physical - layer security schemes exploit channel characteristics , such as noise and fading , to allow a group of nodes to share information in such a way that other unintended receivers ( called eavesdroppers or wiretappers ) can not recover that secret information .physical - layer security has often been studied in the context of the _ wiretap channel _ , which was first introduced by wyner  and later refined by csiszr and krner  . in the wiretap channel, a source tries to send secret information to a destination at the presence of a wiretapper .when the source - to - wiretapper channel is a degraded version of the source - to - destination channel , wyner  showed that the source can transmit a message at a positive ( secrecy ) rate to the destination by taking advantage of the less `` noisy '' channel to the destination .the degradedness condition was removed in  , which showed that a positive secrecy rate is possible for the case where the source - to - destination channel is `` more capable '' than the source - to - wiretapper channel .generalization of wyner s work to the gaussian wiretap channel was considered in  .  in wyner soriginal paper , a code design based on group codes was described for the wiretap channel . in  ,a code design based on coset codes was suggested for the type - ii ( the destination channel is error free ) binary erasure wiretap channel .recently , the authors of  constructed low density parity check ( ldpc ) based wiretap codes for binary erasure channel ( bec ) and binary symmetric channel ( bsc ) .reference  considered the design of secure nested codes for type - ii wiretap channels .more recently , references  and concurrently established the result that polar codes  can achieve the secrecy capacity of the degraded binary - input symmetric - output ( biso ) wiretap channels .note that all these designs are for codes with asymptotically large block lengths .  in some scenarios ,it is sufficient for two nodes to agree upon a common secret ( a key ) , instead of having to send secret information from a source to a destination . under this relaxed criterion, it is shown in  that , with the use of a feedback channel , a positive key rate is achievable when the destination and wiretapper channels are two conditionally independent ( given the source input symbols ) memoryless binary channels , even if the destination channel is not more capable than the wiretapper s channel .this notion of secret sharing is formalized in  based on the concept of _ common randomness _ between the source and destination .a three - phase process of achieving secret sharing over a wiretap channel with an additional public channel between the source and destination is suggested in  .the three phases are respectively advantage distillation , information reconciliation , and privacy amplification . advantage distillation aims to provide the destination an advantage over the wiretapper .information reconciliation aims at generating an identical random sequence at both the source and destination .privacy amplification is the step that extracts a secret key from the identical random sequence agreed upon by the source and destination .information reconciliation is probably the most studied and most essential part of any secret sharing scheme .perhaps the most well known practical application of reconciliation protocols is quantum cryptography , where nonorthogonal states of a quantum system provide two terminals with observations of correlated randomness which are at least partially secret from a potential eavesdropper .many works  have been devoted to the study of reconciliation for both discrete and continuous random variables in quantum key distribution schemes . for the case of discrete random variables, cascade is an iterative reconciliation protocol first proposed by brassard and salvail in  .recently , bsc - optimized ldpc codes have been employed in  to reduce the interactivity and improve the efficiency of cascade . on the other hand ,the work on slice error correction  , which converts continuous variables into binary strings and makes use of interactive error correcting codes , is the first reconciliation protocol for continuous random variables .modern coding techniques like turbo codes  and ldpc codes  have also been directly applied within information reconciliation protocols for continuous random variables .another application of reconciliation protocols is secret key agreement over wireless channels .an ldpc code - based method of extracting secrecy from jointly gaussian random sources generated by a rayleigh fading model has been studied in  . in , multilevel coding / multistage decoding - like reconciliation with ldpc codes has been proposed for the quasi - static rayleigh fading wiretap channel . in  ,punctured ldpc codes were employed in a coding scheme for the gaussian wiretap channel to reduce the security gap , which expresses the quality difference between the destination channel and wiretapper channel required to achieve a sufficient level of security .the main idea of this scheme is to hide the information bits from the wiretapper by means of puncturing . in  ,further reductions in the security gap are achieved using a reconciliation scheme based on non - systematic ldpc codes along with scrambling of the information bits prior to encoding .  in this paper , we consider the problem of secret sharing over the gaussian wiretap channel with the constraints of binary phase - shift keyed ( bpsk ) source symbols and symbol - by - symbol hard - decision quantization at the destination .our main goal is to develop a coding structure based on which practical `` close - to - capacity '' secret sharing ( key agreement ) codes can be constructed .finite block length and moderate encoder / decoder complexity are the two main practical constraints that we consider when designing these codes .the secrecy performance of our designs will be measured by the rate of secret information shared between the source and destination ( which will be referred to as the _ key rate _ ) as well as the rate of information that is leaked to the wiretapper through all its observations of the wiretap and public channels ( which will be referred to as the _ leakage rate _ ) .  to rigorously gauge the secrecy performance of our code designs ,we introduce the notion of relaxed key capacity in section [ sec : relax_keycap ] .the relaxed key capacity is the maximum key rate that can be achieved over the wiretap channel provided that the leakage rate is bounded below a fixed value . in section [ sec : ch_mod ] , we calculate the relaxed key capacities over the bpsk source - constrained gaussian wiretap channel with and without the constraint of hard - decision quantization at the destination . in section [ sec : ssldpc ] , we present a secret sharing scheme employing an ensemble of regular ldpc codes for the bpsk - constrained gaussian wiretap channel with hard - decision quantization at the destination .we prove that the proposed scheme achieves the relaxed key capacity with asymptotically large block length .we note that a similar ldpc - based key agreement scheme employing observations of correlated discrete stationary sources at the source , destination , and wiretapper was studied in  . a more detailed comparison between our scheme and the one proposed in  is also provided in section [ sec : ssldpc ] .the asymptotic result in section [ sec : ssldpc ] provides us a reasonable theoretical justification to design practical secret sharing schemes based on the proposed coding structure .we propose in section [ sec : wz ] to replace the regular ldpc code ensemble in section [ sec : ssldpc ] by fixed ldpc codes that are more amenable to practical implementation . in the same section, we describe a code search algorithm based on density evolution analysis to obtain good irregular ldpc codes for the proposed secret sharing scheme .we also compare the secrecy performance achieved by these irregular ldpc codes , bsc - optimized irregular ldpc codes , and some standard regular ldpc codes against the relaxed key capacity calculated in section [ sec : ch_mod ] .finally , conclusions are drawn in section [ sec : con ] .we start by reviewing the framework of secret sharing proposed in .the objective of secret sharing is for the source and destination to share secret information , which is obscure to the wiretapper , by exploiting _common randomness _ available to them through the wiretap channel . here, we consider the wiretap channel to be memoryless and specified by the conditional probability density function ( pdf ) .when the symbol is sent by the source , and denote the corresponding symbols observed by the destination and wiretapper , respectively .in addition , we restrict ourselves to cases in which and are conditionally independent given , i.e. , .this restriction is satisfied by the gaussian wiretap channel considered in section [ sec : ch_mod ] and some other wireless wiretap channels . for convenience, we will refer to the wiretap channel by the triple .in addition to the wiretap channel , there is an interactive , authenticated , public channel with unlimited capacity between the source and destination .the source and destination can communicate via the public channel without any power or rate restriction .the wiretapper can perfectly observe all communications over the public channel but can not tamper with the transmitted messages .the aforementioned common randomness is to be extracted by a proper combination of transmission from the source to the destination through the wiretap channel and information exchanges between them over the public channel . to this end, we consider the class of permissible secret sharing strategies suggested in  .consider time instants labeled by , respectively .the wiretap channel is used times during these time instants at . set .the public channel is used during the other ( ) time instants . before the secret sharing process starts ,the source and destination generate , respectively , independent random variables and .then a permissible strategy proceeds as follows : stands for the sequence of symbols , and is null . ]  * at time instant , the source sends message to the destination , and the destination sends message to the source .both transmissions are carried over the public channel . * at time instant for , the source sends the symbol to the wiretap channel .the destination and wiretapper observe the corresponding symbols and .there is no message exchange via the public channel , i.e. , and are both null . * at time instant for , the source sends message to the destination , and the destination sends message to the source . both transmissions are carried over the public channel .  