any society contains individuals who are carriers of an ideology or fad ( e.g. , a religious or political party affiliation ) that they desire to spread to the rest of the society .thus , for a given ideology , a society can be partitioned into a set of people that represent the ideology and want to spread it , and the complement of this set .for example , the ideology could correspond to the views of a particular political party , with the party members desiring to recruit new members to improve their positions in the government .other areas of recruitment have been proposed as mechanisms for fads which appear as a rapid rise above some threshold , as in music , management technologies , economics , and even science .slower recruitment based on social networks has been postulated in biology and the spread of alcoholism .slowing the rise of a fad or eliminating the spread of an ideology has been also been proposed through the control of critical nodes in a social network .  in today s world where collisions between ideologies can lead to radicalization of society, the problem of existence and formation of terrorist networks or insurgency movements becomes important .recently , mathematical modeling of various radical groups , such as terrorist organizations , has been done to explore their structure and dynamics .in addition to dynamical approaches that measure rates of attacks of radical groups , operations research theories have also been applied to the formation of radical groups . another class of works  focuses on the recruitment dynamics of terrorists within a well - mixed population using compartmental models similar to those used for epidemic spread . in , a systematic analysis of recruitment to hezbollah has been done .data from a political science discipline is used to generate a compartmental model of recruitment .although the modeling is deterministic , it considers the various parameters which explicitly affect the success of recruitment to the radical s cause .finally , a number of recent publications discuss terrorist networks as optimal structures that optimize communication efficiency while balancing the secrecy of the networks  .not considered previously is the combination of the spread of radical ideas plus adaptive changes in social connections to improve recruitment success to the radical cause .some work in this direction is presented in the papers studying voter models in which individuals make connections to influence others opinions , as well as opinion dynamics models in which individuals are influenced by neighbors opinions .we develop a simple model of a society in which some of the members belong to a class that tries to spread an ideology .the ideology spreads as a result of contact of recruiting members with nonrecruiting members .the recruiting members may improve their chances to recruit via network adaptation .the purpose of the adaptation proposed here is to improve the spread of the ideology , which is in contrast to network adaptation in epidemiological models  where the purpose is avoidance of contact with the spreading members .in addition to adaptation , the connectivity within the society changes due to birth of new members and death of existing members . using this model , we explore how the existence of stable recruiting classes depends on the adaptation and other parameters .we also consider the network topology of the recruiting class , as it may be important in accessing the quality of the communication channels in the resulting structure .section [ sec : model ] presents the model and a system of mean field equations describing its dynamics .section [ sec : results ] presents mean field analysis of the threshold for successful recruiting and its dependence on the network adaptation .these results are compared with simulations of the full system , and the network geometry of the recruiting class is considered .section [ sec : conclusions ] concludes .we consider a social structure consisting of individuals represented by nodes in a network .an existing relation between any two individuals is represented by a link between the two nodes .new individuals join the society at a constant total rate , and the individuals in the network can leave the network via death at rate per individual . when new individuals join the system , they arrive with links , which are connected to randomly selected nodes in the population . when individuals die , their links are removed .   and respectively .fluxes between the three classes are described by the transition rates , and .link rewiring ( not shown here ) takes place at rate ., width=144 ]  some members of the society , _ recruiters _ ( also referred to as r - nodes ) , are carriers of an ideology that they try to spread to the individuals they come into contact with .the individuals that do not belong to the recruiting class are divided into two groups : those who are _ susceptible _ to the recruitment ( s - nodes ) and those who are _ non - susceptible _( n - nodes ) .we assume that individuals can spontaneously change their state from non - susceptible to susceptible and vice versa ( with rates and respectively ) .an individual in the recruiting class remains in that class until death  .a susceptible individual joins the recruiting class at a rate that is proportional ( with proportionality constant ) to the number of contacts it has with the recruiting class . in order to improve their recruiting capabilities , the recruiting individuals can rewire their links with rate , abandoning a connection to a nonsusceptible individual in favor of a connection to a susceptible one .as the links are rewired , the network topology changes based on the current states of its nodes . a schematic representation of the node dynamics is shown on fig . [ fig : scheme ] .we perform monte carlo simulations of the above system following the continuous time algorithm described in  .we start with an erdos - renyi random network , which then evolves according to the rules of birth , death , and rewiring .the results presented in the rest of the paper are for the systems that have reached a steady state .to describe the dynamics of this system , we construct a mean field model for nodes and links as in  .the time evolution of the nodes of each type is described by the following rate equations :  [ eq : nodes ]   here the functions , , represent the number of nodes of each type .the process of recruitment is captured by the term , where the recruitment is shown to take place at a rate proportional to the number of links between the recruiting class and the susceptible class , .  in order to capture the rewiring process, we follow the evolution of the number of different types of links present in the network :  [ eq : links ]   where the terms correspond to the number of links connecting nodes from classes x and y , e.g. , corresponds to the number of nn links .the terms proportional to correspond to the influx of edges due to the birth of new nodes , where the probability for the new node to attach itself to a node from class x is proportional to the number of nodes in that class .the third order terms , , and , describe the formation of triples of nodes with an s - node at the center , and at least one of the edges terminating at an r - node .these terms describe the rate at which ns- , ss- and rs - links become nr- , sr- and rr- links respectively , due to the interaction of the central s - node with its neighboring r - node . note that our definition of rsr triples includes _ degenerate _ triples ,i.e. , rsr triples where both of the r - nodes correspond to a single r - node , by analogy with degenerate triangles .the resulting system of equations is not closed , as it contains higher order terms .following earlier works in epidemiology  , we introduce the closure based on the assumption of homogeneous distribution of -links :  [ eq : closure ]   we thus obtain a system of nine mean field equations , which we analyze .we first consider the bifurcation point of the recruitment model where the zero recruit ( trivial ) steady state becomes unstable , which we call the _ recruiting threshold_. this is a transcritical bifurcation point , which is analogous to the epidemic onset in epidemic spreading models .it can be found analytically as a function of parameters for the mean field model , and certain asymptotic limits have a simple form .we nondimensionalize eqns .( [ eq : nodes ] ) and ( [ eq : links ] ) to simplify the analysis .we introduce the dimensionless time variable , where time is rescaled by the average node lifetime .further , the node variables are rescaled by the expected population size at steady state , , while the link variables are rescaled by the expected number of links at steady state , .let \delta \mu^{-1} ] is a vector of all the system parameters .( recall that is a dimensionless integer . )the full system of dimensionless equations is given in eqs .( [ eq : dimensionless ] ) .we can now find the trivial steady state solution , where the number of nodes is zero .this restricts the state to be of the form $ ] , where the number of links involving nodes is zero as well .this guarantees that for . since this subset of equations represents an invariant manifold , we concentrate on solving the equations for the rest of the 5 components , , , , and .the first observation is that the and equations are solvable when the number of recruiting nodes is zero , and they yield steady state values of    where .the nonzero link variables may be expressed in terms of , and they are given by the following :   now that we have the full trivial solution , we can examine its stability . in order to do this , we linearize the vector field about andexamine where it has a one dimensional null space .that is , at the bifurcation point , there is only one real eigenvalue passing through zero .this is equivalent to examining where the determinant of the jacobian vanishes ; i.e. , we compute those parameters where the following relation describes the location of the bifurcation : (\lambda_1+\lambda_2 + 1)}\right ] \nonumber\\ + \lambda_2 \frac{1}{\gamma ( 2-\sigma^{-1})-1 } + \frac{2(\gamma \sigma^{-1}+1)}{\gamma ( 2-\sigma^{-1})-1 }  \label{eq : w }  \end{aligned}\ ] ]  we next examine the limits of eq .