Related papers: Certifying rings of integers in number fields
Zero-knowledge circuits are sets of equality constraints over arithmetic expressions interpreted in a prime field; they are used to encode computations in cryptographic zero-knowledge proofs. We make the following contributions to the…
We suggest using Fully Homomorphic Encryption (FHE) to be used, not only to keep the privacy of information but also, to verify computations with no additional significant overhead, using only part of the variables length for verification.…
Over the past decade, machine learning has demonstrated impressive results, often surpassing human capabilities in sensing tasks relevant to autonomous flight. Unlike traditional aerospace software, the parameters of machine learning models…
We prove that the theory of all modules over the ring of algebraic integers is decidable.
Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to…
Local certification consists in assigning labels (called \emph{certificates}) to the nodes of a network to certify a property of the network or the correctness of a data structure distributed on the network. The verification of this…
A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…
Formal verification techniques based on computer algebra have proven highly effective for circuit verification. The circuit, given as an and-inverter graph, is encoded as a set of polynomials that automatically generates a Gr\"obner basis…
This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…
We explain the algebra needed to make sense of the log signature of a path, with plenty of examples. We show how the log signature can be calculated numerically, and explain some software tools which demonstrate it.
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
We present two verification protocols where the correctness of a "target" computation is checked by means of "trap" computations that can be efficiently simulated on a classical computer. Our protocols rely on a minimal set of noise-free…
Formal verification guarantees proof validity but not formalization faithfulness. For natural-language logical reasoning, where models construct axiom systems from scratch without library constraints, this gap between valid proofs and…
Quantum program generation demands a level of precision that may not be compatible with the statistical reasoning carried out in the inference of large language models (LLMs). Hallucinations are mathematically inevitable and not addressable…
Quantum computers promise to efficiently solve not only problems believed to be intractable for classical computers, but also problems for which verifying the solution is also considered intractable. This raises the question of how one can…
Coding theory is very useful for real world applications. A notable example is digital television. Basically, coding theory is to study a way of detecting and/or correcting data that may be true or false. Moreover coding theory is an area…
We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several equivalent statements} borrowed from algebra, real analysis, general topology, and non-standard analysis. We also…
If $R$ is the ring of integers of a number field, then there exists a polynomial parametrization of the set $\text{SL}_2(R)$, i.e., an element $A \in \text{SL}_2(\mathbb{Z}[x_1,\ldots,x_n])$ such that every element of $\text{SL}_2(R)$ is…