Antipodal distance regular covers of complete and complete bipartite graphs give rise to projective planes and Hadamard matrices. Inspired by this, we investigate antipodal distance regular covers of strongly regular graphs which are not complete bipartite. We use geometric properties of antipodal distance regular covers to show that infinite families of strongly regular graphs (Steiner graphs and latin square graphs) have no antipodal distance regular covers. This implies that for fixed integer $m$ only finitely many strongly regular graphs with the smallest eigenvalue $-m$ can have distance regular intipodal covers. We search for feasible parameters of strongly regular graphs with small valency (up to 100) and feasible parameters of their antipodal distance regular covers.

It is an object of the invention to provide a processor that combines finite field arithmetic and integer arithmetic for providing operations required for Elliptic Curve (EC) cryptography. It is a further object of the invention to provide an arithmetic processor design that may be scaled to different field or register sizes. A still further object of the invention to provide an arithmetic processor that may be used with different field sizes. A stil further object of the invention to provide an arithmetic processor that is capable of being scaled to provide an increase in spead when performing multi-sequence operations by simultaneously executing multiple steps in the sequence.

We use geometric properties of antipodal distance regular covers to show that infinite families of strongly regular graphs (Steiner graphs and latin square graphs) have no antipodal distance regular covers.

Purpose of P363: The transition from paper to electronic media brings with it the need for electronic privacy and authenticity. Public-key cryptography offers fundamental technology addresing this need. Many alternative public-key techniques have been proposed, each with its own benefits. However, there has been no single, comprehensive reference defining a full range of common public-key techniques covering key agreement, public-key encryption, digital signatures, and identification from several families , such as discrete logarithms, integer factorization, and elliptic curves. It is not purpose of this project to mandate any particular set of public-key techniques such as key sizes. Rather, the purpose is to provide a reference for specifications of a variety of techniques from which applications may select. Specifications of common public-key cryptographic techniques supplemental to those considered in IEEE P1363, including mathematical primitives for secret value (key) derivation, public-key encryption, digital signatures, and identification, and cryptographic schemes based on those primitives. Specifications of related cryptographic parameters, public keys and private keys. Class of computer and communications systems is not restricted.

A connected graph $G$ is called distance-regular graph if for any two vertices $u$ and $v$ of $G$ at distance $i$, the number $c_i$ (resp. $B-i$) of neighbours of $v$ at distance $i-1$ (resp. $i+1$) from $u$ depends only on $i$ rather than on individual vertices. A distance regular graph $G$ of diameter $d \in \{2m,2m+1\}$ is antipodal ($r$-cover of its folded graph) if and only if $b_i = c_{d-i}$, for $i = 0,...,d$, $i \ne m$ (and $r = 1+b_m/c_{d-m}$). For example, the dodecahedron is a double-cover of the Petersen graph, the cube is the double-cover of the tetrahedron. Most finite objects of sufficient regularity are closed related to certain distance-regular graphs, in particular, antipodal distance-regular graphs give rise to projective planes, Hadamard matrices and other interesting combinatorial objects. Distance-regular graphs serve as an alternative approach to these objects and allow the use of graph eigenvalues, graph representations, association schemes and the theory of orthogonal polynomials. We start with cyclic covers and spreads of generalized quadrangles and find a switching which uses some known infinite families of antipodal distance-regular graphs of diameter three to produce new ones. Then we examine antipodal distance-families of antipodal distance-regular graphs of diameter three to produce new ones. Then we examine antipodal distance-regular graphs of diameter four and five. Together with J. Koolen we use representations of graphs to extend in the case of diameter four the result P. Terwilliger who has shown, using the theory of Krein modules, that in a $Q$-polynomial antipodal diatance-regular graph the neighbourhood of any vertex is a strongly regular graph. New nonexistence conditions for covers are derived from that. This study relates to the above switching and to extended generalized quadrangles. In an imprimitive association scheme there always exists a merging (i.e., a grouping of the relations) which gives a new nontrivial association scheme. We determine when merging in an antipodal distance-regular graph produces a distance-regular graphs with regular near polygons containing a spread. In case of diameter three we get a Brouwer's characterization of certain distance-regular graphs with generalized quadrangles containing a spread. Finally, antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. In most cases the structure of short cycles provides a tool to determine the existence of an antipodal cover. A relationship between antipodal covers of a graph and its line graph is investigated. Antipodal covers of complete bipartite graphs and their line graphs (lattice graphs) are characterized in terms of weak resolvable transversal designs which are, in the case of maximal covering index, equivalent to affine planes with a parallel class deleted. We conclude by mentioning two results which indicate the importance of antipodal distance-regular graphs. The first one is a joint work with M. Araya and A. Hiraki. Let $G$ be a distance-regular graph of diameter $d$ and valency $k>2$. If $b_t=1$ and $2t \le d$, then $G$ is an antipodal double-cover. Consenquently, if $m>2$ the multiplicity of an eigenvalue of the adjacency matrix of $G$ and if $G$ is not an antipodal double-cover then $d \le 2m-3$. This result is an improvement of Godsil's diameter bound and it is very important for the classification of distance-regular graphs with an eigenvalue of small multiplicity (as opposed to dual classification of distance-regular graphs with small valency). The second result is joint work with C. Godsil. We show that distance-regular graphs that contain maximal independent geodesic paths of short length are antipodal. A new infinite family of feasible parameters of antipodal distance-regular graphs of diameter four is found. As an auxiliary result we use equitable partitions to show that the determinant of a Töplitz matrix can be written as a product of two determinants of approximately half size of the original one.