a mesh is a partition of a domain into smaller parts , typically with simpler geometry , called cells . in two dimensions , both on the plane and on curved surfaces , cells are usually triangles or quadrilaterals. the shapes of the cells may be important ; for many applications , cells with shapes similar to an equilateral triangle or a square are preferred .the problem of mesh generation can then be seen as an optimization problem : to find a partition of a domain into well - shaped cells , possibly under additional demands , such as cell size requirements .the mesh generation problem has been the subject of extensive research .many techniques for creating meshes exist , especially in two dimensions ,, . nevertheless , some of the most popular techniques , which create good meshes in many cases , are heuristic in nature , and may create less than optimal meshes for some inputs .two of the inherent characteristics of the mesh generation problem seem to make it very difficult to solve :  ( characteristic 1 ) : :  the constraints on cells shapes are _global_. that is , the shape of  one cell is constrained by the possible shapes of its neighbors , which  in turn are constrained by the shapes of their neighbors , and so on .thus , at least in principle , the constraints on the mesh layout extend  over the whole domain ( or more precisely , over each connected  component of the domain ) .( characteristic 2 ) : :  the problem combines _ discrete and continuous aspects_. the number of  cells and the mesh connectivity ( i.e. : which cells are neighbors ?  which faces do they share ? ) are of discrete nature , whereas the  locations of the vertices can vary continuously .these aspects are  closely intertwined , preventing the sole use of purely discrete  techniques ( algebraic , graph - theoretic , etc . ) , or techniques designed  for use in problems of continuous nature .[ ptb ]  the meshes created by mesh generation algorithms can be divided into structured meshes , and unstructured meshes . a structured mesh is a mesh whose connectivity is that of a regular grid , see fig .[ mapping_vs_unstructured ] , ( iii ) . by assuming the connectivity of the mesh beforehand ,the problem of creating such a mesh is considerably simplified ( see characteristic 2 above ) , and reduces to the problem of assigning locations to the mesh vertices .one way of doing that is by finding a _ mapping function _ that maps the domain of the regular grid to the domain to be meshed , and using it to map the vertices .such techniques are known as _mapping techniques_. for small enough cells , the differential properties of the mapping function at the cell s location dictate its shape . if , for example , a mapping is angle preserving , then the inner angle of a small enough cell will be approximately preserved under the mapping . for a survey ,mapping techniques have also been offered for unstructured meshes , ( in the context of smoothing a given mesh see , and references therein ) , as long as the connectivity of the mesh is given beforehand .  just as a structured mesh can be imagined as created by mapping of a region of the plane to the domain to be mapped , an unstructured mesh in two dimensions can be imagined as a surface , that is mapped onto the domain to be meshed .the simplest example is that of an unstructured mesh with just one irregular vertex ( a vertex that has more or less than four cells incident upon it ) . the surface to be mapped in this case is a conethis can be visualized using the volterra construction : consider a piece of paper with an angular section cut out , see fig .[ 3_cone_all],(i ) . if the two edges of the section are identified , i.e. glued together , the paper will assume the shape of a cone , see fig .[ 3_cone_all],(ii ) .if the cone is then mapped to the plane , a regular grid drawn on the cone would be mapped to an unstructured mesh such as the one shown in fig . [ 3_cone_all],(iii ) . the mapping shown in fig . [3_cone_all],(iii ) has the special property of being _ conformal _ : a small square on the cone is approximately mapped to a square on plane .this creates well shaped cells in the resulting mesh : in the limit of an increasingly smaller cell , its square shape is preserved under the conformal mapping . a similar construction can be imagined for creating an unstructured mesh with more than one irregular vertex ; each irregular vertex will then correspond to one cone tip of the surface .the approach of the present work to the problem of creating unstructured meshes can be expressed as follows : given a domain to be meshed , _ what surface , with a square grid drawn upon it , can be mapped conformally into this domain ?_ thus , it is not just the mapping function that is sought after , but rather the surface to be mapped together with the mapping function . unlike mapping techniques, however , the mapping function is required to be conformal .[ tbh ]  a mathematical framework highly suitable for dealing with such questions is riemannian geometry .given the domain to be meshed , riemannian geometry allows one to define a new geometry for that domain .this includes a redefinition of the distances between points of the domain , and it is used here to redefine distances such that _ a cell edge have unit length_. thus , instead of defining the surface that is mapped and the mapping function separately , both are treated together , as the mapping induces a new distance definition on the domain to be meshed .for example , the cone in fig . [ 3_cone_all],(ii ) and the surface in fig .[ 3_cone_all ] , ( iii ) have the same geometry ( i.e. are isometric ) if the distances in fig . [3_cone_all],(iii ) are _ defined _ such that cell - edges have unit length . using the terminology of riemannian geometry, the problem can be restated as assigning a new metric to the domain being meshed , having the following properties :  ( property 1 ) : :  the metric is locally _ flat _ everywhere , except at some points , called  _cone points_. a flat region can be imagined as a bent , but not  stretched , piece of paper .the cone points are the tips of the  cones as described above .( property 2 ) : :  a single real function is defined in the domain  ( except at cone points ) , such that at any given point ,  the new metric at   is proportional to the original metric   :   , with   , .( on the plane with cartesian  coordinates . )we stress that the  proportionality can vary throughout the  domain .the quantity is the local change in  the distance definition and will be associated with the local cell  size .( in manifold theory terminology , the metric   is _ conformally _ _ _ _ related _ to   ) .these two properties are not sufficient for our purposes .first of all , as the volterra construction implies , for a grid of squares to be drawn around a cone tip , the total angle as measured around the tip must be a multiple of radians .secondly , the mesh must be aligned along the boundary , see fig .[ boundary_aligment ] . to formalize these demands, we define the direction of mesh - edges at each point on the surface .since the edges are assumed to form right angles at incidence , this direction is defined up to an addition of radians . ] .the four directions at each point will hence be called a _ cross _ , and the field of directions on the entire domain being meshed will be called a _ cross - field_. the cross - field is related to the new metric by requiring that the curves , generated by following the directions of the cross - field , along which the edges will be laid , will be geodesics of the new metric .( geodesics are the generalizations of straight lines for surfaces and manifolds . )the boundary alignment is formalized by requiring that the crosses be aligned with the boundary .we therefore add the following property to the required properties of the new metric :  ( property 3 ) : :  a cross - field exists .( exact definition is given in section  [ sec : cross_field ] . )property 1 states that the gaussian curvature of the manifold with metric be identically zero everywhere , except at cone points .combined with the equation of property 2 , the two formulas give a remarkably simple result , viz .that must obey the poisson equation everywhere exept at cone points , where is the gaussian curvature of the surface .this is a well known result in conformal geometry , see section [ sec : cross_field ] .[ tbh ]  the main result of the present work is that for a cross - field to exist the function , which uniquely defines the manifold , has to obey the poisson equation , with the gaussian curvature of the surface to be meshed ( vanishes for the plane ) , and a sum of localized charges ( dirac delta functions ) .the charges are placed at the cone points locations , thus corresponding to the irregular vertices of the mesh .the charge strength is equal to the cone excess angle ( the difference between the cone angle and ) which corresponds to the number of cells incident on the irregular vertex .for example , in the manifold shown in fig .[ 3_cone_all],(iii ) , the cone point has an excess angle of .the charge of the cone point , located at a point , corresponding to such a excess angle is , where is the dirac delta function in two dimensions .this is the charge of any singularity corresponding to an irregular vertex surrounded by three cells . along with the boundary alignment conditions on , the problem of finding an appropriate manifoldis thus reduced to an _ inverse poisson _ problem , of finding a charge distribution adhering to these conditions .the reduction gives _ exact , global _ relations that the function must obey .the inverse poisson problem on planar domains is of interest in many fields of science and engineering ( see references in section [ sec : case_c ] ) ; the relevance of existing techniques to the present application remains to be examined .another possible application is to the problem of creating a surface mesh aligned with predefined directions , which has recently attracted much attention ,,, .the algorithm described in creates conformal parametrizations and meshes approximately aligned with predefined directions . in that work ,the local size and the local direction are treated as independent variables . in order to reduce the number of singularities , a preprocessing step that modifies the cell size demand ( the curl correction process ) can be applied .in contrast , the theory developed in the present work uses conformality to a - priory link the cell - size and cell - direction fields , leaving just one field to work with , either the cell - size or the cell - direction .the applications of this theory to surface parameterization and surface quadrangulation problems are an interesting subject for future work .conformal parametrization of manifolds with conical singularities have been used in surface parametrization problems . for reviews and related work ,see - .mesh generation with boundaries and surface parametrization are different problems , because of the boundary alignment requirement in mesh generation .for example , the parametrization problem for planar domains is trivial : the coordinates of the plane form a good parametrization , but do not solve the mesh generation problem .the rest of the article is organized as follows : in section [ sec : parallel_transport_and_geodesics ] some elements of differential geometry are shortly reviewed . in section [ sec : cross_field ] the cross - field and -manifold are defined .section [ sec : relation_to_mesh_generation ] discusses the relation of the definitions in previous sections to mesh generation .cone points are analyzed in section [ sec : cone_points ] .the necessary and sufficient conditions for cross - field existence are developed in section [ sec : boundary_alignment_conditions ] .the set of conditions derived in sections [ sec : cone_points],[sec : boundary_alignment_conditions ] forms the core result of the work . in section [ sec : discussion ]the theory is discussed through four case studies .case studies a , b are used to discuss the meaning of the conditions derived before . in case study c the possible structure of a mesh generation algorithm is discussed .an example quadrilateral mesh problem , though admittedly simple and artificial , demonstrates how , given an input , a manifold with cone points is constructed , adhering to the imposed conditions . this manifold allows the construction of meshes with irregular vertices ; at any given point in the domain that is not an irregular vertex , at the limit of increasingly finer meshes , the cells shapes tend to a square .case study d gives an example of a curved surface meshing problem , and how it is solved .in this section basic facts from differential geometry , required in subsequent sections , are shortly reviewed .more complete accounts can be found in any textbook on differential geometry , such as , .  