Coppersmith's attack

Coppersmith's attack describes a class of cryptographic attacks on the public-key cryptosystem RSA based on the Coppersmith method. Particular applications of the Coppersmith method for attacking RSA include cases when the public exponent e is small or when partial knowledge of a prime factor of the secret key is available.
RSA basics
The public key in the RSA system is a tuple of integers , where N is the product of two primes p and q. The secret key is given by an integer d satisfying ; equivalently, the secret key may be given by and if the Chinese remainder theorem is used to improve the speed of decryption, see CRT-RSA. Encryption of a message M produces the ciphertext , which can be decrypted using by computing .
Low public exponent attack
In order to reduce encryption or signature verification time, it is useful to use a small public exponent. In practice, common choices for are 3, 17 and 65537 . These values for e are Fermat primes, sometimes referred to as and respectively . They are chosen because they make the modular exponentiation operation faster. Also, having chosen such , it is simpler to test whether and while generating and testing the primes in step 1 of the key generation. Values of or that fail this test can be rejected there and then. (Even better: if e is prime and greater than 2, then the test can replace the more expensive test .)
If the public exponent is small and the plaintext is very short, then the RSA function may be easy to invert, which makes certain attacks possible.
Padding schemes ensure that messages have full lengths, but additionally choosing the public exponent is recommended. When this value is used, signature verification requires 17 multiplications, as opposed to about 25 when a random of similar size is used. Unlike low private exponent (see Wiener's attack), attacks that apply when a small is used are far from a total break, which would recover the secret key d.
The most powerful attacks on low public exponent RSA are based on the following theorem, which is due to Don Coppersmith.
Coppersmith method
Theorem 1 (Coppersmith)
Let N be an integer and be a monic polynomial of degree over the integers. Set for . Then, given , attacker (Eve) can efficiently find all integers satisfying . The running time is dominated by the time it takes to run the LLL algorithm on a lattice of dimension O with .
This theorem states the existence of an algorithm that can efficiently find all roots of modulo that are smaller than . As gets smaller, the algorithm's runtime decreases. This theorem's strength is the ability to find all small roots of polynomials modulo a composite .
Håstad's broadcast attack
The simplest form of Håstad's attack is presented to ease understanding. The general case uses the Coppersmith method.
Suppose one sender sends the same message in encrypted form to a number of people , each using the same small public exponent , say , and different moduli . A simple argument shows that as soon as ciphertexts are known, the message is no longer secure: Suppose Eve intercepts , and , where . We may assume for all (otherwise, it is possible to compute a factor of one of the numbers by computing .) By the Chinese remainder theorem, she may compute such that . Then ; however, since for all , we have . Thus holds over the integers, and Eve can compute the cube root of to obtain .
For larger values of , more ciphertexts are needed, particularly, ciphertexts are sufficient.
Generalizations
Håstad also showed that applying a linear padding to prior to encryption does not protect against this attack. Assume the attacker learns that for and some linear function , i.e., Bob applies a pad to the message prior to encrypting it so that the recipients receive slightly different messages. For instance, if is bits long, Bob might encrypt and send this to the -th recipient.
If a large enough group of people is involved, the attacker can recover the plaintext from all the ciphertext with similar methods. In more generality, Håstad proved that a system of univariate equations modulo relatively prime composites, such as applying any fixed polynomial , could be solved if sufficiently many equations are provided. This attack suggests that randomized padding should be used in RSA encryption.
Theorem 2 (Håstad)
Suppose are relatively prime integers and set . Let be polynomials of maximum degree . Suppose there exists a unique satisfying for all . Furthermore, suppose . There is an efficient algorithm that, given for all , computes .
Proof
Since the are relatively prime the Chinese remainder theorem might be used to compute coefficients satisfying and for all . Setting , we know that . Since the are nonzero, we have that is also nonzero. The degree of is at most . By Coppersmith’s theorem, we may compute all integer roots satisfying and . However, we know that , so is among the roots found by Coppersmith's theorem.
This theorem can be applied to the problem of broadcast RSA in the following manner: Suppose the -th plaintext is padded with a polynomial , so that . Then is true, and Coppersmith’s method can be used. The attack succeeds once , where is the number of messages. The original result used Håstad’s variant instead of the full Coppersmith method. As a result, it required messages, where .
Franklin–Reiter related-message attack
Franklin and Reiter identified an attack against RSA when multiple related messages are encrypted: If two messages differ only by a known fixed difference between the two messages and are RSA-encrypted under the same RSA modulus , then it is possible to recover both of them. The attack was originally described with public exponent , but it works more generally (with increasing cost as grows).
Let be Alice's public key. Suppose are two distinct messages satisfying for some publicly known polynomial . To send and to Alice, Bob may naively encrypt the messages and transmit the resulting ciphertexts . Eve can easily recover , given , by using the following theorem:
Theorem 3 (Franklin–Reiter)
Let be an RSA public key. Let satisfy for some linear polynomial with . Then, given , attacker (Eve) can recover in time quadratic in .
Proof
Since , we know that is a root of the polynomial . Similarly, is a root of . Hence, the linear factor divides both polynomials.
Therefore, Eve may calculate the greatest common divisor of and , and if the turns out to be linear, is found. The can be computed in quadratic time in and using the Euclidean algorithm.
Coppersmith’s short-pad attack
Like Håstad’s and Franklin–Reiter’s attacks, this attack exploits a weakness of RSA with public exponent . Coppersmith showed that if randomized padding suggested by Håstad is used improperly, then RSA encryption is not secure.
Suppose Bob sends a message to Alice using a small random padding before encrypting it. An attacker, Eve, intercepts the ciphertext and prevents it from reaching its destination. Bob decides to resend to Alice because Alice did not respond to his message. He randomly pads again and transmits the resulting ciphertext. Eve now has two ciphertexts corresponding to two encryptions of the same message using two different random pads.
Even though Eve does not know the random pad being used, she still can recover the message by using the following theorem, if the random padding is too short.
Theorem 4 (Coppersmith)
Let be a public RSA key, where is bits long. Set . Let be a message of length at most bits. Define and , where and are distinct integers with . If Eve is given and the encryptions of (but is not given or ), she can efficiently recover .
Proof
Define and . We know that when , these polynomials have as a common root. In other words, is a root of the resultant . Furthermore, . Hence, is a small root of modulo , and Eve can efficiently find it using the Coppersmith method. Once is known, the Franklin–Reiter attack can be used to recover and consequently .
See also
ROCA attack
References
Cryptographic attacks
Attacks on public-key cryptosystems