficult to be decrypted without knowing 
the decryption key.When the length of message is 
changed then the length of encrypted message will 
proportionally change, hence larger chunks are 
selected to obtained larger encrypted message to 
increase the security of the data in use[5]. RSA -1024 
bits is good for last 20 years but now Bernstain 
described circuitry for fast factorization. It is entirely 
possible that an organization with suffientely deep 
pockets can build a large scale version of his circuits 
and effectively crack an RSA 1024 bit message in a 
relatively short period of time, which could range any 
where from a number of minutes to some days [7,8]. 
Time analysis of RSA algorithm performed by varing 
its parametes[9].We use natural numbers in pair of 
keys in addition to existing parametes of RSA.Then 
after simulations of results on basis of speed and 
security we compare the RSA and new algorithm . 
We use fast modulation method in RSA for big 
exponential calculation. 
The RSA algorithm is described here 
fig 3. how RSA works 
fig 4.encrypt ion and decrypt ion 
When n is a product of two primes, in arithmetic 
operations modulo n, the exponents behave modulo 
the totient φ(n) of n. 
For example, consider arithmetic modulo 
15. since 15 = 3 × 5, 
for the totient of 15, we have φ(15) = 2 × 4 = 8. We 
can easily verify the following: 
57 . 54푚표푑 15 = 5(7+4)푚표푑 8푚표푑 15 = 53푚표푑 15 
= 125 푚표푑 5 
(43 )5푚표푑 15 = 4(3∗5)푚표푑 8푚표푑 15 = 47mod 15=4 
Considering arithmetic modulo n, let’s say that e is 
an integer that is coprime to the totient φ(n) of n. 
Further, say that d is the multiplicative inverse of e
modulo φ(n). These definitions of the various 
symbols are listed below for convenience: 
n = a modulus for modular arithmetic 
φ(n) = the totient of n 
e = an integer that is relatively prime to φ(n) 
[T his guarantees that e will possess a 
multiplicative inverse modulo φ(n)] 
d = an integer that is the multiplicative 
inverse of e modulo φ(n) 
Now suppose we are given an integer M, M < n, that 
represents our message, then we can transform M 
into another integer C that will represent our cipher 
text by the following modulo exponentiation: 
C = 푀푒 mod n 
At this point, it may seem rather strange that we 
would want to represent any arbitrary plaintext 
message by an integer. But, it is really not that 
strange. Let’s say you want a block cipher that 
encrypts 1024 bit blocks at a time. Every plaintext 
block can now be thought of as an integer M of value 
0 ≤ M ≤ 2102 4 − 1. 
We can recover back M from C by the following 
modulo operation 
M = 퐶 푑 mod n 
since 
(푀푒 )푑 (mod n) = 푀푒푑(푚표푑 φ (푛)) ≡ M (mod n) 
1. The RSA Algorithm — Putting To Use The 
Basic Idea 
The basic idea described in the previous subsection 
can be used to create a confidential communication 
channel in the manner described here. 
An individual A who wishes to receive messages 
confidentially will use the pair of integers {e, n} as 
his/her public key. At the same time, this individual 
can use the pair of integers {d, n} as the private key. 
The definitions of n, e, and d are as in the previous 
subsection. 
Another party B wishing to send a message M to A 
confidentially will encrypt M using A’s public key 
{e, n} to create cipher text C. Subsequently, only A 
will be able to decrypt C using his/her 
private key {d, n}. 
If the plaintext message M is too long, B may choose 
to use RSA as a block cipher for encrypting the 
message meant for A. When RSA is used as a block 
cipher, the block size is likely to be half the number 
of bits required to represent the modulus n. If the 
modulus required, say, 1024 bits for its 
representation, message encryption would be 
based on 512-bit blocks. [While, in principle, RSA 
can certainly be used as a block cipher, in practice it 
is more likely to be used just for exchanging a secret 
session key and, subsequently, the session key used 
for content encryption using symmetric-key 
cryptography based on, say, AES.] 
2. How To Choose The Modulus For The Rsa 
Algorithm? 
With the definitions of d and e, the modulus n must 
be selected in such a manner that the following 
is guaranteed: 
(푀푒 )푑 ≡ 푀푒푑 ≡ M (mod n) 
We want this guarantee because C = 푀푒mod m is the 
encrypted form of the message integer M and 
decryption is carried out by 
퐶 푑mod n. 