at the end of the timeinstants , the source generates its secret key , and the destination generates its secret key , where and take values from the same finite set .  slightly extending the achievable key rate definition in , for ,we call an _ achievable key - leakage rate pair _ through the wiretap channel if for every , there exists a permissible secret sharing strategy of the form described above such that  1 . , 2 . , 3 . , 4 . , and 5 . ,  for sufficiently large .condition 2 restricts that the public messages ( the messages conveyed through the public channel ) contain a negligible rate of information about the key , while condition 3 limits to the rate of key information that the wiretapper can extract from its own channel observations and the public messages . note that condition 3 is trivially satisfied if of the achievable key rate definition in  when , if the combined condition is employed instead of conditions 2 and 3 , then it is easy to see that if is an achievable key - leakage rate pair , is also achievable , for any , by simply transmitting the additional key information ( of rate ) through the public channel .separating the two conditions as suggested avoids such artificial consequence of the combined condition . ] . for the cases in which the alphabet of is not finite , we also impose the following power constraint to the symbol sequence sent out by the source : with probability one ( w.p.1 ) for sufficiently large .we note that the idea of key - leakage rate pair is similar to that of the secrecy - equivocation rate pair originally defined in  .-relaxed key capacity _ is defined as the maximum value of such that is an achievable key - leakage rate pair .the main reason for us to introduce the notion of relaxed key capacity is to employ it as a gauge to measure the performance of practical codes that will be presented in .since these codes have finite block lengths and are to be decoded by the belief propagation ( bp ) algorithm , they do not achieve zero leakage rate . thus using the relaxed key capacity provides a more suitable comparison than using the original `` straight '' key capacity in  .also , since these practical codes do not give zero leakage rate , their use could be considered as an information - reconciliation step .the secrecy performance could be further improved by additional privacy amplification .  for wiretap channels that satisfy the aforementioned conditional independence requirement , we have the following result , whose proof is sketched in appendix [ app : pf_keycapgen ] :  [ thm : keycapgen ] the -relaxed key capacity of the memoryless wiretap channel with conditional pdf is given by \leq p } \left [  \min \{i(x;y ) - i(y;z ) + r_l , i(x;y)\ } \right].\ ] ]  we employ this result to calculate the relaxed key capacities of the bpsk - constrained gaussian wiretap channel in the next section .the achievability proof of theorem [ thm : keycapgen ] ( cf .appendix [ app : pf_keycapgen ] ) employs random wyner - ziv coding , in which the received symbols at the destination need to be quantized due to the fact that the channel alphabet at the destination in the gaussian wiretap channel is continuously distributed . in this paper, we consider a simple symbol - by - symbol hard - decision quantization scheme in which the quantized destination symbol , where is the signum function .note that this quantization is suboptimal and leads to a loss in key capacity .we quantify this loss by applying theorem [ thm : keycapgen ] to the bpsk - constrained gaussian wiretap channel with hard - decision quantization at the destination to calculate the relaxed- key capacity . using the standard notation , it is not hard to establishthat , \label{eqn : relaxkeycap_bpsk_constr}\ ] ] where ^{-2\alpha \tilde \beta z}}{1+e^{-2 \alpha \tilde \beta z}}\right)[1+e^{-2 \alpha \tilde \beta z } ] e^{-\frac{(z-\alpha\tilde \beta)^2}{2}}dz .\label{eqn : cw}\end{aligned}\ ] ] are respectively the capacities of the quantized - destination - to - source and quantized - destination - to - wiretapper channels at the normalized gain . likebefore , is achieved when is equiprobable ; but it is not necessarily achieved by transmitting at the maximum allowable power .to visualize the loss in key capacity , shows and versus the maximum allowable snr ( ) for different values of .we can see that the loss in key capacity due to the hard - decision quantization is no more than bits per ( wiretap ) channel use for the cases shown .the achievability proof of theorem [ thm : keycapgen ] in appendix [ app : pf_keycapgen ] employs a secret sharing scheme with random wyner - ziv coding . for the bpsk - constrained gaussian wiretap channel with destination hard - decision quantization , we show in this section that a secret sharing scheme that employs a properly constructed ensemble of regular ldpc codes can also asymptotically achieve the -relaxed key capacity. we will design practical secret sharing schemes for the bpsk - constrained gaussian wiretap channel in section [ sec : wz ] based on the ldpc coding structure proposed here .to start describing the proposed secret sharing scheme , let us consider an binary linear block code with distinct codewords of length and an -dimensional subspace in .the pair defines what we call an _ secret sharing binary linear block code_. given any such pair , let be the quotient of by . then is a linear space of distinct cosets of the form , where .we will use the coset index in as the secret key .we will see later that the ordering of the cosets in is immaterial .the ratios and will be referred to as the _ code rate _ and _ key rate _ of the secret sharing binary linear block code , respectively .next , we consider the following random ensemble of secret sharing binary linear block codes :  * the linear block code is chosen uniformly from the ensemble of -regular ldpc codes considered in . that is , we consider that is chosen uniformly from the set of all bipartite graphs with degree- variable nodes and degree- check nodes . * the subspace is chosen uniformly over the set of all possible -dimensional subspaces in .  note that a realization of the randomly chosen may actually have distinct codewords , where . in such case, will be of dimension ; so the actual key rate will be larger than .hence , we can conservatively assume is always an linear code with distinct codewords to simplify the notation below .consider the following secret sharing scheme :  1 .* random source transmission and destination quantization : * the source randomly generates a sequence of i.i.d . equally likely bpsk symbols and transmits them consecutively over the gaussian wiretap channel .the destination receives the sequence and obtains the quantized sequence by performing symbol - by - symbol hard - decision quantization on , i.e. , .this quantization effectively turns the source - to - destination channel into a bsc , whose cross - over probability depends on the snr of the original source - to - destination channel .we note that the wiretapper also observes through the source - to - wiretapper channel .* syndrome generation through ldpc encoding at destination : * the next step is for the destination to feed a compressed version of back to the source through the public channel so that the source can resolve the differences between and .this is similar to the problem of compressing an equiprobable memoryless binary source with side information using ldpc codes considered in  .more precisely , the destination selects randomly from the ensemble of secret sharing -regular ldpc codes described above .it then generates the syndrome sequence , where is a parity check matrix of .we note that each uniquely corresponds to a coset .further , the destination determines which coset in that belongs .denote that coset by .finally , the destination sends , , and back to the source via the public channel .3 .  * decoding at source : * the source then tries to decode for from observing and according to . treating as a noisy version of , it performs maximum likelihood ( ml ) decoding to obtain a codeword in and then determines which coset in that the decoded codeword belongs .denote that coset by .