( [ eq : w ] ) when either the recruitment rate or rewiring rate is large . the minimum amount of recruitment required to maintain nonzero values of recruited population , when approaches infinity , can be found by setting the least common denominator of eq .( [ eq : w ] ) to zero , which yields a simple expression for : if the recruitment rate is below this value , additional rewiring is not sufficient to enable spreading of the recruiting class .similarly , we find the smallest amount of rewiring required for existence of a nontrivial steady state solution , conditioned on our ability to control , with all the other parameters fixed : +\\ & + 2\frac{\sigma^{-1}}{2-\sigma^{-1 } }  \end{aligned}\ ] ] for lower rewiring rates , the recruiting class can not spread even if the recruiting rate is large .the value of this asymptote is greatest , meaning rewiring is most necessary , when approaches zero . in this casefew non - susceptible nodes become susceptible to recruiting , and the necessary rewiring value approaches   [ fig : simt0001 ]   we next compare the mean field predictions for the spread of recruiters with simulations of the full stochastic network system .thus , we compare the average size of the recruited portion of the population in statistical steady state to the solution of the mean field equations at steady state , which we solve exactly in the appendix [ app : exact ] .we assume that the parameters and ( rates for gaining and losing susceptibility ) and ( determines average degree ) depend on details of the society . on the other hand , we assume that the parameters and can be controlled by the recruiters , i.e. , the recruiters may choose to be more or less aggressive in their recruitment , as well as in how quickly they rewire their links to susceptible members of society .therefore , we investigate the recruiting effectiveness for a given choice of , and , while varying and .we distinguish two parameter regimes , corresponding to two qualitatively different behaviors of the system : small values of ( ) and large values of ( ) .the small regime corresponds to the case where nodes become susceptible to recruiting only rarely , on average much less than once in their lifetime ( where the average lifetime of a node is in the original time units and in the dimensionless units ) .the large regime corresponds to the case where , on average , nodes become susceptible to recruitment at least once in their lifetime .  in figs .[ fig : simt1000 ] and [ fig : simt0001 ] we show the results of simulating the system in these two regimes .the density plots in figs .[ fig : t1000 ] and [ fig : t0001 ] show the fraction of recruited nodes as a function of recruiting rate and rewiring rate in networks with and respectively .the black curves in the two figures represent the location of the recruiting threshold as derived from the mean field equations and given by eq .( [ eq : w ] ) . here mean field allows us to accurately predict the onset of the stable nontrivial solution . in figs .[ fig : tx1000 ] and [ fig : tx0001 ] we compare the simulation results to the mean field predictions .even though there is some discrepancy between the direct simulations and the mean field near the bifurcation , the recruiting threshold and the asymptotic behavior for large show excellent agreement with the simulations .we observe in simulations that the steady state trivial solution undergoes a forward transcritical bifurcation in the recruiting rate for all values of for which a nontrivial solution exists .this is in contrast with epidemiological models where the purpose of the rewiring is avoidance of the nodes spreading infection  , which can undergo a backward transcritical bifurcation and exhibit bistability .note that , unlike those models , the purpose of the rewiring in the recruitment model is attraction of the susceptible population by the recruiters .we would like to draw the reader s attention to an important difference in the system behavior in the two regimes . in the regimewhere , the nontrivial solution exists for all values of as long as is large enough . on the other hand , in the regime where , the nontrivial solution may fail to exist for any value of unless rewiring is aggressive enough . in fig .[ fig : t0001 ] there is a range of rewiring rates for which only trivial solutions exist . the horizontal dashed line , given by eq .( [ eq : asympt_horiz ] ) , indicates the mean field level of rewiring that is _ sufficient _ for the nontrivial solution to exist for rapid recruiting ( large ) .also , we can see that the rewiring level that is _ necessary _ for the emergence of the nontrivial solution can be very close to the predicted sufficient level of eq .( [ eq : asympt_horiz ] ) , for small values of .  , but same ratio of .squares ( red online ) , right axis : , .crosses ( green online ) , left axis : , .both of the simulations were performed with the following parameters : , , , . ]another important difference between the two regimes of values is seen when we compare two systems with different values of and while the ratio is kept fixed .note that in the absence of recruitment , such systems have identical fractions of susceptible nodes .in other words , in the presence of recruitment , the pool of individuals available for recruitment would appear to be the same . however , as we can see from fig .[ fig : l1l2 ] , there is a significant difference in the size of the recruited population as well as the recruiting threshold .the difference appears to be caused by the difference in the dynamics in the two regimes .thus , for the large values of and a node can become susceptible several times during its life , while as the value of decreases , only some nodes will ever become susceptible , with low likelihood of doing so more than once .  , on rescaled rates of rewiring , , and recruiting , . solid line ( green online )indicates the value of where the degree is at a maximum for a given value of , found using analytic expression in eq .( [ eq : gamma_m ] ) .the white diamonds correspond to the location of the maximum in simulations for a given value of .the simulations are performed with the following parameters : , , , , . ]we now investigate the structure of the portion of the network ( later referred to as the r - subnetwork ) consisting only of r - nodes and links between them .in particular , we are interested in the mean degree of its nodes when the system reaches a steady state .the mean field approach , in addition to predicting the fraction of recruited nodes ( which corresponds to the size of the r - subnetwork ) , also provides information about topology of the subnetwork in the form of mean degree of the nodes .the mean degree of a node in the subnetwork serves as a low order description of how well the nodes in the network are connected , which may be important for communication within the established subnetwork . the density plot in fig .[ fig : tt1000 ] shows the dependence of the mean degree on and . the nonmonotonic behavior as a function of for fixed predicted by the mean field model .the solid line corresponds to the analytical prediction of the maximum s location as given by eq .( [ eq : gamma_m ] ) derived in appendix [ app : extremum ] .note that the analytic description of the maximum s location would allow the recruiting class to optimize connectivity within the subnetwork , if that happens to be an important goal for that class .the mean field approximation fails to capture the nonmonotonic behavior of the mean degree in the limit where , and we leave this issue to a future study .we develop a toy model that describes recruitment of new members by an interested class within a society .the members of the recruiting class can improve their recruiting capabilities by via adaptation .thus , they may choose to abandon their relations with those members of society that are not prone to recruitment .this model assumes that once a node joins the recruiting class , it itself becomes a recruiter and it remains a member of this class until death .in contrast to avoidance rewiring used to reduce infection spreading in epidemic models  , network adaptation in our model promotes spreading .additionally , the population is open with birth and death modeled explicitly , while most previous adaptive network models have been of closed populations ( e.g. , ) .  in this paper, we develop and analyze a mean field description of our model .thus , we are able to accurately predict the onset of the stable nontrivial solution of the system at steady state . furthermore , we are able to accurately predict the size of the recruited class for a given set of system parameters , as well as the mean degree of the subnetwork formed by the recruited members .we compare the predictions made by the mean field description with the direct simulations of our model .we generally observe a good agreement between the model and its mean field approximation .analyzing the mean field model , we find two parameter regimes with very distinct qualitative behavior .thus , we show that in the society where the particular ideology is unpopular ( perhaps corresponding to radical ideology ) and individuals rarely become susceptible to it , adaptation is necessary in order to observe stable nonzero levels of the recruiting class . on the other hand , if the idea is sufficiently popular ( e.g. , recruitment into a moderate political party ) , the adaptation improves the recruiting capabilities and may affect the ultimate topology of the recruited , but adaptation is not a necessary condition for the existence of a nonzero stable solution .furthermore , we speculate that if the model were changed to describe a society with two competing recruiting classes ( think two party system ) , the adaptation may be a mechanism by which one competing class gets an edge over the other class .  as evidenced by the results presented in figs . [fig : tx1000 ] and [ fig : tx0001 ] , the mean field approximation has some inaccuracies in the parameter regime between the bifurcation point and the asymptotic saturation .we predict that this inaccuracy is due to errors in the homogeneous closure assumption ( eqs .