COMMONWEALTH ASSOCATION FOR
EDUCATION, ADMINISTRATION AND
MANAGEMENT
VOLUME 2 ISSUE 3
ISSN NO 2322- 0147
MARCH
2014
ANALYSIS OF RSA AND MSSRSA USING
KEYFACTORING ALGORITHM
Excellence International Journal of Education and
Research (Multi- subject journal)
Excellence International Journal Of Education And Research
VOLUME 2
ISSUE 3
ISSN 2322-0147
ANALYSIS OF RSA AND MSSRSA USING
KEYFACTORING ALGORITHM
S.Suganya,
PG-Scholar ME (CSE)-2ndyear,
Department of computer science and engineering
Kathir College of Engineering,
Neelambur,Coimbatore.
Email id: suganya19.cse@gmail.com
Timna.P.Elizabeth,
PG-Scholar ME (CSE)-2ndyear,
Department of computer science and engineering
Kathir College of Engineering,
Neelambur,Coimbatore.
timna615@gmail.com,
Kabil Dev
PG-Scholar ME (CSE)-2ndyear,
Department of computer science and engineering
Kathir College of Engineering,
Neelambur,Coimbatore.
kabil444dev@gmail.com
ABSTRACT
RSA is the asymmetric cryptography system. The security of RSA public key cryptosystem is based
on the assumption that factoring of a large number (modulus) is difficult. In RSA if one can factor modulus
into its prime numbers then the private key is also detected and hence the security of the cryptosystem is
broken. The Subset-Sum cryptosystem (Knapsack Cryptosystem) is also an asymmetric cryptographic
technique. The Merkle-Hellman system is based on the subset sum problem (a special case of the knapsack
problem): given a list of numbers and a third number, which is the sum of a subset of these numbers,
determine the subset. In general, this problem is known to be NP-complete. However, if the set of numbers
(called the knapsack) is super increasing, that is, each element of the set is greater than the sum of all the
numbers before it, the problem is 'easy' and solvable in polynomial time with a simple greedy algorithm. So a
Modified Subset-Sum over RSA Public key cryptosystem (MSSRPKC) is presented which is secure against
Mathematical and brute-force attacks on RSA as well as Shamir attacks. This project also presents
comparison between MSSRPKC and RSA cryptosystems in respect of security and performance.
1. INTRODUCTION
Wireless network refers to any type of computer network that is not connected by cables of any kind. It is a
method by which homes, telecommunication network and enterprise (business) installations avoid the costly process
of introducing cables into a building, or as a connection between various equipment locations. Wireless
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telecommunications networks are generally implemented and administered using a transmission system called radio
waves. This implementation takes place at the physical level (layer) of the OSI model network structure.
In the security threats of WLAN despite the productivity, convenience and cost advantage that WLAN
offers, the radio waves used in wireless networks create a risk where the network can be hacked. This section
explains three examples of important threats: Denial of Service, Spoofing, and Eavesdropping.
Denial of Service is a kind of attack; the intruder floods the network with either valid or invalid messages
affecting the availability of the network resources. Due to the nature of the radio transmission, the WLAN are very
vulnerable against denial of service attacks. The relatively low bit rates of WLAN can easily be overwhelmed and
leave them open to denial of service attacks. By using a powerful enough transceiver, radio interference can easily
be generated that would enable WLAN to communicate using radio path. 2F94 998D FDB5 DE3D F8B5 06E4
A169 Spoofing and Session Hijacking is a type of attacker could gain access to privileged data and resources in the
network by assuming the identity of a valid user. This happens because 802.11 networks do not authenticate the
source address, which is Medium Access Control (MAC) address of the frames. Attackers may therefore spoof
MAC addresses and hijack sessions. Moreover, 802.11 do not require an Access Point to prove it is actually an AP.
This facilitates attackers who may masquerade as AP’s. In eliminating spoofing, proper authentication and access
control mechanisms need to be placed in the WLAN.