It was shown by Rivest, Shamir, and Adleman that 
we have this guarantee when n is a product of two 
prime numbers: 
n = p × q for some prime p and prime q 
(1) 
The above factorization is needed because the proof 
of the algorithm, presented in the next subsection, 
depends on the following two properties of primes 
and coprimes: 
1. If two integers p and q are coprimes (meaning, 
relatively prime to each other), the following 
equivalence holds for any two integers a and b: 
{a ≡ b (mod p) and a ≡ b (mod q)} ⇔ {a ≡ b (mod 
pq)} (2) 
This equivalence follows from the fact a ≡ b (mod p) 
implies a − b = k1p for some integer푘1 . But since we 
also have a ≡ b (mod q) implying a−b = 푘2푞 , it must 
be the case that 푘1= 푘3× q for some 푘3. Therefore, 
we can write 
a−b = 푘3×p×q, which establishes the equivalence. 
(Note that this argument breaks down if p and q have 
common factors other than 1.) 
2. In addition to needing p and q to be coprimes, we 
also want p and q to be individually primes. It is only 
when p and q are individually prime that we can
decompose the totient of n into the product of the 
totients of p and q. That is 
φ (n) = φ (p) × φ (q) = (p − 1) × (q − 1) 
(3) 
So that the cipher cannot be broken by an exhaustive 
search for the prime factors of the modulus n, it is 
important that both p and q be very large primes. 
Finding the prime factors of a large integer is 
computationally harder than determining its 
primality. 
We also need to ensure that n is not factorizable by 
one of the modern integer factorization algorithms. 
IV. APPLICATIONS 
When it comes to assymetric cryptography the most 
popular and widely used application that comes to 
anyone's mind is PGP. PGP stands for “Pretty Good 
Privacy” and is the standard public key cryptography 
application used today. In the examples of this 
project we chose to use PGP Desktop. The reason for 
this choice is that PGP Desktop is easier to use than 
other text-based versions of PGP such as gnuPGP. 
PGP Desktop provides us with a very intuitive GUI 
accessible from the Windows Start Menu ,the PGP 
taskbar icon and from Windows explorer (shell 
integration). So from now on, every time we mention 
PGP, we will be referring to the PGP Desktop 
version. 
V. ADVANTAGES 
1. Convenience: It solves the problem of 
distributing the key for encryption.Everyone 
publishes their public keys and private keys 
are kept secret. 
2. Provides for message authentication: Public 
key encryption allows the use of digital 
signatures which enables the recipient of a 
message to verify that the message is truly 
from a particular sender. 
3. Detection of tampering: The use of digital 
signatures in public key encryption allows 
the receiver to detect if the message was 
altered in transit. A digitally signed message 
cannot be modified without invalidating the 
signature. 
4. Provide for non-repudiation: Digitally 
signing a message is akin to physically 
signing a document. It is an 
acknowledgement of the message and thus, 
the sender cannot deny it. 
VI. Disadvantages 
1. Public keys should/must be 
authenticated: No one can be absolutely 
sure that a public key belongs to the person 
it specifies and so everyone must verify that 
their public keys belong to them. 
2. Slow: Public key encryption is slow 
compared to symmetric encryption. Not 
feasible for use in decrypting bulk messages. 
3. Uses up more computer resources: It 
requires a lot more computer supplies 
compared to single-key encryption. 
4. Widespread security compromise is 
possible: If an attacker determines a person's 
private key, his or her entire messages can 
be read. 
5. Loss of private key may be irreparable: The 
loss of a private key means that all received 
messages cannot be decrypted. 
VII. CONCLUSION 
We have proposed a method for implementing a 
public-key cryptosystem whose security rests in part 
on the difficulty of factoring large numbers. If the 
security of our method proves to be adequate, it 
permits secure communications to be established 
without the use of couriers to carry keys. 
The security of this system needs to be examined in 
more detail. In particular, the difficulty of factoring 
large numbers should be examined very closely. 
Once the method has withstood all attacks for a 
sufficient length of time it may be used with a 
reasonable amount of confidence. 
VIII. REFERENCES 
1. Frederick J. Hirsch. "SSL/TLS Strong 
Encryption: An Introduction". Apache HTTP 
Server. Retrieved 2013-04-17.. The first two 
sections contain a very good introduction to 
public-key cryptography. 
