Evaluating the Degree of Randomness in Optical Quantum Random Number Generators

Transcription

1 Evaluatig the Degree of Radomess i Optical Quatum Radom Number Geerators Adreas Möller molleresa@gmail.com uder the directio of Prof. Mohamed Boureae Quatum Iformatio ad Quatum Optics Departmet of Physics Stockholm Uiversity Research Academy for Youg Scietists July 13, 2016

2 Abstract Quatum radom umber geerators are used today i a variety of applicatios; maily for stochastic simulatio ad secure commuicatios. As these fields grow, fast ad coveiet radom umber geerators that geerate truly radom umber sequeces are sought for. The aim of this study is therefore to build differet kids of optical quatum radom umber geerators ad evaluate their degree of radomess. The setups studied all deped o quatum idetermiism. A laser beam was divided i differet ways, ad the path the photos took was registered ad coverted to a biary sequece. Four differet experimetal setups were studied. To divide the beam, a 50/50 beam splitter, a H/V polarized beam splitter ad a two path fiber couplig was used. The two path fiber couplig was further divided ito two separate experimets, oe usig a field programmable gate array for data collectio ad oe usig a time taggig uit. The two correlatio algorithms used to test for radomess i the strigs fout that the two path fiber couplig method with a time taggig uit for data collectio geerates radom umber sequeces with the highest degree of radomess i the study. Its radom umber sequeces also passes the radomess test suite of The America Natioal Istitute of Stadards ad Techology (NIST).

3 Cotets 1 Itroductio Radomess True Radomess Pseudo Radomess Radom Number Geeratio (RNG) Pseudo Radom Number Geeratio (PRNG) Quatum Radom Number Geeratio (QRNG) Quatum Superpositio Previous Research Aim of the Study Method QRNG Usig a 50/50 Beam Splitter QRNG Usig a H/V-Polarized Beam Splitter QRNG Usig a Two-Path Fiber Couplig Data Collectio Data Evaluatio Methods Results The 50/50 Beam Splitter Method The Polarized Beam Splitter Method The Two Path Fiber Couplig Method The Two Path Fiber Couplig Method Employig a Time Taggig Uit for Data Collectio The NIST Radomess Test P -Value Summarizatio Geerated Bitmap

4 4 Discussio QRNG Through Beam Splitters QRNG Through Two Path Couplig with a FPGA QRNG Through Two Path Couplig with a TTU The NIST Radomess Test Bitmap Compariso Coclusios Ackowledgemets 19 Refereces 20

5 List of Abbreviatios BS FPGA HWP MMF NDF PBS PC PRNG QRNG RNG SMF SPD TTU Beam splitter Field programmable gate array Half wave plate Multi mode fiber Neutral desity filter Polarized beam splitter Polarizatio cotroller Pseudo radom umber geeratio Quatum radom umber geeratio Radom umber geeratio Sigle mode fiber Sigle photo detector Time taggig uit 1 Itroductio 1.1 Radomess Radomess is described as a occurrece whose outcome caot be predicted, i.e the couterpart to what is predictable [1]. This study will maily focus o radom umber sequeces, whose mai applicatios are stochastic simulatio ad ecryptio [1, 2]. Today, differet methods are used to geerate sequeces of radom umbers i a fast ad coveiet way, maily pseudo radom umber geerators (PRNGs) [1]. However, these always show repeatig patters after a certai amout of geeratios [1]. The mai limitatio as of today though, is the rate at which the sequeces of radom umbers are created [1]. The rate at which radom umber sequeces are eeded is costatly growig, thus creatig a commercial iterest to develop a fast ad coveiet source of truly radom umbers [1, 2] True Radomess Number sequeces whose past etries caot be used to predict upcomig etries are said to be truly radom. This implies that there caot exist a way to predict what umber is ext i a truly radom umber sequece, ot eve i theory. To achieve true radomess, 1

