An Efficient Binary Tree Algorithm for Batch GCD Computation to Identify Shared Prime Factors in RSA Keys

Core Concepts

This study proposes a novel binary tree-based batch GCD algorithm that is more efficient than the current best batch GCD algorithm (the remainder tree batch GCD algorithm) for identifying shared prime factors in a set of RSA public keys.

Abstract

The study focuses on the problem of efficiently processing and analyzing a set of RSA public keys to identify any shared prime factors, which can be exploited in a batch GCD attack to break the security of the RSA implementation.
The key highlights and insights are:

RSA security relies on the hardness of integer factorization, and a batch GCD attack can efficiently recover shared prime factors between RSA keys, breaking the security.
The proposed binary tree batch GCD algorithm is more efficient than the current best remainder tree batch GCD algorithm, with a roughly 6x speedup in practice.
The binary tree approach recursively performs GCD operations on the product tree of the input RSA moduli, extracting the product of all common divisors. This is followed by enumerating the GCD between this product and each input modulus to recover the shared factors.
Asymptotically, the binary tree and remainder tree batch GCD algorithms have similar complexity, but the binary tree approach avoids the overhead of the remainder tree computation.
Computational complexity analysis and timing results on simulated datasets of RSA moduli with shared prime factors demonstrate the performance advantages of the binary tree batch GCD algorithm.

Stats

The number of RSA moduli being analyzed ranges from 5,000 to 120,000.
The number of RSA moduli with shared prime factors is varied between 2, 100, and 1000.
The RSA key sizes used are 1024 bits and 2048 bits.

Quotes

"The binary tree batch GCD algorithm has better runtime than the existing remainder tree batch GCD algorithm, although asymptotically it has nearly identical scaling and its complexity is dependent on how many shared prime factors exist in the set of RSA keys."
"In practice, the implementation of the proposed binary tree batch GCD algorithm has a roughly 6x speedup compared to the standard remainder tree batch GCD approach."

Key Insights Distilled From

An Efficient All-to-All GCD Algorithm for Low Entropy RSA Key Factorization

by Elijah Pelof... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03166.pdf

Deeper Inquiries

How could the binary tree batch GCD algorithm be further optimized, for example by reducing the size of the product of shared factors during the GCD enumeration

To further optimize the binary tree batch GCD algorithm, one approach could involve reducing the size of the product of shared factors during the GCD enumeration in Step 3. By dynamically updating the aggregate shared factor integer B as non-trivial divisors are found, the computational cost of subsequent GCD operations can be minimized. This optimization would involve dividing out the discovered divisors from B as they are identified, ensuring that unnecessary computations are avoided. This adjustment would streamline the process and potentially enhance the algorithm's efficiency, especially in scenarios where multiple shared factors are present within the RSA moduli set.

What are the potential limitations or edge cases of the binary tree batch GCD approach, and how could it be extended to handle more complex scenarios involving repeated shared prime factors

The binary tree batch GCD approach, while efficient, may encounter limitations and edge cases when handling scenarios with repeated shared prime factors. In cases where an RSA modulus contains multiple factors shared with other moduli, additional computations may be required to uniquely identify and handle these repeated shared factors. To address this complexity, the algorithm could be extended by incorporating a mechanism to track and manage multiple instances of shared factors across different RSA keys. This extension could involve implementing a more sophisticated data structure or algorithm to differentiate and process repeated shared prime factors effectively, ensuring accurate factorization results without compromising efficiency.

Beyond RSA, what other cryptographic systems or applications could benefit from the efficient batch GCD computation enabled by the binary tree algorithm, and how could it be adapted for those use cases

Beyond RSA, the efficient batch GCD computation facilitated by the binary tree algorithm can benefit various cryptographic systems and applications that require factorization of large integers. Cryptosystems such as ElGamal, Diffie-Hellman, and DSA, which rely on the generation and utilization of large prime numbers, could leverage this algorithm for identifying shared prime factors within their key sets. By adapting the binary tree batch GCD approach to these systems, it becomes possible to detect vulnerabilities arising from weak keys with shared factors, enhancing overall security. Additionally, the algorithm can be applied to scenarios where public configurations of cryptographic systems need to be analyzed for shared prime factors, enabling comprehensive security evaluations and risk assessments in diverse cryptographic contexts.