CRYPTANALYSIS ON RSA
Mrinalini, Bhoomika Luthra, Anupriya Dixit
Student, Department Of CSE
RKGITW, Ghaziabad
Student, Department Of CSE
RKGITW, Ghaziabad
Student, Department Of CSE
RKGITW, Ghaziabad
msaini727@gmail.com
luthrabhoomika@gmail.com
anupriyadixit252@gmail.com
Priyanka Mehta
Assistant Professor,Department Of CSE
RKGITW ,Ghaziabad
priyankamehta@rkgitw.edu.in
Abstract— The most widely developed public –
key cryptography system is the RSA. Mainly the
RSA used in the e-commerce and e-mail and
also used for both encryption and digital
signature. In this paper, the main methods used
in attacks against the RSA cryptosystem. This
paper describe the algorithm, mathematical
function and attacks and also defines the
countermeasures that can be used to prevent
many of the attacks .It include also cryptanalysis
means related study of breaking ciphers. It
provides the secure transmission of the messages
in the network.
(CRT) used for RSA against fault attacks [2]. This
theorem mainly breaks the practical attack .The
RSA is used for encryption, digital signatures and
key establishment. In fact, RSA is usually
providing the security of digital data .The study of
the mathematical techniques related to the security
of transmission. Cryptography is an important tool
in today’s information security. For decades,
cryptographers have analyzed the security of
cryptosystems by treating them as mathematical
entities and analyzing the properties of the
underlying cryptographic algorithm [2].
Type of the cryptography:
Keywords—RSA, Encryption, Decryption, Plain
text, Cipher test, Keys.
I.
INTRODUCTION
Cryptanalysis is the art and science of
analyzing information systems in order to study the
hidden aspects of the systems [1]. Cryptanalysis is
used to break cryptographic security systems and
gain access to the contents of encrypted messages.
Cryptanalysis is the complementary of the
Cryptography .Cryptography is the combination of
two words crypto + graphy crypto means secret
graphy means writing means secret writing.
Cryptography and RSA algorithms used for the key
generation. Protect Chinese remainder theorem
There are the two type of cryptography.
1. Symmetric – Key Cryptography
2. Public Key Cryptography
1. Symmetric – Key Cryptography- If users want to
securely communicate with each other, they must
share a key, which is used to both encrypt and
decrypt messages. Symmetric-key schemes are
usually fast. Key establishment is one of the main
problems. When considering a network of user
wishing to communicate securely, each pair of
user must share a secret key, which makes it
impractical for any medium-size network.
2. Public Key Cryptography-Each user has a key
pair (e, d), which consist of a public key e and a
private key d [4]. User A can make e publicly
available and keep d only secret any one can
encrypt information with e while only user A
decrypt it with d.
II.
7. Decryption- It is the processes in which cipher
text convert in to the plain text using the private
keyor Method of obtaining the encrypted message
back to its original form.
Plain Text
CRYPTOGRAPHY TERMINOLOGY
Cipher Text
1. Plaintext- The format of the data before being
encrypted. For example user A send the message
HELLO to the user B.HELLO is the actual form of
the data.
2. Cipher Text- The “scrambled” form of data after
being encrypted. For example if we add three in
each alphabet then makes the scrambled form is
KHOOR this is the scrambled form of the data.
3. Key- A secret value used during the encryption
and decryption process. There are two types of the
key


Encrypt
Key
Cipher Text
Decrypt
Plain Text
Public key
Private key
III.
HELLO WORLD
HELLO WORLD
#%GIUGRWTMN, S (?
RSA (RIVEST, SHAMIR, ADLEMAN)
They are the three inventors which develop the
RSA algorithm. RSA is most widely deployed
public key cryptosystem and is used for both
encryption and digital signature. It is implemented
in most web server and browsers and present in
most commercially available security product [4].
RSA ALGORITHM

Let p and q be two large prime numbers

Let n= p q be the modulus

Findф(n)=(p-1).(q-1)

Choose e such that it is relatively prime to
ф (n).

Choose d such that: e x d modф(n)=1

Public key is(n, e)

Private key is (d , n)
5. Private Key-private key is used when receiver
receive the message from the sender

To encrypt message M compute

C=M e mod N
6. Encryption- It is the processes in which plain
text convert in to the unreadable form using the
public key or Method of obtaining the original
message transfer in to its encrypted form.

To decrypt C compute

M=C d mod
ENCRYPTION
Key
DECRYTION
(Secret shared key)
Key
4. Public Key-public key is used when sender send
the message to the receiver.
IV.
ATTACK
It is hard to break the system but generally some
conditions due to attacker can break the system
such as if