2. N. Ferguson; B. Schneier (2003). Practical 
Cryptography. Wiley. ISBN 0-471-22357-3.
3. J. Katz; Y. Lindell (2007). Introduction to 
Modern Cryptography. CRC Press. ISBN 1- 
58488-551-3. 
4. A. J. Menezes; P. C. van Oorschot; S. A. 
Vanstone (1997). Handbook of Applied 
Cryptography. ISBN 0-8493-8523-7. 
5. IEEE 1363: Standard Specifications for 
Public-Key Cryptography

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RSA - ENCRYPTION ALGORITHM CRYPTOGRAPHY

  • 1. PUBLIC KEY CRYPTOGRAPHY-RSA ENCRYPTION ALGORITHM Meenakshi Shetti GENESYS Receipt No:79 Department Of Computer Science And Engineering K.L.S.Gogte Institute Of Technology Belgaum, Karnataka, India. meenakshishetti_11@yahoo.com Muthu Gomathy V GENESYS Receipt No:130 Department Of Computer Science And Engineering K.L.S.Gogte Institute Of Technology Belgaum, Karnataka, India. muthugomathy1003@gmail.com
  • 2. Abstract—Practice and study of techniques for secure communication in the presence of third parties is the cryptography. In this paper we are explaining about public key cryptography also called as asymmetric key cryptography where two different keys are used. No other key can decrypt the message – not even the original (i.e. the first) key used for encryption. The beauty of this scheme is that every communicating party needs just a key pair for communicating with any number of other communicating parties. Once some one obtains a key pair, he /she can communicate with any one else. Here we explain about the wide used encryption algorithm the RSA algorithm developed in 1977. Which is developed on the basis of Diffie Hellman key exchange algorithm due to its shortcoming in one sender and many receiver. In this paper we discuss about working of RSA algotithm , its application in various sectors and its weekness and limitations. I. INTRODUCTION Cryptography (from Greek means "hidden, secret" and ,graphein, "writing",respectively) is the practice and study of techniques for secure communication in the presence of third parties (called adversaries). Cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions , making such algorithms hard to break in practice by any adversary. It is theoretically possible to break such a system but it is infeasible to do so by any known practical means. These schemes are therefore termed computationally secure; theoretical advances, e.g., improvements in integer factorization algorithms, and faster computing technology require these solutions to be continually adapted. There exist information-theoretically secure schemes that provably cannot be broken even with unlimited computing power—an example is the one-time pad—but these schemes are more difficult to implement than the best theoretically breakable but computationally secure mechanisms. More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries[3] and which are related to various aspects in information security such as data confidentiality, data integrity, authentication, and non-repudiation.[4] Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce. fig 1. cryptography II. PUBLIC KEY CRYPTOGRAPHY(PKC) Public-key cryptography is used where each user has a pair of keys, one called the public key and the other private key. Each user’s public key is published while the private key is kept secret and thereby the need for the sender and the receiver to share secret information (key) is eliminated. The only requirement is that public keys are associated with the users in a trusted (authenticated) manner using a public key infrastructure (PKI) . The public key cryptosystems are the most popular, due to both confidentiality and authentication facilities. PKC depends upon the existence of one-way functions, or mathematical functions that are easy to compute whereas their inverse function is relatively difficult to compute. Generic PKC employs two keys that are mathematically related although knowledge of one key does not allow someone to easily determine the other key. One key is used to encrypt the plaintext and the other key is used to decrypt the ciphertext. The important point here is that it does not matter which key is applied first, but that both keys are required for the process to work . Because pair of keys are required, this approach is also called asymmetric cryptography fig 2. publickey cryptography
