Motivated by cryptographic applications, we investigate two machine learning
approaches to modular multiplication: namely circular regression and a
sequence-to-sequence transformer model. The limited success of both methods
demonstrated in our results gives evidence for the hardness of tasks involving
modular multiplication upon which cryptosystems are based.

通过密码应用为动机，我们研究了两种机器学习方法用于模块乘法：即循环回归和序列到序列变换器模型。我们结果中两种方法都取得的有限成功为基于模块乘法的密码系统中任务的困难性提供了证据。

模块乘法的机器学习

Machine learning for modular multiplication

Circular variables arise in a multitude of data-modelling contexts ranging
from robotics to the social sciences, but they have been largely overlooked by
the machine learning community. This paper partially redresses this imbalance
by extending some standard probabilistic modelling tools to the circular
domain. First we introduce a new multivariate distribution over circular
variables, called the multivariate Generalised von Mises (mGvM) distribution.
This distribution can be constructed by restricting and renormalising a general
multivariate Gaussian distribution to the unit hyper-torus. Previously proposed
multivariate circular distributions are shown to be special cases of this
construction. Second, we introduce a new probabilistic model for circular
regression, that is inspired by Gaussian Processes, and a method for
probabilistic principal component analysis with circular hidden variables.
These models can leverage standard modelling tools (e.g. covariance functions
and methods for automatic relevance determination). Third, we show that the
posterior distribution in these models is a mGvM distribution which enables
development of an efficient variational free-energy scheme for performing
approximate inference and approximate maximum-likelihood learning.

本文介绍了一些标准概率建模工具在圆形区域中的扩展，包括引入了一个新的多元分布，称为多元 GvM 分布；提出了一个受高斯过程启发的圆形回归概率模型和一种进行圆形隐变量的概率主成分分析方法。此外，我们展示了这些模型中的后验分布是 mGvM 分布，可以使用基于变分自由能的近似推断和近似最大似然学习方法。