for a surface in three - dimensional space , distances and angles on the surface can be defined by the embedding . in some coordinate system ,denote by the metric given by the embedding .for example , on the plane with cartesian coordinates , , the kronecker delta .another metric , , is said to be _ conformally related _ to , if there exists a real function on the surface such that   given two vectors defined at some point on a surface or riemannian manifold , the angle between them is given by where summation over repeated indices ( the einstein convension ) is assumed .the angle as measured with the metric is substituting in eq .( [ eq : cos_theta_tilde ] ) , and comparing with eq .( [ eq : cos_theta ] ) it is found that , so the measurement of angles using the two conformally related metrics agrees .given a vector field on the plane ( with the euclidean metric ) the question of whether two vectors at two different points are parallel has a definite answer .this is not the case on curved surfaces , and more generally , for riemannian manifolds .there , the definition of parallel vectors at different locations generally depends on the path chosen between the points . for a curve connecting points and , the parallel transport of a vector from to along be defined .if is such a vector at we denote its parallel translate to along by , see fig .[ parallel_transport_basic ] .[ ptb ]  the geodesic curvature of a curve is the amount by which a curve turns . on the plane ( with the euclidean metric ) turningis measured by the change of the angle of the tangent vector to the curve . on a surface( and , more generally , on a riemannian manifold ) the angle is defined relative to a vector that is parallel translated along the very same curve . for a curve ,denote the tangent vector at by .define .then where is the length parameterization of , see fig .[ parallel_transport ] .[ ptb ]  the parallel transport depends on the metric , so a curve can have different geodesic curvatures under different metrics .let be the geodesic curvatures of a curve at some point , with the metrics respectively , related as in eq .( [ eq : conformal_relation ] ) .( henceforth , all quantities relating to the metric will be marked with a tilde . )then are related by where is the derivative of along the normal vector , which is defined such that form a right hand system .( [ kappa_g_relation ] ) is derived in appendix [ app : k_g_different_metrics ] .a geodesic is a generalization of a straight line .it is the curve whose geodesic curvature vanishes .the equation for a geodesic of the metric is found by substituting  in eq .( [ kappa_g_relation ] ) :   two integral theorems are used throughout the paper .the first is the gauss - bonnet theorem , which relates the change in a vector undergoing parallel transport along a closed loop , to the total gaussian curvature inside the loop .let be a closed curve ,  , enclosing a region .the junction angles , , measure the change in direction of the tangent at junction points . as a conventionwe assume that is traversed in a counter - clockwise direction .let be a vector at .then the angle is equal to where is the gaussian curvature .the second integral theorem is green s theorem , also known as the divergence theorem , or gauss theorem .suppose is a curve enclosing a region , transversed in a counter - clockwise manner , and a function defined on the surface .then the ( unconventional ) minus sign appears because the normal direction to was defined such that form a right hand system , so points _inwards_. for a surface the laplacian operator denotes the laplace - beltrami operator .  if two metrics are conformally related as in eq .( [ eq : conformal_relation ] ) , and the manifold with the metric is flat , i.e. a differential equation for can be derived .this can be done using the expression for the curvature in terms of .it is derived here in a different way , using the integral theorems quoted above , as this technique is used again in subsequent sections .suppose that a region equipped with the metric is flat , i.e. eq . ( [ eq : flatness_assumption ] ) holds in . then according to the gauss - bonnet theorem , eq .( [ gauss_bonnet ] ) , the length element for the two metrics discussed are related by substitute eq .( [ kappa_g_relation]),([ds_vs_ds_tilde ] ) in eq .( [ derive1:lap_phi_eq_k ] ) to get here green s theorem , eq .( [ greens_theorem ] ) , was used .subtracting eq .( [ derive1:lap_phi_eq_k ] ) from ( [ derive2:lap_phi_eq_k ] ) , where gauss - bonnet was used . sincethis result is true for an arbitrary flat region , the integrand must vanish , i.e. , for any point in where eq .( [ eq : flatness_assumption ] ) holds : eq .( [ lap_phi_eq_k ] ) is a differential equation for .it is a well - known result in conformal geometry , see e.g. , .the input to the mesh generation problem is assumed to be a surface embedded in three dimensional euclidean space , such that :  ( i ) : :  its boundary is a union of a finite number of  connected components . (ii ) : :  every component is a piecewise   closed curve .  the points where a boundary component is not differentiable will be called_ junction points_. the set of all junction points will be denoted by .the edges of the final mesh are to be laid along geodesic curves of a manifold that will be defined below , the -manifold__. to create a high quality mesh , these curves should cross each other , and reach the boundary , at certain angles . in the case of a quadrilateral mesh ,a cell with right inner angles is preferred for many applications . in the case of a triangular mesh ,the preferred inner angle is radians .it is therefore natural to define the direction of edges at a point to within an addition of radians in the quadrilateral case , and radians in the triangular case .this difference between the two cases leads to slightly different results ; for clarity of presentation , the quadrilateral case is presented first , and the results for the triangular case are defered to appendix [ triangular_mesh ] .the directions of edges at a point will be called a _cross_. the field of edge directions will be called a _ cross - field_.  [ cross]first , define an equivalence of vectors in : for 2 vectors , if and only if , are parallel or perpendicular .cross _ is an element of .thus , a given cross is a set of vectors , every pair of which are either perpendicular or parallel .if two vectors , defined at some point , undergo parallel transport along the same curve , the angle between two vectors is preserved : the angle before the parallel transport is equal to the angle after the parallel transport . therefore , the cross equivalence class structure is preserved , and the parallel transport of crosses is well - defined .let be a finite set of points in .denote the metric on by .supoose a metric is defined on .the following definition of a cross - field assures that the flow lines of the cross - field are geodesics , and that the crosses on the boundary are aligned with boundary .[ cross - field][cross_field_definition ] a _ cross - field _ on a given manifold with metric is a mapping such that :  ( i ) : :  for points ,  the parallel transport under the metric   of to along a curve   is independent of , and is  equal to :   .( ii ) : :  for   , the  tangent belongs to the cross there :   .[ ptb ]  the flow - lines curves of a cross - field are geodesics of the metric .that is , a geodesic aligned with the cross - field at one point ( i.e. , whose tangent belongs to the cross at that point ) , is aligned with the cross - field everywhere else on the curve .this is because both the cross - field and the tangent to a geodesic are parallel - translated along the geodesic , see definition [ cross_field_definition],(i ) , and the discussion preceding eq .( [ phi_manifold_geodesic ] ) .geodesic curves aligned with the cross - field will be called _ cross - field geodesics_. cross - field geodesics and their relation to the cross - field are illustrated in fig .[ c - frame_illustration ] .  as before ,let be a finite set of points in .we now define the -manifold__.  [ -manifold][def : phi_manifold]a -manifold _ _ is a riemannian manifold defined on , with metric such that :  ( i ) : :   is conformally related to   , i.e.   where   is a real function on . (ii ) : :  the -manifold is locally flat , i.e. its gaussian  curvature tensor for all points in   .( iii ) : :  for every boundary point ,   , the limits   and   exist and are continuous  along at .( iv ) : :  a cross - field with the metric exists .the points in will be called _cone points_. this name is justified in section [ sec : cone_points ] , see also section [ sec : overview ] . fornow , the points in are just points where is undefined .  in the following sections ,the requirement that a -manifold exists is translated into conditions on the function .the cross - field formulates the demand that mesh cells have certain inner angles . in order to have square - like shapes, the cells should further have edges of similar lengths .this is where the conformal metric property of the -manifold comes into play .  integrating over eq .( [ ds_vs_ds_tilde ] ) , the length of a curve on a -manifold is given by   the area of a region of a manifold is given by , where  .the tilde denotes , as before , a quantity with respect to -manifold metric .the following claim is a consequence of the isometry of a flat manifold to the euclidean plane .it is the manifold version of the properties of a rectangle .[ claim : rectangle]let be 4 geodesic segments in a flat region of a manifold , organized as in fig .