6 there caot exist ay repetitive patters i the sequece whatsoever. [3] Pseudo Radomess All pseudo radom sequeces repeat themselves after a umber of geeratios [3]. Eve though moder PRNGs have bee highly refied ad remais urepeated after up to geeratios [4], they are ot truly radom [3]. This is maily depedet o the way the umbers are created, which meas that these sequeces might seem radom to humas, ad i some degree eve to computers, but still fails the criterio of beig truly radom [3]. 1.2 Radom Number Geeratio (RNG) There are several processes i which radom sequeces of umbers are created. These processes ca be divided ito two mai categories based o how the radom umbers are geerated. Either the RNG utilizes determiistic algorithms for its geeratios, or it uses other methods, such as measurig quatum states, to achieve radomess. [3] Pseudo Radom Number Geeratio (PRNG) Pseudo radom umber sequeces are ofte created by computers, which ca be doe by usig a algorithm whose outcome is very sesitive to the startig value. I this way, two almost idetical startig values geerate widespread results. The startig value ca be take from a large variety of places, for example chaotic user iput, but sice the same iput always produces the same output, the system as a whole is pseudo radom. [5] The idea with PRNGs is that the sequece of umbers is supposed to seem radom to someoe who has o isight i the algorithm. This is sufficiet for most applicatios, but for some applicatios, such as secure ecryptio, truly radom umbers are required. [5] 2

7 1.2.2 Quatum Radom Number Geeratio (QRNG) True radomess does ot exist i classical physics, sice it describes a determiistic system ad does ot provide upredictable variatios. Quatum physics does the opposite, with truly radom properties ad true upredictability. Therefore QRNGs are the oly RNGs that provide true radomess, which ca be take advatage of i the process of radom umber geeratio. [5] 1.3 Quatum Superpositio Quatum physics differ from classical physics by itroducig quatum pheomea; these are uique to the quatum world, thus makig quatum physics useful i areas where classical physics caot describe reality. Oe quatum pheomeo that is utilized i this study is quatum superpositio. Particles have various quatum properties, such as polarizatio ad spi, ad have oe quatum state for each property, such as H- polarizatio ad couterclockwise spi. A superpositio is essetially a state icludig two eigestates, where the particle does ot take o a eigestate before a measuremet is take. Quatum states ca be deoted usig Dirac otatio, which for example expresses a superpositio regardig Horizotal/Vertical-polarizatio as see i the equatio below: ψ = 1 2 H V, (1) where ψ represets the quatum state, H represets horizotal polarizatio ad V represets vertical polarizatio. I this example, both outcomes have a 50 % probability of occurrig. [6] 1.4 Previous Research Several studies have bee focusig o QRNG, ad some of them have used methods similar to the oes used i this study [3, 4]. The America Natioal Istitute of Stadards ad Techology (NIST) has developed 3

8 a test suite for radomess i umber sequeces, which is recommeded by them i order to test RNGs [7]. The suite has bee used to test the results of this study as well, mostly for compariso agaist curretly available QRNGs. 1.5 Aim of the Study The aim of this study is to test ad evaluate four differet QRNGs. Those addressed i this study all use photos ad quatum idetermiism to acquire a truly radom data set. This is doe by measurig quatum superpositios ad thereby gettig a radom distributio betwee two eigestates. 2 Method I this study, a Helium-Neo laser with a wavelegth of λ = m ad with a specified output power of P = 2.0 mw was used as a photo source. The idea behid all methods that follow is to split the laser beam ito two equally powerful beams. The ew beams are the coupled to sigle photo detectors (SPDs) which, whe coected to a coutig uit, geerate biary sequeces whe sigle photos choose differet paths based o differet coditios, for example which eigestate their superpositio decomposes ito. What differs betwee three of the four various methods is how the divisio of the beam is achieved. The fourth method differs from the rest by the meas of processig the data, with a time taggig uit (TTU) itroduced ito the setup. For all experimetal setups, the beam was atteuated to a poit where the SPDs maaged to detect sigle photos whilst retaiig a large flow of data. The power of the atteuated beam varied betwee 10 9 W ad W for the various setups. The atteuatio was coducted i part usig a polarizatio filter, which cosists of a half wave plate (HWP) that polarizes light, ad a PBS that splits the beam with regards to its polarizatio. It reflects all photos except for H-polarized oes off to the side, which 4