* key generation at source and destination : * the destination sets its key to be index of in . similarly , the source sets its key to be the index of in .it is clear that this secret sharing scheme is permissible .indeed , under the notation of , for the proposed secret sharing scheme , , for , , , and is the only message sent via the public channel .hence , we can evaluate the secrecy performance of the scheme in the context of its achievable key rate defined in as follows .first , based on the linearity of ldpc codes , the memoryless nature of the gaussian wiretap channel , the chosen distribution of , and the symbol - by - symbol hard decision performed to obtain at the destination , it is easy to check that , , , and . then , hence , , , and .if the decoding process at the source achieves the ensemble average error probability , then we have .thus , and by fano s inequality .that in turn implies \\  & \leq &   \frac{1}{n}i(k;e_s^{n},\mathcal{c},\mathcal{w}|l ) \leq \frac{1}{n}h(k|l )   \leq r_k\bar \epsilon_s+\frac{1}{n}\end{aligned}\ ] ] and   \geq r_k - r_k\bar \epsilon_s - \frac{1}{n}. \label{eqn : c4}\ ] ] hence , conditions 2 and 5 in are satisfied when is sufficiently large if can be made arbitrarily small .similarly , where the last line is due to the fact that . here , where the last equality follows from the fact that is independent of .also because of the memoryless nature of the channel from and and of the fact that the achieves the capacity of this channel .moreover , consider a fictitious receiver at wiretapper trying to decode for from observing , , and ( or equivalently ) .suppose that the ensemble average error probability achieved by this receiver , employing ml decoding , is .then we have again by fano s inequality . putting all these and  back into , we obtain the preceding secrecy analysis of the proposed secret sharing scheme based on the secret sharing regular ldpc code ensembles allow us to arrive at the following result :  [ thm : relax_keyach ] fix .suppose that .for any , choose .then is an achievable key - leakage rate pair through the bpsk - constrained gaussian wiretap channel with symbol - by - symbol hard - decision destination quantization .moreover , this rate pair can be achieved by the aforementioned secret sharing scheme using the secret sharing -regular ldpc code ensemble described before when increases .first , suppose that and . since , . then .thus , by , if we can show that there is a pair such that , and both and in the preceding discussion vanish as increases , then condition 3 in section [ sec : relax_keycap ] will be satisfied when is sufficiently large . from the preceding discussion, conditions 1 , 2 , and 5 will also be satisfied .comparing and condition 4 , we see then that will be an achievable key - leakage pair .the existence of such pair results from the following lemma , whose proof is an adaptation of the arguments in ( * ? ? ?* theorem 3 ) to the proposed secret sharing -regular ldpc code ensemble .the details are presented in appendix [ app:2 ] .[ lm : existldpc ] consider the ensemble average error probabilities and achieved by the respective ml decoders at the source and wiretapper of the secret sharing -regular ldpc code ensemble mentioned above . for any fixed , suppose that and .then , there exists a choice of such that  1 . , 2 . decreases exponentially ( polynomially ) with increasing for ( for ) , and 3 . decreases polynomially with increasing .finally , note that the before - imposed restrictions and can be removed since the key - leakage rate region is closed .a comparison of theorem [ thm : relax_keyach ] and shows that the restriction to the secret sharing regular ldpc code ensemble described in this section does not reduce the relaxed key capacity of the bpsk - constrained gaussian wiretap channel with destination hard - decision quantization .as mentioned in , a similar ldpc - based secret - key agreement scheme employing observations of correlated discrete stationary sources at the source , destination , and wiretapper was studied in  .after step 1 ) of our proposed secret sharing scheme , the observations , , and at the three terminals can be viewed as generated from correlated sources ; thus reducing our model to the one considered in  , except that the wiretapper alphabet is continuous in our case . as in our scheme , the scheme in  has the syndrome of sent to the source . on the other hand ,the key in  is obtained by calculating the syndrome of with respect to another independently selected ldpc code .the scheme in  is shown to achieve key capacity via a similar approach as ours .first , the consideration of leakage information is converted to that of the error probabilities achieved by decoders at the source and wiretapper by an upper bound similar to  for a pair of fixed ldpc codes ( cf .eqn . ) .then , the existence of a fixed code pair with vanishing error probabilities is shown via an ml decoding error analysis of the code ensemble based on the method of types  .because of the continuous wiretapper alphabet , the ml decoding error analysis in  does not directly apply to our case .hence , we have opted for the combined union and shulman - feder bounding technique in  , which does however require the biso nature of the channel from the ( quantized ) destination to the wiretapper .obviously , lemma [ lm : existldpc ] also implies the existence of a fixed from the secret sharing regular ldpc ensemble with vanishing decoding errors in our design , and hence the use of this fixed is also sufficient to achieve the relaxed key capacity in our case.  expressed in our notation , elements in the ldpc code ensemble of are also of the form .for our ensemble , is ( conditionally ) uniformly distributed over the set of all subspaces of a given . for the ensemble of , is ( conditionally ) uniformly distributed over the set of subspaces of specified by the concatenation of the parity matrices of and another properly chosen regular ldpc code .while each element in the ensemble of is also an element of our ensemble , the two ensembles are different since the respective ( conditional ) uniform distributions for are defined over two different sets of subspaces . in a sense , the ensemble of is more restrictive since also needs to be an ldpc code .the discussion in this section shows that the ldpc structure needs to be imposed only on but not on .this bears significance in the design of practical codes because the design based on one ldpc structure derived from our ensemble is much simpler , as will be illustrated in the following section .in practice , it is not realistic to employ the secret sharing regular ldpc code ensemble and ml decoding at the source as suggested in , for even moderate values of . in this section ,we investigate the secrecy performance of a secret sharing scheme similar to the one suggested in , but with fixed choices of from the secret sharing regular ldpc code ensemble and more - practical bp decoding .in addition , from the proof of lemma [ lm : existldpc ] in appendix [ app:2 ] , the values of and need to be large in order for the ensemble average error probabilities and to decrease with , and hence to achieve the relaxed key capacity . as large values of and increase the graph complexity of a ldpc code , and hence the complexity of bp decoding , we have to limit ourselves to small values of and . to alleviate the shortcoming of regular ldpc codes with small and , we also consider the use of more - efficient irregular ldpc codes in the proposed secret sharing scheme .we consider the secret sharing scheme described in , except that the secret sharing code is fixed and is known to the source and destination ( and also the wiretapper ) beforehand . here, we consider the ( fixed ) code chosen from ensembles of regular and irregular ldpc codes .the details will be discussed later . for convenience in the key generation step ( and later in the search of good irregular ldpc codes ) , the subspace is chosen as follows . referring back to step 2 ) of the scheme , choose a lower triangular version to be of full rank as discussed before .alternatively , an approximate lower triangular version of as described in  can also be used if efficient encoding is needed . ]of , for example by performing gaussian elimination on the connection matrix of the bipartite graph of as discussed in  . hence ,  ] where and are row vectors containing and elements , respectively. then the syndrome , codeword  ] .note that contains the systematic bits of the codeword while contains the parity bits .the subspace is chosen to be the set of codewords obtained by setting the first bits bits in for the bp decoders described below . ] in the vector above to zero .the quotient space is isomorphic to the set of codewords obtained by setting the last bits in the vector to zero .hence we can use the first bits in as the key . since is known to the source beforehand, there is no need to feed it back to the source via the public channel in step 2 ) of the secret sharing scheme .step 3 ) of the scheme is modified to replace ml decoding by the practical bp decoding .first , it is unlikely that the above fixed choice of results in an ldpc code .hence , the fixed coding scheme suggested here is different from that of .second , the secrecy analysis of can be easily modified to reflect the use of the fixed secret sharing code mentioned above .in particular , the upper bound on the leakage rate in becomes where and are now the error probabilities achieved by the bp decoders at the source and wiretapper , respectively .since the bound above is derived from fano s inequality , it applies for any decoder ( ml , bp , _ etc . _ ) , and