Related papers: Cryptanalysis of PLWE based on zero-trace quadrati…

Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in…

Fully homomorphic encryption (FHE) allows an untrusted party to evaluate arithmetic cir- cuits, i.e., perform additions and multiplications on encrypted data, without having the decryp- tion key. One of the most efficient class of FHE…

We discuss the advantages and limitations of cyclotomic fields to have fast polynomial arithmetic within homomorphic encryption, and show how these limitations can be overcome by replacing cyclotomic fields by a family that we refer to as…

We establish necessary and sufficient conditions for a quadratic polynomial to be irreducible in the ring $Z[[x]]$ of formal power series with integer coefficients. For $n,m\ge 1$ and $p$ prime, we show that $p^n+p^m\beta x+\alpha x^2$ is…

Let $k$ be a perfect field of characteristic $p > 0$, and let $K = k((u))$ be the field of Laurent series over $K$. We study the skew polynomial ring $K[T, \Phi]$, where $\Phi$ is an endomorphism of $K$ that extends a Frobenius endomorphism…

By the classic results of Fricke and Klein, for every word $w$ in the free group $F(a,b)$ there exists a unique integer \it{trace polynomial} $f_w(x,y,z)\in Z[x,y,z]$ such that $Tr(w(A,B))=f_w(Tr A,Tr B,Tr AB)$. for all $A,B\in SL(2,C)$. We…

Multiplicative relations between the roots of a polynomial in $\mathbb{Q}[x]$ have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other…

Let $S$ be a domain and $R=S[t;\sigma,\delta]$ a skew polynomial ring, where $\sigma$ is an injective endomorphism of $S$ and $\delta$ a left $\sigma$ -derivation. We give criteria for skew polynomials $f\in R$ of degree less or equal to…

Quantum money is the cryptographic application of the quantum no-cloning theorem. It has recently been instantiated by Montgomery and Sharif (Asiacrypt '24) from class group actions on elliptic curves. In this work, we propose a concrete…

The Rank metric decoding problem is the main problem considered in cryptography based on codes in the rank metric. Very efficient schemes based on this problem or quasi-cyclic versions of it have been proposed recently, such as those in the…

Let U(L) be the enveloping algebra of a finite dimensional Lie algebra L over a field k of characteristic zero, Z(U(L)) its center and Sz(U(L)) its semicenter. A sufficient condition is given in order for Sz(U(L)) to be a polynomial algebra…

A rooted tree module (RTM) $M:=M(T,F)$ over a zero-relation algebra $\Lambda:=\mathcal KQ/\langle\rho\rangle$ over a field $\mathcal K$ is given by the data of a quiver morphism $F:T\to Q$ from a rooted tree $T$ (either with a source or a…

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials…

The Learning with Errors (LWE) problem is a hard math problem in lattice-based cryptography. In the simplest case of binary secrets, it is the subset sum problem, with error. Effective ML attacks on LWE were demonstrated in the case of…

We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater…

Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate…

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the discriminant valuation, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater than…

Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (\emph{absolutely irreducibles}) and irreducible elements where some…

We consider the problem of recovering (that is, interpolating) and identity testing of a "hidden" monic polynomial $f$, given an oracle access to $f(x)^e$ for $x\in{\mathbb F_q}$ (extension fields access is not permitted). The naive…