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By analogy with the developed cryptographic theory of discrete logarithm problems, we define several hard problems in Entropoid based cryptography, such as Discrete Entropoid Logarithm Problem (DELP), Computational Entropoid Diffie-Hellman…
Large language models (LLMs) have exhibited impressive competence in various tasks, but their internal mechanisms on mathematical problems are still under-explored. In this paper, we study a fundamental question: how language models encode…
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…
The assumed hardness of the Linear Code Equivalence problem (LCE) lies at the core of the security of the LESS signature scheme and other signature schemes with advanced functionalities. The LCE problem asks to determine whether two linear…
We develop a $D-$module approach to various kinds of solutions to several classes of important differential equations by long divisions of different differential operators. The zeros of remainder maps of such long divisions are handled by…
The problem of iterated partial summations is solved for some discrete distributions defined on discrete supports. The power method, usually used as a computational approach to finding matrix eigenvalues and eigenvectors, is in some cases…
Encryption schemes often derive their power from the properties of the underlying algebra on the symbols used. Inspired by group theoretic tools, we use the centralizer of a subgroup of operations to present a private-key quantum…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…
A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory solutions can be defined in algebraically closed p-adic fields.…
Differential privacy is the dominant standard for formal and quantifiable privacy and has been used in major deployments that impact millions of people. Many differentially private algorithms for query release and synthetic data contain…
Being able to compute efficiently a low-weight multiple of a given binary polynomial is often a key ingredient of correlation attacks to LFSR-based stream ciphers. The best known general purpose algorithm is based on the generalized…
We revisit the reduction of Cheng and Wan, which transforms instances of the discrete logarithm problem (DLOG) over finite fields into a decoding problem for Reed--Solomon codes, and study how Regev's reduction can be used to solve these…
Consider the classical action of ${\rm GL}_n$ on a sum of $q$ copies of the defining representation and $p$ copies of its dual; by Howe duality, the polynomial functions on this space decompose under the joint action of ${\rm GL}_n$ and…
Let $p$ be a odd prime such that 2 is a primitive element of finite field $F_p*$. In this short note we propose a new algorithm for the computation of discrete logarithm in $F_p*$. This algorithm is based on elementary properties of finite…
Integral linear systems $Ax=b$ with matrices $A$, $b$ and solutions $x$ are also required to be in integers, can be solved using invariant factors of $A$ (by computing the Smith Canonical Form of $A$). This paper explores a new problem…
We present a comprehensive programme analysing the decomposition of proof systems for non-classical logics into proof systems for other logics, especially classical logic, using an algebra of constraints. That is, one recovers a proof…
Lattice-based cryptography relies on generating random bases which are difficult to fully reduce. Given a lattice basis (such as the private basis for a cryptosystem), all other bases are related by multiplication by matrices in…
Quantum algorithms for factoring and discrete logarithm have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden…
We design polynomial time schemes for secure message transmission over arbitrary networks, in the presence of an eavesdropper, and where each edge corresponds to an erasure channel with public feedback. Our schemes are described through…