[ rectangle ] .suppose that the inner angles at the vertices are right angles , and is the manifold - length of the i - th side of the rectangle .then the inner angle at , , is a right angle , and .the area of the rectangle is * * * * .[ ptb ]  on an euclidean plane , the properties of a rectangle allow one to lay a grid on a region of the plane . a grid can be regarded as two families of mutually perpendicular straight lines , with equal spacing between lines of each family . the same can be done for a -manifold using the rectangle properties stated in claim [ claim : rectangle ] , see fig .[ lay_grid ] .the grid divides the space into square - like regions , each bounded by four geodesic segments of manifold length , that will be called _ manifold - edges_. the regions enclosed by the manifold - edges will be called _ manifold - cells_.  [ ptb ]   is a single , fixed number for the mesh .the overall size on the surface or plane of the manifold - cells can be controlled by adding a constant to , so the freedom in choosing is redundant , and we set .according to claim [ claim : rectangle ] such a cell has unit -manifold area . according to eq .( [ eq : curve_manifold length ] ) , for small enough manifold cells ( large enough ) , a manifold edge of length has length , and is interpreted as the _ local edge length _ , or _local cell size_.  a geometrical interpretation of the relation , eq . ( [ phi_manifold_geodesic ] ) , can now be given .for a mesh with approximately square cells , the curving of manifold - edges is related to the changes in cell size _ in the perpendicular direction _ , see fig .[ fig : lay_grid_interpretation ] .this is quantified in eq .( [ phi_manifold_geodesic ] ) : is the curvature of the lines defining the edges , and is the change in , which is related to local cell size by .the existence of a cross - field restricts the possible behavior of in the vicinity of cone points .let be a cone point , and a point that is not a cone point .let be a simple closed curve starting from that encloses , and only of , see fig .[ loop_around_charge ] .the junction angles of are denoted by . according to cross - field definition ,the cross is parallel translated to along , that is , a vector is parallel translated along , to . by the defintion of a cross , are either parallel or perpendicular , andthe angle is , . using eq .( [ kappa_g_relation]),([gauss_bonnet ] ) : where is the region enclosed by .[ ptb ]  on a -manifold a manifold - cell has unit area , so in order to have a finite number of cells , the the area of the -manifold must be finite. this will further restrict the type of singularity allowed at a point , as is now shown .  for a cone point on the surfacelet be a neighborhood of in which there exist isothermal coordinates .in such coordinates , which can always be found locally , the metric takes the form : , where is a real function of .note that on the plane , the standard cartesian coordinates satisfy , and hence are isothermal . in isothermalcoordinates the laplace - beltrami operator can be written as into the definition of the laplace - beltrami operator : , see e.g. . ] eq .[ lap_in_isothermal ] can be seen as a generalization of the usual planar laplacian , given by .let be a disk of radius  in , i.e. the region for which . in isothermal coordinates eq .( [ lap_phi_eq_k ] ) reads in the _ punctured _ disk . in polarisothermal coordinates , ( that is , the polar coordinates corresponding to ) , the general form of this solution can be written as ( see e.g. ) : where is a solution to the poisson equation in ( _ including _ ) , and are all real numbers .let be a curve tracing the circle , in the counter - clockwise direction . for given by eq .( [ jackson_solution ] ) , the flux of through is where green s theorem , eq .( [ greens_theorem ] ) was used to convert the first term to a surface integral . in the limit eq .( [ eq : flux_from_cone_poine ] ) becomes : , and eq .( [ charge_possible_values ] ) with becomes : for some .comparing the two limiting values for we find for some .the quantity will be called the _ charge _ of the cone - point .( [ eq : charge_is_quantized ] ) states that the charges must be multiples of .  in order to have a finite number of unit - area manifold cells , the manifold - area of a neighbourhood of be finite .the manifold area of the disc of radius around is obtained by substituting eq .( [ jackson_solution ] ) into eq .( [ manifold_area ] ) : since a solution of is bounded in a compact region, does not effect the convergence of the integral , nor does , which is bounded away from zero . to converge , it is required that for every , and that .this restricts to the form with .this is a solution of the poisson equation for one point - source in a neighborhood of . for many cone points eq .( [ one_defect_solution ] ) becomes :  * condition 1 : *   _ _ obeys the equation _ _ _ the poisson equation with point sources ( delta functions ) _ _ _ , with _ _ _ _ , _ _ . _ _  [ ptb ]  for a planar domain , condition 1 can be rewritten as  * condition 1 ( planar domain ) : *  _ for _ _ _ , _ _ _ _ can be written as _ _ _ with _ _ _ , and _ _ _ _ on _ _ . _ _  fig .[ defects_3_5_inf ] shows selected cross - field geodesics for a planar domain around a singularity ( with ) , spaced at manifold - distances of  from each other , for different singularity strengths . in practice, good candidates for meshes will only have singularities of charge , due to the inner - angles of cells incident on the singularity , see also remark [ star_geodesics ] in section [ sec : case_c ] below .  to summarize this section, it has been shown that the existence of a grid of geodesics that follow a cross - field and create a finite number of cells , restrict the form that the function can take .this is formulated in condition 1 , stating that must obey the poisson equation , with delta function charges corresponding to cone points .this section examines the conditions for boundary alignment of a cross - field , see definition [ cross_field_definition],(ii ) .three conditions will shown to be necessary .sufficiency of the conditions is discussed in section [ sec : sufficient_conditions ] .we start by developing an equation that will be used in the derivation of the conditions below .let be two points on two boundary curves , respectively .note that may be the same boundary curve : .let be some curve from to , see fig .[ condition4_2_boundaries_surface ] . for by . by definition of a cross - field, the cross in must be parallel - translated along to the cross in .the cross in contains , and the cross in contains so   [ ptb ]  where is defined as the last equality is due to the preservation of angle in parallel transport , together with eq .( [ kappa_is_dtheta_ds ] ) .( [ kappa_g_relation ] ) and ( [ ds_vs_ds_tilde ] ) can now be used with eq .( [ eq : cond_4_proof_1 ] ) substituting this into eq .( [ eq : cond_4_proof_0 ] ) we find the relation eq .( [ eq : general_boundary_cond_equation ] ) is used below to develop the next three conditions .[ ptb ]  let be a segment of a boundary curve , between points that are not junction points : , and suppose does not contain junction points . choose some smooth curve from to ,whose tangents at coincide with the tangents to , see fig .[ kappa_eq_d_phi_ds ] , i.e. , and such that the region enclosed by does not include any cone points . according to eq .( [ eq : general_boundary_cond_equation ] ) , according to gauss - bonnet , eq . ( [ gauss_bonnet ] ) , where is the area enclosed by .green s theorem gives substituting eq .( [ cond_2_proof2]),([cond2_proof3 ] ) and recalling that  in ( see eq .( [ lap_phi_eq_k ] ) ) , eq .( [ cond_2_proof1_1 ] ) now reads   when , the integrals both tend to zero , and the equality is possible only if . then and with it follows that  * condition 2 : *  _ for a point _ _ with boundary curvature _ _ _ , _ _ _ must satisfy _the equation in condition 2 is the same as eq .( [ phi_manifold_geodesic ] ) .this is not incidental : condition 2 causes cross - field geodesics , that are close and parallel to the boundary , to follow the shape of the boundary , as shown in fig .[ boundary_aligment ] .for a point , that is , a junction or cone point of the boundary , let be two points on both sides of the junction point , and the boundary segment from to , see fig .[ condition3_boundary_singularity ] .let be a curve from to .the gauss - bonnet theorem , eq .( [ gauss_bonnet ] ) , for the curve  ] .( note that eq .( [ eq : 1 need to show in theorem ] ) is equivalent to the requirement that the cross in be parallel - translated to itself along . )we now prove eq .( [ eq : 1 need to show in theorem ] ) . the region enclosed by can contain singularities and boundary curves .surround them by additional curves , as shown in fig .[ sufficient_theorem1 ] .denote the region between the curve and the and curves as ( here the assumption that has genus zero enters ) . since , eq .( [ lap_phi_eq_k ] ) , everywhere in , and are finite on the boundary of ( due to -manifold definition , ( iii ) ) , green s theorem , eq .( [ greens_theorem ] ) , can be applied in this case , giving where are the number of points in , and inner - curves enclosed by .consider a single loop around an inner curve , . is composed of curves around points of on the boundary , and curves between junctions , i.e. $ ] , see fig .[ sufficient_theorem1 ].   the total flux through is given by with .the second equality uses conditions 2,3 .the flux through is given by condition 1 : after substituting eq .( [ eq : total_flux_in_theorem ] ) , ( [ eq : total flux singularity ] ) , eq .( [ eq : total flux in theorem ] ) becomes and using gauss - bonnet theorem for a domain with more then one boundary component we find for some .this proves eq .( [ eq : 1 need to show in theorem ] ) .we now turn to prove the property ( ii ) of a cross - field , its boundary alignment .let be a boundary point , , .as was shown above , the cross at , , is well - defined .it is left to show that  .denote .the assumptions of the theorem , together with condition 4 , assure that  . let * * be the section of between and . define a curve from to composed of smooth curves around the points of in the image of , and curves between junctions , see fig .[ sufficient_theorem2 ] .the curves avoid the sigularities that the function might have at points in . let be the junction angle at the point of bypassed by the curve ( zero for points that are not juction points ) . then, if follows closely , the total turn of is equal to : as can be formally verified by applying the gauss - bonet theorem to the area between and .note furthermore that , the inner angle at that point .the variation of along is with .conditions 2 and 3 were applied for the and curves , respectively .thus  , which completes the proof of the boundary properties of the cross - field .           m. s. floater , and k. hormann , surface parameterization : a tutorial and survey , in _advances on multiresolution in geometric modelling_. m. s. f. n. dodgson and m. sabin , eds .( springer - verlag , new york , 2004 ) .m. hmlinen , r. hari , r. j. ilmoniemi , j. knuutila and o. v. lounasmaa , magnetoencephalography - theory , instrumentation , and applications to noninvasive studies of the working human brain , _ reviews of modern physics _ * 65 * issue 2 ( 1993 ) 413 - 497 .

