the motivation for our work is to study asteroid capture missions for a specific population of near earth objects .the targets , temporarily captured orbiters ( tco ) , are small asteroids that become temporarily - captured on geocentric orbits in the earth - moon system .they are characterized as satisfying the following constraints :  * the planetocentric keplerian energy ; * the planetocentric distance is less than three earth s hill radii ( e.g. , au ) ; * it makes at least one full revolution around the earth in the earth - sun co - rotating frame , while satisfying the first two constraints .  in regard to the design of a round trip mission , the main advantage of the tcos lies in the fact that those objects have been naturally redirected to orbit the earth .this contrasts with recently proposed scenarios to design , for instance , a robotic capture mission for a small near - earth asteroid and redirect it to a stable orbit in the em - system , to allow for astronaut visits and exploration ( e.g. the asteroid redirect mission ( arm ) ) .  in this paperwe focus on , a few meters diameter near earth asteroid , officialy classified as a tco .was discovered by the catalina sky survey on september 2006 .its orbit from june 1st 2006 to july 31st 2007 can be seen on figure [ rh120fulltraj ] , generated using the jet propulsion laboratory s horizons database which gives ephemerides for solar - system bodies .the period june 2006 to july 2007 was chosen to include the portion of the orbit within the earth s hill sphere with a margin of about one month .we can also observe that comes as close as 0.72 normalized units from earth - moon barycenter ., title="fig:",width=226 ] , title="fig:",width=226 ]  in , the authors investigate a population statistic for tcos .their work is centered on the integration of the trajectories for 10 million test - particles in space , in order to classify which of those become temporarily - captured by the earth s gravitational field over eighteen - thousandof which do so .this suggests that is not the only tco and that it is relevant to compute a rendezvous mission to this specific asteroid to get insight whether tcos can be regarded as possible targets for transfers with small fuel consumption , and thus cost .  in this paper, our goal is to solve the optimal control problem that consists in performing a rendezvous mission from a halo orbit around the earth - moon libration point to the asteroid , while minimizing the fuel consumption .the choice of targets for our rendezvous mission sets us apart from the existing literature where transfers are typically designed between elliptic orbits in the earth - moon or other systems and , or to a libration point and .rendezvous missions to asteroids in the inner solar system can be found in but they concern asteroids on elliptic orbits which is not the case for us since tcos are presenting complex orbits and therefore require a different methodology .our assumption on the hibernating location for the spacecraft , a halo orbit around the earth - moon unstable libration points , is motivated in part from the successful artemis mission and in part from the constraint on the duration of the mission , mostly impacted by the time of detection of the asteroid .indeed , the arthemis mission demonstrated low delta - v ( ) station keeping on halo orbits around and .while this first study focuses on similar halo orbits to the artemis mission , the study needs to be extended to analyze a variation of halo orbits both in shape ( to include the eight - shaped lissajous orbits ) and also in -excursion or energy . as a first approach , in this work we search the best departure point on the prescribed halo orbit without practical consideration related to synchronization with the asteroid orbit .the primary objective for the mission is to maximize the final mass , but since the earth capture duration of the asteroid is limited it imposes a time constraint which need to be addressed eventually . in this paperhowever , we consider rendezvous missions possibly over long period of time to determine what is the correlation between the required and the duration of the mission. a forthcoming paper will bridge our results to the tcos detection aspect .many methodologies have been developed over the past decades to design optimal transfers in various scenarios .due to the complexity of the tco orbits and the nature of the mission , techniques based on analytical solutions such as in for circular earth orbits are not suitable and we use a numerical approach . a survey on numerical methodscan be found in , and for reasons related to the specifics of our problem we choose to use a deterministic approach based on tools from geometric optimal control versus an heuristic method such as in and .more precisely , we use an indirect method based on the maximum principle , as well as a direct method and continuation techniques to address the difficulties of initialization for our numerical scheme .additionally , we fix the structure of the control norm to be a piecewise constant function with the magnitude of the thrust either zero or maximum with at most three maximum magnitude arcs .our techniques are illustrated on the tco and four others coming from the database of .validation of our approach can be seen by comparing the work to , in which the authors develop a low asteroid rendezvous mission that makes use of a halo orbit around earth - moon .their situation is , however , different from ours , in that they have carefully chosen a specific asteroid for rendezvous . with a one - year transfer time, the value they realize is 432 m / s , which is comparable to the values presented here in a less - ideal scenario .in this paper , the circular restricted three - body problem is used to approximate the spacecraft dynamics .this is justified by the fact that a tco can be assumed of negligible mass , and that the spacecraft evolving in the tco s temporary capture space is therefore attracted mainly by two primary bodies , the earth and the moon .the cr3bp model is well known , let us however go through some notations fo the sequel of the paper .we denote by the spatial position of the spacecraft at time . in the rotatingcoordinates system , and under proper normalization , the primary planet identified here to the earth , has mass and is located at the point ; while the second primary , identified to the moon , has a mass of and is located at , where .the distances of the spacecraft with respect to the two primary bodies are given by , respectively .the potential and kinetic energy , respectively and of the system are given by we assume a propulsion system for the spacecraft is modeled by adding terms to the equations of motion depending on the thrust magnitude and some parameters related to the spacecraft design . the mass of the spacecraft is denoted by . under those assumptions , we have the following equations of motion : where is the control , and satisfies the constraint , and with normalized .a first integral of the free motion is given by the energy of the system .we will later use this energy value to analyze the choice of the rendezvous point and the parking orbit for the spacecraft .it is well know that the uncontrolled motion of the dynamical system has five equilibrium points defined as the critical points of the potential .three of them and are aligned with the earth - moon axis and have been shown to be unstable , while the two others are stable and are positioned to form equilateral triangles in the plane of orbit with the two primaries . since our goal is to maximize the final mass we must include the differential equation governing the variation of the mass along the transfer : where the parameter , the thruster characteristic of our spacecraft , is given by ( it is the inverse of the ejection velocity ) , with the specific impulse of the thruster and the acceleration of gravity at earth sea level .let where represents the position variables and the velocity ones . from section [ section - circular ] ,our dynamical system is an affine control system of the form : where the drift is given by , and the control vector fields are with forming the orthonormal basis of .we consider the rendezvous transfer from an initial point on a parking orbit to a final position and velocity on the orbit .note that the initial and final positions and velocities are variables of the global optimization problem .the criterion to maximize is the final mass which is equivalent to minimizing the fuel consumption or the .since the mass evolves proportionally to the norm of the thrust , our criterion is equivalent to the minimization of the -norm of the control : where , and are respectively the initial and final time .remark that since we choose to be a halo orbit around a libration point , it is uniquely determined by a single point of the orbit using the uncontrolled cr3bp dynamics .this fact will play an important role for one of the necessary optimality conditions below .the large number of variables in our problem significantly adds complexity to the search for a solution .in particular , in the case of free final time we expect an infinite time horizon with a control structure that mimics impulsive maneuvers . to simplify our optimal problem we have two options , either we fix the transfer time or we fix the structure of the control .if we fix the final time , the sensitivity of the shooting method can be addressed by using the solution of a smoother criterion than the -norm , for instance the -norm , and linking it to the target criterion by a continuation procedure , see . in this paper , however , we take a different approach and decide to fix the structure of the control . in the sequel we focus on designing transfers associated to controls with a piecewise constant norm with value in and four switchings .more precisely , we consider control functions that are piecewise continuous such that there exists times with   , width=264 ]  here the final time is free .note that our numerical method will be able to select the best control strategy even if it has less than three boosts .the maximum principle provides first order necessary conditions for a trajectory to be optimal .details regarding the application of the maximum principle to orbital transfers can be found in many references , including .we denote by the state , where is the position and velocity of the spacecraft with its mass .the maximum principle applied to our optimal control problem states that if is an optimal solution defined on  ] such that :  *  ] and we have \cup[t_2;t_3]\cup[t_4;t_f] ] , and the feedback control expressed using the maximization condition .then , we have where comes from equations ( [ hdyn1 ] ) and ( [ hdyn2 ] ) .the goal is to find , , , , and such that the following conditions are fulfilled :  1 . for ; 2 . is the prescribed rendezvous point ; 3 . and the initial transversality condition is verified ; 4 .  