the value of the bound depends on the choices of decoders only through and .below , we perform computer simulation to estimate and and then employ to bound the leakage rates achieved by constructed from different choices of finite block length ldpc codes as described above . more specifically ,suppose that the key rate of a secret sharing ldpc code is and obtained from simulation is small . by setting to be the value of the bound  obtained as described ,then will be considered a key - leakage rate pair achievable by .we start by evaluating the secrecy performance of using regular ldpc codes with small and in the secret sharing scheme described above .first , we pick from the rate- -regular ldpc code ensemble by realizing the random bipartite graph experiment described in  and then remove all length- loops in the realization .the block length of the ldpc code is set to . as mentioned above, we need to estimate the values of and from computer simulation . to get , bp decodingis implemented at the source .similarly , a bp decoder is implemented for the fictitious receiver at the wiretapper to obtain . in order to provide information about to the latter decoder , the intrinsic log - likelihood ratios ( llrs ) of the first elements in , which are associated with ,are explicitly set to according to the true bit values . while this method may not be the optimal way to feed information of to the bp decoder , we choose to employ it because of its simplicity and the fact that this method also allows simple density evolution analysis , which will be used to search for good irregular ldpc codes in below .shows the trajectory of achievable by the rate- secret sharing -regular ldpc code when the maximum allowable snr is limited to  db and  db . different values of on the trajectory shown are obtained by varying the value of ( i.e. , the dimension of also changes ) .when obtaining each shown pair , we choose , up to , such that , and the bound in  is minimized . for any so - obtained pair located to the right of the line in fig .[ fig : ck_rl_regular ] , the bound becomes too loose , and the pair is not plotted . from , we observe that the pair gives the smallest ( bound on ) leakage rate that is achievable by the rate- secret sharing -regular ldpc code in the proposed scheme .next , we try to compare the secrecy performance of our secret sharing scheme to that of  .as discussed near the end of , the scheme of  requires a pair of independently chosen regular ldpc codes .since no practical code designs or examples are provided in  , we choose an ldpc code pair for the scheme of  that is similar to the choice of our secret sharing code above for comparison . for the scheme of , the first ldpc code is set to be above ( i.e. , the rate- -regular ldpc code ) .the other code ( from which the secret key is generated ) is chosen independently from another regular ldpc code ensemble such that a desired key rate is resulted ( cf . ) . note that only a few values of are possible if and are restricted to have small values .again , as discussed near the end of , the pair can be expressed in our notation . as such, the ldpc subcode is obtained from concatenating parity - check matrices of and .note that is in general an irregular ldpc code . to clearly distinguish between our scheme and the one of  in the discussion below, we will employ the notation when referring to the latter .the bound  is employed to determine the rate pairs that can be achieved by , same as described before .  under the parameter setting above (  db ,  db , and ) ,we are not able to find a choice of ( with small and ) that satisfies the requirement . in order to illustrate the comparison between the two schemes, we increase the value of to  db . for this case , we pick to be a rate- -regular ldpc code .the -trajectory achieved by our secret sharing scheme with is overlaid in .we see that the lowest leakage rate achieved by this choice of is at the pair . for the scheme of , picking to be an -regular ldpc code , the pair achieves the key - leakage rate pair as shown by the square symbol in .this value of is the lowest that we can obtain from picking many different with small and .  summarizing the above results, our secret sharing scheme outperforms the scheme of  when the respective code employed in each scheme is restricted among the choices of regular ldpc codes with small node degrees and finite block lengths .however , we can observe that there is a significant gap between the pairs achieved by the proposed scheme and the maximally achievable key - leakage pair boundary .this illustrates that regular ldpc codes with small and and finite block length do not provide good secret sharing performance .   to improve secret sharing performance , we search for `` good '' irregular ldpc codes to be used as in the proposed schemethe structure of secret sharing code described in the beginning of this section facilitates the code search process because only the ldpc structure of needs to be optimized .such optimization can be performed by employing the density - evolution based linear programming technique suggested in  .the search objective is to find an irregular ldpc secret - sharing code with maximum , given a fixed such that both the decoding error probabilities and in are vanishing as the bp decoders iterate . by , this results in minimization of the bound on for the fixed .using standard notation , let the variable and check node degree distribution polynomials of an irregular ldpc code ensemble be , respectively , and , where represents the fraction of edges emanating from the variable ( check ) nodes of degree .we are to design an irregular ldpc code and its subcode that work well for the channel from the ( quantized ) destination to source and the channel from the ( quantized ) destination to wiretapper , corresponding to the error probabilities and , respectively .fix , and let and denote the bit error probabilities obtained by the bp decoders at the source and wiretapper , respectively , at the density evolution iteration  when an initial is used .now , let denote the bit error probability obtained at the source by running the density evolution for iterations , in which is used as the variable node degree distribution for the first iterations and the variable node degree distribution with a singleton of unit mass at degree is used for the final iteration .let denote the similar quantity for bit error probability obtained at the wiretapper .then , we have and .note that the values of and are obtained via density evolution . to account for the availability of perfect information of the bits corresponding to the key at the wiretapper s bp decoder , the intrinsic llr distribution entered into the density evolution analysis for the wiretapper s decoderis set to be a mixture of the distribution of the channel outputs at the wiretapper ( with the quantized destination symbols as the channel input ) and an impulse at .the weights of the two components in the mixture are determined by the value of .the secret sharing irregular ldpc codes presented below are obtained from the code search procedure described above starting with bsc - optimized ldpc codes , which are available from urbanke s website  .shows the -trajectory achieved by a rate- secret sharing irregular ldpc code obtained by performing the above search with set to for the bpsk - constrained gaussian wiretap channel when  db and  db .the degree distribution pair of this secret sharing irregular ldpc code is shown in .we obtain an instance of the irregular code by randomly generating a bipartite graph which satisfies the two given degree - distribution constraints .similar to the case of regular codes , the block length , and all length- loops are removed .each shown pair is obtained in the same manner as described in by using . from , we observe that the pair gives the lowest leakage rate achievable by this secret sharing irregular ldpc code . for comparison ,we also plot in  the -trajectory achieved by the proposed secret sharing scheme using a rate- bsc - optimized irregular ldpc code in place of the secret sharing irregular ldpc code obtained from the code search described above .note that since the channel from the ( quantized ) destination to the source is a bsc , the use of the bsc - optimized ldpc code is essentially the same as the reconciliation method proposed in  .for the bsc - optimized code , the pair gives the lowest achievable leakage rate .similarly ,  shows the secrecy performance of the proposed scheme when  db and  db .a rate- secret sharing irregular ldpc code is obtained by fixing to in the code search .the degree distribution pair of this secret sharing irregular ldpc code is also shown in table [ tb : deg_dist_pair ] .we observe that the lowest leakage rate achieved by this code is given by the pair .again , for comparison , the -trajectory achieved by replacing the secret sharing irregular ldpc code obtained from the code search with a rate- bsc - optimized irregular ldpc code is also shown in .for the bsc - optimized irregular ldpc code , the pair gives the lowest achievable leakage rate . in conclusion ,the secret sharing irregular ldpc codes obtained from the proposed code search procedure significantly outperform , in terms of secrecy performance , secret sharing regular ldpc codes with small node degrees as well as irregular ldpc codes that are optimized just for information reconciliation .in this paper , we developed schemes based on ldpc codes to allow a source and a destination to share secret information over a bpsk - constrained gaussian wiretap channel . in the proposed secret sharing schemes , the source first sends a random bpsk symbol sequence to the destination through the gaussian wiretap channel .then , the destination generates a syndrome of its quantized received sequence using an ldpc code and sends this syndrome back to the source via the public channel .finally , the source performs decoding to recover the quantized destination sequence based on its transmitted sequence , as well as the syndrome that it receives from the destination .the secret key is obtained as the index of a coset in a quotient space of the ldpc code .  