Abstract: a continuum description of unstructured meshes in two dimensions , both for planar and curved surface domains , is proposed . the meshes described are those which , in the limit of an increasingly finer mesh ( smaller cells ) , and away from irregular vertices , have ideally - shaped cells ( squares or equilateral triangles ) , and can therefore be completely described by two local properties : local cell size and local edge directions . the connection between the two properties is derived by defining a riemannian manifold whose geodesics trace the edges of the mesh . a function , proportional to the logarithm of the cell size , is shown to obey the poisson equation , with localized charges corresponding to irregular vertices . the problem of finding a suitable manifold for a given domain is thus shown to _ exactly _ reduce to an inverse poisson problem on , of finding a distribution of localized charges adhering to the conditions derived for boundary alignment . possible applications to mesh generation are discussed .  unstructured mesh generation , differential geometry .


the 1969 the internet ( then arpanet ) was first established as a distributed packet communications network that would not only reliably operate if some of its nodes were destroyed in an enemy attack , but allow easier communications of computer research results by universities .today the internet has grown to become a sprawling network of every aspect of humanity dwarfing previous technological mediums in both complexity and behavior .it was therefore only a matter of time that advanced statistical techniques , such as those developed by physicists in statistical mechanics were applied to investigate it .  since the late 1990s, the internet has been of interest to the physics community , becoming aware to most in the seminal nature paper of watts and strogatz .this was continued or paralleled by the work of countless others .however , until recently this research has focused mostly on the topological aspects of networks and much less on dynamics . a particularly fertile area on network dynamics , and one related to this paper is the study of phase transitions from free flow to congestion in computer networks .most results give a critical packet flow on networks which separates free flow from congested traffic .there have been some investigations of dynamics aspects such as synchronization of coupled oscillators and some metabolic dynamics as well thus dynamics is rapidly moving from being a peripheral to a primary discussion about networks .a very interesting reverse situation is visible in the studies of the statistical mechanics of vehicular traffic .vehicle traffic on roads has been investigated , also using statistical mechanics , but focusing on the dynamics and flow of traffic versus the topology of the road network .though there is some overlap between these two topics , traffic flow has been described in many ways with the most common description using the fundamental diagram of vehicle flow vs. vehicle density .most models propose a two or three phase model of traffic .the first phase is free flow , where cars drive near the speed limit with little congestion and influence on each other s velocity .the final phase is congested traffic where traffic flow becomes spontaneously congested after reaching a critical density . in the three phase model, there is an intermediate phase called synchronized flow where traffic is not congested but cars match their speed at a reduced speed level effectively increasing the correlation length of the system as a prelude to congestion . in gabor and csabaiconduct a similar study comparing vehicle traffic flow to data packet flow using the number of tcp connections as the variable for flow density .they find that a fundamental diagram like pattern appears in data traffic when the flow density is modeled as the number of tcp connections between two different endpoints .in investigating internet dynamics , one is struck by how similar the dynamics of the network can be superficially similar to vehicle traffic .internet networks are made of the flow of countless packets across network links and can be prone to the same free flow or congestion that vehicle traffic can be . like traffic data , data on internet trafficis easily obtainable either by setting up packet sniffers and traffic analyzers between certain nodes or using public data sets such as the wide project s mawi data traces from trans - pacific us - japan t1 lines or the stanford linear accelerator center ( slac ) s pinger project which has monitored icmp ping response against different nodes across the internet on a continuous basis for years .  however , internet traffic data analysis is complicated by many factors that are not easily accountable for in theoretical models .first , since the internet is a decentralized network based on dynamic routing , the route of the traffic is not completely transparent .the path from one point to another can be fairly fluid and changing even within a continuous flow of transmitted data .many of the intermediate routers or autonomous systems ( as ) , which are basically internet service providers , have private and constantly changing routing rules and configurations that alter traffic in unpredictable ways to obtain certain quality of service metrics or traffic shaping priorities .this makes analysis of data collected from internet traffic fraught with questions and confusion as far as disambiguating the effect of network topology on dynamics .in addition to topological constraints , the dynamics of internet traffic itself can affect dynamics studies . under the internet protocol ( ip ) suite, there are many transport level protocols such as tcp and udp and countless application level protocols such as http among others .protocols at all levels , including transport and application levels , can influence data traffic . for example, tcp has a congestion control algorithm which will actually throttle network speed given the feedback it receives from packet loss data on the network , requires periodic acknowledgements from the destination before sending more data , and will buffer data to send depending on the round trip time ( rtt ) of the connection in order to guarantee delivery .therefore , measurements of tcp / ip network speeds , even in relatively `` clean '' networks , can actually be extremely complicated and dependent on much more than the topology of the network or volume of traffic .finally , the statistical nature of internet traffic is still poorly understood .internet traffic volumes and packet interarrival times to not follow typical distributions in other communications networks such as poisson or erlang distributions but rather exhibit bursty , self - similar traffic patterns which have been very difficult to model and predict . like in measurements of internet topology where a relatively small amount of nodes have many edges , almost everything in the internet that can be measured seems to have a long - tailed distribution as a matter of course .tcp and udp traffic flows follow this trend where a relatively small number of flows carry to bulk of data transferred ( so - called `` elephant flows '' ) .the origins of these patterns of traffic are still a matter of research and debate .add to this other inherent uncertainties and patterns in internet traffic flow such as trimodal distributions of packet sizes , traffic spikes to due to malicious code such as viruses or trojan directed distributed denial - of - service attacks , and periodicities in the volume of traffic caused by 12 hour , 24 hour , and 7 day cycles ( with a 3.5 day harmonic ) .there is also an issue of long - range correlations of internet traffic between different routers which can be relatively uncorrelated or very correlated depending on the nature of traffic and the level of congestion .all of these factors are mentioned to demonstrate that modeling and understanding internet traffic dynamics is a problem likely of greater magnitude than topological analysis . given these difficulties and more , understanding the basics of traffic dynamics and the interactions between topology and dynamics in computer networks is essential to understanding theoretical aspects , creating accurate simulations , and conducting useful experiments .the purpose of the experiment described by this paper is to ask a basic question about traffic dynamics : how is the throughput ( speed ) of a link affected by three fundamental variables that determine the nature of network traffic : the average packet size , the average packet flow rate , and the bandwidth of the link .the bandwidth of the network , in this case 100 mbps ( megabits per second ) , is the theoretically maximum possible throughput .the throughput itself can be described in terms of the average packet size and average flow rate by where is the average throughput of the link , p is the average size of packets in a transmission over a given period of time and is the average flow rate in packets per second . in this experimentsince all packets will be off the same size in each sample we can say two computers , both running windows xp , were connected using an cat 5e cable that connected to the ethernet network adapter ( nic ) of both computers .the ethernet flow control was disabled to ensure that the flow of data , and not signaling between the computers to prevent dropped packets that flow control entails , determines the throughput .traffic was generated using the program iperf which generates a stream of identically sized packets at a throughput inputted by the user .the throughput in iperf was designated at 100 mbps in order to test what the maximum throughput the network would actually demonstrate given an attempt at maximum throughput .the transport protocol udp was chosen over tcp for several reasons .udp is a connectionless protocol , meaning it does not guarantee delivery , and will solely submit a string of packets .tcp in a connection based protocol whose delivery guarantee requires frequent `` handshaking '' between the source and destination and whose congestion control algorithm can affect throughput in a non - trivial fashion delivering a lower throughput due to protocol software , not the maximum throughput of the link . finally , tcp can give different performance over links with different latencies , as measured by packet rtt , so the results may not be subject to larger generalization .  