Eavesdropping involves attack against the confidentiality of the data that is being transmitted across the
network. By their nature, wireless LANs intentionally radiates network traffic into space. This makes it impossible
to control who can receive the signals in any wireless LAN installation. In the wireless network, eavesdropping by
the third parties is the most significant threat because the attacker can intercept the transmission over the air from a
distance, away from the premise of the company.
CRYPTOGRAPHY IN WIRELESS NETWORK
Cryptography is the science of writing in secret code and is an ancient art; the first
documented use of cryptography in writing dates back to circa 1900B.C. when an Egyptian scribe used non-standard
hieroglyphs in an inscription. Some experts argue that cryptography appeared spontaneously sometime after writing
was invented, with applications ranging from diplomatic missives to war-time battle plans. It is no surprise, then,
that new forms of cryptography came soon after the widespread development of computer communications. In data
and telecommunications, cryptography is necessary when communicating over any untrusted medium, which
includes just about any network, particularly the Internet. Within the context of any application-to-application
communication, there are some specific security requirements, including:
Authentication: The process of proving one's identity. (The primary forms of host-to-host authentication on the
Internet today are name-based or address-based, both of which are notoriously weak.)
Privacy/confidentiality: Ensuring that no one can read the message except the intended receiver.
Integrity: Assuring the receiver that the received message not been altered in any way from the original.
Non-repudiation: A mechanism to prove that the sender really sent this message.
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Cryptography, then not only protects data from theft or alteration, but can also be used for user
authentication. There are, in general, three types of cryptographic schemes typically used to accomplish these goals:
secret key (or symmetric) cryptography, public-key (or asymmetric) cryptography, and hash functions, each of
which is described below. In all cases, the initial unencrypted data is referred to as plaintext. It is encrypted into
cipher text, which will in turn (usually) be decrypted into usable plaintext.
RSA Security Enhances Security for Wireless LAN Environments
RSA Security , the most trusted name in e-security®, today announced the availability of new
functionality for the industry-leading RSA Secured two factor authentication
solution that enables enhanced
protection in wireless local area network (WLAN) environments. Without the proper security measures in place,
wireless connectivity will continue to be a major risk until more robust authentication protocols are adopted. Now,
by securing WLANs with the proven solution for two-factor user authentication, organizations can realize the full
Benefits of WLANs – including cost reduction and productivity enhancement – while reducing the risk of exposing
mission-critical data and resources to unauthorized access.
WLAN is viewed as a critical business tool for many organizations because the technology
represents an ideal way to link users – of PCs, laptops, PDAs and other mobile devices – to the Internet and internal
networks without hard-wired connections. However, with the enhanced mobility of wireless networks also comes
serious security threats, because any confidential data that is flowing over these networks – including financial
transactions, credit card numbers and proprietary company information – can be easily exposed or compromised if
not properly secured. User authentication in a WLAN environment is typically done through the use of weak, static
passwords that are particularly vulnerable to hacking. This prevents organizations from using their WLANs for
critical business purposes, limiting their effectiveness. With the RSA SecurID solution, organizations can now
replace weak passwords with strong, two-factor user authentication protecting their WLANs environments for more
strategic or sensitive applications.
RSA Security has been actively working with leading vendors – such as Funk Software (Odyssey and
Steel-Belted Radius solutions) and Proxim (ORiNOCO solution) – who support the 802.1x standard to enable
seamless integration between RSA SecurID two-factor user authentication and their WLAN solutions. This
advancement allows their respective customers to take advantage of the functionality provided by RSA SecurID
software to protect data, resources and business applications. Through the RSA Secured® Partner Program, RSA
Security has worked with these vendors to test and certify their WLAN solutions, ensuring interoperability with
RSA SecurID two-factor authentication. To ensure customer success, implementation guides are available to provide
step-by-step instructions on deploying and configuring these popular WLAN solutions with RSA SecurID software.