the problem has been transformed into solving a multiple points boundary value problem .more specifically , we must find the solution of a nonlinear equation , where is usually called the shooting function .the evaluation of the shooting function is performed using the high order numerical integrator _ dop853 _ , see .the search for a zero of the shooting function is done with the quasi - newton solver _ hybrd _ of the fortran _ minpack _ package .since is nonlinear , the newton - like method is very sensitive to the initial guess .this leads us to consider heuristic initialization procedures .we combine two types of techniques , a direct method and a continuation method .the discretization of the tco s orbit requires the study of thousands of transfers , we use a direct method for a dozen of rendezvous and expand to other points on the orbit using a continuation scheme .the motivation is that direct methods are very robust but time consuming while continuation method succeeded for our problem in most cases very rapidly .the direct method uses the modeling language _ ampl _ , see , and the optimization solver _ ipopt _ , see , with a second order explicit runge - kutta scheme .details on advanced continuation methods can be found in and , but we use a discrete continuation which is enough for our needs .moreover , note that in order to fulfill the initial transversality condition , we first prescribe on the parking orbit and find a zero of the shooting function satisfying all the other conditions .afterward , we do a continuation on along the departing parking orbit in the direction that increases the final mass .once a framing of the best with respect to the final mass is found , we perform a final single shooting to satisfy the initial transversality condition along with the other conditions .this decoupling in the search of an extremal is motivated by the fact that we could very likely find a local maximum on rather than a local minimum because of the periodicity of the initial parking orbit .we avoid this fact by first manually ensuring that the we find will be the one for the best final mass and not the worst . however , our continuation procedure on does not always succeed , mainly because of the high nonlinearity of the shooting function , as the trajectories we find can be very long . even if some of the extremals we find do not satisfy the initial transversality condition with the aimed accuracy ( typically a zero of the shooting functionis deemed acceptable if ) , they are still rather close to satisfy it ( of the order of ) .the objective of this section is to present an analysis of the evolution of the fuel consumption with respect to the location of the rendezvous point on the orbit . as mentioned before, we restrict the study to transfers with at most three boosts . the primary goal is to obtain insights on the variation of fuel consumption based on the features of the rendezvous point such as its distance from , or its energy .the spacrecraft characteristics are assumed to be an initial mass of , a specific impulse of and a maximum thrust of .the halo orbit from which the spacecraft is departing is chosen to have a of 5000 km around the em libration point , see figure [ halo ] .the choice for this specific halo orbit is motivated by the successful artemis mission that used halo orbits with similar characteristics .the point corresponding to the positive is  , and the period of this particular halo orbit is in normalized time units or approximatively .excursion of 5000 km around the em libration point .,width=529 ]  during the period represented in figure [ rh120fulltraj ] , asteroid does 17 clockwise revolutions around the origin of the cr3bp frame , and 3.6 revolutions in earth inertial reference frame .the evolution of the energy of and its distance to the earth - moon libration point as the asteroid evolves on its orbit are given in figure [ rh120_nrg_dist ] .libration point ( right ) for . for energy , the horizontal line represents the energy of the halo periodic orbit around which is about -1.58.,title="fig:",width=226 ]  libration point ( right ) for . for energy ,the horizontal line represents the energy of the halo periodic orbit around which is about -1.58.,title="fig:",width=313 ]  to analyze the variations of the fuel consumption with respect to the rendezvous point on the orbit , we discretize uniformly the orbit of using 6 hours steps . for each rendezvous point of this discretization , we compute an extremal transfer ( i.e. a solution of the maximum principle ) with free final time using the techniques explained in section [ opti ] .figure [ halol2_mfevo ] shows the evolution of the final mass , the and of the duration of the transfer with respect to the rendezvous point on the orbit for a spacecraft departing from the halo orbit and corresponding to the three boosts control strategy .as explained in section [ s_shoot ] , some of the departure point on the halo orbit are not fully optimized this is the case for about two thirds of them .also note that a departure point different from implies a drift phase whose duration is not included in the transfer duration .it can be observed that the final mass has many local extrema and that the variation of the duration of the mission is not continuous ( contrary to what we would expect ) .it is most likely due to local minima or to the fact that the value function of the problem can present discontinuities .evolution of the final mass ( top left ) , of the ( top right ) and transfer duration ( bottom ) from the halo orbit around with respect to rendezvous points on the orbit of .,title="fig:",width=226 ]  evolution of the final mass ( top left ) , of the ( top right ) and transfer duration ( bottom ) from the halo orbit around with respect to rendezvous points on the orbit of .,title="fig:",width=226 ]  evolution of the final mass ( top left ) , of the ( top right ) and transfer duration ( bottom ) from the halo orbit around with respect to rendezvous points on the orbit of .,title="fig:",width=226 ]  figure [ angle_vs_rdv ] shows the evolution of the departure point on the hibernating orbit for the spacecraft with respect to the rendezvous point on ( left ) as well as the three most frequent departure points on the initial halo orbit ( right ) .more precisely , we represents the optimized argument of ( up to the quadrant : ) .note that the initial position on the halo orbit directly depends on the optimized initial drift time .since this initial drift phase has not always been successfully optimized , this figure has to be interpreted with caution .however , we can see that the departure position on the initial periodic orbit seems to always be close to a multiple of .the left picture represents the evolution of argument with respect to time of rendezvous on .the right picture represents the three most frequent departure points on the initial halo orbit.,title="fig:",width=188 ]  the left picture represents the evolution of argument with respect to time of rendezvous on .the right picture represents the three most frequent departure points on the initial halo orbit.,title="fig:",width=302 ]  comparing the evolution of the final mass from figure [ halol2_mfevo ] and the evolution of energy from figure [ rh120_nrg_dist ] , we can see that the best final masses are obtained on the first half of trajectory , that is for rendezvous points which energies are closer to the departing energy . for the best transfer ,the energy difference between the rendezvous point on and the departing point on the halo orbit is about 0.046 .the rendezvous point with the closest energy to the initial orbit is only slightly before the optimal rendezvous point .its final mass is 319.3 kg which is only 0.5 kg worst than the best final mass .this remark suggests strongly that a small difference in energy between the rendezvous point on orbit and the departing point for the spacecraft on the halo orbit is advantageous .the best transfer is represented on figures [ halol2_traj1 ] , and some relevant data is also presented in table [ table - best - halo2-transfer ] .* table [ table - best - halo2-transfer ] : best transfer data from halo - l *  .[table - best - halo2-transfer ] data for the best transfer from to asteroid . [ cols="<,<,<,<",options="header " , ]   [ 4tco ]  forthcoming work will also add the sun as a perturbation into the model , such as in , to determine its impact on the transfers .moreover , for practical reason it would be very interesting to consider the spacecraft parked on a halo orbit around the es libration point .we would like to specially thank robert jedicke and mikael granvik for their generous access to the database of synthetic tcos , and in general , their help and support for this research .geoff patterson and monique chyba were partially supported by the national science foundation ( nsf ) division of mathematical sciences , award # 1109937 .e. belbruno and j. miller , _ sun perturbed earth - to - moon transfers with ballistic capture _ , journal of guidance and control , vol .4 , pp . 770 - 775 , 1993 .besette , d.b .spencer , _ optimal space trajectory design : a heuristic - based approach _ , adavances in the astronautical sciences , vol .124 , pp.1611 - 1628 , 2006 .  j.t .betts , _ practical methods for optimal control using nonlinear programming ._ society for industrial and applied mathematics , 2001 .caillau , b. daoud , j. gergaud , _ minimum fuel control of the planar circular restricted three - 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zazzera , _ a method to design sun - perturbed earth - to - moon low - thrust transfers with ballistic capture _ , xix congresso nazionale aidaa , forle , italia , 17 - 21 september 2007 .sweetser , s.b .broschart , v. angelopoulos , g.j .whiffen , d.c .folta , m - k .chung , s.j .hatch , m.a .woodard , _ artemis mission design . _ space science reviews , volume 165 , issue 1 - 4 , pp .27 - 57 , december 2011 .m. ozimek , k. howell , _ low - thrust transfers in the earth - moon system including applications to libration point orbits _ , journal of guidance , control , and dynamics , vol .2 , march - april 2010 .g. picot , _ shooting and numerical continuation method for computing time - minimal and energy - minimal trajectories in the earth - moon system using low propulsion _ , discrete cont .b , 17 , pp .245 - 269 , 2012 .l. s. pontryagin , v. g. boltyanskii , r. v. gamkrelidze and e. f. mishchenko , _ the mathematical theory of optimal processes _ , john wiley & sons , new york , 1962 .a. waechter , l.t .biegler , _ on the implementation of an interior - 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Abstract: missions to rendezvous with or capture an asteroid present significant interest both from a geophysical and safety point of view . they are key to the understanding of our solar system are as well stepping stones for interplanetary human flight . in this paper , we focus on a rendezvous mission with , an asteroid classified as a temporarily captured orbiter ( tco ) . tcos form a new population of near earth objects presenting many advantages toward that goal . prior to the mission , we consider the spacecraft hibernating on a halo orbit around the earth - moon s libration point . the objective is to design a transfer for the spacecraft from the parking orbit to rendezvous with while minimizing the fuel consumption . our transfers use indirect methods , based on the pontryagin maximum principle , combined with continuation techniques and a direct method to address the sensitivity of the initialization . we demonstrate that a rendezvous mission with can be accomplished with low delta - v . this exploratory work can be seen as a first step to identify good candidates for a rendezvous on a given tco trajectory .   * keywords : * temporarily captured objects , three - body problem , optimal control , indirect numerical methods + * msc2010 : * 49m05 , 70f07 , 93c10