to test the performance of the network under different sized packets ,the udp packet payload was varied from 25 bytes up to 1450 bytes in 25 byte increments . in order to ensure that equation [ eq2 ] holds, you must ensure that all packets are the same size .the structure of packets under ethernet / ip is shown in figure [ overhead ] .the payload , a variable selected in the iperf software , is encapsulated by a header for udp ( 8 bytes ) , ip ( 20 bytes ) , and a header and footer in the ethernet frame ( total 18 bytes ) .frame is just a general term for an ethernet packet .these headers mostly provide routing data , priorities , checksums , and other information important to packet logistics .in addition , in standard ethernet the maximum frame size , minus ethernet headers and footers , is 1500 bytes . with the ethernet overheadthe total maximum size for an ethernet frame is 1518 bytes .therefore , at 1475 bytes payload , the total would be 1503 bytes and you would have packet fragmentation - instead of one frame you would have two , one with a frame payload of 1500 bytes and a second with a frame payload of 3 bytes .this would affect throughput by showing a sudden change since average frame payload size would drop to about 750 .given the phenomenon of fragmentation , an actual analysis of the effect of packet size on throughput is only useful up the fragmentation size limit .in the experiment , iperf delivered a packed stream of udp traffic from the client to the server computer , trying to send as close to bandwidth as possible , and outputted the average throughput in kbps ( kilobits per second ) .it also gave the packet loss , and a measure called jitter which is not used but measures the deviation in packet interarrival times versus interdeparture times .for throughput , iperf measures a related measured called goodput which measures data speed in terms of the payload size , not including any packet overhead in the calculation of bytes transferred .however , the packet flow rate is accurate and is calculated from equation [ eq2 ] . therefore the actual throughput was recalculated using packet sizes that include both payload and packet overhead .at all packet sizes , packet loss was small , much less than 1% .the results of the experiment are shown in figure [ graphs1 ] .first , it is clear that throughput decreases with decreasing packet size .this first fact is well - known in the network engineering community .this is an inherent property of all network adapters , ethernet or otherwise .in fact , one of the key requirements in next generation networks is the network capability to send `` jumbo frames '' where fragmentation limits at the network layer are much larger than 1500 bytes , up to 9000 bytes in some cases .these larger packet sizes cause higher throughput and more efficient networking because within the network adapter and computer hardware , there is a per - packet processing overhead .  despite the bandwidth rating of network adapters , be it 10,100 , or 1000 mbps , there is a maximum packet flow that they can effectively handle . given equations [ eq1 ] and [ eq2 ] it is clear that to maintain any given throughput , by lowering the packet size you are increasing the packet flow . because of the packet flow processing bottleneck in the hardware , however , this can make high throughput impossible at low packet sizes .as seen in figure [ graphs1 ] , for large packet sizes , the throughput is very close to bandwidth and can be roughly equivalent to free flow traffic . at a specific critical packet size , however , , the throughput begins to rapidly degrade to the point it is only about 6% of bandwidth at 25 bytes .this is the saturated state . heresaturation is used instead of congestion since congestion is usually a network wide phenomenon while this packet slowdown in throughput is due to overwhelming the processing power at the nic . in a superficial waythis behavior is similar to the fundamental diagram flow - density curve in vehicle traffic .packet flow increases with smaller packet size until saturation forces packet flow to begin to slow its increase and approach a maximum value . comparisons between vehicle traffic should be qualified though . in vehicle traffic , there is interaction between cars on the road giving rise to the collective dynamics which justify a statistical mechanics interpretation . on data networks , packets do not interact with each other and packet collisions are errors , not intrinsic aspects of the packet flow .therefore , as will be shown below , the transition from free flow to saturation should be viewed as a bifurcation in the system dynamics , not as a phase change . on the network level where there are many interacting nodes , perhaps congestion can be seen as a phase change but this perspective is not appropriate at the single link level .  as stated earlier , in free flowthe throughput is nearly bandwidth and comparatively , though not completely , steady state . in this region, there is a mutual relationship between packet size and packet flow .differentiating equation [ eq2 ] we have    assuming is 0 in free flow regardless of the packet size or flow we can conclude    so there is a tradeoff curve , like the production possibility frontiers in economics , between packet size and flow in free flow traffic .also ,    demonstrating that every increase in packet flow is matched by a corresponding decrease in packet size and vice versa .because the networking and computer equipment have various processes and imperfections our free flow region never reaches bandwidth and steadily erodes with smaller packet size , however , the high throughput feature is relatively constant compared to the saturated state .  at a packet size , we have an increasingly rapid breakdown in throughput .this corresponds approximately to the maximum flow to the network adapter and the breakdown into saturation .we calculate the theoretical maximum flow as    where in the theoretically ideal situation where is the bandwidth . therefore in the saturated region ,the throughput given by    so the ratio of packet size to the critical packet size determines the throughput under saturation compared to the maximum possible throughput . in the first graph of figure [ graphs1 ] , is a comparison of this prediction with the observed data in the saturated region where is about 96 mbps .given data from the saturated region for , , and we can estimate the critical packet size .in this case , the critical packet size is approximately 451 bytes at around 25k packets / s .though this model seems to accurately predict the throughput values for almost the entire region of saturation , there is a seemingly glaring contradiction in the second graph of figure [ graphs1 ] where the packet flow keeps increasing with decreasing packet size .though the packet flow does not stagnate , as it should according to ideal theory , the third graph shows that the decline in throughput over a fairly short range of packet flow demonstrates packet flow is the limiting factor in the saturated region . in the congested region the packet flow increases from about 25 to 30 thousand flows per second which is a definite increase but relatively small compared to the region of free flow when it increased from 7 to about 25 without affecting throughput .note , is specific to the equipment and configuration used and does not have a general value of 451 bytes .the fact that the stable throughput of the system changes with the parameter of packet size leads us to suspect a possible bifurcation .the exchange of stability between quasi - bandwidth throughput and packet flow limited throughput leads to the hypothesis of a transcritical bifurcation with stability changing at . assuming the two stable throughput regimes are and , so    and    this can be seen as the traditional form of a transcritical bifurcation where , however , this is made more clear by making the independent variable :    which fits the normal form for a transcritical bifurcation , .the transcritical bifurcation can be more clearly seen in the graph of vs. the reverse order axis with in figure [ bifurcationgraph ] .as mentioned before , the transition from free flow to saturated traffic here is a bifurcation not a phase change given the lack of interaction among the constituent particles in the system .this conclusion can also give pause to extrapolations of `` classical '' flow on network theory to complex networks such as computer networks .if looking at the weights and capacities of the network from the perspective of throughput , typical maximum flow algorithms such as max - flow min - cut may give incorrect answers if the limiting aspect of the flow is the packet flow rate , size of the packets , or another issue that is not plainly visible .flows in computer networks do not behave like incompressible fluid or similar flows assumed in most flow models where any flow rate freely flows up to the capacity minus costs incurred by friction , etc .another point is other similarities to vehicle traffic dynamics .interestingly , the first graph of figure [ graphs1 ] is similar to a flow - density curve seen in traffic models .even equation [ eq1 ] has similarity to such traffic models where   where is the traffic flow in cars / h and is traffic density in cars / km and is flow velocity in km / h . there is a critical traffic density that separates free flow from synchronized flow or congestion that plays a similar role to packet size in data networks .this paper does not address the larger problem of dynamics on networks , however , it is the contention of this paper that understanding the simple dynamics at the network level is essential to understanding the wider implications of network sized dynamics . given the problems with using internet traffic data described earlier , more understanding in the basics of traffic dynamic may be obtained through experimental setups or computer network simulations .future research should look at how topological invariants in networks affect dynamics and how dynamics may affect the evolution of networks and changes in their topological invariants .  