arXiv subjects

Rémy Tuyéras

Publications and source records attributed to Rémy Tuyéras.

7 recordsLinked to original sources

arXiv2025

Constructing a fully homomorphic encryption scheme with the Yoneda Lemma

This paper redefines the foundations of asymmetric cryptography's homomorphic cryptosystems through the application of the Yoneda Lemma. It demonstrates that widely adopted systems, including ElGamal, RSA, Benaloh, Regev's LWE, and NTRUEncrypt, are directly derived from the principles of the Yoneda Lemma. This synthesis leads to the creation of a holistic homomorphic encryption framework, the Yoneda Encryption Scheme. Within this framework, encryption is modeled using the bijective maps of the Yoneda Lemma Isomorphism, with decryption following naturally from the properties of these maps. This unification suggests a conjecture for a unified model theory framework, offering a foundation for reasoning about both homomorphic and fully homomorphic encryption (FHE) schemes. As a practical demonstration, the paper introduces the FHE scheme ACES, which supports arbitrary finite sequences of encrypted multiplications and additions without relying on conventional bootstrapping techniques for ciphertext refreshment. This highlights the practical implications of the theoretical advancements and proposes a new approach for leveraging model theory and forcing techniques in cryptography, particularly in the design of FHE schemes.

cs.CR↗

arXiv2023

Category theory for genetics II: genotype, phenotype and haplotype

The overarching goal of this paper is to solve the word problem for a class of idempotent commutative monoids whose elements model population haplotypes. More specifically, we design an algebraic framework in which it is possible to unravel population stratification relationships and infer linkage disequilibrium in terms of algebraic equations of haplotypes expressed in idempotent commutative monoids. We show how these relations can be used to clarify haplotype-phenotype associations through the consideration of intermediate phenotypes and genetic mechanisms such as segregation and homologous recombination. The present work paves the way for the implementation of combinatorial GWAS in the study of complex traits, and for a framework in which one can infer genetic variants interactions along with the corresponding regulatory circuitry. Throughout the paper, we formalize concepts such as genotypes, haplotypes and haplogroups, and model segregation and homologous recombination in the language of pedigrads (introduced in previous work). The benefit of using pedigrads is that they provide a computational framework in which one can reason about haplotype dynamics over several generations.

math.CT↗

arXiv2020

A category theoretical argument for causal inference

The goal of this paper is to design a causal inference method accounting for complex interactions between causal factors. The proposed method relies on a category theoretical reformulation of the definitions of dependent variables, independent variables and latent variables in terms of products and arrows in the category of unlabeled partitions. Throughout the paper, we demonstrate how the proposed method accounts for possible hidden variables, such as environmental variables or noise, and how it can be interpreted statistically in terms of $p$-values. This interpretation, from category theory to statistics, is implemented through a collection of propositions highlighting the functorial properties of ANOVA. We use these properties in combination with our category theoretical framework to provide solutions to causal inference problems with both sound algebraic and statistical properties. As an application, we show how the proposed method can be used to design a combinatorial genome-wide association algorithm for the field of genetics.

math.ST↗

arXiv2020

Category theory for genetics I: mutations and sequence alignments

The present article is the first of a series whose goal is to define a logical formalism in which it is possible to reason about genetics. In this paper, we introduce the main concepts of our language whose domain of discourse consists of a class of limit-sketches and their associated models. While our program will aim to show that different phenomena of genetics can be modeled by changing the category in which the models take their values, in this paper, we study models in the category of sets to capture mutation mechanisms such as insertions, deletions, substitutions, duplications and inversions. We show how the proposed formalism can be used for constructing multiple sequence alignments with an emphasis on mutation mechanisms.

math.CT↗

arXiv2019

Backprop as Functor: A compositional perspective on supervised learning

A supervised learning algorithm searches over a set of functions $A \to B$ parametrised by a space $P$ to find the best approximation to some ideal function $f\colon A \to B$. It does this by taking examples $(a,f(a)) \in A\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.

math.CT↗

arXiv2018

Category Theory for Genetics

We introduce a categorical language in which it is possible to talk about DNA sequencing, alignment methods, CRISPR, homologous recombination, haplotypes, and genetic linkage. This language takes the form of a class of limit-sketches whose categories of models can model different concepts of Biology depending on what their categories of values are. We discuss examples of models in the category of sets and in the category of modules over the Boolean semi-ring $\{0,1\}$. We identify a subclass of models in sets that models the genetic material of living beings and another subclass of models in modules that models haplotypes. We show how the two classes are related via a universal property.

math.CT↗

arXiv2017

Elimination of quotients in various localisations of premodels into models

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$Ω$-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allows one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak $(ω,n)$-categories, weak $ω$-groupoids and higher moduli stacks.

math.CT↗