the density - matrix renormalisation group ( dmrg ) , developed by white in 1992 , is one of the most precise numerical algorithms for the investigation of low - dimensional strongly correlated systems .originally the dmrg was introduced to compute the ground state and low energy spectrum of a quantum hamiltonian .meanwhile there are a number of variants using the basic dmrg idea of numerical renormalisation in other physical fields .an important progress was made by applying the dmrg to the transfer - matrix of 2d classical systems , a method known as transfer - matrix dmrg ( tmrg ) .this even allows to analyse the thermodynamics of 1d quantum systems by mapping the partition function to a 2d classical model using a trotter - suzuki decomposition .a highly efficient realisation of a tmrg algorithm , which uses corner - transfer - matrices ( ctms ) , was proposed in and is called corner - transfer - matrix dmrg ( ctmrg ) .an upcoming new application field of the dmrg algorithm are 1d stochastic systems .the dynamics of such models are described by a master equation which has the form of a schrdinger equation with a `` stochastic '' hamiltonian  .the dmrg algorithm can be used to compute the stationary limit of the stochastic process , which corresponds to the ground state of .in contrast to quantum systems is in general not hermitian since there is genuinely no detailed balance .carlon first applied the stochastic dmrg algorithm to various reaction - diffusion models and discussed in detail the influence of non - hermitian operators on the numerics .an alternative approach to stochastic models using a tmrg algorithm was proposed in . in complete analogy to quantum systems the dynamics of 1d stochastic models can be mapped to a 2d classical model .therefore it was quite natural to apply the tmrg to the corresponding `` stochastic '' transfer - matrix . even though this so - called stochastictmrg is similar to the quantum case in many respects , some important differences appear .enss and schollwck discussed in detail properties of the stochastic transfer - matrix focussing on the choice of the density - matrix .unfortunately , the stochastic tmrg shows an unsatisfactory convergence caused by inherent numerical problems which are related to the structure of the stochastic transfer - matrix .the present work proposes a new approach to analyse the dynamics of stochastic problems , a method which we refer to as stochastic light - cone ctmrg ( lctmrg ) .as suggested by its name , the lctmrg combines ideas of the stochastic tmrg and ctmrg algorithms , adjusted by a causality argument which demands a number of modifications for an adaption of the ctmrg to stochastic problems .we show that the lctmrg is a considerable improvement of the stochastic tmrg with respect to numerical stability and performance .stochastic models have gained a large interest in statistical physics .they are used not only in physical but many interdisciplinary research fields to describe processes far away from thermal equilibrium .the bandwidth of applications reaches from the description of social behaviour and biological processes to traffic flow ( see e.g.  ) .typically , stochastic models start from an initial state which evolves in time according to ( local ) probabilistic rules .  in the present work we focus on one - dimensional stochastic problems .we consider a chain of length , where each site can either be occupied by a particle ( ) or empty ( ). in stochastic physics one is interested in the dynamic evolution of a probability distribution of states . can be denoted as a vector where is the probability of finding the chain in the configuration   depending on the type of dynamics , can evolve in continuous or discrete time . assuming continuous dynamics ,stochastic processes can be described by a master equation  is called `` stochastic hamiltonian '' because has the form of a schrdinger equation in imaginary time .the matrix elements are given by where denote the probabilistic rates of the transition .although the master equation suggests a close analogy to quantum systems , as an important difference the stochastic hamiltonian is in general _ not _ hermitian .a formal solution of is given by where denotes the initial probability distribution at .obviously , the stationary limit is a ( right ) eigenvector of with eigenvalue 0 .stochastic models can show a rich phase diagram and interesting critical phenomena .the simplest situation is an absorbing phase transition into an empty state . introducing the occupation number operator of site , the average local density of particles ( with arbitrary due to translational invariance ) is an order parameter that distinguishes the phases : all particles can either vanish ( ) and the system falls into the ( absorbing ) state , or a certain number of particles stay `` active '' ( ) . in the critical region according to a power law whereas a non - critical behaviour is characterized by an exponential decay :   like quantum systems , stochastic processes show universal behaviour at criticality . the most prominent universality class , which is typical for phase transitions to an absorbing state , is the directed percolation class ( dp ) .but in general non - equilibrium phase transitions are by far not so well understood as those in equilibrium physics .we focus on the calculation of the dynamic evolution of the local density of a stochastic process in the thermodynamic limit and assume that consists of local `` stochastic interactions '' as in the conventional quantum tmrg algorithm , the stochastic system is first mapped to a 2d statistical model by using a trotter - suzuki decomposition of .the resulting classical 2d lattice is shown in ( a ) which local plaquette interactions are given by thus , the spatial dimension of the stochastic process is expanded by a virtual trotter dimension which corresponds to the time direction and is split into ( discrete ) steps of size . has to be chosen sufficiently small to obtain a good approximation of .formally , the trotter decomposition becomes exact for . as we measure the local density at finite time , but in the thermodynamic limit of the stochastic chain , the space dimension of the 2d lattice is infinite , whereas the trotter dimension is finite .note that in contrast to the quantum tmrg the boundary conditions are fixed in trotter direction and given by the vectors and , cf .( a ) .  in complete analogy to the quantum case one can apply a tmrg algorithm to the 2d lattice . using column transfer - matrices , shown here pictorially for the example of ( a ) , the local density for the thermodynamic limit can be calculated by  labels the leading right / left eigenvector of , having the eigenvalue 1 .the stochastic tmrg algorithm is used to compute and for successively increasing trotter numbers .unfortunately , various computations show numerical problems that limit  , which is far from enough to compete with other methods like monte - carlo simulations ( mcs ) .  herewe propose a different tmrg algorithm based on corner - transfer - matrices ( ctm ) .such a corner - transfer - matrix dmrg algorithm ( ctmrg ) is known to be numerical more stable and faster than tmrg .before we explicitly construct these ctms for the stochastic case , we discuss some physical properties of the 2d lattice relevant for the development of the new algorithm .  due to probability conservationwe have for any state .thus , trivializes by summing out the `` future '' indices , i.e. the effect of to the 2d lattice is discussed in detail in . for the computation of is found that a huge number of plaquette interactions can be omitted , because they `` trivialize '' .the remaining non - trivial plaquettes form a 2d lattice of _ finite _ dimension which is shown in ( b ) .the trivialisation process can easily be understood by a causality argument : only a `` light - cone '' of plaquette interactions can influence the site where the local density is measured .we now construct a ctmrg algorithm which genuinely fits to the triangle structure of the 2d lattice .as shown in , four cuts are set to separate the lattice into four parts .the cuts are somewhat native to our model , because they form the boundaries of the `` future '' and `` past light - cone '' of the center point of the triangle .the four parts \left(c_d\right)_{n_s}^{\bar n_s}= \begin{minipage}{2.3 cm } \pandocstartinclude{cd.pstex}\pandocendinclude{input}{331}{17 } \end{minipage}&,\quad & \left(c_u\right)_{n_s}^{\bar n_s}= \begin{minipage}{2.3 cm } \pandocstartinclude{cu.pstex}\pandocendinclude{input}{335}{17 } \end{minipage}\end{aligned}\ ] ] are interpreted as ctms whereby and label block - spins .we next show how these ctms can be treated within a ctmrg algorithm analogous to e.g. . however , a number of modifications are necessary to adapt the ctmrg to the light - cone of plaquettes .hence , we call this ctmrg variant light - cone ctmrg ( lctmrg ) .   