Blocking Sets of Certain Line Sets Related to a Conic

In this paper we classify point sets of minimum size of two types (1) point sets meeting all secants to an irreducible conic of the desarguesian projective plane PG(2,q), q odd; (2) point sets meeting all external lines and tangents to a given irreducible conic of the desarguesian projective plane PG(2,q), q even.     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

The polynomial degree of the Grassmannian G(1,n,q) of lines in finite projective space PG(n, q)

Let G: = G(1,n,q) denote the Grassmannian of lines in PG(n,q), embedded as a point-set in PG(N, q) with For n = 2 or 3 the characteristic function of the complement of G is contained in the linear code generated by characteristic functions of complements of n-flats in PG(N, q). In this paper we prove this to be true for all cases (n, q) with q = 2 and we conjecture this to be true for all remaining cases (n, q). We show that the exact polynomial degree of is for δ: = δ(n, q) = 0 or 1, and that the possibility δ = 1 is ruled out if the above conjecture is true. The result deg( for the binary cases (n,2) can be used to construct quantum codes by intersecting G with subspaces of dimension at least      

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve then either or . In the former case is projectively equivalent to the curve with equation X^q+1Y+Y^q+1+X=0 studied by Pellikaan. Furthermore, the curve has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its -rational points can be partitioned into finite projective planes . In this paper, the full automorphism group of such curves is determined. It turns out that is the normalizer of a Singer group in .     

Local Schur’s Lemma and Commutative Semifields

A set of linear maps , V a finite vector space over a field K, is regular if to each there corresponds a unique element such that R(x)=y. In this context, Schur’s lemma implies that is a field if (and only if) it consists of pairwise commuting elements. We consider when is locally commutative: at some μ ∈V*, AB(μ)=BA(μ) for all , and has been normalized to contain the identity. We show that such locally commutative are equivalent to commutative semifields, generalizing a result of Ganley, and hence characterizing commutative semifield spreads within the class of translation planes. This enables the determination of the orders |V| for which all locally commutative on V are (globally) commutative. Similarly, we determine a sharp upperbound for the maximum size of the Schur kernel associated with strictly locally commutative . We apply our main result to demonstrate the existence of a partial spread of degree 5, with nominated shears axis, that cannot be extend to a commutative semifield spread. Finally, we note that although local commutativity for a regular linear set implies that the set of Lie products consists entirely of singular maps, the converse is false.     

On Some Sets with Even Valued Partition Function

Let IN be the set of positive integers, = {b₁ < ⋅s < b_h}⊂IN, N∊IN and N≥ b_h. = ₀( ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy) such that {1,..., N} = and p( ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( ,n) denotes the number of partitions of n with parts in . Let us denote by σ ( ,n) the sum of the divisors of n belonging to . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( ,2^k n))_{n≥ 1}mod 2^k+1 is periodic with an odd period q_k. In this paper, we will characterize for any fixed odd positive integer q, the sets and the integers N such that q₀ = q, and those for which q_k = q for all k≥ 0. Moreover, a set = ₀( ,N) is constructed with the property that its period, i.e. the period of (σ( ,n))_{n≥ 1}mod 2, is 217, and for which the counting function is asymptotically equal to that of ₀({1,2,3,4,5},5) which is a set of period 31.Dedicated to Professor J.-L. Nicolas on the occasion of his 60th birthday2000 Mathematics Subject Classification: Primary—11P81, 11P83Research supported by MIRA 2002 program n^o 0203012701, Number Theory, Lyon-Monastir.     

Square-Free Non-Cayley Numbers. On Vertex-Transitive Non-Cayley Graphs of Square-Free Order

A complete classification is given of finite primitive permutation groups which contain a regular subgroup of square-free order. Then a collection of square-free numbers n is obtained such that there exists a vertex-primitive non-Cayley graph on n vertices if and only if n is a member of .     

On the security of stepwise triangular systems

In 2003 and 2004, Kasahara and Sakai suggested the two schemes RSE(2)PKC and RSSE(2)PKC, respectively. Both are examples of public key schemes based on ultivariate uadratic equations. In this article, we first introduce Step-wise Triangular Schemes (STS) as a new class of ultivariate uadratic public key schemes. These schemes have m equations, n variables, L steps or layers, r the number of equations and new variables per step and q the size of the underlying finite field . Then, we derive two very efficient cryptanalytic attacks. The first attack is an inversion attack which computes the message/signature for given ciphertext/message in O(mn ³ Lq ^r + n ² Lrq ^r ), the second is a structural attack which recovers an equivalent version of the secret key in O(mn ³ Lq ^r + mn ⁴) operations. As the legitimate user also has a workload growing with q ^r to recover a message/compute a signature, q ^r has to be small for efficient schemes and the attacks presented in this article are therefore efficient. After developing our theory, we demonstrate that both RSE(2)PKC and RSSE(2)PKC are special instances of STS and hence, fall to the attacks developed in our article. In particular, we give the solution for the crypto challenge proposed by Kasahara and Sakai. Finally, we demonstrate that STS cannot be the basis for a secure ultivariate uadratic public key scheme by discussing all possible variations and pointing out their vulnerabilities.     

