## RSA release, the RSA has been analyzed for vulnerabilities.

RSA Attacks

Introduction

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Invented
by Ron Rivest, Adi Shamir, and Len Adleman, the RSA cryptosystem was first revealed
in the August 1977 issue of Scientific American. The RSA is most commonly used
for providing privacy and ensuring authenticity of digital data. RSA is used by
many commercial systems. It is used to secure web traffic, to ensure privacy and
authenticity of Email, to secure remote login sessions, and it is at the heart
of electronic credit-card payment systems.

Since
its initial release, the RSA has been analyzed for vulnerabilities. Twenty
years of research have led to a number of intriguing attacks, none of them is
devastating. They mostly show the danger of wrong use of RSA. RSA encryption in
its simple form is explained as follow.

Let,
N = p*q be the product of two large primes of the same size (n/2 bits
each).  A typical size for N is n=1024
bits, i.e.309 decimal digits.

Let
e, d be two integers satisfying

e*d
= 1 mod ? (N)

Where,
? (N) = (p-1) (q-1).

N = RSA modulus,

e = encryption exponent, and

d = called the decryption exponent.

The
pair (N, e) is the public key. The pair (N, d) is called the secret key and
only the recipient of an encrypted message knows it.

A
message M is encrypted by computing,

C =
Memod N.

To
decrypt the ciphertext C, the authentic receiver computes

Cdmod N.Cd = Med = M
(mod N)

The
last equality is based on Euler’s theorem.

Factoring Large Integers

This
is known as the first attack on RSA public key (N, e). After getting the
factorization of N, an attacker can easily construct ? (N), from which the
decryption exponent d = e-1mod? (N) can be found. Factoring the modulus is
referred to as brute-force attack. Although factorizing the modulus has been
improving, the current state of the art of this attack is unable to post a
threat to the security of RSA when RSA is used properly.

At
the moment RSA seems to be extremely secure. It has survived over 20 years of
scrutiny and is in widespread use throughout the world. The attack that is most
often considered for RSA is the factoring of the public key. If this can be
achieved, all messages written with the public key can be decrypted. The point
is that with very large numbers, factoring takes an unreasonable amount of time.
It has not been proven that breaking the RSA algorithm is equivalent to
factoring large numbers (there may be another, easier method), but neither has
it been proven that factoring is not equivalent.

Searching
the Message Space

One
of the seeming weaknesses of public key cryptography is that one has to give
away to everybody the algorithm that encrypts the data. If the message space is
small, then one could simply try to encrypt every possible message block, until
a match is found with one of the ciphertext blocks. In practice this would be
an insurmountable task because the block sizes are quite large.

Guessing d

Another
possible attack is a known ciphertext attack. This time the attacker knows both
the plaintext and ciphertext (they simply has to encrypt something). They then
try to crack the key to discover the private exponent, d. This might
involve trying every possible key in the system on the ciphertext until it returns
to the original plaintext. Once d has been discovered it is easy to find
the factors of n. Then the system has been broken completely and all
further ciphertexts can be decrypted.

The
problem with this attack is that it is slow. There are an enormous number of
possible ds to try. This method is a factorizing algorithm as it allows
us to factor n. Since factorizing is an intractable problem we know this
is very difficult. This method is not the fastest way to factorize n.
Therefore one is suggested to focus effort into using a more efficient
algorithm specifically designed to factor n.

Cycle Attack

This
attack is very similar to the last. The idea is to encrypt the ciphertext
repeatedly, counting the iterations, until the original text appears. This
number of re-cycles will decrypt any ciphertext. Again this method is very slow
and for a large key it is not a practical attack. A generalization of the
attack allows the modulus to be factored and it works faster the majority of
the time. But even this will still have difficulty when a large key is used.

Improvement
in algorithm is to use the public exponent of the public key to re-encrypt the
text. However any exponent should work so long as it is coprime to (p-1).(q-1)
(where p, q are factors of the modulus). So using an exponent such as 216
+ 1. This number has only two 1s in its binary representation. Using binary
fast exponentiation, we use only 16 modular squarings and 1 modular
multiplication. This is likely to be faster than the actual public exponent.
The trouble is that we cannot be sure that it is coprime to (p-1).(q-1).
In practice, many RSA systems use 216 + 1 as the encrypting exponent
for its speed.