as a final note regarding research on dynamics in networks , particularly data networks and the internet, the author believes that more interaction and cross - referencing between the engineering and physics communities will help promote better understanding and advancement .although there is research at an almost feverish pitch in both the physics and engineering community on networking , both sets of publications seem to be almost entirely ignorant of each other . in physics ,one of the only such publications commonly cited is the faloutsos team s work on the router topology of the internet .engineering researchers , in journals such as those published by ieee and acm , also have rarely quoted physics literature besides the most well - known papers of barabsi , watts , or strogatz and seem largely unaware of the later results being reached by physics in the topology of networks . without a detailed understanding of the network protocols and engineering literature regarding the function of data networks and the internet , this paper would not have been possible .a similar situation is seen in sociology where a wealth of research has been done on aspects of social network dynamics such as the diffusion of trends along networks , in a quantitatively rigorous fashion , but again both communities do not usually inform each other of progress or motivate collaboration . as the study of networks expands and matures , it will become necessary to read and reach across interdisciplinary boundaries to share tools and knowledge that will allow the most profound and predictive insights to be reached .r. albert , h. jeong , and a.l .barabsi , `` diameter of the world - wide web '' , _ nature _ , vol .401 , pp.130 - 131 , 1999 m.e.j .newman , `` scientific collaboration networks . i. network construction and fundamental results '' , _ phys .64 , 016131 , 2001 m.e.j .newman , `` scientific collaboration networks .ii . shortest paths , weighted networks , and centrality '' , _ phys .e _ , vol .64 , 016132 , 2001 s.n .dorogovtsev and j.f.f .mendes , `` exactly solvable small - world network '' _ europhys .50 , pp . 1 - 7 , 2000 m. faloutsos , p. faloutsos , and c. faloutsos , `` on power - law relationships of the internet topology '' , _ comp ._ , vol . 29 , pp.251 - 262 , 1999 r.pastor - satorras r and a. vespignani , `` epidemic spreading in scale - free networks '' , _ phys .86 , pp.3200 - 3203 , 2001 t. ohira and r. sawatari , `` phase transition in a computer network traffic model '' , _ phys .58 , pp.193 - 195 , 1998 m. takayasu , h. takayasu , and k. fukuda , `` dynamic phase transition observed in the internet traffic flow '' , _ physica _ a , vol .277 , pp.248 - 255 , 2000 r.v .sol and s. valverde , `` information transfer and phase transitions in a model of internet traffic '' , _ physica a _ , vol .289 , pp.595 - 605 , 2001          c.f .daganzo , m.j .cassidy , and r.l .bertini , `` causes and effects of phase transitions in highway traffic ( research report ) '' , university of california , berkeley institute of transportation studies research report , report no .ucb - its - rr-97 - 8 , 1997      h. chang , s. jamin , z.m .mao , and w. willinger , `` an empirical approach to modeling inter - as traffic matrices '' in _ proc . of the 5th acm sigcomm conference on internet measurement _ , berkeley ,ca 2005 , pp.139 - 152         t. mori , r. kawahara , s. naito , and s. goto , `` on the characteristics of internet traffic variability : spikes and elephants '' , in _ proc . of the 2004 international symposium on applications and the internet, tokyo , japan , 2004 , pp.99 - 106   j. cowie and a. ogielski , `` global routing instabilities during code red 2 and nimda worm propagation ( conference report ) '' , caida internet statistics and metrics analysis workshop , san diego , ca , december 2001    k. papagiannaki , n. taft , z.l .zhang , and c. diot , `` long - term forecasting of internet backbone traffic : observations and initial models '' in _ proc . of infocom 2003 _ ,san francisco , ca , vol . 2 , 2003 , pp.1178 - 1188

Abstract: the dynamics of user datagram protocol ( udp ) traffic over ethernet between two computers are analyzed using nonlinear dynamics which shows that there are two clear regimes in the data flow : free flow and saturated . the two most important variables affecting this are the packet size and packet flow rate . however , this transition is due to a transcritical bifurcation rather than phase transition in models such as in vehicle traffic or theorized large - scale computer network congestion . it is hoped this model will help lay the groundwork for further research on the dynamics of networks , especially computer networks .


in wireless ad hoc networks the participating nodes typically engage in group communication such as broadcasting , multiple - access , or relaying operations , where more than two nodes may be simultaneously involved .these sophisticated protocols are motivated by mission - critical goals such as achieving high - throughput , or supporting resource - constrained nodes ( such as bandwidth and power ) in the network . in order to secure group communication from external eavesdroppers , the nodes use group secret keys ( gsks ) to exchange confidential data , and are also involved in key management tasks such as key generation , distribution and maintenance .while strengthening the security of key management is an evolving research field , researchers are concurrently focussing on methods to dynamically generate secret keys by exploiting randomness available in the environment .the purpose of dynamic key generation is either to use it as an independent recipe of generating keys , or to use it to update the existing keys .dynamic key generation techniques are well known , and have been particularly well studied in the context of wireless two - user physical - layer key generation ( pkg ) . in pkg , the two participants observe a common source of randomness to generate a common key without leaking information to an external passive eavesdropper .although , the sources of randomness are innumerable , easily accessible ones are the variations in the wireless channel strengths , such as the received signal strength indicator ( rssi ) values , complex baseband signal samples , and the channel estimates .other than the concept of generating secret - keys for two nodes that are within radio frequency ( rf ) range , some works - have also addressed using third - party relay(s ) to assist the secret - key generation between nodes that are out of range .  a generalization of relay - assisted secret - key sharing is the concept of wireless physical - layer group secret key ( gsk ) generation , wherein a group of nodes in the network generate a secret key by observing a common source of randomness , , .unlike two - user wireless pkg , in gsk generation , nodes in the group exchange additional messages apart from the pilots so that all the nodes witness the common source of randomness .gsk generation is applicable in ad hoc wireless networks that operate in mobile environments , e.g. , manets and vanets .existing works on gsk have focused on developing protocols for exchanging messages so as to ( i ) reduce the latency , and ( ii ) to secure the generated keys from an external eavesdropper , , . also , the models assume trusted group members and secure the communication against passive eavesdroppers and active attacks from external nodes . in this work , we deviate at this point , to not to discard the possibility of the presence of insiders that involve in active attacks , that disturb the key generation process . in particular , we incorporate the possibility that some legitimate nodes might be compromised ( for e.g. , as in ) , and the attackers might enter the system as insiders before the commencement of key generation .our specific contributions are :  * we introduce a novel threat model with insider - compromise , that prior works have not considered .some crude attacks in our model are to stop relaying signals in the key - generation process , or to inject noise into the system , or to prevent the neighbouring nodes in achieving consensus on the group secret key .we , however , consider more sophisticated attacks that enable the attacker to manipulate the key generation process and yet go undetected .more specifically , we have proposed the following two types of attacks : ( i ) different - key attacks , wherein an insider attempts to force different realizations of key at different nodes , especially in a network when the legitimate nodes do not have private channels to verify the attack ( for e.g. , in chain - topology based networks ) , and ( ii ) low - rate key attacks , wherein an insider attempts to slow down the channel variations over time , in order to reduce the _ secret - key rate _ , which is a measure of how fast the generated keys can be updated ( section [ sec : threat_model ] ) . * to detect the above attacks , one can envision a scheme that uses a separate communication channel built on neighbour discovery and explicit pair - wise communication with the legitimate neighbours ; however , such a straightforward scheme introduces new sets of vulnerabilities and can be expensieve in terms of communication - overhead .therefore , we apply detection schemes that are based on standard phy - layer techniques such as measuring average power levels and doppler spread ( variations of channel over time ) to identify anomalies and inconsistencies .we show that the proposed insider - attacks can go undetected on the state - of - the - art gsk generation schemes with a non - negligible rate despite using off - the - shelf channel measurement techniques .this implies that gsk generation schemes are vulnerable to insider - threats , especially on topologies that can not support additional secure links between neighbouring nodes to verify the attacks .( section [ sec : sim_results ] )consider a chain - topology based wireless network of nodes , denoted by the set , where the nodes want to securely communicate among each other using a gsk .we assume that a gsk is dynamically generated at the nodes by exploiting common randomness in the wireless channels . the wireless channel between node and node is denoted by a discrete random process , where the index captures the realization of the channels at different time instants .all the channels are assumed quasi - static with identical coherence interval of seconds , and thus can be viewed as the output of sampling a continuous random process at for . here, denotes the set of channel estimates derived from the complex baseband samples at the receivers .we assume that nodes communicate on a single frequency but in time division duplex ( tdd ) fashion , i.e. , no two nodes transmit at the same time . also , by the virtue of channel reciprocity , we assume that nodes and observe the same channel environment between them , i.e. , .further , the network topology is assumed to be fixed over several blocks of coherence intervals .  