9 together with the HWP makes for a adjustable filter. I order for this to fuctio, the icomig photos must all be of the same polarizatio. This was obtaied by couplig the beam ito a sigle mode fiber (SMF), i.e a fiber i which the light propagates homogeeously, alog with a polarizatio cotroller (PC) that provides adjustability to the output polarizatio of the couplig. The atteuatio was also icreased by reducig the couplig efficiecy for the SMF ad by itroducig eutral desity filters (NDFs) ito the setup. 2.1 QRNG Usig a 50/50 Beam Splitter I this setup, the divisio of the beam is coducted usig a 50/50 BS. I theory, a beam splitter (BS) orthogoal to the icomig beam should divide the beam ito two equally powerful orthogoal beams. Figure 1: A schematic represetatio of the first experimetal setup i this study, where a 50/50 BS is used to split the beam. For abbreviatios, see List of Abbreviatios. 2.2 QRNG Usig a H/V-Polarized Beam Splitter Similar to the previous method, this setup also utilizes a type of BS; a polarized beam splitter (PBS), which divides the beam with regards to the polarizatio of idividual photos. This setup also requires a HWP set at 22.5 before the PBS i order to set the photos i the superpositio described i equatio (1), see Figure 2. This should, i 5

10 theory, split the beam ito two equally powerful beams with H-polarized photos beig trasmitted ad V-polarized photos beig reflected. Figure 2: A schematic represetatio of the experimetal setup usig a PBS to divide the icomig beam ito two equally powerful oes. For abbreviatios, see List of Abbreviatios. HWP 2 is set at 22.5 to brig all the icomig H-polarized photos from PBS 1 ito the superpositio described i equatio (1). 2.3 QRNG Usig a Two-Path Fiber Couplig This experimetal setup was used for two experimets, which differed by the meas of data collectio istead of beam divisio; oe experimet used the same data collectio system as previous experimets i this study with a field programmable gate array (FPGA), ad oe experimet used a TTU for data collectio. Istead of employig beam splitters, this setup divides the icomig beam by offerig two paths for the photos to travel. This is doe by aligig a stripped SMF to two stripped multi mode fibers (MMFs) i free space; ot usig ay coectios as covetioal joiig of fibers does. I this way the beam propagates out of the SMF ad, whe aliged properly, couples evely ito the two MMFs. 6

11 Figure 3: A schematic represetatio of the two-path fiber couplig setup. For abbreviatios, see List of Abbreviatios. 2.4 Data Collectio The outputs from the SPDs were registered by a computer ad were deoted i a text file as biary digits, where the output from oe detector represeted 0 ad the output from the other detector represeted 1. I all setups except the oe usig a two-path couplig alog with a TTU, a FPGA was used to retrieve sigals from the SPDs ad process them ito a biary file. For the secod setup usig a two-path fiber couplig, a TTU was itroduced to process the sigals from the SPDs. The TTU also registers the time of arrival for each detectio, makig processig more accurate by avoidig couts from both detectors registerig at the same time, which would result i useless data. 2.5 Data Evaluatio Methods The data from all experimetal setups were aalyzed with a couple of methods described below. The raw data was first aalyzed, thereafter a ubiasig algorithm was applied to remove classical iterferece oise from the data before it was aalyzed agai. Oe method that was used to aalyze the geerated sequeces for repetitive correlatio was the autocorrelatio test. The test ivestigates whether bits with distace from each other are correlated or ot. That is, if you by kowig the first elemet i ay way ca determie the -th elemet. The autocorrelatio fuctio Γ autocorrelatio () 7