[ eq : closure ] ) . in a subsequent work ,we develop and analyze a new closure of the mean field equations that more accurately describes the full system .mss and lbs were supported by the army research office , air force office of scientific research , and by award number r01gm090204 from the national institute of general medical sciences .ibs was supported by the office of naval research , the air force office of scientific research , and the national institutes of health .the content is solely the responsibility of the authors and does not necessarily represent the official views of the national institute of general medical sciences or the national institutes of health .in this appendix , we derive the exact solution to the system of equations in eqs .( [ eq : nodes ] ) and ( [ eq : links ] ) when the system is at the nontrivial steady state , i.e. , when the left hand side of the equations is zero and the size of recruiting class is nonzero .we begin by deriving the dimensionless equations , as defined in section [ subsec : res_1 ] :  [ eq : dimensionless ]   at steady state , the left hand side of the above equations is zero .we proceed in our derivation by dividing all equations in the steady state by and introducing a new variable , obtaining the following system of equations :    we have used the fact that in the steady state , as can be shown by adding together eqs .( [ eq : node_n])-([eq : node_r ] ) . in the rest of the derivation we will find a closed equation for andexpress the other s in terms of .we can immediately express , , and in terms of by solving eqs .( [ eq : z2 ] ) , ( [ eq : z3 ] ) and ( [ eq : z9 ] ) respectively : we substitute eq .( [ eq : z1_sol ] ) into eq .( [ eq : z4 ] ) , and eq . ( [ eq : z3_sol ] ) into eq .( [ eq : z7 ] ) .the resulting two equations , together with eqs .( [ eq : z5 ] ) , ( [ eq : z6 ] ) and ( [ eq : z8 ] ) , form a closed system of five equations with five unknowns - .  we solve eq .( [ eq : z6 ] ) for in terms of and : and substitute the above result into eq .( [ eq : z4 ] ) and ( [ eq : z7 ] ) , to solve for and in terms of and as follows : ^{-1}[1+\lambda_2+\gamma z_8][1+\lambda_2 z_6]\label{eq : z4_sol}\\ z_7&= [ \gamma  \lambda_1^{-1}(2\gamma z_8 + 2(\lambda_2 + 1))z_6 + 2\gamma + \lambda_2 ] \times\nonumber\\ & \times(\lambda_1 + 2+w)^{-1}z_8.\label{eq : z7_sol }  \end{aligned}\ ] ] substituting the expressions for and from eq .( [ eq : z5_sol ] ) and ( [ eq : z4_sol ] ) into eq . ( [ eq : z5 ] ) , and solving for we obtain finally , substituting results of eqs .( [ eq : z6_sol ] ) and ( [ eq : z7_sol ] ) into eq .( [ eq : z8 ] ) we obtain an equation for in a closed form : } { [ \gamma(\lambda_1 + 1)z_8+(\lambda_1+\lambda_2 + 1)]}=0 \label{eq : z8_sol }  \end{aligned}\ ] ] where .  \end{aligned}\ ] ] note that the physically relevant solutions are positive , and therefore finding a nontrivial solution of is a simple matter of solving a quadratic equation :   solving eq .( [ eq : z1 ] ) for , we can now express in terms of the newly found : ^{-1}.  \end{aligned}\ ] ] the rest of the original variables can be found using .thus , for example , is ^{-1}.  \end{aligned}\ ] ]in steady state , the mean degree of the nodes within the r - subnetwork is found by taking a ratio of twice the number of -links to the number of -nodes in the subnetwork : where the values of and are obtained by solving eqs .( [ eq : node_r ] ) and ( [ eq : link_rr ] ) in steady state . in this appendixwe determine the value of , , that for a given rewiring rate will maximize the degree in the resulting r - subnetwork .we do this by maximizing .differentiating eq .( [ eq : quad ] ) with respect to and evaluating the resulting equation at , the value where extremum is attained , we obtain note that the derivative of with respect to evaluated at is equal to zero because is an extremum there , and the are independent of . multiplying the above equation by and subtracting it from eq .( [ eq : quad ] ) evaluated at allows us to solve for :   substituting the value of into eq .( [ eq : quad_der ] ) and solving for we obtain \label{eq : gamma_m}\\ & /[a_5 ( a_1a_5-a_3  a_2)],\nonumber  \end{aligned}\ ] ] the location of the maximum for the given rewiring rate .

Abstract: we model recruitment in adaptive social networks in the presence of birth and death processes . recruitment is characterized by nodes changing their status to that of the recruiting class as a result of contact with recruiting nodes . only a susceptible subset of nodes can be recruited . the recruiting individuals may adapt their connections in order to improve recruitment capabilities , thus changing the network structure adaptively . we derive a mean field theory to predict the dependence of the growth threshold of the recruiting class on the adaptation parameter . furthermore , we investigate the effect of adaptation on the recruitment level , as well as on network topology . the theoretical predictions are compared with direct simulations of the full system . we identify two parameter regimes with qualitatively different bifurcation diagrams depending on whether nodes become susceptible frequently ( multiple times in their lifetime ) or rarely ( much less than once per lifetime ) .


faraday rotation is a physical phenomenon where the position angle of linearly polarized radiation propagating through a magneto - ionic medium is rotated as a function of frequency .the work on astrophisical faraday rotation has been initiated in , and since then several important contributions have been added to this topic . recently ,faraday rotation measure ( rm ) synthesis has been re - introduced as an important method for analyzing multichannel polarized radio data , where multiple emitting regions are present along the single line of sight of the observations . in practice , the method requires the recovery of the faraday dispersion function from measurements restricted to limited wavelength ranges , which is an ill - conditioned deconvolution problem , raising important computational difficulties .since then , three different approaches have been proposed to solve this problem .a first approach uses an adaptation of the hogbom clean algorithm to the rm deconvolution .the second approach is wavelet - based , and assumes field symmetries in order to project the observed data onto  .the third approach is based on the compressed sensing paradigm .all these methods are more or less successful in the case of mixed problems , i.e. when both thin and thick components are included in the model . for example , in a recent paper it has been shown that rm synthesis may yield an erroneous faraday structure in the presence of multiple , interfering rm components , even when cleaning of the faraday spectrum is performed .also , to our knowledge these methods have not been evaluated in the presence of noise added to the observed data , a situation that makes the deconvolution problem even more difficult .thus , the development of robust deconvolution methods for the recovery of the faraday dispersion function in a given spectral range becomes crucial for the rm synthesis applications .inspired by the above mentioned contributions , in this paper we discuss the case of sparse approximation of the complex faraday dispersion function , i.e. we assume that can be approximated by a small number of discrete components , which can be both thin or thick . also , we present the implementation of a greedy deconvolution algorithm , and we illustrate the described method with several numerical simulations which emphasize the effect of the covered range and sampling resolution in the faraday depth space , and the effect of noise on the observed data .the numerical results show that the described method performs quite well for simple component mixtures , at typical sampling resolution values and coverage range in the faraday depth space , and it is quite robust in the presence of noise .we show that the described technique is well suited for exploratory data analysis , where prior information about the component distributions is not available , and it can be used as a complement to the previously proposed methods .although a sparse solution is an idealized model of a complex astrophysical system , the potential complexity of the solutions is adequate for a wide range of astrophysical situations .the sparseness requirement steers the solution to include the smallest number of components required to fit an observed faraday depth spectrum .double - lobed radio galaxies that are not resolved by the telescope may experience different faraday rotation in each lobe because the differences in the foreground on scales smaller than the beam .the lobes themselves may be extended and experience differential faraday rotation as well .a sparse solution may consist of two discrete faraday components representing each lobe .if the data are good enough to detect differential faraday rotation across the source , the solution may include one or more components with a finite extent in faraday depth .complex source structure may be built up out of a dictionary of basic thin and thick faraday components , subject to the requirement that the solution remains sparse .  in the diffuseinterstellar medium , a case where faraday rotation of galactic synchrotron emission is dominated by a single hii region along the line of sight is an example of a system that is well approximated with two components in faraday depth , e.g. the circular faraday screen discussed by and .as in the case of double lobed radio sources , the sparse solution is not limited by two delta functions in faraday depth , as it can increase in complexity if warranted by the data .the assumption of sparseness may fail in case there is a power on a large range of faraday depths , defined by the minimum and maximum faraday depth detectable in a survey .this may occur in some supernova remnants with complex structure and strong magnetic fields .in this section we give a brief description of the faraday rm synthesis problem , following the formulation introduced in .the faraday rotation is characterized by the faraday depth ( in ) , which is defined as: where is the electron density ( in ) , is the magnetic field ( in ) , and is the infinitesimal path length ( in parsecs ) .we also define the complex polarization as: where is the fractional polarization , , , are the stokes parameters , and is the polarization angle observed at wavelength : the faraday rm is defined as the derivative of the polarization angle , with respect to :   we now identify rm with the faraday depth , and we assume that the observed polarization originates from the emission at all possible values of , such that : where is the complex faraday dispersion function ( the intrinsic polarized flux , as a function of the faraday depth ) .thus , in principle is the inverse fourier transform of the observed quantity : however , this operation is ill - defined since we can not observe for , and also in practice the observations are limited to an interval  ] , and otherwise .therefore , we obtain the reconstructed dispersion function: where ^{-1},\ ] ] is the normalization constant for the observation window . the reconstructed dispersion functioncan also be written as: where is the convolution operator , and is the rm spread function ( rmsf ) .  