arXiv subjects

Junaid Hasan

Publications and source records attributed to Junaid Hasan.

3 recordsLinked to original sources

arXiv2026

Multiplication Beyond Groups: Stratified Fourier Mechanisms in Transformer Circuits

Transformers have demonstrated a remarkable ability to learn algorithmic reasoning, yet mechanistic analyses have mostly focused on globally invertible operations such as cyclic addition and group composition. In this work, we investigate how small transformers learn modular integer multiplication over composite moduli, a fundamentally non-invertible operation due to the presence of zero-divisors. We propose the monoid extension: a localized generalization of Group Composition via Representation (GCR) that suggests the learned computation does not rely on a single global representation space. Instead, the model partitions the input space into local hierarchical algebraic regions, where group-like structure survives and Fourier mechanisms can be applied. In transformers trained on square-free modular multiplication, we find that embeddings organize around these regions, attention exhibits class-sensitive routing and low-rank write directions, and local character features explain a large fraction of the model's output logits. Our results suggest that representation-theoretic mechanisms previously identified for group operations can extend beyond groups to more general structures.

cs.LG↗

arXiv2024

Integral aspects of Fourier duality for abelian varieties

We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas's work on integral Grothendieck-Riemann-Roch. If $S$ is smooth quasi-projective of dimension $d$ over a field and $\pi \colon X\to S$ is a $g$-dimensional abelian scheme, we prove, under very mild assumptions on $X/S$, that all classical results about Fourier duality, including the existence of a Beauville decomposition, are valid for the Chow ring $\mathrm{CH}(X;\Lambda)$ with coefficients in the ring $\Lambda = \mathbb{Z}[1/(2g+d+1)!]$. If $X$ admits a polarization $\theta$ of degree $\nu(\theta)^2$ we further construct an $\mathfrak{sl}_2$-action on $\mathrm{CH}(X;\Lambda_\theta)$ with $\Lambda_\theta = \Lambda[1/\nu(\theta)]$, and we show that $\mathrm{CH}(X;\Lambda_\theta)$ is a sum of copies of the symmetric powers $\mathrm{Sym}^n(\mathrm{St})$ of the $2$-dimensional standard representation, for $n=0,\ldots,g$. For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in $\mathrm{CH}^i(X;\Lambda_\theta)$ for every $i\in \{1,\ldots,g\}$.

math.AG↗

arXiv2022

Finding twin smooth integers by solving Pell equations

Any pair of consecutive B-smooth integers for a given smoothness bound B corresponds to a solution (x, y) of the equation x^2 - 2Dy^2 = 1 for a certain square-free, B-smooth integer D and a B-smooth integer y. This paper describes algorithms to find such twin B-smooth integers that lie in a given interval by using the structure of solutions of the above Pell equation. The problem of finding such twin smooth integers is motivated by the quest for suitable parameters to efficiently instantiate recent isogeny-based cryptosystems. While the Pell equation structure of twin B-smooth integers has previously been used to describe and compute the full set of such pairs for very small values of B, increasing B to allow for cryptographically sized solutions makes this approach utterly infeasible. We start by revisiting the Pell solution structure of the set of twin smooth integers. Instead of using it to enumerate all twin smooth pairs, we focus on identifying only those that lie in a given interval. This restriction allows us to describe algorithms that navigate the vast set of Pell solutions in a more targeted way. Experiments run with these algorithms have provided examples of twin B-smooth pairs that are larger and have smaller smoothness bound B than previously reported pairs. Unfortunately, those examples do not yet provide better parameters for cryptography, but we hope that our methods can be generalized or used as subroutines in future work to achieve that goal.

cs.CR↗