to evaluate the performance of the proposed secret sharing scheme, we employed an upper bound on the leakage information rate that depends on the decoding error probabilities of the decoder at the source and of a fictitious decoder at the wiretapper , which observes the wiretapper received sequence , the syndrome in the public channel as well as the secret key .the design was then converted to making these error probabilities small . for a suitably chosen ensemble of regular ldpc codes, we showed that these error probabilities can indeed be made vanishing , as the block length increases , by ml decoding . as a result , this established that the key capacity of the bpsk - constrained gaussian wiretap channel can be achieved by employing the secret sharing regular ldpc code ensemble in the proposed scheme .considering the practical constraints of finite block length and using bp decoding instead of ml decoding , we employed a density - evolution based linear program to search for good irregular ldpc codes that can be used in the secret sharing scheme .simulation results showed that the secret sharing irregular ldpc codes obtained from our search can get relatively close to the relaxed key capacity of the bpsk - constrained gaussian wiretap channel , significantly outperforming regular ldpc codes as well as irregular ldpc codes that are optimized just for information reconciliation .finally , we point out that the arguments in the proof of can be modified to show the existence of an ldpc code ( from the same regular ldpc code ensemble considered in ) that achieves the secrecy capacity  of the gaussian wiretap channel with the bpsk source - symbol constraint . the code search approach described incan also be employed to find irregular ldpc codes that give secrecy performance close to the boundary of the secrecy - equivocation rate region of that channel .the proof of ( * ? ? ? * theorem 2.1 ) , which corresponds to the case when , can be easily extended to accommodate conditions 2 and 3 in the definition of achievable key - leakage rate pair .  first , consider the converse proof . any permissible secret sharing strategy that achieves the key - leakage rate pair must satisfy ( cf .( 7 ) ) ) + \varepsilon.\ ] ] from conditions 2 , 3 , and the chain rule , we have where the last inequality is due to the bound which is shown in  .similarly , using the chain rule and condition 2 , we also have where the last inequality is due to the bound , which again can be shown by a simple modification to  .as in , let be a uniform random variable that takes value from and is independent of all other random quantities . define if . then . combining the two upper bounds on above , we have the power constraint ( [ eqn : powerconstr ] ) implies that \leq p ] . * if is even , \cdot  \left[\frac{1+\left(1-\frac{2m}{n}\right)^{d_c}}{2}\right]^{n - l} d_c d_c$ } \end{cases}\\ & \leq & \frac{1}{n } \left\ { 1 + \log_2 [ ( n - l)d_c+1 \right]\ } + ( 1-r_c)\log_2[1+(1 - 2\gamma)^{d_c}].\end{aligned}\ ] ]  for bounding , note that the channel with as input and as output is a memoryless bsc and is specified by the conditional pmf . again , since is a coset and the channel is memoryless biso , it suffices to assume . with this identification ,the resulting bound on follows the same line of arguments as above , and is essentially given in .we summarize the bound below for later reference : where \ } \\ & & ~+ \max_{\bar\beta\leq x\leq \gamma}\left\ { x\log_2 d_s + h_2(x)+ ( 1-r_c)\left(\log_2[1+(1 - 2x)^{d_c}]-1\right)\right\},\end{aligned}\ ] ] and for even ,  and \ } + ( 1-r_c)\log_2[1+(1 - 2\gamma)^{d_c}],\ ] ] with , and is the random coding error exponent of the channel of interest based on ^{1+\rho } \\ & & ~~~~~~~~~~~~+ \left [ q(1)p_{x|\tilde y}(-1|1)^{1/(1+\rho ) } + q(-1)p_{x|\tilde y}(-1|-1)^{1/(1+\rho ) } \right]^{1+\rho}\bigg\}.\end{aligned}\ ] ]    with this choice of , we have - 1\right)\right\ } } \\  & \leq &  h_2(\gamma ) + ( 1-r_c)\left\{\log_2[1+(1 - 2\bar\beta)^{d_c}]-1\right\ } \\  & \leq &  h_2(\gamma ) + ( 1-r_c)\left[\log_2 \left(1+e^{-2d_c\bar\beta}\right)-1\right ] \\  & \leq & h_2(\gamma ) + ( 1-r_c)\left[\log_2   \left(1+e^{-4e^{-12-\varepsilon}}\right)-1\right]\end{aligned}\ ] ] for any , where the second inequality follows from the inequality and the last inequality follows from the definition of .hence , we can make - 1\right)\right\ } < 0\ ] ] by choosing small enough since .thus for sufficiently large , we get the following results ,         a. thangaraj , s. dihidar , a. r. calderbank , s. mclaughlin , and j .-merolla , `` applications of ldpc codes to the wiretap channel , '' _ ieee trans .inform . theory _53 , no .29332945 , aug .2007 .nguyen , g. van assche , and n. j. cerf , `` side - information coding with turbo codes and its application to quantum key distribution , '' in _ proc .2004 ieee int .inform . theory and applicat ._ , param , italy , oct . 2004 .j. muramatsu , `` secret key agreement from correlated source outputs using low density parity check matrices , '' _ ieice transactions on fundamentals of electronics , communications and computer sciences _ , vol .e89-a , pp .20362046 , july 2006 .       t. f. wong , m. bloch , and j. m. shea , `` secret sharing over fast - fading mimo wiretap channels , '' _ eurasip journal on wireless communications and networking , special issue on wireless physical layer security _2009 .s. chung , g. d. forney , jr ., t. j. richardson , and r. urbanke , `` on the design of low - density parity - check codes within 0.0045 db of the shannon limit , '' _ ieee commun ._ , vol . 5 , no . 2 ,5860 , feb .2001 .       and for different values of maximum allowable leakage rate over the bspk - constrained gaussian wiretap channel .for , symbol - by - symbol hard - decision quantization is imposed at the destination.,scaledwidth=57.0% ]  -trajectories achieved by the proposed secret sharing scheme employing secret sharing regular ldpc codes with block length of .two cases are shown in the figure .the green curve corresponds to the case of  db ,  db , and is a rate- -regular ldpc code .the brown curve corresponds to the case of  db ,  db , and is a rate- -regular ldpc code . for comparison , the corresponding boundary of the region for each case is also included in the figure .for the second case , the rate pair achieved by the scheme proposed in  is denoted by the square symbol .the code used in that scheme is obtained by concatenating the -regular ldpc parity - check matrix and another -regular ldpc parity - check matrix.,scaledwidth=57.0% ]  -trajectory achieved by the proposed secret sharing scheme employing the rate- secret sharing irregular ldpc code obtained from the code search process described in .the block length is set to .the channel parameter setting of  db and  db is assumed .the boundary of the region for this set of channel parameters is included in the figure .the -trajectory achieved by the proposed secret sharing scheme employing a standard rate- bsc - optimized irregular ldpc code instead is also plotted ( with square markers ) for comparison.,scaledwidth=57.0% ]  -trajectory achieved by the proposed secret sharing scheme employing the rate- secret sharing irregular ldpc code obtained from the code search process described in .the block length is set to .the channel parameter setting of  db and  db is assumed .the boundary of the region for this set of channel parameters is included in the figure .the -trajectory achieved by the proposed secret sharing scheme employing a standard rate- bsc - optimized irregular ldpc code instead is also plotted ( with square markers ) for comparison.,scaledwidth=57.0% ]

Abstract: the problem of secret sharing over the gaussian wiretap channel is considered . a source and a destination intend to share secret information over a gaussian channel in the presence of a wiretapper who observes the transmission through another gaussian channel . two constraints are imposed on the source - to - destination channel ; namely , the source can transmit only binary phase shift keyed ( bpsk ) symbols , and symbol - by - symbol hard - decision quantization is applied to the received symbols of the destination . an error - free public channel is also available for the source and destination to exchange messages in order to help the secret sharing process . the wiretapper can perfectly observe all messages in the public channel . it is shown that a secret sharing scheme that employs a random ensemble of regular low density parity check ( ldpc ) codes can achieve the key capacity of the bpsk - constrained gaussian wiretap channel asymptotically with increasing block length . to accommodate practical constraints of finite block length and limited decoding complexity , fixed irregular ldpc codes are also designed to replace the regular ldpc code ensemble in the proposed secret sharing scheme .