using the shift theorem , we can also write: and where is the mean of the sampled values in  ] , , with the given weights .since the measured values are discrete ( each value constitutes an integral over the channel centered at ) , we should consider the discrete versions of the above equations , i.e. : and respectively the reconstructed function depends on the window , which acts as a filter , and improves substantially by increasing its coverage in the space .obviously , is a `` dirty '' reconstruction of , i.e. the convolution of with , and and a deconvolution step is necessary to recover .in general , the number of data points is limited by the number of independent measurement channels , and therefore there are many different potential faraday dispersion functions consistent with the measurements . the usual approach to resolving such ambiguities ,is to impose some extra constraints on the faraday dispersion function .our approach is based on the recently introduced framework of compressive sensing .compressive sensing relies on the observation that many types of signals can be well - approximated by a sparse expansion in terms of a suitable basis , or dictionary of functions .the main idea of compressive sensing is that if the signal is sparse , then a small number of measurements contain sufficient information for its approximate or exact recovery . in our case ,the problem is to reconstruct a sparse from a relatively small number of measurements .therefore , we assume that the model of is sparse in an over - complete dictionary of functions . by over - completewe understand that the number of functions in the dictionary is larger than the number of independent observation channels .thus , the dictionary functions may be redundant ( linearly dependent ) , and therefore non - orthogonal . in order to give a proper formulation of this approach we need to introduce a discrete representation of the space .it is known that , for a discrete sampled faraday dispersion function , the full width at half maximum of the main peak of the rmsf is given by : where is the width of the observation interval .also , using a uniform grid in space one can estimate the maximum observable faraday depth by : where is the width of an observing channel .this estimation of is only an approximation , since in reality only the frequency is sampled linearly .therefore , in our discrete representation we consider a nonlinear grid in the space : , where is the centered frequency of the channel , and is the speed of light .also , we consider a linear grid in the space , where the computational window , the sampling resolution , and the number of points are set to : where is the integer part of .  the model of is therefore characterized by a uniform grid , , , and a vector \in\mathbb{\mathbb{c}}^{m} ] , and taking into account that : we obtain the following minimization problem: where is a complex matrix . in this more general case ,the goal is to find the sparse vector in the dictionary space .obviously , when the dictionary is reduced to the dirac basis we have , and , where is the identity matrix , and therefore reduces to the weighted fourier matrix , .   we should note that an over - complete dictionary that leads to sparse representations can be chosen as a pre - specified set of analysis functions ( wavelets , gaussian packets , gabor functions etc . ) , or designed by modeling its content to a given set of signal examples .the success of such dictionaries in applications depends on how suitable they are to sparsely describe the signals in question .a general family of analysis functions can be obtained by scaling and translating a single normalized window function , with .therefore , for any scale and translation we define the atom of the dictionary as following: therefore , the index of the atom function depends on both and parameters : .thus , in order to represent in the dictionary , we need to select an appropriate countable subset of atoms , , such that can be written as a linear expansion . depending on the choice of the atoms , the expansion coefficients will give explicit information about the behavior of .for example , we should note here that different wavelet transforms correspond to different families of atoms . in our definition ,we do not limit the dictionary to a single wavelet basis , on contrary we consider an over - complete set , which also may contain different concatenated families ( sub - dictionaries ) of such analysis functions . in order to illustrate numerically this approach ,let us consider the boxcar dictionary , defined as : an important characteristic of the boxcar dictionary is that it can capture sources with arbitrary thickness .another advantage is its easy discretization . in our case , the discretization grid has points with the sampling resolution .thus , assuming that the maximum width of a boxcar atom is , where , then for each scale , , and translation , we can define a boxcar function with the index , such that: therefore , one can define maximum boxcar functions on such a grid , and we can easily build a discrete dictionary matrix of size . in this paperwe limit our discussion to the boxcar dictionary defined above , since it is simple enough to illustrate the approach , and to provide meaningful results .also , this dictionary includes by construction the dirac set of functions , which in this case are the first functions with .a similar approach can be used to build sub - dictionaries corresponding to other families of analysis functions .the sparse decomposition can also be used to perform a multi - scale analysis , by considering all the dictionaries , where . also , let us assume that is the solution obtained for the scale , i.e. the recovered discrete representation of with the dictionary .we consider a matrix , where each line with the index corresponds to the solution obtained for the scale , i.e. .obviously , the solution will depend on the maximum scale used in each dictionary , and by visualizing the matrix , we obtain a representation of the behavior of the solution at different scales .the sparse optimization problem , defined in the previous section , is known as basis pursuit denoising ( bpd ) , and if written in a lagrangian form : ,\ ] ] it can be thought of as a least squares problem with an regularizer , where is a parameter that controls the trade - off between sparsity and reconstruction fidelity .thus , bpd solves a regularization problem with a trade - off between having a small residual and making the solution simple in the sense .the solutions of bpd are often the best computationally tractable approximation of the under - determined system of equations . in our case , since the direct space and the inverse fourier space are perfectly incoherent , the problem can be solved using linear programming techniques whose computational complexities are polynomial . however , for the sparse rm approximation problem , the bpd approach requires the solution of a very large convex , non - quadratic optimization problem , and therefore suffers from high computational complexity . due to the complexity of the linear programming approach, several other optimization methods have been proposed to solve the bpd problem . here, we consider a method based on sub - optimal greedy algorithms , which requires far less computation .our goal is not only to obtain a good sparse expansion , but also to provide a fast computational method , therefore here we focus our attention on the greedy matching pursuit ( mp ) algorithm , which is the fastest known algorithm for the bpd problem .mp has many applications in signal and image coding , shape representation and recognition , data compression etc .one of its main features is that it can be applied to arbitrary dictionaries .starting from an initial approximation and residual , the algorithm uses an iterative greedy strategy to pick the column vectors which best reduce the residual . at every time step the current residual can be decomposed as following : where is the future residual , and is the standard inner product operator in the complex hilbert space . since and are orthogonal , , we have: in order to minimize the norm of the future residual , the algorithm should choose the column vector which maximizes the projection on the current residual : therefore , after choosing the best column one can update the solution and the residual as following: where thus , after iteration steps the resulted solution is a sparse vector with the non - zero coefficients .the algorithm stops when the maximum number of iterations has been reached ( which usually is set to ) , or when the norm of the residual becomes comparable with the standard deviation of the noise .the reconstruction of the target signals is then given by :  the pseudo - code of the rm - mp algorithm is listed in the appendix .in order to illustrate the described deconvolution method , we have considered two different experiment configurations , corresponding to two different ranges of observed frequencies .the first one is consistent with the observations with the westerbork synthesis radio telescope ( wsrt ) in the frequency range 315 mhz to 375 mhz , as described in .the second one is consistent with the observations with the arecibo telescope in the frequency range 1225 mhz to 1525 mhz , for the galactic alfa continuum survey ( galfacts ) , as described in .the separation between the frequency windows is roughly 1 ghz , and therefore the maximum observable faraday depth and the half maximum of the main peak of the rmsf are quite different . herewe will show that the rm - mp method provides very good results in both cases .  