[Paper Review] Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU

Bahi, Jacques M., Raphaël Couturier+2 more|arXiv (Cornell University)|Dec 22, 2011

Chaos-based Image/Signal Encryption2 references4 citations

TL;DR

This paper presents a GPU-accelerated pseudorandom number generator (PRNG) based on chaotic iterations that achieves high performance—up to 20 billion random numbers per second—while passing the stringent BigCrush statistical tests and maintaining cryptographic security when the input PRNG is secure. The method leverages Devaney’s chaos definition and a post-treatment via chaotic iterations to enhance statistical quality and security without sacrificing speed.

ABSTRACT

In this paper we present a new pseudorandom number generator (PRNG) on graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations. It is firstly proven to be chaotic according to the Devaney's formulation. We thus propose an efficient implementation for GPU that successfully passes the BigCrush tests, deemed to be the hardest battery of tests in TestU01. Experiments show that this PRNG can generate about 20 billion of random numbers per second on Tesla C1060 and NVidia GTX280 cards. It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically secure. A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is finally proposed.

Motivation & Objective

• To design a fast, statistically robust, and cryptographically secure PRNG suitable for GPU architectures.
• To address the performance degradation often seen when parallelizing traditional PRNGs on GPUs.
• To formally prove that the proposed PRNG satisfies Devaney’s definition of chaos.
• To demonstrate that cryptographic security of the input PRNG is preserved under the proposed chaotic post-treatment.
• To adapt the Blum-Goldwasser cryptosystem using the new chaotic PRNG for enhanced security and efficiency.

Proposed method

• The PRNG is constructed by applying a chaotic iteration post-treatment to a base PRNG, such as an Xorshift generator, to induce topological chaos.
• The chaotic iteration process is defined by a state update rule that depends on both the current state and a bit extracted from the input PRNG.
• The method ensures Devaney’s chaos by satisfying sensitivity to initial conditions, topological transitivity, and dense periodic orbits.
• GPU implementation uses thread-level parallelism, with each thread generating a sequence of pseudorandom numbers via the chaotic iteration process.
• The algorithm is optimized for GPU memory access patterns and uses lightweight operations (XOR, bit shifts) to maximize throughput.
• Cryptographic security is preserved under the assumption that the input PRNG is cryptographically secure, as proven via reduction to the input generator’s security.

Experimental results

Research questions

• RQ1 Can a PRNG based on chaotic iterations be efficiently implemented on GPU while preserving statistical quality and chaos properties?
• RQ2 Does the chaotic iteration post-treatment preserve the cryptographic security of a cryptographically secure input PRNG?
• RQ3 Can the proposed GPU-based PRNG achieve high throughput (e.g., >10 billion numbers per second) without compromising statistical randomness?
• RQ4 How does the proposed PRNG perform against the most stringent statistical test battery, BigCrush?
• RQ5 Can the Blum-Goldwasser cryptosystem be enhanced by replacing the BBS generator with the proposed chaotic PRNG?

Key findings

• The proposed GPU-based PRNG generates approximately 20 billion pseudorandom numbers per second on Tesla C1060 and NVIDIA GTX280 GPUs.
• The PRNG successfully passes the BigCrush test battery, the most rigorous statistical test suite available in TestU01.
• The PRNG is mathematically proven to be chaotic under Devaney’s definition, satisfying sensitivity, transitivity, and dense periodic orbits.
• When the input PRNG is cryptographically secure, the resulting chaotic PRNG inherits this property, ensuring resistance to cryptanalytic attacks.
• A modified Blum-Goldwasser cryptosystem is proposed, replacing the BBS generator with the new chaotic PRNG, preserving security and enhancing performance.
• The method enables the construction of fast, secure, and chaotic PRNGs suitable for high-performance computing and cryptographic applications.