Question

Alice and Bob use RSA public key encryption in order to communicate between them. Trudy finds out that Alice and Bob shared one of the primes used to determine the number \(n\) of their public key pairs. In other words, Trudy found out that \(n_{a}=p_{a} \times q\) and \(n_{b}=p_{b} \times q\). How can Trudy use this information to break Alice's code?

Short answer

Trudy finds the GCD of \(n_a\) and \(n_b\) to get the shared prime \(q\), then factors \(n_a\) to find \(p_a\).

Step-by-step solution

Understand the Given Problem

Alice and Bob use RSA encryption, and their public keys are determined by products of primes: \(n_a = p_a \times q\) and \(n_b = p_b \times q\). Trudy knows that both Alice's and Bob's public keys share one prime number, \(q\). Her goal is to use this information to factor \(n_a\), thus breaking Alice's RSA encryption.

Identify Shared Prime

Given that both \(n_a\) and \(n_b\) share the prime number \(q\), Trudy only needs to find \(q\) to break both RSA encryptions. To find \(q\), Trudy will compute the greatest common divisor (GCD) of \(n_a\) and \(n_b\). The GCD will be \(q\) since it is the shared prime factor.

Calculate the GCD

Trudy computes the GCD of \(n_a\) and \(n_b\) using the Euclidean algorithm: \(q = \text{gcd}(n_a, n_b)\). Because \(q\) is a common factor, the GCD calculation directly yields \(q\).

Factor Alice's \(n_a\)

Once \(q\) is known, Trudy can easily factor \(n_a\) by dividing \(n_a\) by \(q\) to find \(p_a\): \(p_a = \frac{n_a}{q}\). With \(p_a\) and \(q\), Trudy can recompute Alice's private key and decrypt any message encrypted with Alice's public key.

Key concepts

RSA algorithm

The RSA algorithm is a basis of modern cryptographic security, utilized widely for securing sensitive data. This algorithm involves two main steps: key generation and encryption/decryption. To generate keys:

• Select two large prime numbers, say, \( p \) and \( q \).
• Compute their product \( n = p \times q \), which forms part of the public key.
• Calculate \( \phi(n) = (p-1)(q-1) \), known as Euler's totient function.
• Select a public exponent \( e \), such that \( 1 < e < \phi(n) \) and \( \, e \, \) is coprime to \( \phi(n) \).
• Determine the private key \( d \), where \( d = e^{-1} \mod \phi(n) \).

To encrypt data, convert the plaintext message to an integer \( m \), then compute the ciphertext \( c \) using \( c = m^e \mod n \). To decrypt, the formula \( m = c^d \mod n \) is used, reverting \( c \) back to the original message. RSA's security relies heavily on the difficulty of factoring the product of two large primes, \( n \).

prime factorization

Prime factorization is an essential part of RSA encryption, as it involves expressing a number as a product of prime numbers. For example, if you have a number \( n \), it can be written as \( n = p_1 \times p_2 \times \cdots \times p_k \), where each \( p \) is a prime factor.
Within RSA, the difficulty of breaking the encryption arises from the challenge of deducing these prime factors from \( n \). When large primes are selected for RSA, the product \( n \) becomes computationally expensive to factorize with current technology, thus ensuring the security.
In situations like in the exercise, knowing one shared prime factor \( q \) allows a bypass that simplifies the factorization immensely. Once \( q \) is known, algebraically solving for the remaining primes becomes straightforward, thereby compromising the security.

greatest common divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is a fundamental concept used to determine the largest positive integer that divides two numbers without leaving a remainder. In mathematical terms, the GCD \( \text{gcd}(a, b) \) of two integers \( a \) and \( b \) is the largest integer \( d \) such that \( d \vert a \) and \( d \vert b \).
In the context of the RSA problem, the GCD becomes a powerful tool when two public keys share a prime factor. By calculating the GCD of these keys, the shared prime factor is quickly identified, as is the case with \( q \) in the exercise. Since the GCD method reveals common divisors, it facilitates breaking down complex products into simpler components, paving the way for revealing the originally selected prime numbers.

Euclidean algorithm

The Euclidean algorithm is an efficient method for computing the GCD of two integers. The core idea is to apply a recursive process that reduces the problem size significantly at each step. Here's how it works:

• For two given integers \( a \) and \( b \), check if \( b \) is 0. If yes, then \( a \) is the GCD.
• If \( b eq 0 \), replace \( a \) by \( b \) and \( b \) by \( a \mod b \).
• Repeat the process until \( b \) becomes zero.

Through continual division and remainder operations, the Euclidean algorithm swiftly identifies the greatest divisor common to both numbers. In RSA encryption, when the public key products such as \( n_a \) and \( n_b \) are used, employing this algorithm can reveal the shared factor \( q \), effectively breaking the encryption if unprotected. This method highlights the elegance of mathematical techniques in solving seemingly cryptic encryption puzzles.