Matrix Groups Related to the Quaternion Group and Spherical Orbit Codes

We consider a finite matrix group with 3⁴· 2¹⁶ elements, which is a subgroup of the infinite group , where is the regular representation of the quaternion group and C is a matrix that transforms the regular representation Q to its cellwise-diagonal form. There is a number of ways to define the matrix C. Our aim is to make the group similar in a certain sense to a finite group. The eventual choice of an appropriate matrix C done heuristically. We study the structure of the group and use this group to construct spherical orbit codes on the unit Euclidean sphere in R⁸. These codes have code distance less than 1. One of them has 3²· 2⁸ = 2304 elements and its squared Euclidean code distance is 0.293. Communicated by: V. A. Zinoviev     

The Polynomial Degree of the Grassmannian {\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}

For a subset ψ of PG(N, 2) a known result states that ψ has polynomial degree ≤ r, r≤ N, if and only if ψ intersects every r-flat of PG(N, 2) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when ψ is the Grassmannian , to show that for n <8 the polynomial degree of is .     

Structure of Degenerate Block Algebras

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map : A A F, we define a Lie algebra = (A, ) over F with basis {e_x | x A/{0}} and Lie product [e_x,e_y] = (x,y)e_x+y. We show that is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der of is a complete Lie algebra. We describe the double extension D( , T) of by T, where T is spanned by the locally finite derivations of , and determine the second cohomology group H²(D( , T),F) using anti-derivations related to the form on D( , T). Finally, we compute the second Leibniz cohomology groups HL²( , F) and HL²(D( , T), F).2000 Mathematics Subject Classification: 17B05, 17B30This work was supported by the NNSF of China (19971044), the Doctoral Programme Foundation of Institution of Higher Education (97005511), and the Foundation of Jiangsu Educational Committee.     

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

We consider the quotient of the Hermitian curve defined by the equation y^q + y = x^m over where m > 2 is a divisor of q+1. For 2≤ r ≤ q+1, we determine the Weierstrass semigroup of any r-tuple of -rational points on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, we prove that there are r-point codes, that is codes of the form where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which is obtained by taking m=q +1 in the above equation Communicated by: J.W.P. Hirschfeld     

Contact and Conformal Maps in

When the action of the conformal group O(1, n+1) on may be characterized in simple differential geometric terms, even locally: a theorem of Liouville states that a C⁴ map between domains and in whose differential is a (variable) multiple of a (variable) isometry at each point of is the restriction to of a transformation x g·x, for some g in O(1,n+1). In this paper, we consider the problem of characterizing the action of a more general semisimple Lie group G on the space G/P, where P is a minimal parabolic subgroup.     

(6,3)-MDS Codes over an Alphabet of Size 4

An (n,k)_q-MDS code C over an alphabet (of size q) is a collection of q^k n–tuples over such that no two words of C agree in as many as k coordinate positions. It follows that n ≤ q+k−1. By elementary combinatorial means we show that every (6,3)₄-MDS code, linear or not, turns out to be a linear (6,3)₄-MDS code or else a code equivalent to a linear code with these parameters. It follows that every (5,3)₄-MDS code over must also be equivalent to linear.     

Tubes in three–dimensional projective spaces

A partial tube in PG(3, q) is a pair , where is a collection of mutually disjoint lines of PG(3, q) with the property that for each plane π of PG(3, q) through L, the intersection of π with the lines of is an arc. Here, we generalize the notion of partial tube allowing the ground field to be any algebraically closed field. To a generalized partial tube we will associate an irreducible surface of degree d in providing upper bounds on d. The authors were partially supported by MIUR and GNSAGA of INdAM (Italy).     

Partial t-Spreads in PG(2t+1,q)

This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q ²-1-r lines, then r=s( +1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3, ). We also discuss maximal partial spreads in PG(3,p ³), p=p ₀ ^h , p ₀ prime, p ₀ 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p ²+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p ³) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p ³). In PG(3,p ³),p square, for maximal partial spreads of deficiency p ²+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1, ), which implies that 0 (mod +1) and, when (2t+1)( -1) <q-1, that 2( +1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2, ) and this implies 0 (mod q ^2/3+q ^1/3+1). A more general result is also presented.     

Elliptic Curves Suitable for Pairing Based Cryptography

For pairing based cryptography we need elliptic curves defined over finite fields whose group order is divisible by some prime with where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields with arbitrary embedding degree k are given. Unfortunately, p is of size .We give a method to generate ordinary elliptic curves over prime fields with p significantly less than which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with where or depending on k. For special values of k even better results are obtained.We present several examples. In particular, we found some curves where is a prime of small Hamming weight resp. with a small addition chain.AMS classification: 14H52, 14G50     

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q_4,q of PG(4, q) mapping suitably PG(3, q) over Q_4,q. More precisely the points of PG(3, q) are the lines of Q_4,q; the lines of PG(3, q) are the tangent cones of Q_4,q and the reguli of the hyperbolic quadrics hyperplane section of Q_4,q. A plane of PG(3, q) is the set of lines of Q_4,q meeting a fixed line of Q_4,q. We remark that this representation is valid also for a projective space over any field K and we apply the above representation to construct maximal partial spreads in PG(3, q). For q even we get new cardinalities for For q odd the cardinalities are partially known.     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

Block Primitive 2-(<Emphasis Type="Italic">v</Emphasis>,<Emphasis Type="Italic">k</Emphasis>,1) Designs Admitting a Ree Group of Characteristic Two

Let be a 2-(v,k,1) design, and let G be a group of automorphisms of . We show that if G is block primitive, then G does not admit a Ree group as its socle.