as a testbed for numerical simulations , we have considered a mixed scenario consisting of three components with different widths , such that the simulation results provide the response of the algorithm to a full range of component widths .the first one is a thin component given by : .the second is a thick component given by : if , and otherwise .the third is a thicker component defined by : if , and otherwise .thus , this scenario can be easily scaled for different computational windows $ ] , where is given in .also , we have considered that all the observational channels are equally weighted : i.e. , , and .the various parameters associated with the wsrt experiment layout are listed bellow :  frequency range : , ;  wave length range : , , ;  number of channels : ;  half maximum of the main peak of the rmsf : ;  maximum observable faraday depth : ;  let us first consider the ideal noiseless case , when the sampling resolution in the space is equal with the half maximum of the main peak of the rmsf , , and the computational window is . in this particular case , as shown in figure 1 , the rm - mp algorithm provides an exact solution , since and therefore no information is lost in the measurement .one can also notice that in this noiseless exact sampling case , the solution is independent of the scale used in the dictionary , as it can be seen on the multi - scale representation for .however , the problem becomes ill - defined in the following situations : the noise is present ; the sampling resolution becomes finer than the half maximum of the main peak of the rmsf , ; and the number of independent observed channels is smaller than the number of points in the space , . in this casethe system becomes under - determined , and therefore some information is lost . in order to exemplify this situation, we consider a scenario in which all these factors are present . we add noise with the standard deviation , to the and values .we limit the computational window to , and we increase the number of points on the grid to , which is double of the number of observation channels .this results in a sampling resolution of .the obtained results ( for , ) are shown in figure 2 .one can see that the phase of some components can not be reliably recovered anymore , since there is not enough information in the signal to be detected properly .we should note that the problem is correctly resolved in the noiseless case ( not shown here ) .thus , the effect of noise addition consists in a partial loss of information about the phase of , which is expected , since the number of solutions compatible with the data increases dramatically with the added noise .also , we should point out that the solution improves by increasing the signal - to - noise ratio , as shown in figure 3 , where we have increased the amplitude of the components by a factor of 1.5 , keeping their phase unchanged .one can see that in this case , the rm - mp method resolves correctly all the components .this result suggests that an adequate signal to noise ratio should be taken into account , in order for the method to be successful .the galfacts survey , carried out with the arecibo telescope , has the following parameters :  frequency range : , ;  wave length range : , , ;  half maximum of the main peak of the rmsf : ;  the maximum observable faraday depth , , is inverse proportional with the width of the observation channel , and by increasing the number of channels , the maximum observable faraday depth becomes unreasonable high .therefore , in order to obtain some meaningful results , we have to limit both the number of observation channels in the space , and the computational window in the space .first we consider that the compuational window is limited to and the number of observation channels is . also , we consider the same testbed as for wsrt case , and we add noise with the standard deviation .in addition , we increase the number of points on the grid to , and therefore we obtain a sampling resolution : .the obtained results ( for , ) are shown in figure 4 .one can see that all the components are relatively well resolved , with a small error in the phase , but with almost exact amplitudes . in the next experiment we zoom more in the space , and we impose a computational window of , keeping the same number of observation channels and number of points on the grid , such that .the results are again reasonable good , as shown in figure 5 , with a small variation of the phase due to the uncertainty introduced by the noise addition . however , if we zoom further in the space the solution is not as good anymore , as it can be seen in figure 6 . in this casewe have a much finer sampling resolution , corresponding to a computational window of , number of observation channels , and the number of points on the grid .this is a consequence of the fact that by increasing , we have increased also the standard deviation of the noise to , such that the signal to noise ratio is smaller than before .again , an improved solution can be obtained by increasing the amplitude of the components , such that the signal to noise ratio is higher .   in the previous numerical experiments we have shown that the rm - mp algorithm is able to resolve correctly the components from the input model , if the separation between the components is higher than the half maximum of the main peak of the rmsf . in order to estimate the response of the rm - mp algorithm at resolutions beyond the rmsf limit , we consider two dirac components , , and respectively , separated by , where is the half maximum of the main peak of the rmsf .the numerical experiments show that the rm - mp algorithm can not resolve correctly the two components , but returns a boxcar function centered at the exact position value , with a width equal with the separation between the two components . in order to illustrate this result we consider the wsrt scenario , with , and , which gives a sampling resolution , in the space . in figure 8we give the width of the output boxcar function as a function of the input separation .one can see that for all performed experiments we have . also , in figure 9 we show a typical example , where the input separation is , or approximatively from .thus , even at resolutions beyond the half maximum of the main peak of the rmsf , the rm - mp algorithm provides some useful information , i.e. the position and the separation width of the two components .the above numerical experiments have shown that the sparse rm - mp method works well for relatively simple sparse problems .we should note that the method can be used to recover more complex dispersion functions .for example , let us consider the situation from figure 7 , where we have two thin components and two thick components .the first thick component is modeled as a gaussian , while the second is modeled as a boxcar function .also , we assume the noisy wsrt experiment configuration , with : , , and .one can see that all the sources are almost exactly recovered , including the thick gaussian , even though the dictionary does not contain any gaussian functions .in fact , the shape of the gaussian is reconstructed from several boxcar functions from the dictionary .thus , the boxcar dictionary can be used to recover more complex functions .however , the success of the method depends on another aspect which has not yet been discussed .more specifically , the performance of the rm - mp method depends on the number of observation channels , the number of points on the grid , and the number of non - zero components in the discrete representation of the faraday depth function .an important question here is that given and , what is the maximum value of , for a faithful recovery of ?in it has been shown that any -sparse signals of length , with , can be recovered from only random measurements ( projections ) , where .the answer to this question is not obvious for the sparse rm synthesis problem , since the reconstruction process will depend on experiment layout , i.e. the observed frequency band and the half maximum of the main peak of the rmsf .this is an important theoretical question which we would like to address in the future development , in order to improve the performance of the method .the recently introduced faraday rm synthesis is becoming an important tool for analyzing multichannel polarized radio data , and derive properties of astrophysical magnetic fields .the method requires the solution of an ill - conditioned deconvolution problem , in order to recover the intrinsic faraday dispersion function , and therefore the development of robust methods has become crucial for the rm synthesis applications . here, we have assumed that the complex faraday dispersion function can be approximated by a small number of discrete components from an over - complete dictionary , and we have developed a greedy algorithm to solve the deconvolution problem .the method uses an over - complete dictionary of functions which can be efficiently used in a multi - scaling context , and it can easily include different types of analysis functions .we also have presented several numerical simulations showing the effect of the covered range and sampling resolution in the faraday depth space , and the effect of noise on the observed data .the numerical results show that the described method performs well at common resolution values and coverage range in the faraday depth space , and it is quite robust in the presence of noise .therefore , the described technique is well suited for exploratory data analysis , and it can be used as a complement to the previously proposed methods .the pseudo - code of the rm - mp algorithm :                                  de bruyn , a. g. , bernardi , g. & the lofar team , in the low - frequency radio universe , proceedings of the conference held 8 - 12 december 2008 , at national centre for radio astrophysics ( ncra ) , tifr , pune , india , ed .d. j. saikia , d. a. green , y. gupta & t. venturi , asp conf .407 , 3 .taylor , a. r. & salter , c. j. 2010 , the dynamic interstellar medium : a celebration of the canadian galactic plane survey .proceedings of a conference held at the naramata centre , naramata , british columbia , canada on 6 - 10 june 2010 . edited by r. kothes , t. l. landecker , and a. g. willis .san francisco : astronomical society of the pacific , 2010 , p.402 .

