Applications of Finite Geometry to Cryptography

IEEE Transactions on Information Theory, 2000

The projective space of order n over the finite field q , denoted here as Pq(n), is the set of all subspaces of the vector space n q . The projective space can be endowed with the distance function d(U; V ) = dim U + dim V 0 2 dim(U \V ) which turns Pq(n) into a metric space. With this, an (n; M; d) code in projective space is a subset of Pq(n) of size M such that the distance between any two codewords (subspaces) is at least d. Koetter and Kschischang recently showed that codes in projective space are precisely what is needed for error-correction in networks: an (n; M; d) code can correct t packet errors and packet erasures introduced (adversarially) anywhere in the network as long as 2t + 2 < d. This motivates our interest in such codes. In this paper, we investigate certain basic aspects of "coding theory in projective space." First, we present several new bounds on the size of codes in P q (n), which may be thought of as counterparts of the classical bounds in coding theory due to Johnson, Delsarte, and Gilbert-Varshamov. Some of these are stronger than all the previously known bounds, at least for certain code parameters. We also present several specific constructions of codes and code families in P q (n). Finally, we prove that nontrivial perfect codes in P q (n) do not exist.

Acta Mathematica Sinica, English Series, 2004

The weight hierarchy of a binary linear [n, k] code C is the sequence (d1, d2, . . . , d k ), where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries.

IEEE Transactions on Information Theory, 2003

In this paper, we consider a new class of unconditionally secure authentication codes, called linear authentication codes (or linear A-codes). We show that a linear A-code can be characterized by a family of subspaces of a vector space over a finite field. We then derive an upper bound on the size of source space when other parameters of the system, that is, the sizes of the key space and the authenticator space, and the deception probability, are fixed. We give constructions that are asymptotically close to the bound and show applications of these codes in constructing distributed authentication systems.

Journal of Pure and Applied Algebra, 2005

In this paper, we present three algebraic constructions of authentication codes with secrecy. The codes have simple algebraic structures and are easy to implement. They are asymptotically optimal with respect to certain bounds.

Des. Codes Cryptogr., 2021

In the history of secret sharing schemes many constructions are based on geometric objects. In this paper we investigate generalizations of threshold schemes and related finite geometric structures. In particular, we analyse compartmented and hierarchical schemes, and deduce some more general results, especially bounds for special arcs and novel constructions for conjunctive 2-level and 3-level hierarchical schemes.

IEEE Transactions on Information Theory, 2000

We present a new application of algebraic curves over finite fields to the constructions of universal hash families and unconditionally secure codes. We show that the constructions derived from the Garcia-Stichtenoth curves yield new classes of authentication codes and universal hash families which are substantially better than those previously known.

CIM Series in Mathematical Sciences, 2015

We give a polynomial time attack on the McEliece public key cryptosystem based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes on the distinguishability of such codes from random codes using the Schur product. Wieschebrink treated the genus zero case a few years ago but his approach cannot be extent straightforwardly to other genera. We address this problem by introducing and using a new notion, which we call the t-closure of a code.

2021

Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values. These varieties give rise to q-divisible linear codes with at most 5 weights. Furthermore, for q odd these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic that is, each participant belongs to the same number of minimal access sets.

Contemporary Mathematics, 2019

Subjects: LCSH: Coding theory-Congresses. | Geometry, Algebraic-Congresses. | Cryptography-Congresses. | Number theory-Congresses. | AMS: Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Curves over finite and local fields. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Curves of arbitrary genus or genus = 1 over global fields. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Dessins d'enfants, Belyi theory. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture. msc | Number theory-Finite fields and commutative rings (number-theoretic aspects)-Algebraic coding theory; cryptography. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Zeta-functions and related questions. msc | Algebraic geometry-Computational aspects in algebraic geometry-Curves. msc | Group theory and generalizations-Representation theory of groups-Modular representations and characters. msc | Group theory and generalizations-Linear algebraic groups and related topics-Exceptional groups. msc Classification:

IEEE Transactions on Information Theory, 1972

Designs, Codes and Cryptography, 2010

We study a class of authentication codes with secrecy. We determine the maximum success probabilities of the impersonation and the substitution attacks on these codes and the level of secrecy. Therefore we give an answer to an open problem stated in Ding et al. (J Pure Appl Algebra 196:149-168, 2005). Our proofs use the number of rational places of a certain class of algebraic function fields. We determine this number by extending the corresponding results of E. Çakçak and F. Özbudak (Finite Fields Appl 14(1): [209][210][211][212][213][214][215][216][217][218][219][220] 2008). Our authentication codes use a map which is not perfect nonlinear in certain subcases. We give an extended and unified approach so that the parameters of our authentication codes are good also when the corresponding map is not perfect nonlinear.

Designs, Codes and Cryptography, 2014

ABSTRACT This special issue of designs, codes, and cryptography is dedicated to Prof. Frank De Clerck, full professor at the Mathematics Department of Ghent University since 1999, who has recently retired. The research articles appearing in this special issue are authored by selected mathematicians active in the field of Finite Geometry and Combinatorics. As a contribution to this special issue in honor of Prof. Frank De Clerck we find it most appropriate to include a short scientific biography. Frank graduated from Ghent University where he remained for his PhD (completed in 1978) during which he studied partial geometries under the supervision of Joseph A. Thas. His thesis was titled “Een combinatorische studie van de eindige partiële meetkunden (en: A combinatorial study of finite partial geometries)”.

Association for Women in Mathematics Series, 2017

Focusing on the groundbreaking work of women in mathematics past, present, and future, Springer's Association for Women in Mathematics Series presents the latest research and proceedings of conferences worldwide organized by the Association for Women in Mathematics (AWM). All works are peer-reviewed to meet the highest standards of scientific literature, while presenting topics at the cutting edge of pure and applied mathematics. Since its inception in 1971, The Association for Women in Mathematics has been a non-profit organization designed to help encourage women and girls to study and pursue active careers in mathematics and the mathematical sciences and to promote equal opportunity and equal treatment of women and girls in the mathematical sciences. Currently, the organization represents more than 3000 members and 200 institutions constituting a broad spectrum of the mathematical community, in the United States and around the world.

In the infancy of Cryptography Mono-alphabetic Substitution Ciphers were considered good enough to baffle any potential attackers but with the advancements in technology & the upsurge of computing power those methods have become trivial. Even the very complex methods of encryption are vulnerable to the brute force attacks of contemporary computers and with Quantum computing on the horizon even the current state of the art cryptosystems are at risk. Lots of research is being done and every possible field is being explored in order to create that elusive unbreakable cipher. Among other subjects, Geometry is also being applied and various ciphers based on the properties of different geometrical figures have been developed. This paper ventures to investigate the recent research applying the concept of geometry to boost the caliber of pre-existing cryptosystems enhance the understanding of the subject.

Designs, Codes and Cryptography, 2019

An outstanding folklore conjecture asserts that, for any prime p, up to isomorphism the projective plane P G(2, F p) over the field F p := Z/pZ is the unique projective plane of order p. Let π be any projective plane of order p. For any partial linear space X , define the inclusion number i(X , π) to be the number of isomorphic copies of X in π. In this paper we prove that if X has at most log 2 p lines, then i(X , π) can be written as an explicit rational linear combination (depending only on X and p) of the coefficients of the complete weight enumerator (c.w.e.) of the p-ary code of π. Thus, the c.w.e. of this code carries an enormous amount of structural information about π. In consequence, it is shown that if p > 2 9 = 512, and π has the same c.w.e. as P G(2, F p), then π must be isomorphic to P G(2, F p). Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.