12 is described below: Γ autocorrelatio () = 1 N N (b i b i+ ), (2) where is the distace for which correlatio is ivestigated, N is the umber of etries i the sequece {b i } N i=1 [3], ad is the xor fuctio. I theory, sice the bits should be equally distributed, the output value from the autocorrelatio fuctio should be 0.5 for all. As metioed i Sectio 1.4, NIST has developed a radomess test for RNGs. For further details o the test, see Referece [7]. The test is applied to the radom sequeces foud i this study to provide material for compariso with commercial QRNGs, ad will amog other geerate a bitmap usig the provided radom umber sequece. The geerated bitmap will be compared to the bitmap show i Figure 4, which is based o a radom umber sequece from a PRNG. Aother correlatio fuctio used, Γ correlatio (), is described below: i=1 Γ correlatio () = 1 N (b i b i+ ) E [b] 2, (3) N usig the same costats ad variables as equatio 2 ad E [x], which is the expected value fuctio. For ifiite radom sequeces, the output of the correlatio fuctio is 0. However, for fiite sequeces, the output fluctuates thereabout due to statistical oise i=1 depedig o the size of the iput sequeces. [1] 8

13 Figure 4: A bitmap based o a pseudo radom umber sequece. 1 3 Results The autocorrelatio ad correlatio for the fuctio Γ () are ivestigated for each method. Both fuctios are represeted as the raw data ad the ubiased data. Observe that the scale of the y-axis varies betwee plots, which ca make for some difficulties i comparig differet plots

14 3.1 The 50/50 Beam Splitter Method (a) m = , σ = The fuctio Γ autocorrelatio () for the raw data retrieved usig the BS method described i Sectio (b) m = , σ = The fuctio Γ autocorrelatio () for the ubiased data retrieved usig the BS method (c) m = , σ = The fuctio Γ correlatio () for the raw data retrieved usig the BS method (d) m = , σ = The fuctio Γ correlatio () for the ubiased data retrieved usig the BS method. Figure 5: Plots for the correlatio tests regardig the 50/50 BS method. 10

15 3.2 The Polarized Beam Splitter Method (a) m = , σ = The fuctio Γ autocorrelatio () for the raw data retrieved usig the PBS method described i Sectio (b) m = , σ = The fuctio Γ autocorrelatio () for the ubiased data retrieved usig the PBS method (c) m = , σ = The fuctio Γ correlatio () for the raw data retrieved usig the PBS method (d) m = , σ = The fuctio Γ correlatio () for the ubiased data retrieved usig the PBS method. Figure 6: Plots for the correlatio tests regardig the data from the PBS method. 11

16 3.3 The Two Path Fiber Couplig Method (a) m = , σ = The fuctio Γ autocorrelatio () for the raw data retrieved usig the two path fiber couplig method described i Sectio (b) m = , σ = The fuctio Γ autocorrelatio () for the ubiased data retrieved usig the two path fiber couplig method (c) m = , σ = The fuctio Γ correlatio () for the raw data retrieved usig the two path fiber couplig method (d) m = , σ = The fuctio Γ correlatio () for the ubiased data retrieved usig the two path fiber couplig method. Figure 7: Plots for the correlatio tests regardig the two path fiber couplig method. 12

17 3.4 The Two Path Fiber Couplig Method Employig a Time Taggig Uit for Data Collectio (a) m = , σ = The fuctio Γ autocorrelatio () for the raw data retrieved usig the two path fiber couplig method described i Sectio 2.3 with a TTU for data collectio (b) m = , σ = The fuctio Γ autocorrelatio () for the ubiased data retrieved usig the two path fiber couplig method with a TTU for data collectio (c) m = , σ = The fuctio Γ correlatio () for the raw data retrieved usig the two path fiber couplig method with a TTU for data collectio. -1 (d) m = , σ = The fuctio Γ correlatio () for the ubiased data retrieved usig the two path fiber couplig method with a TTU for data collectio. Figure 8: Plots for the correlatio tests regardig the two path fiber couplig method with a TTU for data collectio. 13