The IEEE 802.1x standard was proposed to solve authentication challenges in wireless LANs. While
802.1x currently provides a way to move beyond the shared secret approach, the standard does not address the
serious problems resulting from a lack of access point and end user authentication leading to the well publicized
“man-in-the-middle” and session hijacking attacks. Written by Cisco, Microsoft and RSA Security and approved by
the IETF standards board, the new protocol called the Protected Extensible Authentication Protocol (PEAP) standard
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addresses both of these concerns. It is designed to easily plug into products based on 802.1x to provide both strong
user authentication and access point authentication. RSA Security is partnering with many of the key WLAN
vendors to enable them to use this protocol, as well as other protocols such as EAP-TTLS, in a way that
interoperates with RSA Security’s technologies and products, including RSA SecurID two-factor authentication.
RSA SecurID two-factor authentication is designed to provide a fast, simple and highly reliable way to verify a
user’s identity before granting access to a protected resource. Traditionally used to protect access to networks, Web
pages, VPNs, and business applications in a wired environment, RSA SecurID functionality has now been extended
to protect access to WLANs. With more than 13 million devices deployed, the RSA SecurID solution is the de facto
standard in two-factor authentication. The solution uses patented, time-synchronous technology to provide twofactor user authentication by combining a token or smart card with a secret PIN. RSA Security offers its
authenticators in multiple form factors including three hardware styles, software versions that run on PCs, PDAs and
mobile phones, and various smart card options.“Strong security is always the starting point for any wireless business
environment, and by securing wireless LAN environments with RSA SecurID user authentication software
leveraging new protocols like PEAP, wireless LAN vendors will be able to provide the products that allow
businesses to extend a higher level of trust to their wireless business processes without inconveniencing corporate
users or IT staff,” said Bill McQuaide, senior vice president of the Authentication division at RSA Security. “RSA
Security and its strategic partners are committed to developing these secure technologies and implementing strong
user authentication within WLAN infrastructures – a critical requirement for driving profitability and ensuring trust
in today’s economy.”
EXISTING RSA
RSA is an algorithm for public-key cryptography that is based on the presumed difficulty of factoring large
integers, the factoring problem. RSA stands for Ron Rivest, Adi Shamir and Leonard Adleman, who first publicly
described it in 1978. A user of RSA creates and then publishes the product of two large prime numbers, along with
an auxiliary value, as their public key. The prime factors must be kept secret. Anyone can use the public key to
encrypt a message, but with currently published methods, if the public key is large enough, only someone with
knowledge of the prime factors can feasibly decode the message. Whether breaking RSA encryption is as hard as
factoring is an open question known as the RSA problem.The RSA algorithm involves three steps of operations such
as: key generation, encryption and decryption.
Key generation
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. The keys
for the RSA algorithm are generated the following way:
1. Choose two distinct prime numbers p and q.
1a. For security purposes, the integer’s p and q should be chosen at random, and should be of similar bitlength. Prime integers can be efficiently found using a primality test.
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2. Compute n = pq.
2a. n is used as the modulus for both the public and private keys
3. Compute φ(n) = (p – 1)(q – 1), where φ is Euler's totient function.
4. Choose an integer e such that 1 < e < φ(n) and greatest common denominator of (e,φ(n)) = 1, i.e. e and
φ(n) are co prime.
4a. e is released as the public key exponent.
4b. e having a short bit-length and small Hamming weight results in more efficient encryption - most
commonly 0x10001 = 65537. However, small values of e (such as 3) have been shown to be less secure in some
settings.
5.Determine d = e–1 mod φ(n); i.e. d is the multiplicative inverse of e mod φ(n).
5a. This is more clearly stated as solve for d given (d*e)mod φ(n) = 1
5b.This is often computed using the extended Euclidean algorithm.
5c.d is kept as the private key exponent.
The public key consists of the modulus n and the public (or encryption) exponent e. The private key
consists of the modulus n and the private (or decryption) exponent d which must be kept secret.
Encryption
Alice transmits her public key (n,e) to Bob and keeps the private key secret. Bob then wishes to send
message M to Alice.
He first turns M into an integer m, such that 0 < m < n by using an agreed-upon reversible protocol known
as a padding scheme. He then computes the cipher text c corresponding to
c = me (mod n).
This can be done quickly using the method of exponentiation by squaring. Bob then transmits c to Alice.