in a ctmrg algorithmthe ctms are enlarged sequentially by adding transfer - matrices ( tms ) to each cut .we first define the tms in a pictorial way   \label{eq : trans2 }  \left(t_{lu}\right)_{n_s s}^{\bar n_s}=\hspace{-0.8 cm }  \begin{minipage}{3.5cm}\pandocstartinclude{tlu.pstex}\pandocendinclude{input}{359}{43}\end{minipage}&,\quad &   \left(t_{ru}\right)_{n_s s}^{\bar n_s}=\hspace{-0.3 cm }  \begin{minipage}{3.5cm}\pandocstartinclude{tru.pstex}\pandocendinclude{input}{361}{43}\end{minipage}\hspace{-1cm}.\end{aligned}\ ] ] the bullets represent single spin sites and the spins marked with bars will become renormalized block - spins in the dmrg algorithm .exemplarily , and show tms which are used to enlarge the triangle lattice of ( a ).  demonstrates graphically how the tms are used to enlarge the ctms , whereby ctms and tms are `` jointed '' by summing out the adjacent indices ( similar to a matrix multiplication ) . due to the exotic geometry of the lattice we have to distinguish between an upper and lower extension step depending on whether or should be enlarged .  in our lctmrg algorithmboth extension steps are implemented alternately . that way all ctms grow step by step with the crossing point of the cuts always situated in the center of the triangle .after each extension step the ctms have to be renormalized by a density - matrix projection , cf . .the local density can be obtained using and  it is important to notice that is computed in the center of the triangle - lattice . here, influences of boundary effects are expected to be smallest . in terms of a dmrg algorithm , the ctms , and act as an `` environment '' of the `` system '' .the key problem is to find a reasonable density - matrix projection for the renormalization of the ctms after each extension step .we exemplify the construction of the density - matrix by looking at ( a ) . here, one block - spin and two spins of and have to be renormalized into one block - spin .the construction of the optimal density - matrix projection is now discussed in detail .first , we define four vectors : the block - spin belongs to the environment , and to the system block .note that these vectors approximate the left and right eigenvector of the leading eigenvalue of diagonal tms which have a different shape compared to the stochastic tms used in the stochastic tmrg .as and ( ) have not to be computed by any expensive diagonalization routine like in tmrg , the lctmrg algorithm is much faster . and are used to construct a reduced density - matrix for each of the two cuts .the most generic choice would be a symmetric density - matrix } = \tr_{\bar n_s }  \left({\ensuremath{\big|\psi_l^x\big>}}{\ensuremath{\big<\psi_l^x\big|}}+{\ensuremath{\big|\psi_r^x\big>}}{\ensuremath{\big<\psi_r^x\big| } }  \right)\ ] ] which was also used in . here denotes the partial trace over .} ] has rank one and represents a pure projector .a physical explanation for the choice of } ] , can be omitted completely .if additionally the local hamiltonian is parity invariant , i.e.  , the local transfer - matrix becomes symmetric .hence , only the ctms and tms have to be stored . and can be reconstructed by mirroring and .  in order to avoid floating point overflows of the algorithm ,it is recommended to rescale all ctms and tms where is some norm .note that these rescaling factors have to be considered in the computation of .all computations were done on sun workstations ( ultra sparc iii , 900 mhz ) .compared to the old stochastic tmrg the lctmrg algorithm is tremendously more efficient .furthermore , as most parts of the program consist of matrix multiplications of ctms and tms , the lctmrg algorithm can easily be parallized .the cpu time needed for one trotter step ranges from a few milliseconds for states up to a couple of seconds for states .the consumption of computer memory is modest as well , e.g. 10 mb for up to 200 mb for .the numerical studies of the present work focus on reaction - diffusion processes ( rdp ) which are used to model various chemical reactions . we consider a simple rdp with one type of particle which exhibits the following reactions this rdp can be expressed by a stochastic hamiltonian with local interactions  is parity invariant and the local transfer - matrix becomes symmetric .therefore the lctmrg algorithm simplifies , cf ..  as an example , we apply the new lctmrg algorithm to two rdps , the diffusion - annihilation process and the branch - fusion process .these models have also been chosen by carlon to demonstrate the efficiency of the stochastic dmrg algorithm .  the diffusion - annihilation process ( dap ) is exactly solvable . for an unbiased initial probability distribution the dynamic evolution of the local densityis given by where are modified bessel functions .thus we can use analytical results to check the numerical precision of the tmrg data .note that the dap is critical for all with an asymptotic behaviour .the branch - fusion process ( bfp ) is a simple one parameter model which exhibits a non - equilibrium phase transition from an active to an absorbing phase .the bfp is not exactly solvable , but monte - carlo - simulations and stochastic dmrg computations are available .the critical behaviour of the bfp falls into the dp universality class where precise data for the critical exponents have been calculated by series expansions .using the stochastic lctmrg for both models can demonstrate , whether the method is capable to produce reliable results for  * critical and non - critical systems * systems at phase transition points .  the next subsections present numerical results for the two processes .if not stated differently , all computations were performed with .( left ) compares lctmrg calculations with exact data for , keeping various numbers of states .up to more than trotter steps we obtain highly precise data with a deviation of less than from the exact results .the inset of ( left ) plots the lctmrg data in a double - logarithmic plot , which shows that falls of algebraically .the high number of trotter steps is a considerable improvement to the old stochastic tmrg algorithm by at least _ three _ orders . even though the dap is critical, we observe an extremely stable convergence of the lctmrg algorithm .( right ) plots numerical data for different density - matrices } ] , cf . .in all our calculations we observe highly instable numerics , if the conventional density - matrix } ] .thus the arguments given in can be confirmed numerically : }$ ] is not an adequate density - matrix for the stochastic lctmrg .   in this sectionwe focus on the critical phase transition of the bfp at  .( left ) compares numerical data computed by the lctmrg algorithm with conventional monte - carlo simulations .  for sufficiently far away from criticality, we observe a convergence up to more than trotter steps . in the vicinity of the critical point the convergence becomes worse .( right ) plots the logarithmic derivative at the critical point for states , which is very sensitive to numerical errors .up to more than trotter steps the numerics are extremely precise , and one can verify that switches to an algebraic behaviour .it is also possible to determine the critical exponent by extrapolating , cf .inset of ( right ) .thereby , we were able to compute up to a precision of less than 0.1% :   the question arises why the convergence of the lctmrg at is two orders less than in the dap process , yet both models behave critical .we checked various numerical aspects to determine the origin of the worse convergence .in the bfp it is conspicuous that the quality of the results at strongly depends of the number of states that are retained within the lctmrg algorithm .this is demonstrated in ( left ) which plots numerical computations for various at .  however , it is surprising that so many states are needed although there is a very strong fall off of the density - matrix eigenvalues , cf .( right ) .as another possibility we check the influence of the size of the trotter steps on the numerics .the curves of ( left ) belong to various , but are rescaled to for comparison .even though finer trotter decompositions increase the total number of convergent trotter steps , one can not improve the accuracy of the data with respect to the absolute time .if on the other hand becomes too large , the trotter decomposition itself gets worse and is then responsible for unsatisfactory numerical data .  to estimate the effect of numerical errors caused by floating point inaccuracies we implemented the diagonalization routine for the density - matrix alternatively by using higher mantissa bits .this was technically realized by using the gmp library which allows an arbitrary number of mantissa bits . as shown in ( right ) , only a marginal effect on the numerics is observed .overall , it remains an open question what exactly is the limiting factor for the worse convergence at the phase transition point in the bfp . to exclude model specific reasons, we also checked other rdps , e.g. the contact process .qualitatively , the same limited convergence near the critical phase transition point is observed .in the present work we proposed a new variant of the stochastic tmrg by using corner - transfer - matrices which we call stochastic light - cone ctmrg ( lctmrg ) .the lctmrg algorithm fits genuinely to the specific structure of the triangle classical lattice which evolves from the trotter - suzuki decomposition of the stochastic model .we tested the new algorithm by comparing lctmrg data to exact results and monte - carlo simulations of two different reaction - diffusion models .we obtained highly precise numerical results ( ) up to trotter steps , even if the model behaves critical as the diffusion - annihilation process .compared to the old stochastic tmrg algorithm with , this is an enormous increase of the number of reachable trotter steps of three orders .an important observation is that inherent numerical problems of the old stochastic tmrg algorithm do obviously not appear in our new approach . in the vicinity of a critical phase transition point , exemplified by the branch - fusion model, the convergence gets worse , but is nevertheless sufficient to determine precise results for critical exponents .since _ within _ critical phases a much better convergence has been observed , it remains an open question what exactly causes the reduced convergence at critical phase _boundaries _ , which presumably does not originate from purely numerical reasons . therefore our future research is concentrated on further modifications and improvements of the lctmrg algorithm , e.g. the implementation of a finite size algorithm .overall the numerical investigations show that the new lctmrg algorithm is a considerable step towards a general and very efficient method for 1d stochastic problems .compared to the