Punctured combinatorial Nullstellensätze

Chapter 2 - Projective and Affine Geometry over Division Rings

A cap set in a projective or affine geometry over a finite field is a set of points no three of which are collinear.We construct complete cap sets with sizes 274432, 13991936, and 30294016 in affine geometries (, ), (, ), and (, ), respectively.

Targets are matroids that arise from a nested sequence of flats in a projective geometry. This class of matroids was introduced by Nelson and Nomoto, who found the forbidden induced restrictions for binary targets. This paper generalizes their result to targets arising from projective geometries over $GF(q)$. We also consider targets arising from nested sequences of affine flats and determine the forbidden induced restrictions for affine targets.

A new definition for the dimension of a combinatorial t- (v,k,\lambda) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs.

Lattice-Theoretic Approach to Projective and Affine Geometry

Наведено порівняльний аналіз відомих протоколів узгодження таємного ключа через відкритий канал звʼязку. Запропоновано узагальнення відомого протоколу з використанням некомутативного множення матриць над простим скінченним полем на випадок довільного скінченного поля. A symmetric cryptosystem requires a secret key agreed by both parties. The Diffie-Hellman protocol was originally proposed for its exchange via an open communication channel. It is based on the computational complexity of the discrete logarithm problem in certain finite groups (the multiplicative group of a finite field, the group of points of an elliptic curve over a finite field). The availability of a powerful quantum computer will allow solving the discrete logarithm problem in these groups. Therefore, the issue of construction of secret key exchange protocols that will be resistant to attacks using a quantum computer has become urgent. The paper provides a comparative analysis of known protocols for a secret key exchange via an open communication channel. A generalization of the known protocol using non-commutative matrix multiplication over a prime finite field for the case of an arbitrary finite field is proposed. In this protocol, two high order elements from the general linear group over a finite field should be used, which satisfy an additional condition. It consists in the fact that each of the matrices cannot be reduced to a diagonal form by conjugation transformation. This condition follows from the considerations of avoiding an attack on the protocol. It is shown how to construct such elements. For the proposed generalization of the protocol, the sizes of public and private keys, secret key, the size of the finite field, which ensure the appropriate level of security, are calculated. The dimensions of matrices (elements of the general linear group) that we use are also given. The generalization of the protocol to the case of an arbitrary finite field increases the flexibility of choosing protocol parameters to ensure the desired level of security.

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years, the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection, it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation that is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.

Structural Properties of Models of N1-Categorical Theories

Three constructions of perfect authentication codes from projective geometry over finite fields

Dynamical systems with discrete time (discrete systems) are widely used in various applications (e.g., see the monographs [1–4]). Most often, the parameters of such systems are chosen from the field R of real numbers or C of complex numbers. However, if one deals with the so-called sequential machines [5] used in construction of nodes of discrete objects for communication and control, finite fields, in particular, the field GF (p) of residues modulo a prime number p [6], are chosen as the parameter spaces. Further, a number of problems in contemporary theoretical physics require that techniques for studying equations of various types over the field of p-adic numbers be developed [7]. Thus, numerous problems in applied and general mathematics are stated in terms of discrete systems over specific fields; moreover, the range of such fields is fairly broad. Therefore, it seems worthwhile to study discrete equations over arbitrary fields so as to simulate well-known models using a single mathematical object and to investigate them with the help of unified approaches. In this paper, we prove stabilization criteria for linear discrete nonstationary systems whose parameters may belong to an arbitrary field.

Rational functions over finite fields having continued fraction expansions with linear partial quotients

Fast parallel algorithms are presented for the following problems in symbolic manipulation of univariate polynomials: computing all entries of the extended Euclidean scheme of two polynomials over an arbitrary field, gcd and 1cm of many polynomials, factoring polynomials over finite fields, and the squarefree decomposition of polynomials over fields of characteristic zero and over finite fields. For the following estimates, assume that the input polynomials have degree at most n, and the finite field has $p^d $ elements. The Euclidean algorithm is deterministic and runs in parallel time $O(\log ^2 n)$. All the other algorithms are probabilistic (Las Vegas) in the general case, but when applicable to ${\bf Q}$ or ${\bf R}$, they can be implemented deterministically over these fields. The algorithms for gcd and lcm use parallel time $O(\log ^2 n)$. The factoring algorithm runs in parallel time $O(\log ^2 n\log ^2 (d + 1)\log p)$. The algorithm for squarefree decomposition runs in parallel time $O(\log ^2 n)$ for characteristic zero, and in parallel time $O(\log ^2 n+(d - 1)\log p)$ for finite fields. All Las Vegas algorithms have failure probability less than $2^{ - n} $. For all algorithms, the number of processors is polynomial in n.

We determine the asymptotic normalized rank of a random matrix $\vA$ over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge (2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations. We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well. We provide a simplified proof of the random $k$-XORSAT satisfiability threshold theorem. As an extension we also determine the full rank threshold for sparse random matrices over finite fields with precisely $k$ non-zero entries per row. This complements a result from [Ayre, Coja-Oghlan, Gao, Müller: Combinatorica 2020]. The proof combines physics-inspired message passing arguments with a surgical moment computation.

The multiplicative complexity of an S-box over a finite field is the minimum number of multiplications needed to implement the S-box as an arithmetic circuit. In this paper we fully characterize bijective S-boxes with multiplicative complexity 1 up to affine equivalence over any finite field. We show that under affine equivalence in odd characteristic there are two classes of bijective functions and in even characteristic there are three classes of bijective functions with multiplicative complexity 1. Moreover, in (Jeon et al., Cryptogr. Commun., 14(4), 849-874 (2022)) A-boxes where introduced to lower bound the differential uniformity of an S-box over F2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{2}^{n}$$\\end{document} via its multiplicative complexity. We generalize this concept to arbitrary finite fields. In particular, we show that the differential uniformity of a (n, m)-S-box over Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{q}$$\\end{document} is at least qn-l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^{n - l}$$\\end{document}, where ⌊n-12⌋+l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lfloor \\frac{n - 1}{2} \\rfloor + l$$\\end{document} is the multiplicative complexity of the S-box.