  • 3. III. RSA ALGORITHM The Rivest-Shamir-Adleman (RSA) cryptosystem is one of the best known publickey cryptosystems for key exchange or digital signatures or encryption of blocks of data. RSA uses a variable size encryption block and a variable size key. The key-pair is derived from a very large number, n, that is the product of two prime numbers chosen according to special rules; these primes may be 100 or more digits in length each, yielding an n with roughly twice as many digits as the prime factors. The public key information includes n and a derivative of one of the factors of n; an attacker cannot determine the prime factors of n (and, therefore, the private key) from this information alone and that is what makes the RSA algorithm so secure. RSA's safety is due to the difficulty in factoring large prime numbers. The main arithmetic operation in the RSA Cryptosystem is modular exponentiation defined as C = Me mod n for encryption and M = Cd mod n for decryption, where C is the cipher, M is the message, e is the public key, d is the private key and n is the modulus. RSA algorithm has some important parameters affecting its level of security and speed. By increasing the modulus length plays an important role in increasing the complexity of decomposing it into its factors.This will increase the length of private key and hence difficult to be decrypted without knowing the decryption key.When the length of message is changed then the length of encrypted message will proportionally change, hence larger chunks are selected to obtained larger encrypted message to increase the security of the data in use[5]. RSA -1024 bits is good for last 20 years but now Bernstain described circuitry for fast factorization. It is entirely possible that an organization with suffientely deep pockets can build a large scale version of his circuits and effectively crack an RSA 1024 bit message in a relatively short period of time, which could range any where from a number of minutes to some days [7,8]. Time analysis of RSA algorithm performed by varing its parametes[9].We use natural numbers in pair of keys in addition to existing parametes of RSA.Then after simulations of results on basis of speed and security we compare the RSA and new algorithm . We use fast modulation method in RSA for big exponential calculation. The RSA algorithm is described here fig 3. how RSA works fig 4.encrypt ion and decrypt ion When n is a product of two primes, in arithmetic operations modulo n, the exponents behave modulo the totient φ(n) of n. For example, consider arithmetic modulo 15. since 15 = 3 × 5, for the totient of 15, we have φ(15) = 2 × 4 = 8. We can easily verify the following: 57 . 54푚표푑 15 = 5(7+4)푚표푑 8푚표푑 15 = 53푚표푑 15 = 125 푚표푑 5 (43 )5푚표푑 15 = 4(3∗5)푚표푑 8푚표푑 15 = 47mod 15=4 Considering arithmetic modulo n, let’s say that e is an integer that is coprime to the totient φ(n) of n. Further, say that d is the multiplicative inverse of e
  • 4. modulo φ(n). These definitions of the various symbols are listed below for convenience: n = a modulus for modular arithmetic φ(n) = the totient of n e = an integer that is relatively prime to φ(n) [T his guarantees that e will possess a multiplicative inverse modulo φ(n)] d = an integer that is the multiplicative inverse of e modulo φ(n) Now suppose we are given an integer M, M < n, that represents our message, then we can transform M into another integer C that will represent our cipher text by the following modulo exponentiation: C = 푀푒 mod n At this point, it may seem rather strange that we would want to represent any arbitrary plaintext message by an integer. But, it is really not that strange. Let’s say you want a block cipher that encrypts 1024 bit blocks at a time. Every plaintext block can now be thought of as an integer M of value 0 ≤ M ≤ 2102 4 − 1. We can recover back M from C by the following modulo operation M = 퐶 푑 mod n since (푀푒 )푑 (mod n) = 푀푒푑(푚표푑 φ (푛)) ≡ M (mod n) 1. The RSA Algorithm — Putting To Use The Basic Idea The basic idea described in the previous subsection can be used to create a confidential communication channel in the manner described here. An individual A who wishes to receive messages confidentially will use the pair of integers {e, n} as his/her public key. At the same time, this individual can use the pair of integers {d, n} as the private key. The definitions of n, e, and d are as in the previous subsection. Another party B wishing to send a message M to A confidentially will encrypt M using A’s public key {e, n} to create cipher text C. Subsequently, only A will be able to decrypt C using his/her private key {d, n}. If the plaintext message M is too long, B may choose to use RSA as a block cipher for encrypting the message meant for A. When RSA is used as a block cipher, the block size is likely to be half the number of bits required to represent the modulus n. If the modulus required, say, 1024 bits for its representation, message encryption would be based on 512-bit blocks. [While, in principle, RSA can certainly be used as a block cipher, in practice it is more likely to be used just for exchanging a secret session key and, subsequently, the session key used for content encryption using symmetric-key cryptography based on, say, AES.] 2. How To Choose The Modulus For The Rsa Algorithm? With the definitions of d and e, the modulus n must be selected in such a manner that the following is guaranteed: (푀푒 )푑 ≡ 푀푒푑 ≡ M (mod n) We want this guarantee because C = 푀푒mod m is the encrypted form