Abstract: faraday rotation measure synthesis is a method for analyzing multichannel polarized radio emissions , and it has emerged as an important tool in the study of galactic and extra - galactic magnetic fields . the method requires the recovery of the faraday dispersion function from measurements restricted to limited wavelength ranges , which is an ill - conditioned deconvolution problem . here , we discuss a recovery method , which assumes a sparse approximation of the faraday dispersion function in an over - complete dictionary of functions . we discuss the general case , when both thin and thick components are included in the model , and we present the implementation of a greedy deconvolution algorithm . we illustrate the method with several numerical simulations that emphasize the effect of the covered range and sampling resolution in the faraday depth space , and the effect of noise on the observed data .


quantum key distribution ( qkd ) admits two remote parties , known as alice and bob , to establish an unconditional secret key ; even the eavesdropper ( eve ) has unlimited power admitted by the quantum mechanics .the unconditional security of qkd has been proved in theory for both the ideal system and the practical system based on some assumptions .however , the practical qkd system is imperfect . strictly speaking, any deviation between the standard security analysis and the practical qkd system can be exploited by eve to attack the practical system . therefore , in order to guarantee the unconditional security of the final key generated by the practical system , the legitimate parties must survey the practical qkd system carefully and close these loopholes .  in the standard security analysis for the bennett - brassard 1984 ( bb84 ) qkd system with a weak coherent source ( wcs ) ,an important assumption is that the phase of source has been randomized totally .thus , in the view of eve , the state sent by alice is a mixed state of all number states .however , in a practical qkd system , the phase information of source might be accessible to eve . in order to remove the phase randomization assumption from the standard security analysis , lo and preskillhave proved the security of bb84 protocol using the wcs with nonrandom phase .but their proof is at the price of the secret key rate and the maximal security distance is very short .thus the best choice for the legitimate parties is to actively randomize the wcs phase , which can be implemented by modulating a totally random phase  ] and .furthermore , we note that in most of the practical system , no active setup is used to randomize the phase of source .we think there may be two reasons : ( 1 ) when alice actively randomizes the source , an additional active setup is needed which will increase the complexity of the qkd system .( 2 ) more importantly , within the best of our knowledge , until now there is not an effective attack strategy to exploit the phase information of wcs . in other words , evedoes not know how to spy the secret key , even if the source is not randomized or just partially randomized .  in this paper, we propose a partially random phase ( prp ) attack to break the security of the practical two - way qkd system using wcs. then a simple intercept - and - resend attack strategy and experimental arrangement within current technology are proposed to spy the secret key .our analysis shows that the error rate induced by eve can be lower than the tolerable threshold value of error rate , whereas the same range of error rate has been proved secure if the legitimate parties are unaware of our attack .thus when our attack is taken into account , the secret key rate will be compromised .specifically , the numerical simulations show that , in some parameter regime , our attack is immune to the one - decoy - state method which is often used to defeat the pns attack .therefore , the legitimate parties should consider our attack carefully when they use the wcs to implement the bb84 protocol .however , note that we only claim that our attack is immune to the one - decoy - state method in some parameter regimes , but we do not claim that our attack is completely immune to the decoy - state - method .in fact , our attack can be defeated by the two - decoy - state ( weak + vacuum ) method , since the vacuum state is used to estimate the gain and error rate of the background .we note that in ref. , lo and preskill have proposed a simple attack to exploit the nonrandom phase of source . however , our attack performs better than their attack at least in two aspects .first , in their attack eve performs the positive operator valued measure ( povm ) belonging to the photon number state space to distinguish the key bit , which can not be implemented within current technology .our attack needs only a homodyne detector , which can be implemented within current technology .second , in their attack eve needs to know the exact phase of source , which corresponds to the case that the source is not randomized .but our attack is valid as long as the source is partially randomized .therefore , the legitimate parties do not need to consider their attack , especially in practical situations , but they must consider our attack and monitor their system carefully .the paper is organized as follows : in sec .[ sec : attack ] we introduce how to exploit the imperfection of a partially random phase to break the security of the cryptosystem . in sec .[ sec : result ] we introduce an intercept - and - resend attack with our prp attack and then analyze the error rate induced by our attack . in sec .[ sec : decoy ] we show that our attack is immune to the one - decoy - state method which can be used to defeat the pns attack . in sec .[ sec : dis ] we provide some discussion regarding our attack . finally , in sec .[ sec : summary ] we give a brief summary of this paper .in this section , we first introduce the _ plug - and - play _ qkd system briefly .then we discuss how to exploit the partially random phase of source to spy the secret key .a simple diagram of a _ plug - and - play _ qkd system without eve is shown in fig.[fig : exp_arr](a ) . a strong pulse sent by bob s laser will be divided into two parties by a 50:50 beam splitter ( bs ) , noted as a signal pulse ( _ s _ ) and reference pulse ( _ r _ ) .when the two pulses arrive at alice s zone , alice encodes her information on the signal pulse by modulating a phase . in order to ensurethe global phase of pulse is totally random , alice modulates a random phase  ] . for the three cases that the phase is nonrandomized ,partially randomized , and totally randomized , , , and , respectively .note that in eq .we consider only the signal pulse and ignore the reference pulse , since only the signal pulse is modulated by alice .  herewe remark that if alice does not actively randomize the phase of source or she thinks that eve may know the phase of source exactly , she must use the method given by ref . to estimate the key rate .however , the security analysis of ref . can only be used for short - distance quantum cryptography .thus , generally speaking , alice should use an active setup to randomize the phase of source in the long - distance quantum cryptography .however , in the practical qkd system , it is a hard task for the legitimate parties to check whether the phase is really totally randomized or not .in fact , if alice and bob do not check carefully that the phase is truly random , then the range of random phase can be controlled by eve so that it is just partially randomized ( ) . for example , in the two - way qkd system , since the phase modulator has a finite response time , eve can control the practical phase modulated on the pulse by shifting the arrival time of the signal pulse to the rising or falling edge of the phase modulator .it is known as a phase remapping attack , which was proposed by fung __ in theory and then demonstrated by xu _ et al . _ in experiment . thus the practical phase modulated on the source will be lower than the expected phase of alice , if eve shifts the arrival time of the pulse .for instance , alice wants to modulate a random phase , but the practical phase modulated on the source is . therefore , in the practical qkd system , the range of random phase modulated by alice can be controlled by eve so that the pulse is just partially modulated ( ) .we have shown that the random phase can be controlled by eve so that it is just partially modulated . in the following, we will show how eve can exploit this imperfection to spy the secret key .since alice admits the pulse into and out of her zone in the two - way system , it is easy for eve to set her experimental arrangement to load our attack , which is shown in fig.[fig : exp_arr](b ) .the strong pulse sent by eve s laser will be divided into three parts by two bs , signal pulse ( _ s _ ) , reference pulse ( _ r _ ) and local pulse ( _ l _ ) .then eve sends _ s _ and _ r _ to alice and keeps _l _ in her own hand .when the pulse is modulated by alice and resent back to eve , eve modulates randomly _l _ with a phase . then _ s _ will interfere with _l _ at bs and be detected by two photodiodes ( dc and dd ) .  hereeve uses a homodyne detector to analyze alice s information .the homodyne detection is a well - established quantitative method to measure the quadrature - amplitude operator of the signal field or implement the continuous variable quantum cryptography .a simple illustration of the homodyne detection is shown in fig.