in the above network model ,the nodes intend to generate a gsk by observing an accessible and secure source of common randomness . unlike two - node physical - layer key generation , which requires a minimum of two rounds of pilot exchanges , the multi - node generalization requires more exchanges ;the number of times a node transmits depends on the underlying protocol , and also the chosen source of common randomness . for the chain - topology ,the set of all channels for a given is . for this case ,the common source of randomness is some function for .since each node can broadcast its pilot to a maximum of two nodes , some nodes need to forward their observations so that all the nodes witness .we assume that the coherence time is longer than one round of the gsk generation protocol . a generic protocol for exchanging the observations is given below :  * for * * for * * * node broadcasts pilot symbol(s ) , which are used by the rest of the nodes in its rf range to learn the corresponding channel estimates . + end * * depending on the target , the nodes exchange their acquired set of channel estimates , one after the other until each node learns .+ end  in practice , gsk generation protocols have to be repeated times to accumulate the required number of bits for the digital key ( say bits ) .therefore , the number depends on number of bits generated in every round ( say bits ) .also , the nodes go through additional rounds of communication over a public channel to arrive at consensus on the generated digital secret . in this work, we only focus on the protocol for sharing observations , as it is fundamental to establishing the common source of randomness .for some examples of physical - layer gsk generation , we refer the readers to , .consider a small network of three nodes as shown in fig .[ fig : gsk_protocol ] , where the nodes 1 and 3 are out of each other s rf range .this implies that nodes 1 and 3 can communicate only through node 2 .we assume that the topology remains fixed for rounds of the protocol . for a given ,the set of underlying channels of the network is , which is the set of complex channel estimates . for this network ,let the common source of randomness be , which implies and is a product function . to gain insights , we assume that the nodes in the network do not experience additive noise ( the noisy case is treated in section [ sec : sim_results ] ) .to achieve the target at all the nodes , the following protocol is followed during the -th round .   a gsk generation protocol between nodes 1 , 2 and 3 , where denotes the channel between nodes 1 and 2 , and denotes the channel between nodes 2 and 3 . the protocol includes 4 phases ( read clock - wise from top left ) , at the end of which all the 3 nodes learn the common source of randomness .]  * * phase 1 * : node 1 transmits a pilot symbol , which is used by node 2 to learn , i.e , here denotes the observation seen by node at the end of * phase *. * * phase 2 * : node 3 transmits a pilot symbol , which is used by node 2 to learn , i.e , * * phase 3 * : node 2 transmits a pilot symbol , which is used by node 1 to learn , and node 3 to learn , i.e. , * * phase 4 * : finally , node 2 broadcasts the sum to the other nodes .node 1 and node 3 receive and , respectively , i.e , since nodes 1 and 3 already know one of the components in the linear combination , they cancel the self - interference to retrieve , i.e. , meanwhile , since node 2 has the realizations of two channels , it computes the product to acquire the common randomness .  the above protocol is depicted in fig .[ fig : gsk_protocol ] by dropping the index . in this illustrative example, the order of the first three phases * phase 1 * , * phase 2 * and * phase 3 * can be permuted in any of 6 possible ways .we study insider - attacks from one of the nodes in the network ( specifically the middle node ) , and leave open the general problem of multiple - insiders or collaborative insider - attacks for future research .we assume that the attacker in the network has the following capabilities : _ assumption 1 : _ the insider can transmit signals with higher power than the rest of the nodes ._ assumption 2 : _ the insider has large storage size to store the past channel realizations of the network ._ assumption 3 : _ the insider can learn the topology of the network .we envisage the following two types of attacks from an insider during the key generation process .   in gsk generation , the nodes have to securely exchange their observations so that they view correlated source of randomness .a potential threat is the different - key attack , wherein an insider forwards messages to partition the network into two or more subgroups such that nodes across subgroups observe uncorrelated source of randomness . as a result ,the nodes across subgroups are forced to generate a different digital - key than the rest of the nodes in other subgroups .if the insider achieves such an attack , then the nodes believe that they have the correct group key. however , during the data communication phase , they will fail to accurately recover the group messages due to mismatched keys ._ assumption 2 _ and _ assumption 3 _ may have to be invoked to realize this attack .we exemplify a different - key attack on the protocol of section [ subsec : ie ] , by introducing the attack parameters such that the special case of represents the no - attack case .   for the protocol discussed in section [ subsec : ie ] , node 2 gets two chances to sabotage the key - generation process in every round ; first time when it transmits the pilot , and the next time when it transmits the sum of two channels .the most general attack scenario from node 2 is the following :  * in * phase 3 * , node 2 transmits other than . * in * phase 4 * , node 2 transmits instead of , where and are chosen based on the attack - objective .  since in * phase 3* , the nodes 1 and 3 believe that their channels to node 2 are and , respectively . upon completion of * phase 4 * , the observations at node 1 and node 3 after cancelling self - interference are respectively given by ,  the objective of the different - key attack is to make sure that and are uncorrelated , which can be achieved by choosing statistically independent of . in this example , _ assumption 2 _ has to be invoked in order to store the channels , and _ assumption 3 _ has to be invoked for node 2 to learn the chain - topology .  in -, we illustrate the effect of the different - key attack on the digital keys , by quantizing the values of and into two levels .although more sophisticated algorithms exist in literature to generate digital keys , we have used a primitive method to showcase the effect of the proposed attacks on the digital keys .,\\ \label{eq : digital_dka2 } \mathcal{k}^{1}_{\mbox{attack } } & = & \left[0 ~ 1 ~ 0 ~ 1 ~  1  ~   0 ~   0  ~  1 \right],\\ \label{eq : digital_dka3 } \mathcal{k}^{3}_{\mbox{attack } } & = & \left[0 ~ 0 ~  1  ~  1  ~  0  ~  1 ~   0  ~  1\right].\end{aligned}\ ] ]  in the above example , 8-bit digital group keys are generated with ( denoted by and for node 1 and 3 , respectively ) and without ( denoted by ) the different - key attack by node 2 . with the different - key attack ,the keys generated at node 1 and node 3 are different .   in physical - layer gsk generation, the number of bits generated from every round of the protocol determines the key - rate . to illustrate further ,if bits of gsk are generated from each round , and the absolute time difference between successive rounds is seconds , then the key - rate is bits / sec .this implies that to generate a random bit sequence of length , say for application under advanced encryption standard , the key generation method requires seconds .thus , if we wish to update the -bit key frequently , then we should use wireless channels that support high entropy , and/or use protocols that have low - latency .  with the above discussion on key - rate , a possible insider - attack is to lower the key - rate .this can be accomplished either by reducing the entropy of the channel , i.e. , by reducing the number , or by enforcing a slow - varying channel so that the accumulated bits look structured rather than random ._ assumption 1 _ and _ assumption 2 _ may have to be invoked to realize this attack .we illustrate a low - rate key attack on the protocol of section [ subsec : ie ] . in the next subsection, we introduce the attack parameters for the low - rate attack such that the special case of represents no - attack .   to execute a low - rate key attack , the fundamental criterion is to enforce and are identical and vary slowly over time .an attack strategy from node 2 is given below :  * in * phase 3 * , instead of , node 2 transmits . * in * phase 4 * , node 2 transmits instead of , where and suitably chosen .  under the above threat model ,choosing forces although nodes 1 and 3 observe identical source of randomness , their realization has been changed by . with this attack ,the nodes are forced to believe that the individual channels are and , instead of and , respectively . for the above model, the attacker can replace and by a slow - varying channel by choosing as thus , this attack forces the nodes to generate a structured digital key rather than a random one .  in - , we illustrate the effect of low - rate key attack on the digital keys , by quantizing the values of and into two levels .,\\ \label{eq : digital_lrka2 } \mathcal{k}_{\mbox{attack } } & = & \left[1 ~1 ~1 ~1 ~1 ~1 ~1 ~1\right].