The Wiener's attack, named after cryptologist Michael J. Wiener, is a type of cryptographic attack against RSA. The attack uses the continued fraction method to expose the private key d when d is small.
Background on RSA
Fictional characters Alice and Bob are people who want to communicate securely. More specifically, Alice wants to send a message to Bob which only Bob can read. First Bob chooses two primes p and q. Then he calculates the RSA modulus N = pq. This RSA modulus is made public together with the encryption exponent e. N and e form the public key pair (e,N). By making this information public, anyone can encrypt messages to Bob. The decryption exponent d satisfies , where denotes the Carmichael function, though sometimes , the Euler’s phi function, is used (note: this is the order of the multiplicative group , which is not necessarily a cyclic group). The encryption exponent e and also must be relatively prime so that there is a modular inverse. The factorization of N and the private key d are kept secret, so that only Bob can decrypt the message. We denote the private key pair as (d, N). The encryption of the message M is given by and the decryption of cipher text is given by (using Fermat's little theorem).
Using the Euclidean algorithm, one can efficiently recover the secret key d if one knows the factorization of N. By having the secret key d, one can efficiently factor the modulus of N.
Small private key
In the RSA cryptosystem, Bob might tend to use a small value of d, rather than a large random number to improve the RSA decryption performance. However, Wiener’s attack shows that choosing a small value for d will result in an insecure system in which an attacker can recover all secret information, i.e., break the RSA system. This break is based on Wiener’s Theorem, which holds for small values of d. Wiener has proved that the attacker may efficiently find d when .
Wiener's paper also presented some countermeasures against his attack that allow fast decryption. Two techniques are described as follows.
Choosing large public key: Replace by , where for some large of . When is large enough, i.e. , then Wiener’s attack can not be applied regardless of how small is.
Using the Chinese Remainder Theorem: Suppose one chooses d such that both and are small but itself is not, then a fast decryption of can be done as follows:
1. First compute and .
2. Use the Chinese Remainder Theorem to compute the unique value of which satisfies and . The result of satisfies as needed. The point is that Wiener’s attack does not apply here because the value of can be large.
How Wiener's attack works
there exists an integer K such that
Defining and , and substituting into the above gives:
Divided by :
- , where .
So, is slightly smaller than , and the former is composed entirely of public information. However, a method of checking and guess is still required. Assuming that (a reasonable assumption unless is large) the last equation above may be written as:
By using simple algebraic manipulations and identities, a guess can be checked for accuracy.
Let with . Let .
Given with , the attacker can efficiently recover .
Suppose that the public keys are
The attack shall determine .
By using Wiener's Theorem and continued fractions to approximate , first we try to find the continued fractions expansion of .
Note that this algorithm finds fractions in their lowest terms.
We know that
According to the continued fractions expansion of , all convergents are:
We can verify that the first convergent does not produce a factorization of . However, the convergent yields
Now, if we solve the equation
then we find the roots which are . Therefore we have found the factorization
Notice that, for the modulus , Wiener's Theorem will work if
Proof of Wiener's theorem
The proof is based on approximations using continued fractions.
Since , there exists a such that . Therefore
Let , note that if is used instead of , then the proof can be replaced with and replaced with .
Then multiplying by ,
Hence, is an approximation of . Although the attacker does not know , he may use to approximate it. Indeed, since
and , we have:
Using in place of we obtain:
Now, , so . Since , so , then we obtain:
Since and .
Hence we obtain:
Since then , we obtain:
- , so (2)
From (1) and (2), we can conclude that
If , then is a convergent of , thus appears among the convergents of . Therefore the algorithm will indeed eventually find .