[fig : exp_arr](b ) . the signal pulse ( _ s _ ) andthe local pulse ( _ l _ ) interfere at a 50/50 beam splitter ( bs ) .then the two output modes of bs , _ c _ and _ d _ , will reach two photodiodes ( dc and dd ) respectively . the measured output signal of homodyne detection is determined by the difference of dc and dd , which is given by  where is the average photon number of the local pulse , is the annihilation operator of signal pulse , is the difference of phase between signal pulse and local pulse .here we assume the local pulse is strong coherent state . is known as the normalized quadrature amplitude of the signal . generally speaking, takes a random value for each pulse due to the quantum fluctuations .theoretically , the probability of is given by integrating the wigner distribution , which is given by where is the wigner function in space .it is easy to check that , when the signal pulse is coherent state , the probability distribution of is given by ,\ ] ] where is the average photon number of signal pulse , is the difference of phase modulated by eve and alice ,  ] , thus eq .should be rewritten as .\ ] ]  fig.[fig : theo_dis ] shows the theoretical probability distribution of when the total phase shifts are , , and .it shows clearly that measured result of the homodyne detection can be used to distinguish and by setting a suitable threshold value .for example , eve can set two threshold value and . when the measured quadrature amplitude , eve judge that ; when , eve judge that .but when , eve ca nt give a judgement about the phase .although eve ca nt distinguish the four phases deterministically , it is easy to check that , the conditional probability that eve obtain a valid outcome given that or can be much larger than that of and . here, a valid outcome means the measured outcome satisfies or .obviously , the smaller the probability eve makes a wrong judgement , the larger is . here _wrong judgement _ means eve obtains ( or ) but the state sent by alice is nt ( or ) . in order to maximize eve s information for a given system , we make two remarks about the optimal intensity of signal pulse ( ) that is accessible for eve ._ remark one _ : generally speaking , alice will monitor the power of light coming into her zone , thus eve should set the intensity of light sent to alice carefully .however , note the fact that , alice does nt monitor the intensity of _ s _ and _ r _ respectively in most practical qkd systems .thus eve only needs to ensure the total of them entering alice s zone , denoted as , is unchanged , but no needs to keep the intensities of both _s _ and _ r _ to be the same as their expected values respectively .in other words , eve can change the proportion of average photon number between signal pulse and reference pulse ( denoted as and respectively ) , but keeps constant . in the following ,we let and .note that , if alice monitors the intensity of signal pulse and reference pulse at the same time , eve ca nt change the proportion between and .then eve must keep the intensity of and unchanged , which means ._ remark two _ : note that only _ s _ will be modulated by alice , thus eve should increase the coefficient to maximize her information .the maximal value of that can be set by eve , denoted as , depends on the way that alice and bob synchronize their clock in the practical qkd system . generally speaking ,there are two ways to synchronize the clock of alice and bob .one way is that alice triggers her setups according to the arrival time of _r_. thus is the minimal intensity of _ r _ that can trigger alice s setups , which depends on the efficiency of alice s photodiode .the other way is that alice synchronizes her clock with bob according to another optical or electric signal , such as the wavelength division multiplexing . in this case , alice does nt know whether _r _ arrives at her zone or not .thus , eve can block the reference pulse and only send _s _ to alice , which means . in this paperwe assume that alice and bob use the second way to synchronize their clock , thus .therefore , the intensity of signal pulse outgoing alice s zone is , where is the transmittance of alice s attenuator .we show that the partially random phase of source may leave a loophole for eve to spy the secret key . generally speaking , when an imperfection is found by eve, she can combine all imperfections of the system and take advantage of all attack strategies to maximize her information of the key .however we only consider a simple intercept - and - resend attack in this paper , which clearly shows that the generated key will be compromised due to the partially random phase .  according to the analysis above, we can consider the following attack : eve intercepts all the pulse from alice and measures the quadrature amplitude ( ) of the signal pulse with the experimental arrangement of fig.[fig : exp_arr](b ) . in order to judge alice s phase information, eve modulates the local pulse randomly and equiprobably with one of the two phases .then she sets two threshold value and .simply , we consider the symmetrical case that in this paper .when , eve judges that the phase modulated by alice is the same as her , thus she resends a state with phase to bob .when , eve judges that the difference of phase modulated by alice and her is , thus she resends a state with phase to bob . when , she blocks this pulse and resends a vacuum state to bob . note that these invalid judgements will not affect our attack , since the channel between alice and bob is lossy .obviously , sometimes , eve may make a wrong judgement .however , it is easy to check that , when eve uses the same basis as alice , the probability that eve obtains a valid outcome is much higher than that eve uses the different basis with alice . herea valid outcome means that the measured outcome of eve satisfies or . generally speaking , the larger eve sets , the lower the error rate .but , at the same time , the probability that eve obtains a valid outcome will decrease .thus eve should make a tradeoff between the error rate and efficiency when she loads her attack in a practical situation .  in order not to be discovered , eve should ensure that the error rate induced by her attack is smaller than the tolerable threshold value of alice and bob . in the following ,we analyze the error rate induced by eve s attack and show that eve can load our attack without being discovered by the legitimate parties . without loss of generalization , we assume that the phase modulated by alice is .according to eq ., when eve modulates the local pulse with a phase , the probability that she obtains a valid outcome are given by  ,\ ] ]  .\ ] ]  obviously , when eve obtains , she will not induce any error .but when eve obtains , she will induce an error event with a probability up to 1 . at the same time , eve may modulate the local pulse with a phase , then the probability that she obtains a valid outcome are given by  ,\ ] ]  .\ ] ]  obviously , she will induce an error event with a probability 1/2 for this case , no matter which outcome is obtained by her . therefore , the error rate induced by eve can be written as /2 } { p_0^+ + p_0 ^ - + p_{\pi/2}^+ + p_{\pi/2}^-}.\ ] ] here we assume eve modulates randomly or with equal probability . since the four states sent by alice are symmetrical , the total error rate induced by eve s attack is the same as eq .. at the same time , the probability that eve obtains a valid outcome is given by :   the error rate induced by our attack is shown in fig.[fig : error_rate ] .it shows clearly that when the phase of source is just partially random , the generated key will be compromised . in part(a ) we show that the error rate changes with and .as expected , for given , the higher the threshold value is set by eve , the lower the error rate will be .for example , when and , the error rate is 9.21% and 13.79% for and respectively . at the same time, the smaller the range of partially random phase is , the lower the error rate will be .for example , when eve sets , the error rate is 8.01% , 9.21% and 12.65% for , and respectively .according to eq.[p_x ] , the error rate induced by eve will depend on the intensity of signal pulse , which is shown in fig.[fig : error_rate ] ( part(b ) ) .for example , when eve sets and , the error rate will be 6.06% , 9.21% and 18.65% for respectively .in other words , if eve wants to keep the error rate is smaller than 20% , she only needs to set for , but will increase to 1.86 for   according to the analysis above , we know that eve can reduce the error rate induced by her attack by increasing the threshold value .however , the higher is , the lower eve obtains an unambiguous result that or . although eve can send vacuum state to bob when she obtains the ambiguous result that , she should ensure that the expected count rate of bob is unchanged . in other words , eve should ensure that the equation that holds . here is the transmittance of bob s setups . is the transmittance of channel . is the intensity of pulse sent to bob by eve .here we assume that eve can send a strong pulse to bob to compensate , which means .therefore , the maximal value of is determined by the transmittance of channel between alice and bob .the equivalent length of channel for a given is given by ,\ ] ] where we assume the channel is fiber and is the typical loss of fiber and is given by eq.[post ] .fig.[fig : equivalent_length ] shows the equivalent length of channel changes with and .it shows clearly that our attack is valid even the channel distance of alice and bob is very short .it can be explained as the maximal that eve can set for a given length of channel .note that the error rate induced by eve is determined by .thus it also can be explained as the minimal error rate induced by our attack for a given length of channel .for example , if the length of channel between alice and bob is 50 km , the maximal that eve can set is 1.97 when and , thus the minimal error rate induced by eve is 9.36% .it is well known that , in the practical qkd systems with the wcs , the decoy state method should be used to beat the pns attack . in this section , we will show that , if alice and bob use the one decoy state method to estimate the final secret key rate , eve still can spy the secret key using our attack in some parameter regime .although the one decoy state is not optimal for alice and bob , it is still adopted in some experimental systems .  