In
the cycle attack, the encrypting exponent could be chosen to make
the system more efficient. Many RSA systems use e=3 to make encrypting
faster. However, there is a vulnerability with this attack. If the same message
is encrypted 3 times with different keys (that is same exponent, different
moduli) then we can retrieve the message. The attack is based on the Chinese
Remainder Theorem.

Common Modulus

The
assumption that generating the same modulus N = p*q for all users of a system,
and user is provided with a unique pair (ei, di) from which user I forms a
public key (N, ei) and a secret key (N, di) may seem to work providing that a
trusted central authority provides the unique pairs.

One
of the early weaknesses found was in a system of RSA where the users within an
organization would share the public modulus. That is to say, the administration
would choose the public modulus securely and generate pairs of encryption and
decryption exponents (public and private keys) and distribute them all the
employees/users. The reason for doing this is to make it convenient to manage
and to write software for. But, the resulting system is insecure since Bob who
is unable to decipher Alice’s cipher due to not having Alice private key d Alice
he however, can factor N using his own exponents.

With
blinding eavesdrop  obtain a valid
signature on a message of his choice by asking user to sign a random
“blinded message”. In that case, user 1 does not know what message is
actually signing and most signature schemes apply a “one-way hash” to
the message prior to signing, thus the attack is not a serious concern. Let (N,
d) be user 1’s  private key and (N, e) be
public key. Assume that an adversary wants user 1’s  signature on a message M ? Z*N. Being a smart move,
user 2 should refuse to sign M. Otherwise eavesdrop can compute S = S’ / ? mod
N and obtains user 1’s signature S on the original M.

Thus,

Se =
(S’)e/ ?e= (M’)ed / ?e ? M’ / ?e= M (mod N)

Faulty Encryption

Consider
channel used by Alice and Bob. In other words, Malory can listen to anything
that is transmitted, and can also change what is transmitted. Alice wishes to
talk privately to Bob, but does not know his public key. She requests by
sending an email, to which Bob replies. But during transmission, Malory is able
to see the public key and decides to flip a single bit in the public exponent
of Bob, changing (e,n) to (e’,n). When Alice receives the faulty key, she encrypts
the prepared message and sends it to Bob (Malory also gets it). But of course,
Bob cannot decrypt it because the wrong key was used. So he lets Alice know and
they agree to try again, starting with Bob re-sending his public key. This time
Malory does not interfere. Alice sends the message again, this time encrypted
with the correct public key.

Malory
now has two ciphertexts, one encrypted with the faulty exponent and one with
the correct one. She also knows both these exponents and the public modulus.
Therefore she can now apply the common modulus attack to retrieve Alice’s
message, assuming that Alice was foolish enough to encrypt exactly the same
message the second time.

4.1Coppersmith
theorem

The
most powerful attacks on low public exponent RSA are based on a Copper-smith
theorem.

The
theorem provides an algorithm for efficiently finding all roots of f modulo N that are less than X = N1/d.
The algorithm’s running time decreases as X gets smaller. The strength of this theorem
is its ability to find small roots of polynomials modulo a composite N.

Application
of Coppersmith’s Theorem:

·
Attack
stereotyped messages in RSA (sending messages whose difference is less than N1/e
can compromise RSA)

·
Security
proof of RSA-OAEP (constructive security proof).

·
Affine

·
Polynomially
related RSA messages (sending the same message to multiple recipients)

·
Factoring
N = p*q if the high bits of pare known.

·
An
algorithm that can get the private key for RSA in deterministic polynomial time
can be used to factor N in deterministic polynomial time.

·
Finding
integers with a large smooth factoring a proscribed interval.·Finding roots of modular multivariate polynomials.

4.2

The
first application of Coppersmith’s theorem and the improvement to an old attack

Suppose,
Bob wishes to send an encrypted message M to a number of parties P1, P2,……Pk.
Each party has its own RSA key (Ni, ei). We assume M is less than all the Ni’s.
Idealistically, to send M, Bob encrypts it using each of the public keys and
sends out of the ith ciphertext to pi. An attacker Eve can
eavesdrop on the connection out of Bob’s sight and collect the k transmitted ciphertexts.

For
simplicity, suppose all public exponents ei, are equal to 3. A
simple arguments shows that Eve can recover M if k ? 3. Indeed, Bob obtains C1,
C2, C3,

Where,                            C1= M3mod
N1, C2= M3mod N2, C3= M3mod
N3.