traditional approach using monte - carlo simulations , there are two fundamental advantages of the lctmrg :  * the lctmrg is not a simulation technique .there is no need of taking random numbers and sample averages .the lctmrg is a numerical renormalisation group based on the quantum formalism for stochastic models where averages are directly accessible . *the algorithm describes the _ exact _ thermodynamic limit of the stochastic model .note that here we even have to deal with a _ finite _classical 2d system only , due to the simplification from the `` light - cone decoupling '' .thus , there are in principle no finite - size effects like in mcs or stochastic dmrg.  even if the number of possible time steps , in particular at phase transition points , can not compete with mcs up to now , we believe that stochastic tmrg algorithms can be an extremely valuable tool for studying 1d stochastic systems .finally we mention , that as in the case of the ctmrg  a generalisation of the lctmrg to more than one dimension is also imaginable .the work of ak , as and jz has been performed within the research program sfb 608 of the _deutsche forschungsgemeinschaft_. ak is supported by _studienstiftung des deutschen volkes_. he thanks ag and tn for their warm hospitality at the university of kobe and the _ klner gymnasial- und stiftungsfonds _ for the financial support of this visit .99 white s r 1992 2863  white sr 1993 b * 48 * 10345 peschel i , wang x , kaulke m and hallberg k ( eds . ) 1998 _ density - matrix renormalization _ ( heidelberg : springer ) nishino t 1995 3598 bursill r j , xiang t and gehring g a 1996 l583    nishino t and okunishi k 1997 3040 alcaraz f c , droz m , henkel m and rittenberg v 1994 250 , schtz g m 2000 in _ phase transitions and critical phenomena _domb c and lebowitz j ( eds . ) vol 19 ( london : academic press ) carlon e , henkel m and schollwck u 1999 _ eur .j. _ b * 12 * 99 kemper a , schadschneider a and zittartz j 2001 l279 enss t and schollwck u 2001 7769 hinrichsen h 2000 _ adv .* 49 * 815 de oliveira s m , de oliveira p m c and stauffer d 1999 _ evolution , money , war and computers _ ( stuttgart : teubner ) chowdhury d , santen l and schadschneider a 2000 _ phys . rep . _ * 329 * , 199 marro j and dickman r 1999 _ nonequilibrium phase transitions in lattice models _( cambridge : cambridge university press ) kinzel w 1983 _ ann ._ vol 5 ( bristol : adam hilger )

Abstract: we develop a new variant of the recently introduced stochastic transfer - matrix dmrg which we call stochastic light - cone corner - transfer - matrix dmrg ( lctmrg ) . it is a numerical method to compute dynamic properties of one - dimensional stochastic processes . as suggested by its name , the lctmrg is a modification of the corner - transfer - matrix dmrg ( ctmrg ) , adjusted by an additional causality argument . as an example , two reaction - diffusion models , the diffusion - annihilation process and the branch - fusion process , are studied and compared to exact data and monte - carlo simulations to estimate the capability and accuracy of the new method . the number of possible trotter steps of more than shows a considerable improvement to the old stochastic tmrg algorithm .


massive multiple - input multiple - output ( mimo ) has been regarded as one of the enabling technologies in next generation wireless communications .considering the large number of antennas at the base station ( bs ) , it becomes almost imperative to rely on the time - division duplexing ( tdd ) channel reciprocity to learn the downlink ( dl ) channel state information ( csi ) from the uplink ( ul ) channel measurements at the bs to avoid the huge overhead for csi feedback as in frequency - division duplexing ( fdd ) systems . even though the antenna array size is large at the bs , each useris only equipped with a few antennas , e.g. only one antenna .a user just needs to send one ul pilot sequence per transmit antenna to facilitate the bss to acquire reliable estimates of the corresponding many ul channels from the user to the large antenna array at the bs .  to ensure best csi estimation quality ,we want to allocate orthogonal ul pilot sequences to different users so that the pilot transmissions do not interfere with each other .but in reality , within a limited time period and a limited bandwidth , there are only a limited number of orthogonal pilot sequences .as the number of users becomes large , non - orthogonal pilot sequences are re - used by the users served by different bss , which gives rise to the so - called pilot contamination and is one limiting factor in multi - cell massive mimo systems .various approaches have been proposed to alleviate the pilot contamination issue in massive mimo .recent works include .see also and references therein for a brief overview . in and ,pilot decontamination was achieved by utilizing the fact that users with non - overlapping angles of arrival ( aoa ) enjoy asymptotic orthogonal covariance matrices . in ,aoa diversity and amplitude - based projection were jointly exploited to null the pilot contamination and achieve better channel estimation . a blind singular value decomposition ( svd ) based method was proposed in to separate the signal subspace from the interference subspace . in , least - squares ( ls )channel estimate was derived by treating the blindly detected ul data as pilot symbols and the pilot contamination effect was shown to diminish as the data length grew .unlike previous studies where a single narrow - band channel was typically assumed , in this paper , we look into the issue of intra - cell pilot orthogonalization and schemes for mitigating the inter - cell pilot contamination with a realistic massive mimo ofdm system model .first , we examine how to align the power - delay profiles ( pdp , a.k.a .delay power spectrum in ) of different users served by one bs so that the pilots sent within one common ofdm symbol are orthogonal . from the aligning rule, we can see much more users can be sounded in the same ofdm symbol as their channels are sparse in time . in the case of massive mimo , to alleviate the pilot contamination when the schemes in do not apply well due to interfering users overlapping aoas , we further propose to exploit the fact that different paths in time are associated with different aoas and the pilot contamination can be significantly reduced by aligning the pdps of the users served by different bss appropriately .thus , pdp aligning can serve as the new baseline design philosophy for massive mimo ul pilots .this paper is organized as follows : section [ secsysmodel ] describes the massive mimo ofdm system model and the channel model .section [ orthopilot ] provides a sufficient condition for orthogonal pilots design through pdp aligning , which is also applicable to conventional mimo systems .then we explain how to mitigate the pilot interference in the case of massive mimo by pdp alignment in section [ intfreduction ] .low - complexity pilot designs are provided in section [ protocol ] .corroborating simulation results are provided in section [ performance ] and section [ conclusion ] concludes the paper ._ notations : _ denotes the diagonal matrix with diagonal elements defined inside the curly brackets . refers to the entry of matrix and stands for the -th entry of the vector . denotes the identity matrix . ] .the channel frequency response is the discrete fourier transform ( dft ) of the corresponding channel impulse response ( cir ) in time domain , i.e. where stands for the unitary fft matrix defined as and is the cir with the following pdp : ,\ ] ] where we have assumed the channels from one user to the antennas at one bs share a common pdp .assuming uncorrelated scattering as in , i.e. the scattering at two different paths is uncorrelated , the pdp matrix becomes diagonal . combining ( [ sysmodel ] ) and ( [ cfr_cir ] ), we have   in order to characterize the spatial covariance among multiple receive antennas at the bss , we adopt the multi - path spatial channel model ( scm ) defined in and , which is illustrated in fig .[ chmodelfig ] .each resolvable channel path corresponds to one independent scatterer including sub - paths : ^t= \sqrt{\frac{p_{k}^{l , b}(n , n)}{q}}\sum_{q=1}^q { \bm a}(\theta_{k , n , q}^{(l , b)})e^{j\phi_q},\label{channelmodel}\ ] ] where the arriving angles : of the sub - paths are uniformly distributed within the angle spread ( as ) of this path , the phases : are drawn from a uniform distribution over  ] , we can get where denotes the vector of the -th taps in the aggregate channels from the orthogonal users served by bs- to bs- , i.e. ^t ] . since the circulant matrix carries out cyclic shift operation , we have note that for channels of limited delay spread , only a few users will have non - zero contribution toward the aggregated channel tap in ( [ aggtap ] ) .  denoting the spatial covariance matrix of the -th taps in the aggregate channels from users in bs- to bs- across the bs antennas as , i.e.  ] : _ , l=1, .. ,b . \label{constraint3}\end{aligned}\ ] ] in each simplified sub - problem for tone group , we only need to optimize the objective over variables : .the optimal cyclic shifts for the interfering users belonging to different tone groups can be derived independently .additionally , the allocation of the users served by one bs to the tone groups : can also be optimized .in this section , we simulate the system as illustrated in fig .[ chmodelfig ] , where there are two neighboring cell - edge users creating strong inter - cell interference . other simulation parameters and assumptionsare listed as follows :  1 .ofdm configurations : ( ) , , ; 2 .each bs is equipped with a ula of size and antenna spacing .each path toward the bs is generated according to the scm defined in ; 3 .both uniform and exponential pdps are simulated . for the uniform pdp, we have  ] .4 .  user- and user- are close to each other but are served by different bss .they share the same scatterers toward each bs , i.e. their visibility regions ( vr ) are overlapping ; 5 .user- and user- are assigned to the same tone group as defined in section [ scheme2 ] and the optimal pdp alignment is found by minimizing the cost function in ( [ optalignment3 ] ) .  in fig .[ diffpdp ] , we show the cumulative distribution functions ( cdf ) of the normalized mean - square error ( mse ) . ] of the estimated channels with and without pdp alignment . in each one of the - carlo runs , the second - order statistics of the channel taps are randomly generated according to the scm .the pdp alignment in ( [ optalignment3 ] ) is based on the second - order statistics to mitigate the pilot contamination . from fig .