Notes on Small Private Key Attacks on Common Prime RSA

We point out critical deficiencies in lattice-based cryptanalysis of common prime RSA presented in ``Remarks on the cryptanalysis of common prime RSA for IoT constrained low power devices'' [Information Sciences, 538 (2020) 54--68]. To rectify these flaws, we carefully scrutinize the relevant parameters involved in the analysis during solving a specific trivariate integer polynomial equation. Additionally, we offer a synthesized attack illustration of small private key attacks on common prime RSA.Comment: 15 pages, 1 figur

Note on Integer Factoring Methods IV

This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer factorization.Comment: 20 Pages, New Versio

Polynomial Invariants for Affine Programs

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate

Generic interpolation polynomial for list decoding

AbstractWe extend results of K. Lee and M.E. OʼSullivan by showing how to use Gröbner bases to find the interpolation polynomial for list decoding a one-point AG code C=CL(rP,D) on any curve X, where P is an Fq-rational point on X and D=P1+P2+⋯+Pn is the sum of other Fq-rational points on X. We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial

Struktury dostępu i kryptosystemy oparte na krzywych eliptycznych

We develop the theory of access structures and include elliptic curve based cryptosystems applications. Shown are results concerning methods of encrypting monotonic access structures basing on logical formulae and our proposed, extended method with an abstract function, basing on set-theoretic approach. Introduced is an idea of hierarchy in any general access structure and shown are results related to security with respect to the hierarchy. Given are multivariate extensions of secret sharing schemes. Included are considerations on threshold sharing with a multivariate polynomial and a setting for generalized secret sharing. They are based on generalized Chinese Remainder Theorem in multivariate polynomial ring and use methods of the theory of Gröbner bases. Given are elliptic curve based applications in a form of general access structure based signature schemes. The considerations extend to the general access structure based decryption schemes. General access structure in these applications could be given by, apart of method related to a generalized Asmuth-Bloom sequence, by a method based on logical formulae, a method based on extended Blakley’s scheme and our method based on plain set-theoretic approach with an introduced abstract function. The bilinear pairings which are appropriate for the designs of our schemes are for instance modified Weil pairing or modified Tate-Lichtenbaum pairing.Rozwijamy teorię struktur dostępu uwzględniając kryptograficzne zastosowania oparte na teorii krzywych eliptycznych. Uzyskano wyniki związane z metodami szyfrowania monotonicznych struktur dostępu, opartymi na formułach logicznych oraz zaproponowaną przez nas, uogólnioną metodą opartą na podejściu teorio-mnogościowym korzystającą z abstrakcyjnej funkcji. Wprowadzone jest pojęcie hierarchii w dowolnej ogólnej strukturze dostępu i uzyskano wyniki związane z bezpieczeństwem dotyczącym hierarchii w naszym ujęciu. Podane zostały rozszerzenia schematów dzielenia sekretu na wiele zmiennych. Możemy zaliczyć tutaj rozważania dotyczące rozdzielania progowego wykorzystującego wielomian wielu zmiennych oraz w podobnym duchu, rozdzielania w ogólnej strukturze dostępu. Oparte są one na uogólnionym Chińskim Twierdzeniu o Resztach w pierścieniu wielomianów wielu zmiennych i używają metod z teorii baz Grobnera. Podane zostały zastosowania wykorzystujące krzywe eliptyczne w postaci schematów podpisu w ogólnej strukturze dostępu. Rozważania te przenoszą się na schematy deszyfrowania w ogólnej strukturze dostępu. Ogólna struktura dostępu w zastosowaniach tych może być zadana, obok metody związanej z uogólnionym ciągiem Asmutha-Blooma także przez metodę opartą na formułach logicznych,metodę opartą na rozszerzonym schemacie Blakley'a oraz naszą metodę opartą na czystym teorio-mnogościowym podejściu z wprowadzoną funkcją abstrakcyjną. Iloczynem dwuliniowym, użytecznym w konstrukcjach naszych schematów jest zmodyfikowany iloczyn Weila lub zmodyfikowany iloczyn Tate'a-Lichtenbauma

First-Order Tests for Toricity

Motivated by problems arising with the symbolic analysis of steady state ideals in Chemical Reaction Network Theory, we consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a coset of a multiplicative group. That property corresponds to Shifted Toricity, a recent generalization of toricity of the corresponding polynomial ideal. The key idea is to take a geometric view on varieties rather than an algebraic view on ideals. Recently, corresponding coset tests have been proposed for complex and for real varieties. The former combine numerous techniques from commutative algorithmic algebra with Gr\"obner bases as the central algorithmic tool. The latter are based on interpreted first-order logic in real closed fields with real quantifier elimination techniques on the algorithmic side. Here we take a new logic approach to both theories, complex and real, and beyond. Besides alternative algorithms, our approach provides a unified view on theories of fields and helps to understand the relevance and interconnection of the rich existing literature in the area, which has been focusing on complex numbers, while from a scientific point of view the (positive) real numbers are clearly the relevant domain in chemical reaction network theory. We apply prototypical implementations of our new approach to a set of 129 models from the BioModels repository

Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases

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