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Pierre Guillot

Publications and source records attributed to Pierre Guillot.

At least 19 recordsLinked to original sources
arXiv2025

Introducing GRAFHEN: Group-based Fully Homomorphic Encryption without Noise

We present GRAFHEN, a new cryptographic scheme which offers Fully Homomorphic Encryption without the need for bootstrapping (or in other words, without noise). Building on the work of Nuida and others, we achieve this using encodings in groups. The groups are represented on a machine using rewriting systems. In this way the subgroup membership problem, which an attacker would have to solve in order to break the scheme, becomes maximally hard, while performance is preserved. In fact we include a simple benchmark demonstrating that our implementation runs several orders of magnitude faster than existing standards. We review many possible attacks against our protocol and explain how to protect the scheme in each case.

cs.CR↗
arXiv2024

The binary actions of simple groups of Lie type of characteristic 2

Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $\Gamma(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For $t\in \mathcal{C}$, we define the component group of $t$ to be the subgroup of $G$ generated by all vertices in $\Gamma(\mathcal{C})$ that lie in the connected component of the graph that contains $t$. We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic $2$. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic $2$ for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group.

math.GR↗
arXiv2024

The binary actions of simple groups with a single conjugacy class of involutions

We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph $\Gamma(\mathcal{C})$ based on the conjugacy class $\mathcal{C}$ of the group $G$, which was introduced in our previous work, and the notion of a strongly embedded subgroup of $G$. We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.

math.GR↗
arXiv2023

The binary actions of alternating groups

Given a conjugacy class $\mathcal{C}$ in a group $G$ we define a new graph, $\Gamma(\mathcal{C})$, whose vertices are elements of $\mathcal{C}$; two vertices $g,h\in \mathcal{C}$ are connected in $\Gamma(\mathcal{C})$ if $[g,h]=1$ and either $gh^{-1}$ or $hg^{-1}$ is in $\mathcal{C}$. We prove a lemma that relates the binary actions of the group $G$ to connectivity properties of $\Gamma(\mathcal{C})$. This lemma allows us to give a complete classification of all binary actions when $G=A_n$, an alternating group on $n$ letters with $n\geq 5$.

math.GR↗
arXiv2022

Homogeneous coherent configurations from spherical buildings and other edge-coloured graphs

We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build coherent configurations (also known as non-commutative association schemes). The condition we require is that the graph be endowed with a certain distance function, taking its values in the adjacency algebra (itself generated by the adjacency operators). When all the edges are of the same colour, the condition is equivalent to the graph being distance-regular, so our result is a generalization of the classical fact that distance-regular graphs give rise to association schemes. The Bose-Mesner algebra of the coherent configuration is then isomorphic to the adjacency algebra of the graph. The latter is more easily computed, and comes with a "small" set of generators, so we are able to produce examples of Bose-Mesner algebras with particularly simple presentations. When a group acts "strongly transitively", in a certain sense, on a graph, we show that a distance function as above exists canonically; moreover, when the graph is (the chamber system of) a building, we show that strong transitivity is equivalent to the usual condition involving pairs of incident chambers and apartments. We study affine planes in detail. These are not buildings, yet the machinery developed allows us to state and prove some results which are directly analogous to classical facts in the theory of projective planes (which {\em are} buildings). In particular, we prove that an affine plane with a group acting strongly transitively on it must be Desarguesian.

math.CO↗
arXiv2018

Four-fold Massey products in Galois cohomology

In this paper, we develop a new necessary and sufficient condition for the vanishing of 4-Massey products of elements in the mod-2 Galois cohomology of a field. This new description allows us to define a splitting variety for 4-Massey products, which is shown in the Appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such 4-Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.

math.NT↗
arXiv2017

Extensions of unipotent groups, Massey products and Galois cohomology

We study the vanishing of four-fold Massey products in mod p Galois cohomology. First, we describe a sufficient condition, which is simply expressed by the vanishing of some cup-products, in direct analogy with the work of Guillot, Mináč and Topaz for p=2. For local fields with enough roots of unity, we prove that this sufficient condition is also necessary, and we ask whether this is a general fact. We provide a simple splitting variety, that is, a variety which has a rational point if and only if our sufficient condition is satisfied. It has rational points over local fields, and so, if it satisfies a local-global principle, then the Massey Vanishing conjecture holds for number fields with enough roots of unity. At the heart of the paper is the construction of a finite group $\tilde U_5(\mathbb{F}_p)$, which has $U_5(\mathbb{F}_p)$ as a quotient. Here $U_n(\mathbb{F}_p)$ is the group of unipotent $n\times n$-matrices with entries in the field $\mathbb{F}_p$ with p elements; it is classical that $U_{n+1}(\mathbb{F}_p)$ is intimately related to n-fold Massey products. Although $\tilde U_5(\mathbb{F}_p)$ is much larger than $U_5(\mathbb{F}_p)$, its definition is very natural, and for our purposes, it is easier to study.