[ diffpdp ] , we see the exponential pdp can provide better decontamination than the uniform pdp . compared with the case without alignment , optimal pdp aligning can bring about improvement in ul channel estimation at the median point , i.e. in the cdf curve .  in fig .[ diffas ] , we examine the pdp alignment performance for different as values . from the plotted curves , we see the pdp alignment favors the scattering environment with a small as . in fig .[ sumratecdf ] , we compare the sum of dl spectral efficiency to the two users in fig .[ chmodelfig ] when the bss formulate the dl matched - filter precoding vectors with the estimated ul channnels assuming tdd channel reciprocity .it can be observed that the achieved spectral efficiency after pdp aligning is pretty close to that without ul pilot contamination .in this paper , relying on a realistic massive mimo ofdm system model , we have addressed the issue of pilot contamination and proposed to rely on pdp aligning to mitigate this type of inter - cell pilot interference .on one hand , we have shown the ul pilots from the users served by one common bs can be made orthogonal through pdp aligning . on the other hand , due to the massive amount of antennas at the bs , we have shown that pdp alignment can also help to alleviate the pilot contamination thanks to the fact that different paths in time are originating from different aoas .numerical simulations validate that the pilot contamination can be significantly reduced through aligning the pdps of the users served by different bss appropriately .the proposed pdp aligning can serve as the new baseline design philosophy for the ul pilots in massive mimo .f. rusek , d. persson , b. k. lau , e. g. larsson , t. l. marzetta , o. edfors , and f. tufvesson , `` scaling up mimo : opportunities and challenges with very large arrays , '' _ ieee signal process . mag .40 - 60 , jan . 2013 .h. yin , d. gesbert , m. filippou , and y. liu , `` a coordinated approach to channel estimation in large - scale multiple - antenna systems , '' _ ieee . j. sel . areas .commun . _ , vol .264 - 273 , feb . 2013 .h. yin , d. gesbert , and l. cottatellucci , `` dealing with interference in distributed large - scale mimo systems : a statistical approach , '' _ ieee j. sel .topics signal process . _ , vol . 8 , no .942 - 953 , oct . 2014 .h. yin , l. cottatellucci , d. gesbert , and r. r. muller , `` pilot decontamination using combined angular and amplitude based projections in massive mimo systems , '' in _ proc .ieee international workshop on signal processing advances in wireless communications ( spawc ) _ , pp .216 - 220 , stockholm , sweden , jun . 2015

Abstract: in this paper , we look into the issue of intra - cell uplink ( ul ) pilot orthogonalization and schemes for mitigating the inter - cell pilot contamination with a realistic massive multi - input multi - output ( mimo ) orthogonal frequency - division multiplexing ( ofdm ) system model . first , we show how to align the power - delay profiles ( pdp ) of different users served by one bs so that the pilots sent within one common ofdm symbol are orthogonal . from the derived aligning rule , we see much more users can be sounded in the same ofdm symbol as their channels are sparse in time . second , in the case of massive mimo , we show how pdp alignment can help to alleviate the pilot contamination due to inter - cell interference . we demonstrate that , by utilizing the fact that different paths in time are associated with different angles of arrival ( aoa ) , the pilot contamination can be significantly reduced through aligning the pdps of the users served by different bss appropriately . computer simulations further convince us pdp aligning can serve as the new baseline design philosophy for the ul pilots in massive mimo .   massive mimo , pilot contamination , time - division duplexing , power delay profile


developing secure systems is difficult and costly . the high cost of effectively mitigating all vulnerabilities and the far lesser cost of exploiting a single one creates an environment which advantages cyber attackers .new cyber defense paradigms have been proposed to re - balance the landscape and create uncertainty for the attackers .one such paradigm is active defense based on temporal platform diversity .temporal platform diversity ( or simply platform diversity ) techniques dynamically change the properties of a computing platform in order to complicate attacks .platform properties refer to hardware and operating system ( os ) attributes such as instruction set architecture ( isa ) , stack direction , calling convention , kernel version , os distribution , and machine instance .various platform diversity techniques have been proposed in the literature .emulation - based techniques change the calling sequence and instruction set presented to an application ; multivariant execution techniques change properties such as stack direction or machine description using compiler generated diversity and virtualization ; migration - based techniques change the hardware and operating system of an application using containers and compiler - based checkpointing ; server diversification techniques rotate a server across multiple platforms and software stacks using network proxies ; self cleansing techniques change the machine instance by continuously rotating across many virtual machines and re - imaging the inactive ones . the key point to these techniquesis that platform diversity does not remove vulnerabilities , it just makes them more complicated to exploit .  unfortunately littlework has been done on understanding and quantifying the impact of temporal platform diversity on the security of a system .the following important questions often remained unanswered : how often should the platform change ?how diverse are various platforms ?what is the quantitative impact of platform diversity on attacks ?what are the important parameters and factors for the effectiveness of a platform diversity system ? should the next platform or platform configuration be selected uniformly or preferentially ? what is the optimal strategy for controlling a platform diversity system ?  in this work , we take various approaches to quantitatively analyze and study active cyber defenses based on platform diversity .we use a migration - based technique ( i.e. talent ) in our study and quantitatively measure its properties using four approaches .first , we perform an abstract analysis of platform diversity using algebraic models , combinatorics , and markov models .second , we implement the platform diversity system on a testbed and perform experiments with it using real exploits to quantitatively measure its properties .third , we analyze the temporal migration patterns using game theory and derive an optimal strategy for selecting the next platform for five operating systems : centos , freebsd , debian , fedora , and gentoo .fourth , we model the platform diversity system running the above operating systems with and without the optimal strategy and simulate its impact on an attacker trying to compromise the system .  the quantitative results from our four approaches are consistent and in some cases surprising and counter - intuitive .our findings indicate that although platform diversity can help protect a system against some attacks ( specifically , persistent service disruption attacks ) , it can negatively impact security for others ( specifically , fast , single compromise attacks ) .we show that the benefit of platform diversity techniques heavily depends on their threat models and that operational requirements and temporal parameters must be understood and fine tuned for these techniques to be effective .moreover , we prove that preferential selection of platforms can achieve better protection and derive the optimal strategy for determining temporal patterns ( i.e. the next platform ) .the rest of the paper is organized as follows .section [ pdb ] provides a brief overview of platform diversity techniques and talent .section [ tm ] describes the threat model used throughout the paper .section [ aa ] discusses our abstract analysis approach and its results .section [ exper ] discusses our testbed experiments and measurements performed on a real system .section [ gtoc ] provides the game theoretic analysis of temporal patterns and the optimal solution .section [ sim ] explains our simulation approach and results .section [ lld ] enumerates a number of lessons learned and discusses our findings .we discuss the related work in section [ rw ] before concluding the paper in section [ conc ] .in this section , we briefly describe the active defense techniques based on platform diversity .we provide enough background for understanding the rest of the paper .more details about each technique can be found in its original publication .temporal platform diversity techniques change platform properties in order to make attacks more complicated .they often rely on temporal changes ( e.g. vm rotation ) , diversity ( e.g. multivariant execution ) , or both ( e.g. migration - based techniques ) to protect a system .these techniques are often implemented using machine - level or operating system - level virtualization , compiler - based code diversification , emulation layers , checkpoint / restore techniques , or a combination thereof .emulation - based techniques such as genesis often use an application - level virtual machines such as strata or valgrind to implement instruction set diversity . in some cases ,multiple instances are executed and a monitor compares their results .multivariant execution techniques such as reverse stack ( also called n - variant systems ) use compiler - based techniques to create diverse application code by replacing sets of instructions with semantically equivalent ones .migration - based techniques such as talent use operating system - level virtualization ( containers ) to move an application across diverse architectures and operating systems .platform diversity can also be achieved at a higher abstraction level by switching between different implementations of servers .these techniques either do not preserve the state ( e.g. a web server ) or they preserve it using high level configuration files ( e.g. dns server ) . finally , self - cleansing techniques such as scit only change the current instance of the platform without diversifying it . the main goal , in this case , is bringing the platform to its pristine state and removing persistence of attacks .  