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.
Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
Cryptography is the practice and study of techniques for conveying information security.
The goal of Cryptography is to allow the intended recipients of the message to receive the message securely.
The most famous algorithm used today is RSA algorithm
traditional private/secret/single key cryptography uses one key
Key is shared by both sender and receiver
if the key is disclosed communications are compromised
also known as symmetric, both parties are equal
hence does not protect sender from receiver forging a message & claiming is sent by sender
Public Key Cryptography and RSA algorithmIndra97065
Public Key Cryptography and RSA algorithm.Explanation and proof of RSA algorithm in details.it also describer the mathematics behind the RSA. Few mathematics theorem are given which are use in the RSA algorithm.
This report to document the RSA code and how it works from encrypting certain message to how to decrypt it using general and private keys which will be generated in the given code.
This presentation is based on the paper :
"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" by R.L. Rivest, A. Shamir, and L. Adleman
ANALYSIS OF RSA ALGORITHM USING GPU PROGRAMMINGIJNSA Journal
Modern-day computer security relies heavily on cryptography as a means to protect the data that we have become increasingly reliant on. The main research in computer security domain is how to enhance the speed of RSA algorithm. The computing capability of Graphic Processing Unit as a co-processor of the CPU can leverage massive-parallelism. This paper presents a novel algorithm for calculating modulo value that can process large power of numbers which otherwise are not supported by built-in data types. First the traditional algorithm is studied. Secondly, the parallelized RSA algorithm is designed using CUDA framework. Thirdly, the designed algorithm is realized for small prime numbers and large prime number . As a result the main fundamental problem of RSA algorithm such as speed and use of poor or small prime numbers that has led to significant security holes, despite the RSA algorithm's mathematical soundness can be alleviated by this algorithm.
Modern-day computer security relies heavily on cryptography as a means to protect the data that we have
become increasingly reliant on. The main research in computer security domain is how to enhance the
speed of RSA algorithm. The computing capability of Graphic Processing Unit as a co-processor of the
CPU can leverage massive-parallelism. This paper presents a novel algorithm for calculating modulo
value that can process large power of numbers which otherwise are not supported by built-in data types.
First the traditional algorithm is studied. Secondly, the parallelized RSA algorithm is designed using
CUDA framework. Thirdly, the designed algorithm is realized for small prime numbers and large prime
number . As a result the main fundamental problem of RSA algorithm such as speed and use of poor or
small prime numbers that has led to significant security holes, despite the RSA algorithm's mathematical
soundness can be alleviated by this algorithm.
traditional private/secret/single key cryptography uses one key
Key is shared by both sender and receiver
if the key is disclosed communications are compromised
also known as symmetric, both parties are equal
hence does not protect sender from receiver forging a message & claiming is sent by sender
Public Key Cryptography and RSA algorithmIndra97065
Public Key Cryptography and RSA algorithm.Explanation and proof of RSA algorithm in details.it also describer the mathematics behind the RSA. Few mathematics theorem are given which are use in the RSA algorithm.
This report to document the RSA code and how it works from encrypting certain message to how to decrypt it using general and private keys which will be generated in the given code.
This presentation is based on the paper :
"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" by R.L. Rivest, A. Shamir, and L. Adleman
ANALYSIS OF RSA ALGORITHM USING GPU PROGRAMMINGIJNSA Journal
Modern-day computer security relies heavily on cryptography as a means to protect the data that we have become increasingly reliant on. The main research in computer security domain is how to enhance the speed of RSA algorithm. The computing capability of Graphic Processing Unit as a co-processor of the CPU can leverage massive-parallelism. This paper presents a novel algorithm for calculating modulo value that can process large power of numbers which otherwise are not supported by built-in data types. First the traditional algorithm is studied. Secondly, the parallelized RSA algorithm is designed using CUDA framework. Thirdly, the designed algorithm is realized for small prime numbers and large prime number . As a result the main fundamental problem of RSA algorithm such as speed and use of poor or small prime numbers that has led to significant security holes, despite the RSA algorithm's mathematical soundness can be alleviated by this algorithm.
Modern-day computer security relies heavily on cryptography as a means to protect the data that we have
become increasingly reliant on. The main research in computer security domain is how to enhance the
speed of RSA algorithm. The computing capability of Graphic Processing Unit as a co-processor of the
CPU can leverage massive-parallelism. This paper presents a novel algorithm for calculating modulo
value that can process large power of numbers which otherwise are not supported by built-in data types.
First the traditional algorithm is studied. Secondly, the parallelized RSA algorithm is designed using
CUDA framework. Thirdly, the designed algorithm is realized for small prime numbers and large prime
number . As a result the main fundamental problem of RSA algorithm such as speed and use of poor or
small prime numbers that has led to significant security holes, despite the RSA algorithm's mathematical
soundness can be alleviated by this algorithm.
RSA is one of the most popular Public Key Cryptography based algorithm mainly used for digital
signatures, encryption/decryption etc. It is based on the mathematical scheme of factorization of very large
integers which is a compute-intensive process and takes very long time as well as power to perform.
