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In this paper we study the Product Partition Problem (PPP), i.e. we are given a set of $n$ natural numbers represented on $m$ bits each and we are asked if a subset exists such that the product of the numbers in the subset equals the…

Using Semaev's summation polynomials, we derive a new equation for the $\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size representation…

Let $\mathcal{E}/\mathbb{F}_q$ be an elliptic curve, and $P$ a point in $\mathcal{E}(\mathbb{F}_q)$ of prime order $\ell$. V\'elu's formulae let us compute a quotient curve $\mathcal{E}' = \mathcal{E}/\langle{P}\rangle$ and rational maps…

This paper presents an exact and explicit tensor-network equation for the search of nontrivial divisors of a composite integer, together with an algorithm for its computation. The proposed method is based on the MeLoCoToN approach, which…

For a prime $p{\,>\,}3$ and a supersingular elliptic curve $E$ defined over $\mathbb{F}_{p^2}$ with ${j(E)\notin\{0,1728\}}$, consider an endomorphism $\alpha$ of $E$ represented as a composition of $L$ isogenies of degree at most $d$. We…

We show that given the order of a single element selected uniformly at random from $\mathbb Z_N^*$, we can with very high probability, and for any integer $N$, efficiently find the complete factorization of $N$ in polynomial time. This…

The goal of this paper is to develop a numerical algorithm that solves a two-dimensional elliptic partial differential equation in a polygonal domain using tensor methods and ideas from isogeometric analysis. The proposed algorithm is based…

This article proposes a new method to inject backdoors in RSA and other cryptographic primitives based on the Integer Factorization problem for balanced semi-primes. The method relies on mathematical congruences among the factors of the…

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\sqrt{p} \cos…

We present a heuristic algorithm to compute the ideal class group, and a generator of a principal ideal in $\mathbb{Q}(\zeta_{p^s})$ in time $2^{O(n^{1/2+\varepsilon})}$ for $n:= deg(K)$ and arbitrarily small $\varepsilon$. This yields an…

Resolution and superposition are common techniques which have seen widespread use with propositional and first-order logic in modern theorem provers. In these cases, resolution proof production is a key feature of such tools; however, the…

Sparse optimization has seen its advances in recent decades. For scenarios where the true sparsity is unknown, regularization turns out to be a promising solution. Two popular non-convex regularizations are the so-called $L_0$ norm and…

Current algorithms for large-scale industrial optimization problems typically face a trade-off: they either require exponential time to reach optimal solutions, or employ problem-specific heuristics. To overcome these limitations, we…

The availability of working quantum computers has led to several proposals and claims of quantum advantage. In 2023, this has included claims that quantum computers can successfully factor large integers, by optimizing the search for nearby…

Quantum block encoding (QBE) is a crucial step in the development of most quantum algorithms, as it provides an embedding of a given matrix into a suitable larger unitary matrix. Historically, the development of efficient techniques for QBE…

We study recursive algorithm for computing DCT of lengths $N=q 2^m$ ($m,q \in \mathbb{N}$, $q$ is odd) due to C.W.Kok. We show that this algorithm has the same multiplicative complexity as theoretically achievable by the prime factor…

We study the problem of graph clustering where the goal is to partition a graph into clusters, i.e. disjoint subsets of vertices, such that each cluster is well connected internally while sparsely connected to the rest of the graph. In…

Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a "quantum attack" of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation…

This paper studies one of the best known quantum algorithms - Shor's factorisation algorithm - via categorical distributivity. A key aim of the paper is to provide a minimal set of categorical requirements for key parts of the algorithm, in…

The computation of the dominant eigenpair for symmetric positive semidefinite matrices is fundamental in numerical optimization. This work shifts the paradigm from the classical Rayleigh quotient to an unconstrained difference formulation,…