in the one decoy state method , alice will send two kinds of pulses with different intensities to bob , the signal state and the decoy state , whose average photon number are denoted as and ( ) respectively .then they estimate the lower bound for the yield of single photon state , , and the upper bound for error rate of single photon state , , which are given by the eq.(41 ) of ref. :  [ y1_e1 ] ,\ ] ]   where and are the gain of the signal state and the decoy state , is the error rate of the decoy state , is the error rate of background . note that alice and bob will think that the phase of source has been randomized totally ( in fact it is just partially randomized ) , thus they will use the gllp formula to estimate the key rate , which is given by : \},\ ] ] where with the standard bb84 protocol . is the bidirectional error correction efficiency . is the binary shannon information entropy . is the gain of single photon state , is the error rate of the signal state .  note that , in the decoy state method , eve ca nt distinguish the signal state and the decoy state .thus we assume eve sends a single photon state to bob , when she obtains a valid measurement outcome .thus the gain and error rate of bob for the signal state and the decoy state are given by where is the dark count of bob s single photon detector , is the error rate of dark count . under our attack , and are given by eq ., and are given by eq ..  substituting these parameters into eq . , it is easy to estimate the key rate under our attack , which is shown in fig.[fig : key_rate ] .here we also show the equivalent length of channel between alice and bob , which is given by ,\ ] ] where is the gain of signal state . herethe equivalent length is used to ensure the gain of signal state under our attack is same as bob s expectancy .the numerical simulations show clearly that , eve can ensure the key rate between alice and bob is still positive by setting a suitable threshold value .for example , when , the key rate is positive if eve sets , but in fact no secret key can be generated in this range , since eve s intercept - and - resend attack is an entanglement breaking channel .fig.[fig : key_rate ] also shows clearly that our attack can be implemented successfully even the distance of channel between alice and bob is short .for example , if eve sets , the key rate will be positive and the equivalent length of channel is about 50.83 km .note that , in the simulation of fig.[fig : key_rate ] , we assume the homodyne detector of eve is perfect .although eve can make a perfect homodyne detector in theory , a practical eve is still imperfect . in fig.[fig : key_rate_imper ] , we show the key rate changes with the two parameters and which are used to characterize the imperfections of eve s homodyne detector .it shows clearly that our attack is still valid even the homodyne detector of eve is imperfect . in fig.[fig : key_rate_imper ] , we also draw the equivalent length of channel with the same parameters as that of key rate .it shows clearly that even the length of channel between alice and bob is short , our attack is still valid .  in fig.[fig :key_rate_mu_nu ] , we also show the key rate changes with the intensity of signal state and decoy state . generally speaking , for a given practical qkd system , the legitimate parties will optimize the intensity of signal state and decoy state ( and ) to maximize the key rate .for example , in the experimental parameters of gobby - yuan - shield ( gys ) , the optimal and are 0.48 and 0.036 when the length of channel between alice and bob is 50 km . however , under these parameters , the final key rate is still positive under our attack , but in fact no secret key can be generated .in other words , when our attack is taken into account , the estimated optimal parameters that are unaware of our attack may be not secret .thus , in a practical system , the legitimate parties must set their experimental parameters carefully .here we remark that , although our attack is immune to the one decoy state method , it can be defeated by the vacuum+weak decoy state method , in which alice sends three kinds of pulses to bob , the signal state , the decoy state and the vacuum state .the main reason is that eve may obtain valid outcome with a high probability , even alice sends a vacuum state .thus the gain of vacuum state will be much larger than the dark count of single photon detector ( spd ) , and then can be found by alice and bob .we have shown that our attack can be used by eve to break the security of a practical qkd system with wcs , even the one decoy state method is used by alice and bob . in the end of this paper , we give some discussions about our attack .first , it has shown that our attack can beat the bb84 qkd system with wcs and spd .however , it is invalid for the continuous variable qkd ( cv - qkd ) scheme . in the cv - qkd, bob also uses the homodyne detection to measure the signal pulse , thus he can reconstruct the probability distribution of signal pulse and then discover the existence of eve .furthermore , due to the same reason as the cv - qkd , our attack is also invalid for the bb84 qkd system with pulse homodyne detection , which is proposed and demonstrated by hirano _et al_.  second , in this paper , we assume that eve only changes the random phase of source which is a global phase , but not remaps the bit phase which is a relative phase .in other words , in the analysis above , we assume the states sent by alice are still the standard bb84 states . strictly speaking, eve also can change the bit phase .then eve can combine our attack with the phase remapping attack to maximize her information .third , in high - speed qkd system , eve needs a high - speed homodyne detector to detect the signal pulse .it seems an experimental challenge to implement our attack .in fact , the speed of some commercial homodyne detectors produced by picometrix can reach 40ghz .thus our attack is valid even for the high - speed qkd system .further , a well - stable interferometer is needed to ensure that the local pulse and signal pulse can interfere at the bs .thus eve must compensate the phase shift of fiber induced by the environment .in fact , the feedback method developed in the ghz qkd system can also be used in our attack to ensure the interferometer of eve is very stable .for the bb84 qkd protocol , phase randomization is a very important assumption in the standard security analysis .however , it is a hard task to check whether the phase of source is randomized totally or not in the practical situation .in fact , eve can control the range of random phase so that it is just partially randomized .  in this paper , we proposed a partially random phase attack to spy the secret key .our results show that if the phase of source is partially randomized , the error rate induced by our attack can be lower than the tolerable threshold value of error rate whereas the same range of error rate has been proved secure if the legitimate parties are unaware of our attack .thus the secret key generated by a practical qkd system will be compromised .specially , the numerical simulation shows that our attack is immune to the one decoy state method in some parameter regime .therefore , the legitimate parties should consider our attack carefully , when they use the wcs to implement the bb84 protocol .this work is supported by the national natural science foundation of china , grant no .sun is supported by the hunan provincial innovation foundation for postgraduates , grant no .cx2010b007 , and the fund of innovation , graduate school of nudt , grant no .b100203 .here we means that it is hard for alice to monitor the degree of random phase using current qkd system .of course , if she adds additional optical and electrical setups , she can do this easily . for example, she can establish one interferometer to measure the visibility of pulses .if the visibility is zero , the phase of pulse is randomized totally .otherwise , it is just partially randomized . therefore , if our attack is taken into account , alice must redesign her system to monitor carefully the random phase .here , where is the loss of optical setups , is the quantum efficiency ( qe ) of photodiode , is the visibility of interference .thus the crucial imperfection of homodyne detection is the qe of photodiode .although the qe is just 0.84 in the experiment of ref. , the qe of hamamatsu s new product can approach 0.9 ( see the website of hamamatsu : http://jp.hamamatsu.com/products/sensor-ssd/pd128/pd129/pd130/g8370-81/index.html ) .furthermore , is determined by the excess noise of the signal and the phase error .but , as the best of our knowledge , we only find one experimental result that which is given by ref. .however , it does not matter , theoretically , eve can make a perfect homodyne detection with .

Abstract: phase randomization is a very important assumption in the bennett - brassard 1984 quantum key distribution ( qkd ) system with a weak coherent source . thus an active phase modulator is needed to randomize the phase of source . however , it is hard to check whether the phase of source is randomized totally or not in practical qkd systems . in this paper a partially random phase attack is proposed to exploit this imperfection . our analysis shows that eve can break the security of a two - way qkd system by using our attack , even if an active phase randomization is adopted by alice . furthermore , the numerical simulation shows that in some parameter regimes , our attack is immune to the one - deco - state method .