\end{aligned}\ ] ] in the above example , 8-bit digital group key is generated with ( denoted by ) and without ( denoted by ) the low - rate key attack by node 2 . with the low - rate key attack ,the channel realizations are forced to vary slowly over time , thereby ensuring the quantized values to lie in one of the regions .in this section , we discuss possible techniques to detect insider - attacks during the key generation protocol .detection methods can be broadly classified into two types : ( i ) direct detection and ( ii ) indirect detection . in the former class ,the nodes communicate among each other to verify the correctness of the generated key by exchanging either the common source of randomness or the generated key ( preferably over a private channel to shield leakage from an eavesdropper ) . in the latter class of methods ,the nodes do not explicitly exchange keys or common source of randomness , instead they measure some physical characteristics of their channels to identify anomalies or inconsistencies . for the chain - topology based networks ,former class of methods suffer from higher communication - overhead ( e.g. , nodes need high transmit power to reach the farther nodes ) , and hence , we focus on the following standard methods in the latter class : ( i ) measurement of doppler spread ( ii ) measurement of average power levels on the channel estimates .  in the rest of the section , we present a comprehensive analysis on the possibility of detecting an insider - attack by measuring some physical characteristics of wireless channels . in our simulations , we incorporate the existence of additive noise at all the receivers . as a result , the observations made at the nodes are erroneous channel estimates ( not ideal as in section [ subsec : ie ] ) .to demonstrate the impact of the detection schemes , we realize the attacks of section [ sec : threat_model ] on the protocol discussed in section [ subsec : ie ] .   for simulations , all the channels in the network are generated using the -st order ar model as , for , where is the doppler spread of the channel , formally defined as with denoting the zero - th order bessel function of first kind , is the absolute time difference ( in seconds ) between the -th and -th samples , and is the doppler frequency .the doppler frequency is given by , where is the node velocity , is the speed of light and is the carrier frequency of signal transmission . for the simulations , we use .the noise components are such that are independent and distributed as circularly symmetric complex gaussian noise with zero mean and unit variance , denoted by .we assume that the nodes employ the state - of - the - art channel estimation algorithms to estimate the channel gains .  at the end of * phase 3 * , the noisy estimates at nodes 1 and 3 are given by , similarly , at the end of * phase 4 * , the estimates before self - interference cancellation are and after self - interference cancellation are in and , the noise components and , for capture the errors in channel estimates , and they are assumed to be distributed as , where denotes their variance values . in practice , depends on the signal - to - noise ratio ( snr ) of the wireless channel , the number of pilots used to estimate the channel , and also the underlying channel estimation method . for our experiments ,we use such that the normalized mean square error ( nmse ) values during * phase 3 * and * phase 4 * are held fixed at some , i.e. , for , where and .similarly ,   other than considering noisy estimates , we also study the impact of the attacks against the number of observations , which here refers to the number of rounds of the protocol ( denoted by ) . in the following subsections , we share simulation results obtained while detecting the different - key attack and the low - rate key attacks .different - key attack with static values of : probability of detection ( pd ) and probability of false alarm ( pfa ) values as a function of channel estimate quality measured by the normalized mean square error ( nmse ) values in db , and the number of rounds of the protocol .this attack can go undetected even at moderately large values of . ]low - rate key attack from node 2 : pd and pfa values as a function of nmse values and the number of rounds of the protocol .this detection technique requires accurate channel estimates ( with lower nmse ) and more number of protocol rounds . ]we have simulated a different - key attack of section [ subsec : dka_node2 ] on the protocol in section [ subsec : ie ] .we refer to this attack as the static different - key attack , wherein , and , i.e. , are fixed throughout several rounds of the protocol .in our experiments , we measure the average power levels on the signals received during * phase 3 * and * phase 4 * to calculate the probability of detection ( pd ) and probability of false alarm ( pfa ) . in particular, the following decision rule was used to detect an attack : where the expectation is empirically computed over rounds of the protocol .the above decision rule follows due to self - interference cancellation at nodes 1 and 3 .the different - key attack would result in significant difference in the power levels before and after interference cancellation due to non - zero residual terms and .however , the residual terms get fully cancelled in the no attack case . based on the above rule ,the attack is detected if atleast one of the nodes output the flag .the results on pd and pfa were obtained by repeatedly running the protocol for against nmse values of in db .the results are plotted in fig .[ fig : sims_dka_static ] , which show that the static attack gets undetected significant number of times .the plots also show that the probability of false alarm is higher for lower number of observations .this trade - off between pd and pfa can be further optimized by treating the above detection problem as a binary hypothesis problem .we have simulated a low - rate key attack of section [ subsec : lka_node2 ] in order to force a doppler spread of on the common source of randomness and , by replacing by , which otherwise takes the value of ( since and have doppler spread of , the product should have doppler spread of ) . in our experiments , we have gathered doppler measurements to compute the probability of detection ( pd ) and probability of false alarm ( pfa ) .the doppler computations were obtained as    where and denote the measured doppler values on and , respectively at node for .these values are empirically computed over rounds of the protocol .the following decision rule was used on the measured doppler values : the above decision rule is coined based on the rationale that the doppler measurements on and should be less than ( or equal to ) that of and , respectively , in the no attack case .this is because the doppler spread of the product of two independent channels is lower than or equal to that of each channel in the first - order ar model .we noticed that the _ criterion _ on average power levels can not be used to detect this attack since the attack parameters make sure that the self - interference is successfully cancelled at both nodes 1 and 3 .the pd and pfa results were obtained for the number of protocol rounds with nmse values of in db .the corresponding values of pd and pfa are plotted in fig . [fig : sims_lra ] , which shows that the low - rate attack can go undetected unless measured over large values of , along with accurate channel estimate values .the threshold values on the doppler spread can be subject to optimization to further improve the trade - off between pd ad pfa , which we defer for future research .we have chosen a simple 3-node wireless network with chain topology to highlight that gsk generation schemes are vulnerable to insider - threats , especially on topologies that can not support additional secure links to verify the attacks . without delving into the expensive option of establishing secure links, it is an interesting direction for future research to explore other _ low communication - overhead _ methods that can detect attacks at high success rate by keeping the false positive rates to low values .this work was supported by the research grant for the human - centered cyber - physical systems programme at the advanced digital sciences center from singapore s agency for science , technology and research ( a*star ) .suhas mathur , narayan m , chunxuan ye , alex reznik , radio - telepathy : extracting a secret key from an unauthenticated wireless channel , " _mobicom 2008_. h. zhou , l. m. huie and l. lai , secret key generation in the two - way relay channel with active attackers , " in _ ieee tifs _ , vol . 9 , no .476 - 488 , mar .2014 .h. liu , j. yang , y. wang , y. j. chen and c. e. koksal , group secret key generation via received signal strength : protocols , achievable rates , and implementation , " in _ ieee trans . on mobile computing _2820 - 2835 , dec . 2014 .t. yucek , r. m. a. tannious and h. arslan , doppler spread estimation for wireless ofdm systems , " _ ieee / sarnoff symposium on advances in wired and wireless communication _ , princeton , nj , 2005 , pp .233 - 236 .j. j. van de beek , o. edfors , m. sandell , s. k. wilson and p. o. borjesson , on channel estimation in ofdm systems , " _ ieee vehicular technology conference , 1995 _ , chicago , il , pp .815 - 819 , vol . 02 .

Abstract: physical - layer group secret - key ( gsk ) generation is an effective way of generating secret keys in wireless networks , wherein the nodes exploit inherent randomness in the wireless channels to generate group keys , which are subsequently applied to secure messages while broadcasting , relaying , and other network - level communications . while existing gsk protocols focus on securing the common source of randomness from external eavesdroppers , they assume that the legitimate nodes of the group are trusted . in this paper , we address insider attacks from the legitimate participants of the wireless network during the key generation process . instead of addressing conspicuous attacks such as switching - off communication , injecting noise , or denying consensus on group keys , we introduce stealth attacks that can go undetected against state - of - the - art gsk schemes . we propose two forms of attacks , namely : ( i ) different - key attacks , wherein an insider attempts to generate different keys at different nodes , especially across nodes that are out of range so that they fail to recover group messages despite possessing the group key , and ( ii ) low - rate key attacks , wherein an insider alters the common source of randomness so as to reduce the key - rate . we also discuss various detection techniques , which are based on detecting anomalies and inconsistencies on the channel measurements at the legitimate nodes . through simulations we show that gsk generation schemes are vulnerable to insider - threats , especially on topologies that can not support additional secure links between neighbouring nodes to verify the attacks .