_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ `` a device without an oscillator either does nt do anything or expects to be driven by something else ( which probably contains an oscillator ) .'' + horowitz and hill , the art of electronics , 1980 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  the plasmodium of _ physarum polycephalum _ ( order _ physarales _ , class _ myxomecetes _ , subclass _ myxogastromycetidae _ ) is a single cell , visible with the naked eye , with many diploid nuclei .the plasmodium feeds on bacteria and microscopic food particles by endocytosis . when placed in an environment with distributed sources of nutrients the plasmodium forms a network of protoplasmic tubes connecting the food sources .the topology of the plasmodium s protoplasmic network optimises the plasmodium s harvesting of nutrient resource from the scattered sources of nutrients and makes more efficient the transport of intra- cellular components  . in we have shown how to construct specialised and general purpose massively - parallel amorphous computers from the plasmodium ( slime mould ) of _ p. polycephalum _ that are capable of solving problems of computational geometry , graph - theory and logic .plasmodium s foraging behaviour can be interpreted as a computation  : data are represented by spatial of attractants and repellents , and results are represented by structure of protoplasmic network  .plasmodium can solve computational problems with natural parallelism , e.g. related to shortest path  and hierarchies of planar proximity graphs  , computation of plane tessellations  , execution of logical computing schemes  , and natural implementation of spatial logic and process algebra  .  in the framework of our `` physarum chip '' eu project we aim to experimentally implement a working prototype of a physarum based general purpose computer .this computer will combine self - growing computing circuits made of a living slime mould with conventional electronic components .data and control inputs to the physarum chip will be implemented via chemical , mechanical and optical means . aiming to develop a component base of future physarum computers we designed physarum tactile sensor  and undertook foundational studies towards fabrication of slime mould chemical sensors ( physarum nose )  , physarum memristive devices  and insulated physarum wires  .a future physarum chip will be a hybrid living - electronic computing device .being a bio - electronic device the chip will need components to generate waveforms and a clock , the source of regularly spaced pulses , implemented as oscillators .we experimentally demonstrate that it is possible to implement an electronic oscillator a device which converts direct current to alternating current signal with living slime _polycephalum_. experimental setup is outlined in sect .[ methods ] .section [ results ] presents experimental results on physarum resistance oscillations , input to output potential transfer function and current dynamics in physarum oscillators .advantages and limitations of living physarum oscillators are discussed in sect .[ discussion ] .a scheme of experimental setup is shown in fig .[ scheme ] .two blobs of agar 2 ml each ( fig .[ scheme]b ) were placed on electrodes ( fig .[ scheme]c ) stuck to a bottom of a plastic petri dish ( 9 cm ) .distance between proximal sites of electrodes is 10 mm in all experiments .physarum was inoculated on one agar blob .we waited till physarum colonised the first blob , where it was inoculated , and propagated towards and colonised the second blob .when second blob is colonised , two blobs of agar , both colonised by physarum ( fig .[ scheme]a ) , became connected by a single protoplasmic tube ( fig .[ scheme]d ) . in each experiment a resistance , potential and current were measured during 10 min with four wires using fluke 8846a precision voltmeter , test current 1.0013 .direct current potential was applied using gw instek gps-1850d laboratory dc power supply .dynamics of resistance of the blobs connected by a single protoplasmic tube was measured in 22 experiments ; dynamics of electrical potential difference between blobs and current for direct current potential applied in a range of 2 v to 15 v was measured in 6 experiments for each value of potential difference applied .resistance of two agar blobs colonised by physarum and connected with each other by a silver wire exhibits quasi - chaotic oscillations ( fig . [ exampleresistance]a ) with a wide range of dominating frequencies ( fig .[ fftsamples]a ) .when a physarum propagates from one agar blob to another blob it connects two blobs with a single protoplasmic tube ( fig .[ scheme]b ) .average resistance of a 10 mm protoplasmic tube is 3 mohm , standard deviation 0.715 mohm .resistance of the protoplasmic tube exhibits oscillatory behaviour ( fig .[ exampleresistance]bc ) with highly pronounced dominating frequency ( fig .[ fftsamples]bc ) .average dominating frequency is 0.0137606 hz , median dominating frequency is 0.0126953 hz , standard deviation is 0.0033492429 hz . the resistance oscillations have average amplitude 0.59 mohm , standard deviation 0.256 mohm ; minimum amplitude of resistance oscillations observed was 0.11 mohm and maximum amplitude 1 mohm .oscillation in resistance observed are due to peristaltic contractions of the protoplasmic tube  , see details in sect .[ discussion ] .when we apply a potential to a protoplasmic tube oscillations of the tube s resistance result in oscillation of the output potential as shown in fig .[ outputvoltages ] .values of average output potential ; and frequencies , periods and amplitudes of potential oscillations are given in tab .[ statisticstable ] .average output potential and average amplitude of output potential oscillations grow linearly with increase of an input potential ( fig .[ relations]ab ) .frequency of oscillations remains almost constant ( fig .[ relations]c ) , physarum oscillator produces the same frequency oscillations at 2 v and 15 v applied potential .a ratio of average amplitude of output potential oscillations to average output potential decreases by a power low with increase of input potential ( fig .[ relations]d ) .examples of oscillations of output current for for 5 , 10 and 14 v input potential applied as shown in fig .[ current ] and values of current are given in tab .[ tabcurrent ] .stability , accuracy , adjustability and ability to produce accurate waveforms are amongst key desirable features of an ideal electronic oscillator  .  to test physarum oscillators stability we recorded output behaviour of oscillators for 30 min ,examples are shown in fig .[ outputvoltageslongrun ] . in such long runs oscillations of an output potential persisted and frequency of oscillations was stable .we observed drifts of the output potential baseline and , sometimes sudden yet short - living , changes in amplitude of oscillations , e.g. between 300th and 500th seconds in example fig .[ outputvoltageslongrun]a , 1100th and 1500th seconds in example fig .[ outputvoltageslongrun]b , and 1100th and 1500th seconds in example fig .[ outputvoltageslongrun]c .frequency of oscillations is _ stable _ during over 70% time of physarum oscillator functioning .a drift of the electrical potential baseline occurs more likely due to unequal growth of physarum on agar blobs .dynamically changing difference in physarum body mass on reference and recording electrodes leads to a corresponding changes in resistance of the system and subsequent drift of the background potential .this drift is often in a range exceeding the electrical potential oscillation amplitude and therefore must be dealt with . immediate solution would be to involve auxiliary components to independently measure background resistance or potential , calculate and adjust the baseline potential  .further studies will concern how to control growth of physarum mass on the electrodes .this could possibly be done by illuminating blobs with a strong light every time increase of mass is detected .physarum exhibits photo - avoidance and therefore its grows will be limited in the illuminated areas .protoplasmic tubes of the slime mould exhibit periodic propagations of calcium wave along the tubes .a calcium ion flux through membrane triggers oscillators responsible for dynamic of contractile activity  .the calcium waves are reflected in oscillations of external membrane potential of physarum and periodic reversing of cytoplasmic flow in the tubes  .average dominating frequency of resistance oscillation is 0.0137606 hz , i.e. period of oscillations is c. 73 sec .this is consistent with our previous findings on periods of electrical potential oscillation and reversing of cytoplasmic flow .thus , average period of electrical potential oscillation recorded in our experiments is 67 sec  , 97 sec  , 103 sec  , and 115 sec  ; average 95 sec over four above sets of data  .cytoplasmic flow in tubes reverses with period 54 sec  .further experiments might deal with establishing an exact link between oscillations of a protoplasmic tube s resistance and peristaltic oscillations of the tube  .frequency of electrical potential oscillations of physarum can be modified by tactile  and chemical  stimuli , by coloured illumination of a protoplasmic tube  , and by loading physarum with functional nano - particles  .the correlation between resistance frequency and frequency of electrical potential oscillations allows to speculate that frequency of resistance oscillation can be also tuned by chemical , optical and tactile control inputs .thus , we believe the physarum electrical oscillator is _ adjustable_.applicability of physarum oscillators in conventional electronic circuits is limited to none .physarum oscillators produce a very low frequency waveforms and therefore can only be used in computing devices where speed of information processing is not critical .thus a potential application domain of physarum oscillators is in self - growing biological computing devices and hybrid bio - silicon devices , and amorphous bio - inspired robots  .another promising application would be in disposable bio - sensors and bio - circuits : living physarum oscillator can produce stable waveforms for up to 5 - 7 days ; such time frame is sufficient to make reliable measurements and to perform non - time consuming computations .a single protoplasmic tube of an acellular slime mould _ p. polycephalum _ can be employed as a living electronic oscillator which produces stable , accurate , and , in principle , adjustable waveforms .this is because the tube exhibits periodic peristaltic contractions which lead to oscillations of the tube s resistance .de lacy costello b. , adamatzky a. assessing the chemotaxis behavior of _ physarum polycephalum _ to a range of simple volatile organic chemicals .communicative & integrative biology 6:5 , e25030 ; september / october 2013 .hulsmann n. and wohlfarth - bottermann k.e .spatio - temporal relationships between protoplasmic streaming and contraction activities in plasmodial veins of physarum polycephalum .1978 ( 17 ) 317 - 334 .kottke t. , perry d. m. , thomson g. m. a simple digital method for compensation of baseline drift in low - frequency small - signal waveform measurements .army research laboratory preprint , arl - mr-446 june 1999 .melhuish c. , adamatzky a. , kennedy b. biologically inspired robots . in : proc .spie 4329 , smart structures and materials 2001 : electroactive polymer actuators and devices , 16 ( july 16 , 2001 ) ; doi:10.1117/12.432659     t. shirakawa , a. adamatzky , y .- p .gunji , y. miyake , on simultaneous construction of voronoi diagram and delaunay triangulation by _ physarum polycephalum _j. bifurcation chaos 9 ( 2009 ) , 31093117 .sun t. , tsuda s. , zauner k, morgan h. single cell imaging using electrical impedance tomography .nems 09 : proc . of the 2009 4th ieee int .conf , on nano / micro engineered and molecular systems , pp . 858863 .    whitting j. , de lacy costello b. , adamatzky a. towards slime mould chemical sensor : mapping chemical inputs onto electrical potential dynamics of physarum polycephalum .sensors and actuators b : chemical ( 2013 ) , in press .