in this work , we use a migration - based platform diversity technique , talent , in our analyses . since talent provides both temporal changes ( periodic migrations ) and diversity ( multiple operating systems ) , it is a good candidate for capturing the major effects and dynamics in our quantitative analysis .   rusted dynmic ogical hterogeeity sysem ( talent ) is a framework for live - migrating critical applications across heterogeneous platforms .talent has several design goals : i ) heterogeneity at the instruction set architecture level , ii ) heterogeneity at the operating system level , iii ) preservation of the state of the application , including the execution state , open files and sockets , and iv ) working with a general - purpose , system language such as c. talent uses two key concepts , operating - system - level virtualization and portable checkpoint compilation , to address the challenges involved in using heterogeneous platforms , including binary incompatibility and the loss of state and environment .the environment of an application refers to the filesystem , configuration files , open files , network connections , and open devices .many of the environment parameters can be preserved using virtual machine migration .however , virtual machine migration can only be accomplished using a homogeneous operating system and hardware .talent uses operating - system - level virtualization to sandbox an application and migrate the environment . in operating - system - level virtualization, the kernel allows for multiple isolated user - level instances .each instance is called a container ( jail or virtual environment ). this type of virtualization can be thought of as an extended ` chroot ` in which all resources ( devices , filesystem , memory , sockets , etc . )are virtualized .hence , the semantic information that is often lost in hardware virtualization is readily available in operating - system - level virtualization .talent is implemented using the openvz .when migration is requested , talent migrates the container of the application from the source machine to the destination machine .this is done by synchronizing the filesystem of the containers , migrating the ip address of the source container s virtual network interface , and transferring the state of each tcp socket ( ` sk_buff ` structure of the kernel ) .ipc and signal migration is supported in talent through the underlying virtualization layer .migrating the environment is only one step in backing up the system because the state of running programs must also be migrated . to do this , a method to checkpoint running applicationsmust be implemented .talent uses a portable checkpoint compiler ( cppc ) to migrate a process state .cppc allows talent to checkpoint the state of the binary on one architecture and recover the state on a different architecture . to perform checkpointing ,the code is precompiled to find restart - relevant variables .these variables and their memory locations are then registered in the checkpointing tool .when checkpointing , the process is paused and the values of the memory locations are dumped into a file .the checkpointing operation must occur at safe points in the code to generate a consistent view . at restart, the memory of the destination process is populated with the desired variable values from the checkpoint file .some portions of the code are re - executed in order to construct the entire state .the checkpoint file itself must have a portable format to achieve portability across 32-bit/64-bit architectures and little / big endian machines .talent uses the hdf5 portable file format for its checkpoints .talent has been implemented on intel xeon 32-bit , intel core 2 quad 64-bit , and amd opteron 64-bit processors .it has also been tested with gentoo , fedora ( 9 , 10 , 11 , 12 , and 17 ) , centos ( 4 , 5 , and 6.3 ) , debian ( 4 , 5 , and 6 ) , ubuntu ( 8 and 9 ) , suse ( 10 and 11 ) , and freebsd 9 operating systems .we discuss multiple threat models in this paper but analysis shows that they share common features . to make the analysis more precise , we explicitly describe the core threat model in this section .variations upon the core threat model are described in the other sections as appropriate .  in our model, the defender has a number of different platforms to run a critical application .the attacker has a set of exploits ( attacks ) that are applicable against some of these platforms , but not the others .we call the platforms for which the attacker has an exploit `` vulnerable '' and the others `` invulnerable . '' in a strict systems security terminology , vulnerable does not imply exploitable ; without loss of generality , we only consider exploitable vulnerabilities .an alternative interpretation of this threat model is that the vulnerabilities are exploitable on some platforms , but not on the other ones .the defender does not know which platforms are vulnerable and which are invulnerable , nor does he have detection capabilities for the deployed exploits .this scenario , for example , describes the use of zero - day exploits by attackers , for which no detection mechanism exists by definition .since there is little attempt to isolate the inactive platforms in platform diversity systems , we assume that all platforms are accessible by the attacker , and the attacker attempts to exploit each one . moreover , we assume that the defender does not have a recovery or re - imaging capability to restore a compromised platform .the reason for this assumption is two fold : first , typical recovery methods ( e.g. clean installation of the operating system ) can take a long time to complete and unless the defender has many spare platforms , it is hard to accomplish it effectively .a large number of spare platforms also implies large hardware cost for these systems ( e.g. scit ) .second , and more importantly , the goal of this paper is to study the impact of temporal platform changes , not the effectiveness of recovery capabilities being used .hence , once exploited , a platform remains vulnerable and the attacker requires no staging time in the future .  the attacker s goal is what creates the variations in our threat model .for example , one success criteria may be for the adversary to compromise the system for a given period of time to cause irreversible damage ( e.g. crash a satellite ) , while a different success criteria gives the attacker gradual gain the longer the system is compromised ( e.g. exfiltration of information ) . these variations and their impact on the effectiveness of temporal platform diversityare discussed in the subsequent sections .much of the analysis of one system using temporal platform diversity as a security strategy applies to any such system . in this sectionwe will specify an abstraction that generically describes temporal platform diversity , which is useful as a security strategy in protecting against attacks that have a temporal requirement .we analyze a number of cases where this holds . to better convey the meanings , we use simple phrases which are examples of these cases to refer to them : crash the satellite , sneaking past a sensor , and exfiltration over a difficult channel .  1 .crash the satellite - once the attacker has controlled the system continuously for a specified time period , the game is over and the attacker has won .2 .  sneaking past a sensor - the attacker needs control for a specified time period for an attack to succeed .once the attacker has control for the period , the attacker s payoff function starts to increase from zero .the scenario can either be continuous or finite duration .3 .  exfiltration over a difficult channel - the attacker needs control for a specified time period for an attack to succeed .once the attacker has control for the period , the attackers payoff function starts to increase from a positive value .we can categorize the problem space according to a number of properties :  * the attackers temporal requirement can either be aggregate or continuous , * the attackers payoff can be either fractional or binary ( all or nothing ) , * the period of usage ( equivalently attack window ) can either be of fixed duration or ongoing , and * the migration models include random with repeat , random without repeat , and periodic permutation  we will define the abstract model of a temporal platform diversity system as a system that migrates through a finite fixed collection of states .each state either has or does not have a property exploitable by the attacker which we call vulnerable . in the first approximation to the model we assume that the states are fully independent .we will use the notation presented in table [ table : notation ] ..notation describing platform diversity system [ cols= " < , < " , ]   in fig .[ fig : percentage - vulnerable ] we plot the cumulative distribution function ( cdf ) of the fraction of time the selected platform was vulnerable , computed across all 500 mc trials . in this plot , a curve trending towards the upper - right corner is preferred , as it demonstrates that a vulnerable system was presented less often .we see that optimizing diversity generally selects less vulnerable systems than either of the other methods ; the mean rate of vulnerability for the three methods is shown in table [ tab : mean - vulnerability - rate ] . while diversity does show statistically significant improvement over other strategies , the conclusion from this evaluation would be that none of the strategies are particularly viable .indeed , in section [ aa ] we determined that was the optimal number of platforms to use , yet its performance is nearly identical to leveraging more . as discussed earlier , however , this is a flawed metric that does nt represent this attacker model .we now demonstrate the cdfs of the two metrics which are appropriate for evaluating the given threat models in fig .[ fig : evaluation - metrics - sim ] .first , we study the case for which a single compromise is considered success for the adversary and failure for the defender ( e.g. ` crash the satellite ' ) . in fig .[ fig : time - to - first ] , we plot the cdf for the time it takes before the system is fully compromised , that is , vulnerable platforms are presented consecutively . noting that these results are from the same trials as fig .