Learning Pseudorandom Numbers with Transformers: Permuted Congruential Generators, Curricula, and Interpretability

TL;DR

This paper demonstrates that Transformer models can learn to predict sequences generated by complex pseudo-random number generators, revealing insights into their structure, scalability, and interpretability through extensive experiments and analysis.

Contribution

It shows that Transformers can effectively learn PCG sequences, including large moduli, and introduces a curriculum learning approach and interpretability insights into the model's internal representations.

Findings

Transformers successfully predict PCG sequences beyond classical attack capabilities.

Prediction accuracy scales with the square root of the modulus size.

Embedding analysis reveals clustering based on bitwise rotations.

Abstract

We study the ability of Transformer models to learn sequences generated by Permuted Congruential Generators (PCGs), a widely used family of pseudo-random number generators (PRNGs). PCGs introduce substantial additional difficulty over linear congruential generators (LCGs) by applying a series of bit-wise shifts, XORs, rotations and truncations to the hidden state. We show that Transformers can nevertheless successfully perform in-context prediction on unseen sequences from diverse PCG variants, in tasks that are beyond published classical attacks. In our experiments we scale moduli up to $2^{22}$ using up to $50$ million model parameters and datasets with up to $5$ billion tokens. Surprisingly, we find even when the output is truncated to a single bit, it can be reliably predicted by the model. When multiple distinct PRNGs are presented together during training, the model can jointly learn them, identifying structures from different permutations. We demonstrate a scaling law with modulus $m$: the number of in-context sequence elements required for near-perfect prediction grows as $\sqrt{m}$. For larger moduli, optimization enters extended stagnation phases; in our experiments, learning moduli $m \geq 2^{20}$ requires incorporating training data from smaller moduli, demonstrating a critical necessity for curriculum learning. Finally, we analyze embedding layers and uncover a novel clustering phenomenon: the top principal components spontaneously group the integer inputs into bitwise rotationally-invariant clusters, revealing how representations can transfer from smaller to larger moduli.