Note that at least nine values of m will yield a cipher text c equal to m, But this is very unlikely to occur in
practice.
Decryption
Alice can recover m from c by using her private key exponent d via computing
m = cd (mod n).
Given m, she can recover the original message M by reversing the padding scheme.
PROPOSED ALGORITHM FOR FACTORING RSA KEYS
Sattar J Aboud and Mohammad A AL-Fayoumi,proposed a new algorithm for factoring the modulus. The
suggested algorithm aims to obtain the private key of the RSA scheme and then factoring the modulus based on the
public key e of the RSA scheme.The new idea claimed to be more efficient than the already existed algorithms
especially when the public key e is small, since most of public key encryption schemes select a small encryption
exponent e in order to improve the efficiency of encryption.A highly secure cryptography method by RSA Security,
Inc., Bedford, MA , a division of EMC Corporation since 2006. It uses a two-part key. The private key is kept by the
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owner; the public key is published.Data are encrypted by using the recipient's public key, which can only be
decrypted by the recipient's private key.
RSA is very computation intensive, thus it is often used to create a digital envelope, which holds an RSAencrypted DES key and DES-encrypted data. This method encrypts the secret DES key so that it can be transmitted
over the network, but encrypts and decrypts the actual message using the much faster DES algorithm. RSA is also
used for authentication by creating a digital signature. In this case, the sender's private key is used for encryption,
and the sender's public Key is used for decryption. The RSA algorithm is also implemented in hardware. As RSA
chips
get
faster,
RSA
encoding
and
decoding
add
less
overhead.
The secret method uses the same key to encrypt and decrypt. The problem is transmitting the key to the
recipient in order to use it. The public key method uses two keys: one kept private and never transmitted, while the
other is made public. Very often, the public key method is used to safely send the secret key to the recipient so that
the message can be encrypted using the faster secret key algorithm.
The mathematical details of the algorithm used in obtaining the public and private keys are available at
the RSA Web site. Briefly, the algorithm involves multiplying two large prime numbers (a prime number is a
number divisible only by that number and 1) and through additional operations deriving a set of two numbers that
constitutes the public key and another set that is the private key. Once the keys have been developed, the original
prime numbers are no longer important and can be discarded. Both the public and the private keys are needed for
encryption /decryption but only the owner of a private key ever needs to know it. Using the RSA system, the private
key never needs to be sent across the Internet.
The private key is used to decrypt text that has been encrypted with the public key. Thus, if I send you a
message, I can find out your public key (but not your private key) from a central administrator and encrypt a
message to you using your public key. When you receive it, you decrypt it with your private key. In addition to
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encrypting messages (which ensures privacy), you can authenticate yourself to me (so I know that it is really you
who sent the message) by using your private key to encrypt a digital certificate. When I receive it, I can use your
public key to decrypt it. A table might help us remember this.
ENCRYPTION PROCESSS:
Sender A does the following: • Obtains the recipient B's public key (n, e).
• Represents the plaintext message as a positive integer m.
• Computes the cipher text c = me mod n.
• Sends the cipher text c to B.
DECRYTION PROCESS:
Recipient B does the following: • Uses private key (n, d) to compute m = cd mod n.
•Extracts the plaintext from the message representative m.
PROPOSED MODIFIED SUBSETSUM RSA
The Merkle–Hellman knapsack cryptosystem was one of the earliest public key cryptosystems
invented by Ralph Merkle and Martin Hellman in 1978. The Subset-Sum cryptosystem (Knapsack Cryptosystem) is
also an asymmetric cryptographic technique. The Merkle-Hellman system is based on the subset sum problem (a
special case of the knapsack problem): given a list of numbers and a third number, which is the sum of a subset of
these numbers, determine the subset. In general, this problem is known to be NP-complete. However, if the set of
numbers (called the knapsack) is super increasing, that is, each element of the set is greater than the sum of all the
numbers before it, the problem is `easy' and solvable in polynomial time with a simple greedy algorithm.