[ fig : percentage - vulnerable ] , we see a significant difference in performance with the appropriate metric . optimizing for diversity results in system compromise only 2% of the time , while uniform selection eventually results in system compromise in 79% of the trials .we note that because the diversity optimization strategy results in periodic scheduling , if system compromise is going to happen , it does so nearly instantaneously either at or shortly thereafter ( once the deterministic solution is reached ) .this is due to the fact that the system can only be compromised if all of the platforms in the resultant solution are vulnerable ; nothing changes after the solution is reached .the same can be said about the random 3 strategy , but we note that performance is significantly reduced due to the increased probability of drawing 3 platforms that are highly similar .contrarily , the uniform strategy will almost surely result in system compromise as if there exists vulnerable platforms in the fully available set .finally , we study the threat model for which an adversary achieves progressive gains the longer they are present on the system ( e.g. ` exfiltration over a difficult channel ' and sneak past a sensor ) .recall that in these scenarios , the defender loss begins or continues after the adversary is present for consecutive intervals . as such, an appropriate evaluation metric is the fraction of time that the system is in a compromised state , which we plot the cdf for in fig .[ fig : percentage - compromised ] . once again , optimizing diversity yields far superior performance to other methods . in 98% of the trials ,the system was never compromised ( note this matches fig .[ fig : time - to - first ] ) , while in that other 2% it was compromised for the entire duration .meanwhile , uniform migration results in system compromise on average 26% of the time , and only in less than 2% of the trials did it result in less compromise than diversity optimization ( 13% of trials when comparing to random 3 ) .one can see that when evaluating with the appropriate metrics , the difference in performance becomes significant .this is important to take note of and emphasizes the need to develop a deployment strategy that is optimal towards the specific threat model of interest .given the threat model requiring adversary persistence , deterministically optimizing for diversity shows superior performance to attempting to achieve security through randomization .we note that while the effects discovered in these simulations match the theoretical results presented in section [ aa ] as well as in the experimental results in section [ ss : experimentalresults ] .our work in analyzing and testing temporal moving target technologies has provided three main lessons .the first is that the methodology of producing threat derived metrics , as in , can be extended to evaluate base technologies . in the cases ,technologies are designed to address a specific problem or set of problem . as such security technologies implicitly carry the set of threats which they are intended to address .these threats can be used to derive metrics .  while the threats addressed by temporal moving target technologies vary , there remains a certain similarity in the produced metrics and , perhaps more importantly , in the data needed to evaluate them .this allows the transformation of data gathered from one set of , potentially expensive or time consuming , tests in a specific usage scenario to different usage scenarios .the second is the surprising result that security , as measured using the threat derived metrics , is not a strictly increasing as a function of diversity .indeed , the minimal diversity required to cover the attackers required duration provides the optimal solution in the continuous requirement case . in the aggregate case ,the optimal diversity is determined via a process similar to gerrymandering : that is that the attacker just barely loses in the majority of sub - selections .the final lesson is that when we have information concerning diversity between individual platforms , the optimal result is provided by maximizing platform dissimilarity while minimizing the diversity in the sense of platform count . indeed ,even given knowledge of the probability of vulnerability in the various platforms , the most secure ensemble might not be comprised of the most secure individual members .various platform diversity techniques have been proposed in the literature . as mentioned earlier ,the self - cleansing intrusion tolerance ( scit ) project rotates virtual machines to reduce the exposure time .scit - web server and scit - dns preserve the session information and dns master file and keys , respectively , but not the internal state of the application . the resilient web service ( rws ) project uses a virtualization - based web server system that detects intrusions and periodically restores them to a pristine state .certain forms of server rotation have been proposed by blackmon and nguyen and by rabbat , et al . in an attempt to achieve high availability servers .high - level forms of temporal platform changes have been proposed by petkac and badger and min and choic to build intrusion tolerant systems although the diversification strategy is not as detailed in these efforts .compiler - based multivariant and n - variant systems propose another way of achieving platform diversity .holland , et al propose diversifying machine descriptions using a virtualization layer .a similar approach with more specific diversification strategy based on instruction sets and calling sequences has been proposed by williams et al .wong and lee use randomization in the processor to combat side - channel attacks on caches .  on the evaluation side , manadhata and wind a formal model for measuring a system s attack surface that can be used to compare different platforms .evans et al develop models to measure the effectiveness of diversity - based moving target technique .they evaluate the probability of attack success given the time duration of attack probing , construction , and launch cycles and the entropy of randomness in the target system .they evaluate the impact of various attacks on moving target systems including circumvention , deputy , brute force , entropy reduction , probing , and incremental attacks .colbaugh and glass use game theory to analyze strategies for deploying moving target defenses against adaptive adversaries , concluding with the result that uniform randomization is optimal .while these results may seem counter to our own , they are not .their threat model did not require adversary persistence and the defense strategy was designed to minimize predictability .specifically , the more an adversary is able to observe a defense , they are more likely to adapt to it . in the case of temporal platform migration , this implies that the adversary will expend resources towards the development of exploits specific to the systems observed .while their adversary starts with no capabilities and develops them over time , we instead assume the adversary already has access to a full suite of exploits .one can view these analyses as complimentary , where applies to the attacker / defender evolution , while the work we present here applies to the steady state .in this paper , we have quantitatively studied cyber defenses based on temporal platform diversity .although the experiments were focused on a migration - based platform diversity system , much of the analyses apply to any such system .our abstract analysis has studied the major properties in a platform diversity system including the temporal effects , the payoff models , and the window of opportunity using combinatorics and markov models .additionally , the impact of finite duration and fractional effects has been studied .the experiments have collected data from a testbed implementation of such a system in which all of the abstract effects can be observed .the game theoretic analysis has studied the impact of preferential platform selection on the effectiveness of such systems and derived an optimal operational strategy . finally , by simulating the model of the system, we evaluate the impact of different strategies on the effectiveness of temporal platform diversity .our results suggest that while platform diversity is useful for mitigating some attacks , it is of critical importance to understand the threat model one aims to defend against .designing and deploying defensive techniques that have not considered the threat model may result in the illusion of security , when one may actually be increasing their attack surface .moreover , coupling a platform diversity system with a control mechanism that uses optimal strategies can significantly improve its effectiveness .no one technology will defend against all attacks , so understanding the proper operation and strengths of a platform diversity system enables network defenders ensure that other techniques are deployed for areas of weakness .the future work in this domain will focus on performing more experiments with such systems , extending the analysis to other platform diversity techniques and other randomization and diversity approaches , and analyzing the second order behavior such as adaptive adversaries who change tactics based on the deployed defenses .special thanks to mark rabe of mit lincoln laboratory for his help in setting up the experimental test bed .let be the expected length of a sequence beginning with a vulnerable state that requires consecutive vulnerable states to fail .let be the equivalent construct for sequences beginning with an invulnerable state .then the mean time until attacker victory is . due to the markov propertywe know that . also due to the markov property , we can compute the expected lengths of the sequence as     d. arsenault , a. sood , and y. huang .secure , resilient computing clusters : self - 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Abstract: active cyber defenses based on temporal platform diversity have been proposed as way to make systems more resistant to attacks . these defenses change the properties of the platforms in order to make attacks more complicated . unfortunately , little work has been done on measuring the effectiveness of these defenses . in this work , we use four different approaches to quantitatively analyze these defenses ; an abstract analysis studies the algebraic models of a temporal platform diversity system ; a set of experiments on a test bed measures the metrics of interest for the system ; a game theoretic analysis studies the impact of preferential selection of platforms and derives an optimal strategy ; finally , a set of simulations evaluates the metrics of interest on the models . our results from these approaches all agree and yet are counter - intuitive . we show that although platform diversity can mitigate some attacks , it can be detrimental for others . we also illustrate that the benefit from these systems heavily depends on their threat model and that the preferential selection of platforms can achieve better protection .   [ complexity measures , performance measures ]