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Sophie Morier-Genoud
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We show that the classical algebra of quaternions is a commutative $Z_2timesZ_2timesZ_2$-graded algebra. A similar interpretation of the algebra of octonions is impossible.
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We address the problem of which planar sets can be drawn with a pencil and eraser. The pencil draws any union of black open unit disks in the plane $mathbb{R}^2$. The eraser produces any union of white open unit disks. You may switch tools as many times as desired. Our main result is that drawability cannot be characterized by local obstructions: A bounded set can be locally drawable, while not being drawable. We also show that if drawable sets are defined using closed unit disks the cardinality of the collection of drawable sets is strictly larger compared with the definition involving open unit disks.
Cryptography with quantum states exhibits a number of surprising and counterintuitive features. In a 2002 work, Barnum et al. argue that these features imply that digital signatures for quantum states are impossible (Barnum et al., FOCS 2002). In this work, we ask: can all forms of signing quantum data, even in a possibly weak sense, be completely ruled out? We give two results which shed significant light on this basic question. First, we prove an impossibility result for digital signatures for quantum data, which extends the result of Barnum et al. Specifically, we show that no nontrivial combination of correctness and security requirements can be fulfilled, beyond what is achievable simply by measuring the quantum message and then signing the outcome. In other words, only classical signature schemes exist. We then show a positive result: a quantum state can be signed with the same security guarantees as classically, provided that it is also encrypted with the public key of the intended recipient. Following classical nomenclature, we call this notion quantum signcryption. Classically, signcryption is only interesting if it provides superior performance to encypt-then-sign. Quantumly, it is far more interesting: it is the only signing method available. We develop as-strong-as-classical security definitions for quantum signcryption and give secure constructions based on post-quantum public-key primitives. Along the way, we show that a natural hybrid method of combining classical and quantum schemes can be used to upgrade a secure classical scheme to the fully-quantum setting, in a wide range of cryptographic settings including signcryption, authenticated encryption, and CCA security.
In this paper we aim to compare Kurepa trees and Aronszajn trees. Moreover, we analyze the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjaths theorem that asserts the same consistency from two inaccessible cardinals. We will show that our large cardinal assumption is optimal, i.e. if every Kurepa tree has an Aronszajn subtree then $omega_2$ is inaccessible in the constructible universe textsc{L}. Moreover, we prove it is consistent with ZFC that there is a Kurepa tree $T$ such that if $U subset T$ is a Kurepa tree with the inherited order from $T$, then $U$ has an Aronszajn subtree. This theorem uses no large cardinal assumption. Our last theorem immediately implies the following: assume $textrm{MA}_{omega_2}$ holds and $omega_2$ is not a Mahlo cardinal in $textsc{L}$. Then there is a Kurepa tree with the property that every Kurepa subset has an Aronszajn subtree. Our work entails proving a new lemma about Todorcevics $rho$ function which might be useful in other contexts.
Let $R$ be a commutative ring with identity. We define a graph $Gamma_{aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $sigma in aut$ such that $sigma(x)=y$. The idea is to apply graph theory to study orbit spaces of rings under automorphisms. In this article, we define the notion of a ring of type $n$ for $ngeq 0$ and characterize all rings of type zero. We also characterize local rings $(R,M) $ in which either the subset of units ($ eq 1 $) is connected or the subset $M- {0}$ is connected in $Gamma_{aut}(R)$.
We study the notion of $Gamma$-graded commutative algebra for an arbitrary abelian group $Gamma$. The main examples are the Clifford algebras already treated by Albuquerque and Majid. We prove that the Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over $mathbb{R}$ or $mathbb{C}$. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.
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