of the message integer M and decryption is carried out by 퐶 푑mod n. It was shown by Rivest, Shamir, and Adleman that we have this guarantee when n is a product of two prime numbers: n = p × q for some prime p and prime q (1) The above factorization is needed because the proof of the algorithm, presented in the next subsection, depends on the following two properties of primes and coprimes: 1. If two integers p and q are coprimes (meaning, relatively prime to each other), the following equivalence holds for any two integers a and b: {a ≡ b (mod p) and a ≡ b (mod q)} ⇔ {a ≡ b (mod pq)} (2) This equivalence follows from the fact a ≡ b (mod p) implies a − b = k1p for some integer푘1 . But since we also have a ≡ b (mod q) implying a−b = 푘2푞 , it must be the case that 푘1= 푘3× q for some 푘3. Therefore, we can write a−b = 푘3×p×q, which establishes the equivalence. (Note that this argument breaks down if p and q have common factors other than 1.) 2. In addition to needing p and q to be coprimes, we also want p and q to be individually primes. It is only when p and q are individually prime that we can
  • 5. decompose the totient of n into the product of the totients of p and q. That is φ (n) = φ (p) × φ (q) = (p − 1) × (q − 1) (3) So that the cipher cannot be broken by an exhaustive search for the prime factors of the modulus n, it is important that both p and q be very large primes. Finding the prime factors of a large integer is computationally harder than determining its primality. We also need to ensure that n is not factorizable by one of the modern integer factorization algorithms. IV. APPLICATIONS When it comes to assymetric cryptography the most popular and widely used application that comes to anyone's mind is PGP. PGP stands for “Pretty Good Privacy” and is the standard public key cryptography application used today. In the examples of this project we chose to use PGP Desktop. The reason for this choice is that PGP Desktop is easier to use than other text-based versions of PGP such as gnuPGP. PGP Desktop provides us with a very intuitive GUI accessible from the Windows Start Menu ,the PGP taskbar icon and from Windows explorer (shell integration). So from now on, every time we mention PGP, we will be referring to the PGP Desktop version. V. ADVANTAGES 1. Convenience: It solves the problem of distributing the key for encryption.Everyone publishes their public keys and private keys are kept secret. 2. Provides for message authentication: Public key encryption allows the use of digital signatures which enables the recipient of a message to verify that the message is truly from a particular sender. 3. Detection of tampering: The use of digital signatures in public key encryption allows the receiver to detect if the message was altered in transit. A digitally signed message cannot be modified without invalidating the signature. 4. Provide for non-repudiation: Digitally signing a message is akin to physically signing a document. It is an acknowledgement of the message and thus, the sender cannot deny it. VI. Disadvantages 1. Public keys should/must be authenticated: No one can be absolutely sure that a public key belongs to the person it specifies and so everyone must verify that their public keys belong to them. 2. Slow: Public key encryption is slow compared to symmetric encryption. Not feasible for use in decrypting bulk messages. 3. Uses up more computer resources: It requires a lot more computer supplies compared to single-key encryption. 4. Widespread security compromise is possible: If an attacker determines a person's private key, his or her entire messages can be read. 5. Loss of private key may be irreparable: The loss of a private key means that all received messages cannot be decrypted. VII. CONCLUSION We have proposed a method for implementing a public-key cryptosystem whose security rests in part on the difficulty of factoring large numbers. If the security of our method proves to be adequate, it permits secure communications to be established without the use of couriers to carry keys. The security of this system needs to be examined in more detail. In particular, the difficulty of factoring large numbers should be examined very closely. Once the method has withstood all attacks for a sufficient length of time it may be used with a reasonable amount of confidence. VIII. REFERENCES 1. Frederick J. Hirsch. "SSL/TLS Strong Encryption: An Introduction". Apache HTTP Server. Retrieved 2013-04-17.. The first two sections contain a very good introduction to public-key cryptography. 2. N. Ferguson; B. Schneier (2003). Practical Cryptography. Wiley. ISBN 0-471-22357-3.
  • 6. 3. J. Katz; Y. Lindell (2007). Introduction to Modern Cryptography. CRC Press. ISBN 1- 58488-551-3. 4. A. J. Menezes; P. C. van Oorschot; S. A. Vanstone (1997). Handbook of Applied Cryptography. ISBN 0-8493-8523-7. 5. IEEE 1363: Standard Specifications for Public-Key Cryptography