18 3.5 The NIST Radomess Test P -Value Summarizatio These graphs show the p-value for each of the radomess tests icluded i the NIST test suite. The logarithmic y-axis shows the p-value. All data used herei comes from the two path fiber couplig setup with a TTU for data collectio. Figure 9: A diagram showig the output of the NIST radomess test. The iput data comes as raw data from the experimet, ad has ot bee treated i advace. 14

19 Figure 10: A diagram showig the output of the NIST radomess test. The data has bee ubiased before the tests Geerated Bitmap The bitmap created by Wolfram Mathematica usig the radom umber sequece geerated by the two path fiber couplig method with a TTU as coutig system ca be see i Figure Discussio 4.1 QRNG Through Beam Splitters This study has examied how differet QRNGs preform o various radomess tests. It has bee foud that the first two setups, i.e the oes usig beam splitters to divide the beam, preforms iferior o the correlatio tests. The results o the autocorrelatio test after ubiasig were ideed m = ± for the BS ad m = ± for the PBS, however the mea values does ot covey the whole truth; for it is essetial 15

20 Figure 11: A bitmap geerated by Wolfram Mathematica from ubiased data, to be compared with Figure 4. for first order correlatios be miimal i true radomess. Γ (1) should be withi a few stadard deviatios from the mea, however either the BS or the PBS fulfills this criterio with Γ autocorrelatio (1) beig σ apart from m for the BS ad σ apart from m for the PBS. Respectively, Γ correlatio (1) was σ apart from m for the BS ad σ apart from m for the PBS. I theory, a 50/50 BS divides the icomig beam ito two equally powerful beams whe aliged orthogoal to the icomig beam. However, the traslatio ad rotatio of the BS had to be adjusted util the two beams were equally powerful. The positio of the BS with this coditio fulfilled was therefore ot orthogoal to the icomig beam, cotradictig what theory proposes. This is oe plausible explaatio for the iferior results. Although, the explaatio we propose has to do with the data collectio system rather tha the experimetal setup. Our suspicio is that the FPGA used caot hadle high bitrates, sice it retured error messages or empty measuremets whe the measuremet frequecy wet over a certai level. Thus, a FPGA does ot have to be a bad choice for QRNG, although the oe used i this study did ot provide satisfyig results. Other 16

21 causes of ucertaity are laser power fluctuatios, detector precisio, fiber propagatio losses ad detectio rates of the SPDs. It was also foud that quite much classical oise iflueced the measuremets, with dark couts beig resposible for betwee 5 % ad 50 % of the measured data i these setups. 4.2 QRNG Through Two Path Couplig with a FPGA Despite the doubts expressed earlier, the experimet usig a two path fiber couplig with a FPGA coutig uit provided more satisfyig results tha the BS ad the PBS did. Ideed Γ autocorrelatio (1) was σ apart from m, but as σ is more tha 20 times smaller tha for the BS ad the PBS, the sequece is closer to true radomess tha they are. The results from the two correlatio fuctios with the biased raw data were closer to values wished for as well, suggestig it beig a better RNG setup. 4.3 QRNG Through Two Path Couplig with a TTU The two path couplig with a TTU showed to be the best setup for QRNG i this study. Employig a TTU for data collectio made it possible to icrease the bitrate rate sigificatly, makig geeratig sequeces the size of a gigabyte a matter of hours at a geeratio rate of 25 khz, istead of takig approximately 5 days with the FPGA at a detectio rate of 520 Hz. This is mostly depedet o the time taggig, which miimizes the risk of both detectors reportig a photo each i oe measuremet. With the FPGA, approximately 30 % 40 % of measuremets were double detectios, makig them useless i the process of RNG. The rate of double detectios by the TTU were as low as 10 Hz, which represets 0.04 % of all detectios. The results from the correlatio tests were well withi acceptable rages, with Γ autocorrelatio (1) beig σ apart from m = ± , ad Γ correlatio (1) beig σ apart from m = ± This beig said, the two path fiber couplig method with a TTU provided the most radom sequece of umbers i 17