Assume
that gcd(Ni, Nj) = 1 for all i ? j since otherwise Eve
can factor some of the Ni’s. Hence, applying the Chinese Remainder Theorem (CRT)
to C1, C2, C3 gives a C’ ?ZN1N2N3
satisfying C’ = M3mod N1N2N3. Since M is less than all the Ni’s, we
have M3 e. The attack is feasible only when a small e is
used. To stimulate Hastad’s result, if M is m bits long, Bob could send Mi=
i2m+ M to party Pi. Since Eve obtains encryptions of
different messages, he can’t mount the attack.

Unfortunately,
Hastad showed that this linear padding is insecure. In fact, he proved that
applying any fixed polynomial to the message prior to encryption does not
prevent the attack. Suppose that for each of the participants P1…
Pk, Bob has a fixed public polynomial fi ? ZNi x.
To broadcast a message M, Bob sends the encryption of fi (M) to party
Pi. By eavesdropping, Eve learns Ci= fi (M) ei
mod Ni for i =1… k. Hastad showed that if enough parties are
involved, Eve can recover the plaintext M from all the ciphertexts. In more
generality, Hastad proved that a system of univariate equations modulo
relatively prime composites, such as applying any fixed polynomial g1 (M)
= 0 mod Ni, could be solved if sufficiently many equations are provided. This
attack suggests that randomized padding should be used in RSA encryption.

4.4

Generally,
The Franklin-Reiter attack is considered to be an artificial attack because why
should Bob send Alice the encryption of related messages? Coppersmith
strengthened the attack and proved an important result on padding. Coppersmith
showed that if randomized padding suggested by Hastad is used improperly then RSA
encryption is not secure. A naive random padding algorithm might pad a plaintext
M by appending a few random bits to one of the ends. The following attack
points out the danger of such simplistic padding. Suppose Bob sends a
properly-padded encryption of M to Alice. An attacker, Eve, intercepts the ciphertext
and prevents it from reading its destination. Bob notice that Alice did not
respond to his message and decides to resend M to Alice. He randomly pads M and
transmits the resulting ciphertext. Eve now has two ciphertexts corresponding
to two encryptions of the same message using two different random pads.

Implementation
Attacks

5.1
Timing attacks

Let
us consider a smartcard that stores a private RSA key. An attacker may not be
able to see its content and expose the key. However, the precise time of
decryption the card takes can help an attacker find or discover the private
decryption exponent d. Repeated squaring algorithm can be used for this attack
which is explained as follow:

• Let d = dn, dn-1… d0
(binary of d)

• Set z = M and C = 1. For i=0 … n do:

• (1)
if di= 1 set C = C. z mod N

• (2)
z = z * z mod N

• C at the end has the value Mdmod
N

5.2
Random Faults

To
speed up the computation of Md mod N, Implementations of RSA
decryption and signatures frequently use the Chinese Remainder Theorem. Instead
of working modulo N, the signer Bob first computes the signatures modulo p and q
and then combines the result using the Chinese Remainder Theorem.

More
accurately,

Bob
first computes,

Cp= Mdpmod p and Cq=
Mdqmod q

Where dp= d mod (p -1) and dq=
d mod(q-1).

then C = T1Cp+ T2Cq(mod
N)

whereT1= { 1 mod p and T2=
{0 mod p 0 mod q } 1 mod q}

The
running time of the last CRT step is negligible compared to the two
exponentiations. p and q are half the length of  N. Since simple implementations of
multiplication take quadratic time, multiplication modulo p is four times
faster than modulo N. Furthermore, dp is half the length of d and consequently
computing Mdp mod p is eight times faster than computing Mdp
mod N. Overall signature time is thus reduced by a factor of four. Many RSA
implementations use this method to improve performance.

6 Conclusion

Twenty
years of research aimed to break the RSA produced some insightful attacks, but
no serious attack has been found yet. Currently, it appears that proper RSA
implementation can provide the required security in the digital world. Four
main classes of RSA attacks were found: (1) elementary attacks that show the
misuse of the system, (2) low private exponent to show how serious it gets when
a low private is used, (3) low public exponent attacks, and (4) attacks on the
RSA implementation.

Proper
use of RSA and properly padding a message before encryption can defeat the
explained attacks.

Reference:

https://www.utc.edu/center-information-security-assurance/pdfs/course-paper-5600-rsa.pdf

https://crypto.stanford.edu/~dabo/papers/RSA-survey.pdf

http://www.ijcsit.com/docs/Volume%202/vol2issue5/ijcsit2011020578.pdf

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