Regular sets for the affine and projective groups over the field of two elements

In this paper we show that collineation groups of affine and projective spaces over the field of two elements GF (2), except in low dimensions, have regular sets. As an application of this result, we prove that, apart from a finite number of exceptions, any collineation group of affine and projective spaces over GF (2), is geometric. In the exceptional dimensions, all primitive groups are examined.Lavoro eseguito nell'ambito dei finanziamenti del M.P.I. Italia (40%).     

幂函数的一些密码学性质

二元域上n数组空间上的非线性置换在分组码，杂凑函数与流密码等密码学领域中有重要应用.域GF（2n）上的幂函数提供了二元域上n数组空间上的一类非线性置换．本文着重研究幂函数的强完全性、完全性与非线性度等密码学性质．作为结果，本文证明了幂函数具有完全性；证明了具有强完全性的函数必有较高的拓扑非线性度；木文找到一类具有强完全性的幂函数；周时也定出了幂函数的代数非线性度．     

A characterization of classical Minkowski planes over a perfect field of characteristic two

We give a new set of axioms defining the concept of (B^*)-plane (i.e. Minkowski plane without the tangency property) and we show that every (B^*)-plane in which a condition similar to the “Fano condition” of Heise and Karzel (see [5, § 3]) holds, is a Minkowski plane over a perfect field of characteristic two. In particular, every finite (B^*)-plane of even order is a Minkowski plane over a field. Consequences for strictly 3-transitive groups are derived from the preceding results; in particular, every strictly 3-transitive set of permutations of odd degree containing the identity is a protective group PGL₂(GF(2 ⁿ )) over a finite field GF(2 ⁿ , for some positive integer n.     

An interactive program for finite incidence structures

It has been shown (see [3]) that every finite incidence structure can be represented in a suitable way by a polynomial over a convenient finite field. In this paper we present a FORTRAN interactive program which examines the incidence structure associated to a given polynomial over GF(9). Further, a second program is exhibited, which determines a complete system of mutually orthogonal latin squares related to a polynomial associated to a projective plane of order nine.     

Classification of 9-dimensional trilinear alternating forms over GF(2)

Let V be a finite-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms, we introduce two new invariants. Together with a generalized radical polynomial used for classification of forms in dimension 8 over GF(2), they are sufficient to distinguish between all trilinear alternating forms in dimension 9 over GF(2). To prove the completeness of the list of forms, we computed their groups of automorphisms. There are 31 degenerate and 317 nondegenerate forms. We point out some forms with either small or large automorphism group.     

Finite Homogeneous 3-Graphs

By “3-graph” we mean a pair (V, E) such that E ? [V]³. We show that the only non-trivial finite 3-graphs homogeneous in the sense of Fraïssé are those associated with the projective planes over GF(2) and GF(3), and with the projective lines over GF(5) and GF(9). To exclude other possibilities we use the classification of doubly transitive finite permutation groups.     

Uniformly distributed functions inGF[q,x] andGF{q,x}

Summary Let A, B, C, ... denote polynomials over the finite field GF(q). It is shown that the sequence {B_i} is uniformly distributed modulo M if the sequence {B_i+k - B_i} is uniformly distributed modulo M for all integers k>0. A similar result holds for sequences defined by functional values. Also, a result of Weyl concerning uniform distribution modulo 1 is extended to polynomials over finite fields. Entrata in Redazione il 25 febbraio 1972.     

Coboundaries, Flows, and Tutte Polynomials of Matrices

Using matroid duality and the critical problem, we show that certain evaluations of the Tutte polynomial of a matroid represented as a matrix over a finite field GF(q) can be interpreted as weighted sums over pairs f , g of functions defined from the ground set to GF(q) whose difference f – g is the restriction of a linear functional on the column space of the matrix. Similar interpretations are given for the characteristic polynomial evaluated at q. These interpretations extend and elaborate interpretations for Tutte and chromatic polynomials of graphs due to Goodall and Matiyasevich. Received July 14, 2006     

On symmetric BIBDs with the same 3-concurrence

Higher order differentiation was introduced in a cryptographic context by Lai. Several attacks can be viewed in the context of higher order differentiations, amongst them the cube attack of Dinur and Shamir and the AIDA attack of Vielhaber. All of the above have been developed for the binary case. We examine differentiation in larger fields, starting with the field \(\mathrm {GF}(p)\) of integers modulo a prime p, and apply these techniques to generalising the cube attack to \(\mathrm {GF}(p)\). The crucial difference is that now the degree in each variable can be higher than one, and our proposed attack will differentiate several times with respect to each variable (unlike the classical cube attack and its larger field version described by Dinur and Shamir, both of which differentiate at most once with respect to each variable). Connections to the Moebius/Reed Muller Transform over \(\mathrm {GF}(p)\) are also examined. Finally we describe differentiation over finite fields \(\mathrm {GF}(p^s)\) with \(p^s\) elements and show that it can be reduced to differentiation over \(\mathrm {GF}(p)\), so a cube attack over \(\mathrm {GF}(p^s)\) would be equivalent to cube attacks over \(\mathrm {GF}(p)\).     

Error-free computation of a reflesive generalized inverse

The paper develops a method from which algorithms can be constructed to numerically compute an error-free reflexive generalized inverse of a matrix having rational entries. Multiple-modulus residue arithmetic is used to avoid error that is inherent in floating-point arithmetic. Some properties of finite fields of characteristic p, GF(p), are used to find nonsingular minors of the matrix over the field of rational numbers.