22 this study. 4.4 The NIST Radomess Test The NIST radomess test illustrated, amog other, what improvemets came from the ubiasig process. Studyig Figure 9 ad Figure 10, the impact of the ubiasig process is clear. It is also see that the radom umber sequece tested passes the test, with p-values over the sigificace level, which traslates to low correlatio betwee umbers. For a truly radom umber sequece, it is ot expected for all p-values to equal 1, which meas that the umber sequece geerated i this study might be truly radom, for there is o certai way to determie true radomess for fiite umber sequeces. 4.5 Bitmap Compariso Oe bitmap was geerated by the NIST test suite based o the iput radom umber sequece. This is to be compared to the bitmap based o a pseudo radom umber sequece, see Figure 4 ad Figure 11. I the pseudo radom bitmap some patters ca be distiguished, which is due to the repetitive ature of PRNGs. I the bitmap geerated by the QRNG i this study, o such patters portrait. This is a visual represetatio of the differeces betwee PRNG ad QRNG, where QRNGs are urepetitive opposig PRNGs, ad a cofirmatio of the success of this experimetal setup. 4.6 Coclusios To summarize this study, oe ca say that the QRNGs used produced varyig results. The BS ad PBS both geerated iferior umber sequeces i terms of radomess, which might have bee caused by several differet reasos such as classical iterferece ad limitatios of the FPGA. The two path fiber couplig methods were both more successful, especially after the ubiasig procedure. The experimet usig the TTU provided by far 18

23 the most satisfyig result, with large margis o passig the NIST test ad good mea values i the correlatio tests, as well as low stadard deviatios. Compared to commercial RNGs, the QRNGs used i this experimet are disadvataged with regards to the speed of the geeratio. Furthermore, the setups with the BS ad the PBS showed that ot all QRNGs fuctio as wished for all the time. There might therefore be of iterest to evaluate commercial QRNGs regardig degree of radomess i a future study. Ackowledgemets First ad foremost, I would like to thak Prof. Mohamed Boureae at Stockholm Uiversity, who has made this project possible by providig kowledge ad facilities to coduct the experimets i. I would also like to covey my gratitude towards the staff at the istitutio of Quatum Iformatio ad Quatum Optics at Stockholm Uiversity for helpig us with the setup of the experimets ad the aalysis of the data, especially Adria Holzäpfel ad Marco Avesai who have take time from workig with their master theses to help us. I would also like to thak Gustav Arvidsso: the fellow Rays studet with whom I have carried out this project. He has bee a valuable source of iput ad feedback throughout the time of the project. Fially, I thak Rays - for Excellece ad their parters Tekiska Museet ad Tekikföreetage for makig this research project possible. 19

24 Refereces [1] Abella C. Quatum Radom Number Geeratio Based O Vacuum Field Fluctuatios. Barceloa: Polytechic Uiversity of Cataloia; 2013 [2] Nie Y-Q, Huag L, Liu Y, Paye F, Zhag J, Pa J-W. The geeratio of 68 Gbps quatum radom umber by measurig laser phase fluctuatios. Hefei: Review of Scietific Istrumets; vol. 86: o. 6. [3] Stefaov A, Grisi N, Guiard O, Guiard L, Zbide H. Optical quatum radom umber geerator. Geeva: Joural of Moder Optics; vol. 47; o. 4. [4] Jeewei T, Achleiter U, Weihs G, Weifurter H, Zeiliger A. A Fast ad Compact Quatum Radom Number Geerator Wie: Review of Scietific Istrumets; vol. 71: o. 4. [5] James F. A Review of Pseudoradom Number Geerators. Geeva: Computer Physics Commuicatios; vol. 60, o. 3. [6] Nielse M A, Chuag I L. Quatum Computatio ad Quatum Iformatio: 10th Aiversary Editio. Cambridge: Cambridge Uiversity Press; revised 2011, published [7] Ruhki A, Soto J, Nechvatal J, Smid M, Leigh S, et. al. A Statistical Test Suite for Radom ad Pseudoradom Number Geerators for Cryptographic Applicatios. Gaithersburg: Natioal Istitute of Stadards ad Techology; Revisio 1a. 20