CyberSec Research

Ensuring software correctness remains a fundamental challenge in formal program verification. One promising approach relies on finding polynomial invariants for loops. Polynomial invariants are properties of a program loop that hold before and after each iteration. Generating polynomial invariants is a crucial task for loops, but it is an undecidable problem in the general case. Recently, an alternative approach to this problem has emerged, focusing on synthesizing loops from invariants. However, existing methods only synthesize affine loops without guard conditions from polynomial invariants. In this paper, we address a more general problem, allowing loops to have polynomial update maps with a given structure, inequations in the guard condition, and polynomial invariants of arbitrary form. In this paper, we use algebraic geometry tools to design and implement an algorithm that computes a finite set of polynomial equations whose solutions correspond to all loops satisfying the given polynomial invariants. In other words, we reduce the problem of synthesizing loops to finding solutions of polynomial systems within a specified subset of the complex numbers. The latter is handled in our software using an SMT solver.

Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote the subring of $\mathcal{R}$ generated by $g_1, \dots, g_n$, and let $h$ be an element of $\mathcal{S}$. Then, there exists a unique element ${f} \in \mathbb{K}[u_1, \dots, u_n]$ such that $h = f(g_1, \dots, g_n)$. In this paper, we provide an algorithm for computing ${f}$, given $h$ and $g_1, \dots, g_n$. The complexity of our algorithm is linear in the size of the input, $h$ and $g_1, \dots, g_n$, and polynomial in $n$ when the degree of $f$ is fixed. Previous works are mostly known when $f$ is a symmetric polynomial and $g_1, \dots, g_n$ are elementary symmetric, homogeneous symmetric, or power symmetric polynomials.

Let $m,n,d > 1$ be integers such that $n=md$. In this paper, we present an efficient change of level algorithm that takes as input $(B, \mathscr{M}, \Theta_{\mathscr{M}})$ a marked abelian variety of level $m$ over the base field $k$ of odd characteristic and returns $(B, \mathscr{M}^d, \Theta_{\mathscr{M}^d})$ a marked abelian variety of level $n$ at the expense of $O(m^g d^{2g})$ operations in $k$. A similar algorithm allows to compute $d$-isogenies: from $(B, \mathscr{M}, \Theta_{\mathscr{M}})$ a marked abelian variety of level $m$, $K\subset B[d]$ isotropic for the Weil pairing isomorphic to $(\mathbb{Z}/d\mathbb{Z})^g$ defined over $k$, the isogeny algorithm returns $(A, \mathscr{L}, \Theta_{\mathscr{L}})$ of level $m$ such that $A=B/K$ with $O(m^g d^g)$ operations in $k$. Our algorithms extend previous known results in the case that $d \wedge m=1$ and $d$ odd. In this paper, we lift theses restrictions. We use the same general approach as in the literature in conjunction with the notion of symmetric compatible that we introduce, study and link to previous results of Mumford. For practical computation, most of the time $m$ is $2$ or $4$ so that our algorithms allows in particular to compute $2^e$-isogenies which are important for the theory of theta functions but also for computational applications such as isogeny based cryptography.

In difference algebra, summability arises as a basic problem upon which rests the effective solution of other more elaborate problems, such as creative telescoping problems and the computation of Galois groups of difference equations. In 2012 Chen and Singer introduced discrete residues as a theoretical obstruction to summability for rational functions with respect to the shift and $q$-dilation difference operators. Since then analogous notions of discrete residues have been defined in other difference settings relevant for applications, such as for Mahler and elliptic shift difference operators. Very recently there have been some advances in making these theoretical obstructions computable in practice.

This paper introduces an algorithmic approach to the analysis of Jacobi stability of systems of second order ordinary differential equations (ODEs) via the Kosambi--Cartan--Chern (KCC) theory. We develop an efficient symbolic program using Maple for computing the second KCC invariant for systems of second order ODEs in arbitrary dimension. The program allows us to systematically analyze Jacobi stability of a system of second order ODEs by means of real solving and solution classification using symbolic computation. The effectiveness of the proposed approach is illustrated by a model of wound strings, a two-dimensional airfoil model with cubic nonlinearity in supersonic flow and a 3-DOF tractor seat-operator model. The computational results on Jacobi stability of these models are further verified by numerical simulations. Moreover, our algorithmic approach allows us to detect hand-guided computation errors in published papers.

