RSA - ENCRYPTION ALGORITHM CRYPTOGRAPHY
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This document discusses public key cryptography and the RSA encryption algorithm. It provides an overview of public key cryptography, how the RSA algorithm works using a public and private key pair, and some of its applications. The RSA algorithm is based on the difficulty of factoring large prime numbers and allows for secure communication without needing to share secret keys. Some advantages are convenience and enabling message authentication and non-repudiation using digital signatures, while disadvantages include slower performance and the need to authenticate public keys.
![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](https://image.slidesharecdn.com/rsafinals-140906163530-phpapp02/85/RSA-ENCRYPTION-ALGORITHM-CRYPTOGRAPHY-2-320.jpg)
![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](https://image.slidesharecdn.com/rsafinals-140906163530-phpapp02/85/RSA-ENCRYPTION-ALGORITHM-CRYPTOGRAPHY-3-320.jpg)
![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](https://image.slidesharecdn.com/rsafinals-140906163530-phpapp02/85/RSA-ENCRYPTION-ALGORITHM-CRYPTOGRAPHY-4-320.jpg)
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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. [email protected] Muthu Gomathy V GENESYS Receipt No:130 Department Of Computer Science And Engineering K.L.S.Gogte Institute Of Technology Belgaum, Karnataka, India. [email protected]
- 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