In computer science, the subset sum problem is an important problem in complexity theory and
cryptography. The problem is this: given a set of integers, is there a non-empty subset whose sum is zero? For
example, given the set {−7, −3, −2, 5, 8}, the answer is yes because the subset { −3, −2, 5} sums to zero. The
problem is NP-complete. An equivalent problem is this: given a set of integers and an integer s, does any non-empty
subset sum to s? Subset sum can also be thought of as a special case of the knapsack problem. One interesting
special case of subset sum is the partition problem, in which s is half of the sum of all elements in the set.
KEY GENERATION PROCESS FOR
MODIFIED SUBSET SUM RSA CRYPTOSYSTEM
Generate two large random primes, p and q, of approximately equal size such that their product m= p
x q is of the required bit length, e.g. 1024 bits.
(From Big Integer library function of Java)
1. Compute m = p x q and φ = (p-1) x (q-1).
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2. Choose an integer e, satisfying 1 < e < φ, such that gcd (e, φ) = 1.
3. Compute the secret exponent d, 1 < d < φ, such that e x d ≡ 1 (mod φ).
4. Choose a super increasing set A = (a1, ..., an)
5. Choose an integer M with M > SUMi=1...n(ai). M is called the modulus.
6. Choose a multiplier W such that gcd(M, W) = 1 and 1 <= W < M This choice of W guarantees an
inverse element U: UW = 1 (mod M)
To get the components bi of the public key B, perform bi = ai*W mod M, i = 1 .n.The super increasing
property of A is concealed by modular multiplication.
The public key is (B, n, e) and the private key is (A, M, W,n, d). Keep all the values d, p, q and φ secret.
Public key is published for everyone and private key must be kept secret. Then by using these keys encryption and
decryption are performed.
ENCRYPTION:
To encrypt a message, a subset of the hard knapsack is chosen by comparing it with a set of bits (the
plaintext) equal in length to the key, and making each term in the public key that corresponds to a 1 in the plaintext
an element of the subset, while ignoring the terms corresponding to 0 terms in the plaintext. The elements of this
subset are added together and the resulting sum is the hypertext. The length of a message to be encrypted is fixed by
the parameter n prior to encryption; a possibly larger message p has to be divided into n-bit groups.
1. Let p = (p1, p2... pn) the message to be encrypted.
2. The cipher text c is obtained by computing c = b1p1+ b2p2 + ... + bnpn
3. Computes the cipher text c1 = ce mod n.
4. Sends the cipher text c1 to B.
DECRYPTION
Decryption is possible because the multiplier and modulus used to transform the easy, super
increasing knapsack into the public key can also be used to transform the number representing the cipher text into
the sum of the corresponding elements of the super increasing knapsack. Then, using a simple greedy algorithm, the
easy knapsack can be solved using O(n) arithmetic operations, which decrypts the message.
Recipient B does the following: 1. Uses private key, first compute m1 = c1d mod n.
2. First compute c' = Um1 mod M = W-1c mod M
Now solve (A, c'). Because A is super increasing, (A,c') is easily solvable.
Let X = (x1... xn) be the resulting vector and pi = xi and p = (p1... pn) is the plaintext.
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CONCLUSION AND FUTURE ENHANCEMENT
As RSA is a highly secure algorithm, the private key is detected in comparison with the RSA security. For
that the proposed Subset-Sum cryptosystem (Knapsack Cryptosystem) is asymmetric cryptographic techniques that
are based on the subset sum problem introduces an increased level of security and convenience that makes the
private keys not to be transmitted anymore. Thus in the future in order to increase the key size certain approaches
must be carried for increasing the process time and storage requirement.
REFERENCES
[1] Ron Rivest, Adi Shamir and Leonard Adleman “RSA Data Security” in 1982.
[2] In 1995 RSA sent a handful of people across the hall to found Digital Certificates International, better
known as
Verisign.
[3] “Security Dynamics” acquired “RSA Data Security” in July 1996 and DynaSoft AB in 1997.
[4] In January 1997 “ DES Challenges”Data Encryption Standard,In February 2001, “Xcert International,
Inc.,” ,In
August 2001, “Securant Technologies, Inc., “,In April 2006 it acquired PassMark Security.
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