Benji Fisher

New Key Agreement Protocols in Braid Group Cryptography

Key agreement protocols are presented whose security is based on the difficulty of inverting one-way functions derived from hard problems for braid groups. Efficient/low cost algorithms for key transfer/ extraction are presented. Attacks/security parameters are discussed.

Sums of twisted GL(2) L-functions over function fields

Let K be a function field of odd characteristic, and let π (resp.,η) be a cuspidal automorphic representation of GL2(AK ) (resp.,GL1(AK )). Then we show that a weighted sum of the twists of L (s, π) by quadratic charactersχD, ∑ D L(s, π ⊗ χD)a0(s, π, D) η(D) |D|, is a rational function and has a finite, nonabelian group of functional equations. A similar construction in the noncuspidal case gives a rational function of three variables. We specify the possible denominators and the degrees of the numerators of these rational functions. By rewriting this object as a multiple Dirichlet series, we also give a new description of the weight functions a 0( , π, D) originally considered by D. Bump, S. Friedberg, and J. Hoffstein. 0. Introduction Let π be an automorphic representation on GL 2(AK ), whereK is a number field. Then it is a remarkable fact that a weighted sum of the L-functions of quadratic twists of π , ∑ a0(s, π,n) L(s, π, χn) |n| , (0.1) is a meromorphic function of two complex variables and satisfies a group of functional equations. (The sum is best described as a sum over the quadratic twists χn at ached to divisorsn, as formulated by B. Fisher and S. Friedberg in [ FF].) Indeed, this was demonstrated by Friedberg and J. Hoffstein [ FH] using a Rankin-Selberg construction, following earlier work of Bump, Friedberg, and Hoffstein [ BFH1] – [BFH3] giving a different Rankin-Selberg construction. In this paper we present an entirely different way to understand and analyze the series (0.1). We use this method to establish the properties of (0.1) in the function field case, which are rather striking; moreover, the approach applies equally well in the number field case and leads to a substantial simplification of the argument of [ FH]. We also study a three-variable analogue of (0.1), whereπ on GL2(AK ) is replaced by(π1, π2) on GL1(AK )× GL1(AK ). DUKE MATHEMATICAL JOURNAL Vol. 117, No. 3, c

Fourier-space crystallography as group cohomology

We reformulate Fourier-space crystallography in the language of cohomology of groups. Once the problem is understood as a classification of linear functions on the lattice, restricted by a particular group relation and identified by gauge transformation, the cohomological description becomes natural. We review Fourier-space crystallography and group cohomology, quote the fact that cohomology is dual to homology, and exhibit several results, previously established for special cases or by intricate calculation, that fall immediately out of the formalism. In particular, we prove that two phase functions are gauge equivalent if and only if they agree on all their gauge-invariant integral linear combinations and show how to find all these linear combinations systematically.

Cohomology for Anyone

Crystallography has proven a rich source of ideas over several centuries. Among the many ways of looking at space groups, N. David Mermin has pioneered the Fourier-space approach. Recently, we have supplemented this approach with methods borrowed from algebraic topology. We now show what topology, which studies global properties of manifolds, has to do with crystallography. No mathematics is assumed beyond what the typical physics or crystallography student will have seen of group theory; in particular, the reader need not have any prior exposure to topology or to cohomology of groups.

Applications of group cohomology to the classification of quasicrystal symmetries

In 1962, Bienenstock and Ewald described the classification of crystalline space groups algebraically in the dual, or Fourier, space. After the discovery of quasicrystals in 1984, Mermin and collaborators recognized in this description the principle of macroscopic indistinguishability and developed techniques that have since been applied to quasicrystals, including also periodic and incommensurately modulated structures. This paper phrases these techniques in terms of group cohomology. A quasicrystal is defined, along with its space group, without requiring that it come from a quasicrystal in real (direct) space. A certain cohomology group classifies the space groups associated to a given point group and lattice, and the dual homology group gives all gauge invariants. This duality is exploited to prove several results that were previously known only in special cases, including the classification of space groups (plane groups) for lattices of arbitrary rank in two dimensions. Extinctions in x-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of non-zero homology classes.

Group Cohomology and Quasicrystals I: Classification of Two-Dimensional Space Groups

In Fourier-space crystallography, the space group is defined in terms of the point group G, the inverse lattice (or Fourier module) L, and a phase function Φ. We classify the two-dimensional space groups (or plane groups), the major step being the classification of lattices, of all ranks, symmetric under the finite point group G. This step requires new ideas from integer representation theory. Given this classification, the remaining steps can be done easily using techniques of group cohomology.

Group Cohomology and Quasicrystals II: The Three Crystallographic Invariants in Two and Three Dimensions

In Fourier-space crystallography, space groups are classified by their phase functions, Φ, and we can determine Φ (up to a gauge) by its invariants. Usually, the invariants imply necessary extinctions in diffraction, but among the 157 periodic non-symmorphic space groups, two are distinguished not by extinctions—they have none—but by a second type of invariant. We give a non-periodic example of a third type of invariant and assert that all invariants can be expressed as sums of just these three types.