Attacker knows the value of the secret
key.
Attacker knows the format of the plain
text.
Attacker knows the format of the cipher
text
Types Of Attack
In this an attacker typically uses some additional
information leaked by the implementation of the
RSA function or exploit faults in the
implementation [4] Attacks are usually applied
against smart cards and security tokens One usually
tries to reduce the amount of information leaked to
make it irrelevant to the adversary
6. Low Private Exponent Attack- RSA decryption
and signing are very compute intensive operation,
if some low power devices may want to use a d
instead of a random one in order to improve
performance.
1. Key Attack- The principle of the cryptography is
that the security of any cryptographic system based
on the secrecy of the private key. An attack on the
RSA cryptosystem due to the entire private
exponent d. So the Private exponent is consisting of
the large number of bytes because Attacker cannot
find the value of the private exponent.
However an attack due to M. Wiener shows that the
choice of small d can lead to a total break of the
system [4].
2.Message Attacks-When two person communicate
to each other .If Attacker know the scrambled form
of the message then he send same encrypt
message.
These attacks are monitoring the token’s power
consumption based on the fact that the power
consumption varies significantly during different
steps of the cryptographic operations
3. Timing Attack- Timing attacks against RSA were
introduced by P. Kocher in 1995 [4]. Timing
attacks establish the relation between the private
key and the runtime cryptographic operation. RSA
private operation consists of a modular
exponentiation. Modular exponentiation used the
private key d as exponent. If the private key is n
bits long, this consists of a loop running through
the bits of d, with at most 2n modular
multiplications. In each step the data is squared,
with the execution of a modular multiplication [3].
An attacker can recover bits of d one at a time, if a
low public exponent is used.
An attacker can recover the secret information.
4.Adaptive Chosen Cipher Text Attacks-In
1998, Daniel Bleichenbacher described the first
practical adaptive chosen cipher text attack, against
RSA-encrypted messages using the padding
scheme (a padding scheme randomizes and adds
structure to an RSA-encrypted message, so it is
possible to determine whether a decrypted message
is valid). Due to flaws with the padding scheme,
Bleichenbacher was able to practical attack against.
5. Side-Channel Analysis Attacks-A side-channel
attack using branch prediction analysis (BPA) has
been described [5]. Many processors used a branch
predictor to determine whether a conditional
branch in the instruction flow of a program is likely
to be taken or not. Branch prediction analysis
attacks use a spy process to discover (statistically)
the private key when processed with these
processors.Also known as implementation attacks
7. Power Analysis- The new form of attack on
smart cards and cryptographic tokens called power
analysis
Two types of power analysis:
Simple power analysis attack-Work by directly
observing system power consumption
Differentially power analysis attack- are more
power full using statistical analysis and error
correction techniques to extract information co
related to private keys .
8. Fault Analysis- Attacks work by exploiting
errors on key depended cryptography operations
.These errors can be random, latent
There are number of fault analysis attacks against
public key and symmetric key cryptographic
devices
LIPTON introduce RSA private operation is a very
compute intensive operation
RSA use a technique CRT for the improvement in
the performance
V.
CHINESE REMAINDER THEOREM
Around A.D 100, the Chinese mathematician suntsi solved the problem of finding those integers x
that leave remainder 2 when divided by three,
remainder 3 when divided by 5and remainder 2
when divided by 7. One such solution is x=23; all
solutions are the form 23+105k for arbitrary
integers k.
The Chinese remainder theorem provides a
correspondence between systems of equations
module a set of pair wise relatively prime module.
• CRT Algorithm – Gauss -If x = a1 (mod m1) = a2
(Mod m2) = …= a (mod mn) where m1, m2, mn
are relatively prime to each other.
• CRT Algorithm – Gauss where (modulo inverses
can be found using Extended Euclidean
Mi=∏𝒏𝒌=𝟏 𝒎k/mi
(Modulo inverses can be found using Extended
Euclidean Algorithm).
VII.
CONCLUSIONS
RSA cryptosystem is implemented in the most
popular security products and protocols used today
and can be seen as one of the basis for secure
communication in the internet
No devastating attack has ever been found and
most problem appear to be result of misuse of the
system, bad choice of parameters or flaws in
implementations
Years of research have probably increased the trust
the security community has on RSA
We have every reason to believe that it will remain
the most used public key algorithm for year to
come.
VIII.
REFRENCES
[1]Cryptanalysis\signalsanalysisNsa.gov.20090115 retrieves 2013-04-15.
[2]R. Anderson, M. Kuhn, \Low cost attacks on
tamper.
VI.
PREVENTION
Cryptanalysis of RSA provided us some attacks,
and users and developers must be aware when
working with RSA. Special attentions are: key
size, properties of parameters (primes, exponents),
and encoding and implementation details.
1. Key Size – As a rule, RSA keys should be large
enough so Attacker cannot attack the system.
Number of factors can use. Current standards
require a minimum length of 1024 bits for RSA
keys. If the value of the keys is too short then it
must be not used because Attacker easily identity
the value of the key. Factorization method and is
typically used to derive lower bounds for RSA key
sizes.RSA key should be low ,many of the current
standard required a minimum length of1024 bits for
RSA keys
2. Strong prime- Apart from a minimal length some
cryptography standards also require that the prime
used for RSA have some special properties in
particular, that (p-1) needs to have a large prime
factor, called strong primes
3. Public exponent - E is usually chosen to be either
3 or 2^ (16) +1=65537 .We believe 65537 is more
secure and used
4. Private exponent-For the typical 1024 bit key the
private exponent should be at least 300 bits long
5.Encoding- RSA algorithm apply to messages
without any kind of preprocessing known as raw
RSA offers a very week level of security and
should never be used so the message should always
be encoded prior to encryption or signing.
[3]H. Bar-El, H. Choukri, D. Naccache, M.
Tunstall, C.Whelan, \The sorcerer's apprentice
guide to faultattacks," Workshop on Fault
Detection and Tolerance in Cryptography, June
2004.
[4]D. Bleichenbacher, B. Kaliski and J. Staddon.
Recent Results on PKCS #1: RSA Encryption
Standard.RSALaboratories’BulletinNo.7,June1998.
ftp://ftp.rsasecurity.com/pub/pdfs/bulletn7.pdf. D.
Boneh. Twenty Years of Attacks on the RSA
Cryptosystem.