Abstract: we construct electronic oscillator from acellular slime mould _ physarum polycephalum_. the slime mould oscillator is made of two electrodes connected by a protoplasmic tube of the living slime mould . a protoplasmic tube has an average resistance of 3 mohm . the tube s resistance is changing over time due to peristaltic contractile activity of the tube . the resistance of the protoplasmic tube oscillates with average period of 73 sec and average amplitude of 0.6 mohm . we present experimental laboratory results on dynamics of physarum oscillator under direct current voltage up to 15 v and speculate that slime mould _ p. polycephalum _ can be employed as a living electrical oscillator in biological and hybrid circuits .  _ keywords : oscillator , slime mould , bioelectronics _


the periodic orbits of an autonomous hamiltonian system , as well as their stability play a crucial role for the dynamical behavior of the system . orbits that are located near stable periodic orbits are ordered , while , near unstable periodic orbits chaotic motion occurs .therefore , by locating the main periodic orbits of a system and following their stability properties as one of its parameters changes , we obtain valuable information on the ordered or chaotic nature of motion in the system .  the orbital study of barred potentials has provided useful information on the structure of galactic bars ( c.f . * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) . in 2dimensional ( 2d) models , the galactic bar is supported by regular orbits trapped around the so called ` x1 ' periodic orbits , which are elongated along the bar major axis .based on the fact that these orbits do not extend beyond the corotation resonance , predicted that bars should end at , or before , corotation . in 3d models ,the planar x1 family has in general large unstable parts and , thus , its orbits are not sufficient in building the bar .however , other families of periodic orbits that bifurcate from x1 have large stable parts and support the bar .these families build the so called ` x1tree ' ( skokos , patsis & athanassoula 2002a , b ) .specific families of the x1tree are associated with certain morphological features observed in real galaxies ( patsis , skokos & athanassoula 2002 , 2003a ) .although the basic morphological features of barred galaxies are related to the presence of orbits of the x1tree , orbits not belonging to this tree can also influence the galaxy s morphology .recently , it has been shown ( patsis , skokos & athanassoula 2003b ) that inner rings in barred galaxies are associated with specific families of periodic orbits influenced by the 4:1 , 6:1 and 8:1 resonances , which are located just beyond the end of the bar ( for a definition of the resonances see e.g. section 3.1.1 ) .the basic families belonging to the x1tree are usually located very easily , but , as we approach corotation , tracing periodic orbits that are influenced by high order resonances becomes a difficult and challenging problem .the difficulty in finding such orbits is mainly due to the fact that at this region there exist many periodic orbits close to each other , most of which are unstable .thus , one is looking for small or even tiny islands of stability in a region of the phase space of the system , which is mainly characterized by chaotic behavior .usually once a periodic orbit is located , the whole family in which it belongs can be found as a parameter of the system ( e.g. , the jacobi integral ) changes . in this way, we can follow the morphological evolution and the stability transitions of a family and determine its physical importance for the system .the problem of finding the initial conditions of periodic orbits for a given parameter set of a hamiltonian system is the starting point of the present paper .using an appropriate scheme we transform the aforementioned problem into a minimization problem , where the minimizers of a particular function defined on the poincar surface of section ( pss ) of the system correspond to the periodic orbits of the hamiltonian .then , by applying an efficient optimization method , particle swarm optimization ( pso ) , and using deflection techniques , we locate periodic orbits , and follow them as a parameter of the system varies .this procedure is applied successfully on a 3d galactic potential of a ferrers bar .the paper is organized as follows . in sec .[ pot ] we describe the particular model that is used for our orbital calculations . sec .[ alg ] is devoted to the thorough description of the proposed algorithm . in particular , in sec .[ trans ] the procedure of transforming the problem of detecting periodic orbits into a minimization problem is described , in sec .[ pso ] the pso method for addressing it is presented , while sec .[ def ] , is devoted to the deflection technique , which allows us to detect further periodic orbits . in sec .[ res ] , we report the obtained periodic orbits and discuss their physical importance . finally , in sec .[ con ] , we discuss the effectiveness of the proposed numerical scheme and present our conclusions .the 3d barred galaxy model that we use is described in detail in .it consists of a miyamoto disk , a plummer bulge and a ferrers bar .the potential of the miyamoto disk is given by the formula ,    where represents the total mass of the disk , and are scale lengths such that the ratio gives a measure of the flatness of the model , and is the gravitational constant .the bulge is a plummer sphere , i.e. , its potential is given by , where is the bulge scale length and is its total mass . finally , the bar is a triaxial ferrers bar with density , where , in eq .( [ eq : m ] ) , , and are the principal semi - axes , while denotes the mass of the bar component .the corresponding potential , , as well as the forces are given in in a closed form , which is well suited for numerical treatment .regarding the miyamoto disk , we use and , and for the axes of the ferrers bar we set , and .the masses of the three components satisfy . in particular, we have , , and .the length unit is taken as 1 kpc , the time unit as 1 myr and the mass unit as .the bar rotates with a pattern speed , =0.054 , around the -axis , which corresponds to 54 km sec kpc , and places corotation at 6.13 kpc .the hamiltonian governing the motion of a test particle can be written in the form , with , and being the canonical momenta .the numerical value of will be vaguely reported as the ` energy ' , , of the system ._ swarm intelligence _ methods are stochastic optimization , machine learning and classification procedures that model intelligent behavior ( bonabeau , dorigo & theraulaz 1999 ; kennedy & eberhart 2001 ) .they are closely related to the methods of _ evolutionary computation _ , which consists of algorithms motivated from biological genetics and natural selection .a common characteristic of all these algorithms is the exploitation of a population of search points that probe the search space simultaneously .particle swarm optimization ( pso ) belongs to the category of swarm intelligence methods .the development of pso sprang from the simulation of social dynamics of flocking organisms , such as insect swarms , which are governed by fundamental rules like nearest neighbor velocity matching . in nature , information is communicated among the members of bird flocks and fish schools , enhancing their ability to search for food and enabling them to move synchronized without colliding . the social behavior of animals , and in some cases of humans ,is governed by similar rules .there is a general belief , and numerous examples coming from nature enforce it , that social sharing of information among the individuals of a population , provide an evolutionary advantage .the dynamics of population in pso resembles the collective behavior and self organization of socially intelligent organisms .the individuals of the population exchange information and benefit from their discoveries , as well as the discoveries of other companions , while exploring promising areas of the search space . in our case ,the poincar section is our search space , and periodic orbits of a hamiltonian system are computed through the minimization of a function . in the context of function minimization , promising areas of the search space are characterized by low function values .   a suitable way to study the stability of orbits in the 6d phase space of the hamiltonian system ( [ eq : ham ] ) is the well known method of the poincar surface of section ( e.g. * ? ? ?* ) . instead of following the time evolution of an orbit in the whole phase space ,we confine our study on an appropriately chosen subspace of it . in our case , the pss is the subspace of , defined by the conditions , , .the major axis of the bar lies along the axis .thus , for a given value of the energy , an orbit with initial conditions , ( where denotes the transpose of a matrix ) , on the pss is fully defined as and can be obtained by solving eq .( [ eq : ham ] ) keeping only the positive found value of .  in this way ,only the 4 initial conditions of an orbit on the pss are necessary for identifying the orbit .then , the time evolution of the orbit is derived by solving the hamilton s equations of motion .the next intersection of the orbit with the pss , , , is denoted as , which is obviously a point belonging to the 4dimensional pss of the system . by using the notation , the initial conditions , , of a orbit of the system satisfy the equations : thus, finding the initial conditions , , of a orbit is equivalent to solving eq.