Several scientists are working throughout the world to increase the speedup and to decrease the power
consumption of RSA algorithm while keeping the security of the algorithm intact. One popular technique
which can be used to enhance the performance of RSA is parallel programming. In this paper we are
presenting the survey of various parallel implementations of RSA algorithm involving variety of hardware
and software implementations.
Performance evaluation of modified modular exponentiation for rsa algorithmeSAT Journals
Abstract
Authentication is a very important application of public-key cryptography. Cryptographic algorithms make use of secret keys
known to send and receive information. When the keys are known the encryption / decryption process is an easy task, however
decryption will be impossible without knowing the correct key. The shared public key is managed by the sender, to produce a
message authentication code (MAC) for every transmitted message. There are many algorithms to enable security for message
authentication (secret key). RSA is one such best algorithm for public key based message authentication approaches. But it takes
more time for encryption and/or decryption process, when it has large key length. This research work evaluates the performance
of RSA algorithm with modified modular exponentiation technique for message authentication. As a result modified modular
exponent based RSA algorithm reduces execution time for encryption and decryption process.
Key Words: Cryptography, Message authentication, RSA, Modular Exponentiation.
Over this thesis, we did try to optimize tow major challenges of RSA policy:
1# Computational complexity.
2# Apology of unbreakability.
We use here multidimensional random padding scheme (MRPS) as an outer layer protection. RSA policy itself is inner or core layer but not ever unbreakable if additional layers are imposed. Here in this work, our MRPS scheme would able to ensure fully parametrized randomization process.
Bluetooth technology is an emerging wireless networking standard, which is based on chip that provides short-range wireless frequency hopping communication. Now, Bluetooth technology is mainly applied to the communication between mobile terminal devices, such as palm computers, mobile phones, laptops and so on. However, the phenomenon of data-leaking frequently arises in using the Bluetooth technology for data transfer. To enhance the security of data transmission in Bluetooth communication, a hybrid encryption algorithm based on DES and RSA is proposed. The currently used encryption algorithm employed by the Bluetooth to protect the confidentiality of data during transport between two or more devices is a 128-bit symmetric stream cipher called E0. The proposed hybrid encryption algorithm, instead of the E0 encryption, DES algorithm is used for data transmission because of its higher efficiency in block encryption, and RSA algorithm is used for the encryption of the key of the DES because of its management advantages in key cipher. Under the dual protection with the DES algorithm and the RSA algorithm, the data transmission in the Bluetooth system will be more secure. This project is extended with triple des in place of des to enhance more security.
Chaotic Rivest-Shamir-Adlerman Algorithm with Data Encryption Standard Schedu...journalBEEI
Cryptography, which involves the use of a cipher, describes a process of encrypting information so that its meaning is hidden and thus, secured from those who do not know how to decrypt the information. Cryptography algorithms come with the various types including the symmetric key algorithms and asymmetric key algorithms. In this paper, the authors applied the most commonly used algorithm, which is the RSA algorithm together with the Chaos system and the basic security device employed in the worldwide organizations which is the Data Encryption Standard (DES) with the objective to make a hybrid data encryption. The advantage of a chaos system which is its unpredictability through the use of multiple keys and the secrecy of the RSA which is based on integer factorization’s difficulty is combined for a more secure and reliable cryptography. The key generation was made more secure by applying the DES schedule to change the keys for encryption. The main strength of the proposed system is the chaotic variable key generator that chages the value of encrypted message whenever a different number of key is used. Using the provided examples the strength of security of the proposed system was tested and demonstrated.
ENHANCED SECURE ALGORITHM FOR MESSAGE COMMUNICATIONIJNSA Journal
This paper puts forward a safe mechanism of data transmission to tackle the security problem of information which is transmitted in Internet. The encryption standards such as DES (Data Encryption Standard), AES (Advanced Encryption Standard) and EES (Escrowed Encryption Standard) are widely used to solve the problem of communication over an insecure channel. With advanced technologies in computer hardware and software, these standards seem not to be as secure and fast as one would like. In
this paper we propose a encryption technique which provides security to both the message and the secret key achieving confidentiality and authentication. The Symmetric algorithm used has two advantages over traditional schemes. First, the encryption and decryption procedures are much simpler, and consequently, much faster. Second, the security level is higher due to the inherent poly-alphabetic nature of the substitution mapping method used here, together with the translation and transposition operations performed in the algorithm. Asymmetric algorithm RSA is worldwide known for its high security. In this paper a detailed report of the process is presented and analysis is done comparing our proposed technique with familiar techniques
Data Security with Colors using RSA technique that integrates the RGB Color model with the well-known public key cryptographic algorithm RSA (Rivest, Shamir and Adleman). This model provides both confidentiality and authentication to the data sent across the network. RSA algorithm uses public key and private key to encrypt and decrypt the data and thus provides confidentiality. But the public key is known to everyone and so anyone can encrypt the data and send the message. Hence authentication of users is needed. In this technique we use RGB color model to provide authentication. Every user will have a unique color assigned to him. A sender must know the receiver’s color to send a message. The color value is encrypted using a key which is used as a password while decrypting the message. To decrypt the message, the receiver must provide his color values. If the decrypted color values and his color values are equal then the sender and receiver are send to be authentic. The data encryption and decryption follows RSA procedure. Thus both authentication and confidentiality are provided for the data.
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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