math.NT↗
arXiv2016

The Grothendieck-Teichmüller group of $PSL(2, q)$

We show that the Grothendieck-Teichmüller group of $PSL(2, q)$, or more precisely the group $GT_1(PSL(2, q))$ as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, when $q$ is even, we show that it is trivial. We explain how it follows that the moduli field of any "dessin d'enfant" whose monodromy group is $PSL(2, q)$ has derived length less than 4. This paper can serve as an introduction to the general results on the Grothendieck-Teichmüller group of finite groups obtained by the author.

math.GR↗
arXiv2015

The Grothendieck-Teichmüller group of a finite group and $G$-dessins d'enfants

For each finite group G, we define the Grothendieck-Teichmüller group of G, denoted GT(G), and explore its properties. The theory of dessins d'enfants shows that the inverse limit of GT(G) as G varies can be identified with a group defined by Drinfeld and containing the absolute Galois group of the rational field. We give in particular an identification of GT(G), in the case when G is simple and non-abelian, with a certain very explicit group of permutations that can be analyzed easily. With the help of a computer, we obtain precise information for G= PSL(2, q) when q= 4, 7, 8, 9, 11, 13, 16, 17, 19, and we treat A7, PSL(3, 3) and M11. In the rest of the paper we give a conceptual explanation for the technique which we use in our calculations. It turns out that the classical action of the Grothendieck-Teichmüller group on dessins d'enfants can be refined to an action on equivariant dessins, which we define, and this elucidates much of the first part.

math.GR↗
arXiv2015

Cayley graphs and automatic sequences

We study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For $2$-automatic sequences, we find a characterization in terms of what we call homogeneity, and among homogeneous sequences, we single out those enjoying what we call self-similarity. It turns out that self-similar $2$-automatic sequences (viewed up to a permutation of their alphabet) are in bijection with many interesting objects, for example dessins d'enfants (covers of the Riemann sphere with three points removed). For any $p$ we show that, in the case of an automatic sequence produced "by a Cayley graph", the group and indeed the automaton can be recovered canonically from the sequence. Further, we show that a rational fraction may be associated to any automatic sequence. To compute this fraction explicitly, knowledge of a certain graph is required. We prove that for the sequences studied in the first part, the graph is simply the Cayley graph that we start from, and so calculations are possible. We give applications to the study of the frequencies of letters.

math.CO↗
arXiv2014

An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that this absolute Galois group embeds into the Grothendieck-Teichmüller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_0$ by finite groups, and we hope to encourage computations in this area. Our treatment includes a result which has not appeared in the literature yet: the Galois action on the subset of regular dessins - that is, those exhibiting maximal symmetry -- is also faithful.

math.GR↗
arXiv2014

Milnor K-theory and the graded representation ring

Let F be a field, let G be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k_*(F) to the graded ring gr R(G, k) associated to Grothendieck's γ-filtration. We study this map in particular cases, as well as a related map involving the W-group of F rather than G. The latter is an isomorphism in all cases considered. Naturally this echoes the Milnor conjecture (now a theorem), which states that k_*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring gr W(F). The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the real K-theory ring K(X) of the topological space X, again with respect to the γ-filtration, and whose target is a certain subquotient of the mod 2 cohomology of X. In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.

math.KT↗
arXiv2014

Examples of Sweedler cohomology in positive characteristic

In this paper we provide a detailed calculation of the Sweedler cohomology of the algebra of functions on (Z/2)^r, in all degrees, over a field of characteristic 2. The result is strikingly different from the characteristic 0 analog. Then we show that there is a variant in characteristic p of the result obtained by Kassel and the author in characteristic zero, which provides a near-complete calculation of the second lazy cohomology group in the case of function algebras over a finite group.

math.QA↗
arXiv2012

A link invariant with values in the Witt ring

Using Maslov indices, we show the existence of oriented link invariants with values in the Witt rings of certain fields. Various classical invariants are closely related to this construction. We also explore a surprising connection with the Weil representation. This version is identical to the one accepted for publication in Quantum Topology.

math.AT↗
arXiv2011

Twisting algebras using non-commutative torsors

Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and relations for the algebras obtained by such twisting. We give a number of examples, including new constructions of the quantum affine spaces and the quantum tori.

math.QA↗
arXiv2009

Cohomology of invariant Drinfeld twists on group algebras

We show how to compute a certain group of equivalence classes of invariant Drinfeld twists on the algebra of a finite group G over a field k of characteristic zero. This group is naturally isomorphic to the second lazy cohomology group of the Hopf algebra of k-valued functions on G. When k is algebraically closed, the answer involves the group of outer automorphisms of G induced by conjugation in the group algebra as well as the set of all pairs (A, b), where A is an abelian normal subgroup of G and b is a k^*-valued G-invariant non-degenerate alternating bilinear form on the Pontryagin dual of A. We give a number of examples.

math.QA↗
arXiv2009

The computation of Stiefel-Whitney classes

The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here "compute" means to find a presentation in terms of generators and relations, and involves only the underlying (graded) ring. We propose a method to determine some of the extra structure: namely, Stiefel-Whitney classes and Steenrod operations. The calculations are explicitly carried out for about one hundred groups (the results can be consulted on the Internet). Next, we give an application: thanks to the new information gathered, we can in many cases determine which cohomology classes are supported by algebraic varieties.

math.AT↗