([eq : eq ] ) , which in turn is equivalent to computing the global minimizers of the function , this function is called the objective function , and it is actually the square of the euclidean distance on the pss between the initial point of an orbit and its intersection with the pss . obviously , if the studied orbit is this distance is zero .thus , the aforementioned technique transforms the problem of finding periodic orbits into the problem of computing the global minimizers of the function  .let us now describe the procedure of finding a minimizer of the objective function defined on a 4dimensional search space , , which is a subspace of the pss .as already mentioned , pso is a population based method , i.e. , it exploits a population of individuals to probe for promising regions of the search space , simultaneously .the population is called a _ swarm _ andthe individuals ( i.e. , the search points ) are called _particles_. in our case the swarm is a set of initial conditions on the pss and a particle is a point on the pss , which corresponds to an initial condition of an orbit of hamiltonian ( [ eq : ham ] ) . in every iteration of the pso method, we check whether any of the particles is a minimizer of the objective function , , of eq .( [ eq : function ] ) .if it is , the minimizer is recorded , otherwise , each particle is moved to a new position in the search space using an adaptable displacement called _ velocity _ , retaining also a memory of the best position it ever encountered . in our casethe best positions possess lower function values .  the particles of the swarm exchange information among them .there are two variants of the method with respect to the number of particles that share information : the _ global _ and the _ local _ variant . in the global variant of pso , the best position ever attained by all particles of the swarm is communicated among them . in the local variant , each particleis assigned to a neighborhood consisting of a prespecified number of particles and the best position ever attained by the particles that comprise the neighborhood is communicated among them . in the latter case , the information attained by the particles is spread in the swarm slowly , maintaining high diversity of the particles for more iterations of the algorithm than in the global variant , which converges faster to the best position .consequently , the local variant has better exploration abilities while the global variant has better convergence rates ( exploitation ) . in the present paperwe use the local variant of pso as it is more appropriate for the location of periodic orbits in the chaotic region near the corotation of the bar potential . in this regionthere are many periodic orbits , very close to each other , most of which are unstable .thus , in order to locate such orbits , we need an algorithm that performs thorough exploration of the phase space with the cost of slightly slower convergence .let us consider a swarm consisting of particles .each particle is in effect a 4dimensional vector , the velocities of the particles are also 4dimensional vectors , the best previous position encountered by the particle is a point in , denoted by   the positions , , and the velocities , , of the particles are randomly initialized , following a uniform distribution within the search space . the best positions , , are initially set equal to . each particle is evaluated according to the objective function of eq .( [ eq : function ] ) , i.e. , the value is computed for all particles . obviously , at the initialization phase it holds that .let , be a neighborhood of radius of the particle , ( local variant ) .then , is defined as the index of the best particle in the neighborhood of , i.e. , the neighborhood s topology is usually cyclic , i.e. , the first particle is assumed to follow after the last particle , . in the general case we face in the present paper ,the search space is 4dimensional . in this casewe use a swarm consisting of particles , while the neighborhood of every particle has radius , .for example , the neighborhood of the particle consists of the particles , , , , , , and .the specific neighborhood size was selected in order to benefit from the exploitation ability of the local pso variant , while avoiding very slow convergence rates implied by smaller neighborhoods .if there exist a particle , such that , then the initial conditions of a orbit are found . in the opposite case , we proceed to the next iteration of the algorithm , which is to move the particles to new positions , taking into account their history so far .this is done according to the equations  where ; is a parameter called _ constriction factor _ ; and are two fixed , positive parameters called _ cognitive _ and _ social _ parameter respectively ; , , are random vectors with components uniformly distributed in the interval  ] found by the pso method , one should compute all the orbits with initial conditions on a grid of this interval whose grid step should at least be equal to .these procedure requires about 28500 evaluations of , not taking into account the successive iterations of the newton method to converge to the actual initial condition with the desired accuracy .we note that in the case of the ter1 periodic orbit this convergence requires about 10 evaluations of .so it is evident that the computational effort needed by our numerical scheme is significantly lower with respect to the newton iterative method even in the simple case of the 1dimensional search space .  as we pointed out , the main problem that newton like techniques face in cases where many periodic orbits coexist very close to each other , is that they need a very good initial guess of the position of the periodic orbit in order to find it .this difficulty of the newton iterative schemes is well known and some efforts to improve their efficiency have already been done . as an example , we refer to the paper of where the authors evaluate the initial conditions in every successive iteration of the newton scheme by an appropriate least squares technique .we emphasize that our approach is completely different as we do not try to find the roots of the system of eq.([eq : eq ] ) but we transform this problem into a minimization one. we also note that our scheme uses only the values of function ( eq . [ eq : phi ] ) and not its derivatives like newton iterative techniques do .this makes our computation easier and more accurate , since the usual method to evaluate the derivatives of function , which is not known in a close analytical form , is to approximate them by finite differences .  summarizing the main advantages of the proposed algorithm with respect to newton like methods we could mention that our method is faster, it is simple and can be implemented easily , it works using function values solely , and also it has the ability to locate many periodic orbits per run .the orbital behavior of the e and te families and their bifurcations , which were detected by the proposed scheme , as well as the behavior of the s family , helps us to establish the general behavior of the families existing beyond the radial 4:1 resonance gap in our model .the morphology of the basic 2d families are influenced by two successive even resonances : the s family is influenced by the 6:1 and 8:1 resonances , the e family is influenced by the 8:1 and 10:1 resonances and the te family by the 10:1 and 12:1 resonances . from these families , 2d families influenced by the in - between odd resonancesbifurcate : sr1 is influenced by the 7:1 resonance , er1 by the 9:1 resonance and ter1 by the 11:1 resonance .in addition , the main 3d bifurcations of the s , e and te families ( sv1 , ev1 , tev1 families respectively ) have , in general , projections on the plane similar to the main families .this is in accordance with the observed frequency of the various inner rings morphology , as proposed by .we appreciate useful comments from the referee , which helped us improve the clarity of the paper .we also thank prof .g. contopoulos for fruitful discussions on the subject .skokos was supported by the research committee of the academy of athens , the ` karatheodory ' post doctoral fellowship no 2794 of the university of patras , the greek state scholarships foundation ( iky ) and the empeirikeion foundation .

Abstract: we propose the particle swarm optimization ( pso ) as an alternative method for locating periodic orbits in a three dimensional ( 3d ) model of barred galaxies . we develop an appropriate scheme that transforms the problem of finding periodic orbits into the problem of detecting global minimizers of a function , which is defined on the poincar surface of section ( pss ) of the hamiltonian system . by combining the pso method with deflection techniques , we succeeded in tracing systematically several periodic orbits of the system . the method succeeded in tracing the initial conditions of periodic orbits in cases where newton iterative techniques had difficulties . in particular , we found families of 2d and 3d periodic orbits associated with the inner 8:1 to 12:1 resonances , between the radial 4:1 and corotation resonances of our 3d ferrers bar model . the main advantages of the proposed algorithm is its simplicity , its ability to work using function values solely , as well as its ability to locate many periodic orbits per run at a given jacobian constant .   [ firstpage ]  methods : numerical galaxies : kinematics and dynamics galaxies : structure .