We construct optimal secure coded distributed schemes that extend the known optimal constructions over fields of characteristic 0 to all fields. A serendipitous result is that we can encode \emph{all} functions over finite fields with a recovery threshold proportional to the complexity (tensor rank or multiplicative); this is due to the well-known result that all functions over a finite field can be represented as multivariate polynomials (or symmetric tensors). We get that a tensor of order $\ell$ (or a multivariate polynomial of degree $\ell$) can be computed in the faulty network of $N$ nodes setting within a factor of $\ell$ and an additive term depending on the genus of a code with $N$ rational points and distance covering the number of faulty servers; in particular, we present a coding scheme for general matrix multiplication of two $m \times m $ matrices with a recovery threshold of $2 m^{\omega } -1+g$ where $\omega $ is the exponent of matrix multiplication which is optimal for coding schemes using AG codes. Moreover, we give sufficient conditions for which the Hadamard-Shur product of general linear codes gives a similar recovery threshold, which we call \textit{log-additive codes}. Finally, we show that evaluation codes with a \textit{curve degree} function (first defined in [Ben-Sasson et al. (STOC '13)]) that have well-behaved zero sets are log-additive.

The lack of generalization in learning-based autonomous driving applications is shown by the narrow range of road scenarios that vehicles can currently cover. A generalizable approach should capture many distinct road structures and topologies, as well as consider traffic participants, and dynamic changes in the environment, so that vehicles can navigate and perform motion planning tasks even in the most difficult situations. Designing suitable feature spaces for neural network-based motion planers that encapsulate all kinds of road scenarios is still an open research challenge. This paper tackles this learning-based generalization challenge and shows how graph representations of road networks can be leveraged by using multidimensional scaling (MDS) techniques in order to obtain such feature spaces. State-of-the-art graph representations and MDS approaches are analyzed for the autonomous driving use case. Finally, the option of embedding graph nodes is discussed in order to perform easier learning procedures and obtain dimensionality reduction.

We consider dynamical models given by rational ODE systems. Parameter estimation is an important and challenging task of recovering parameter values from observed data. Recently, a method based on differential algebra and rational interpolation was proposed to express parameter estimation in terms of polynomial system solving. Typically, polynomial system solving is a bottleneck, hence the choice of the polynomial solver is crucial. In this contribution, we compare two polynomial system solvers applied to parameter estimation: homotopy continuation solver from HomotopyContinuation.jl and our new implementation of a certified solver based on rational univariate representation (RUR) and real root isolation. We show how the new RUR solver can tackle examples that are out of reach for the homotopy methods and vice versa.

We study how a smooth irreducible algebraic variety $X$ of dimension $n$ embedded in $\mathbb{C} \mathbb{P}^{m}$ (with $m \geq n+2$), which degree is $d$, can be recovered using two projections from unknown points onto unknown hyperplanes. The centers and the hyperplanes of projection are unknown: the only input is the defining equations of each projected varieties. We show how both the projection operators and the variety in $\mathbb{C} \mathbb{P}^{m}$ can be recovered modulo some action of the group of projective transformations of $\mathbb{C} \mathbb{P}^{m}$. This configuration generalizes results obtained in the context of curves embedded in $\mathbb{C} \mathbb{P}^3$ and results concerning surfaces embedded in $\mathbb{C} \mathbb{P}^4$. We show how in a generic situation, a characteristic matrix of the pair of projections can be recovered. In the process we address dimensional issues and as a result establish a necessary condition, as well as a sufficient condition to compute this characteristic matrix up to a finite-fold ambiguity. These conditions are expressed as minimal values of the degree of the dual variety. Then we use this matrix to recover the class of the couple of projections and as a consequence to recover the variety. For a generic situation, two projections define a variety with two irreducible components. One component has degree $d(d-1)$ and the other has degree $d$, being the original variety.

It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem, whether a given D-finite power series is algebraic or transcendental, is notoriously difficult. We prove that this problem is decidable: we give two theoretical algorithms and a transcendence test that is efficient in practice.

We introduce an efficient method for decomposing the circuit variety of a given matroid $M$, based on an algorithm that identifies its minimal extensions. These extensions correspond to the smallest elements above $M$ in the poset defined by the dependency order. We apply our algorithm to several classical configurations: the V\'amos matroid, the unique Steiner quadruple system $S(3,4,8)$, the projective and affine planes, the dual of the Fano matroid, and the dual of the graphic matroid of $K_{3,3}$. In each case, we compute the minimal irreducible decomposition of their circuit varieties.

Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. Once a Groebner basis is certified for the defining ideal I of the quaternionic polynomial algebra, the normal form of a quaternionic polynomial can be computed by routine top reduction with respect to the Groebner basis. In the literature, a Groebner basis under the conjugate-alternating order of quaternionic variables was conjectured for I in 2013, but no readable and convincing proof was found. In this paper, we present the first readable certification of the conjectured Groebner basis. The certification is based on several novel techniques for reduction in free associative algebras, which enables to not only make reduction to S-polynomials more efficiently, but also reduce the number of S-polynomials needed for the certification.

Given a number field with absolute Galois group $\mathcal{G}$, a finite Galois module $M$, and a Selmer system $\mathcal{L}$, this article gives a method to compute Sel$_\mathcal{L}$, the Selmer group of $M$ attached to $\mathcal{L}$. First we describe an algorithm to obtain a resolution of $M$ where the morphisms are given by Hecke operators. Then we construct another group $H^1_S(\mathcal{G}, M)$ and we prove, using the properties of Hecke operators, that $H^1_S(\mathcal{G}, M)$ is a Selmer group containing Sel$_\mathcal{L}$. Then, we discuss the time complexity of this method.

We present a new algorithm for solving the reduction problem in the context of holonomic integrals, which in turn provides an approach to integration with parameters. Our method extends the Griffiths--Dwork reduction technique to holonomic systems and is implemented in Julia. While not yet outperforming creative telescoping in D-finite cases, it enhances computational capabilities within the holonomic framework. As an application, we derive a previously unattainable differential equation for the generating series of 8-regular graphs.

The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical problems critically depends on high-quality, domain-specific datasets. In this paper, we address the computation of Gr\"obner basis using Transformers. While a dataset generation method tailored to Transformer-based Gr\"obner basis computation has previously been proposed, it lacked theoretical guarantees regarding the generality or quality of the generated datasets. In this work, we prove that datasets generated by the previously proposed algorithm are sufficiently general, enabling one to ensure that Transformers can learn a sufficiently diverse range of Gr\"obner bases. Moreover, we propose an extended and generalized algorithm to systematically construct datasets of ideal generators, further enhancing the training effectiveness of Transformer. Our results provide a rigorous geometric foundation for Transformers to address a mathematical problem, which is an answer to Lample and Charton's idea of training on diverse or representative inputs.

Over the years, genetic programming (GP) has evolved, with many proposed variations, especially in how they represent a solution. Being essentially a program synthesis algorithm, it is capable of tackling multiple problem domains. Current benchmarking initiatives are fragmented, as the different representations are not compared with each other and their performance is not measured across the different domains. In this work, we propose a unified framework, dubbed TinyverseGP (inspired by tinyGP), which provides support to multiple representations and problem domains, including symbolic regression, logic synthesis and policy search.

The field of analytic combinatorics in several variables (ACSV) develops techniques to compute the asymptotic behaviour of multivariate sequences from analytic properties of their generating functions. When the generating function under consideration is rational, its set of singularities forms an algebraic variety -- called the singular variety -- and asymptotic behaviour depends heavily on the geometry of the singular variety. By combining a recent algorithm for the Whitney stratification of algebraic varieties with methods from ACSV, we present the first software that rigorously computes asymptotics of sequences whose generating functions have non-smooth singular varieties (under other assumptions on local geometry). Our work is built on the existing sage_acsv package for the SageMath computer algebra system, which previously gave asymptotics under a smoothness assumption. We also report on other improvements to the package, such as an efficient technique for determining higher order asymptotic expansions using Newton iteration, the ability to use more efficient backends for algebraic computations, and a method to compute so-called critical points for any multivariate rational function through Whitney stratification.

In this paper, we extend the work of (Abbondati et al., 2024) on decoding simultaneous rational number codes by addressing two important scenarios: multiplicities and the presence of bad primes (divisors of denominators). First, we generalize previous results to multiplicity rational codes by considering modular reductions with respect to prime power moduli. Then, using hybrid analysis techniques, we extend our approach to vectors of fractions that may present bad primes. Our contributions include: a decoding algorithm for simultaneous rational number reconstruction with multiplicities, a rigorous analysis of the algorithm's failure probability that generalizes several previous results, an extension to a hybrid model handling situations where not all errors can be assumed random, and a unified approach to handle bad primes within multiplicities. The theoretical results provide a comprehensive probabilistic analysis of reconstruction failure in these more complex scenarios, advancing the state of the art in error correction for rational number codes.