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See detailOn basis vectors of lattices
Barthel, Jim Jean-Pierre UL; Müller, Volker UL

E-print/Working paper (n.d.)

Integer lattices enjoy increasing interest among mathematicians and cryptographers. However, there are still many elementary open questions, like finding specific vectors or particular bases of a given ... [more ▼]

Integer lattices enjoy increasing interest among mathematicians and cryptographers. However, there are still many elementary open questions, like finding specific vectors or particular bases of a given lattice. Our study consists in exhibiting which integer vectors may be chosen as basis vectors of a chosen lattice. The compelling part of our development is that this condition is obtained through an unusual application of Dirichlet's Theorem on primes in arithmetic progressions and that it has a surprising consequence for vectors achieving the successive minima. [less ▲]

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See detailA conjecture on primes in arithmetic progressions and geometric intervals
Barthel, Jim Jean-Pierre UL; Müller, Volker UL

E-print/Working paper (n.d.)

We conjecture that any interval of the form [q^t ,q^(t+1) ], where q≥ 2 and t≥1 denote positive integers, contains at least one prime from each coprime congruence class. We prove this conjecture first ... [more ▼]

We conjecture that any interval of the form [q^t ,q^(t+1) ], where q≥ 2 and t≥1 denote positive integers, contains at least one prime from each coprime congruence class. We prove this conjecture first unconditionally for all 2≤q≤45000 and all t≥1 and second under ERH for almost all q≥2 and all t≥2. Furthermore, we outline heuristic arguments for the validity of the conjecture beyond the proven bounds and we compare it with related long-standing conjectures. Finally, we discuss some of its consequences. [less ▲]

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See detailNew Constructions of Verifiable Delay Functions
Barthel, Jim Jean-Pierre UL; Naccache, David; Rosie, Razvan UL

E-print/Working paper (n.d.)

A Verifiable Delay Function (VDF) is a cryptographic protocol thought to provide a proof of elapsed time. At the core of such a protocol lies a sequential task whose evaluation cannot be accelerated, even ... [more ▼]

A Verifiable Delay Function (VDF) is a cryptographic protocol thought to provide a proof of elapsed time. At the core of such a protocol lies a sequential task whose evaluation cannot be accelerated, even in the presence of massive parallel computational resources. We introduce a novel sequentiality assumption, put forth a scheme that achieves this sequentiality constraint by requiring its users to evaluate a function over (levelled) fully homomorphic ciphertexts and provide a heuristic security analysis. [less ▲]

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See detailON THE (NON-)EQUIVALENCE OF INTEGRAL BINARY QUADRATIC FORMS AND THEIR NEGATIVE FORMS
Barthel, Jim Jean-Pierre UL; Müller, Volker UL

E-print/Working paper (n.d.)

Integral binary quadratic forms have been extensively studied in order to compute the class number of real and complex quadratic fields. Many studies restricted the equivalence class of quadratic forms or ... [more ▼]

Integral binary quadratic forms have been extensively studied in order to compute the class number of real and complex quadratic fields. Many studies restricted the equivalence class of quadratic forms or identified specific forms in order to compute the class number through enumeration of equivalence classes. One often used assumption is that a given form and its negative are equivalent or are synthetically identified. This document presents a concise Lagrangian approach to integral binary quadratic forms, outlines the equivalence classes in its broadest sense and uses it to develop a general condition when a given integral binary quadratic form and its negative are equivalent. Then, the existence of integral binary quadratic forms which are non-equivalent to their negatives is proven and an elementary algorithm to determine the (non-)equivalence of a given form and its negative is outlined. [less ▲]

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