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Quantum Random Number Generators and Their Roles

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Quantum Random Number Generators and Their Roles Quantum Random Number Generators Miguel Herrero-Collantes∗ Instituto Nacional de Ciberseguridad, Avenida Jos´eAguado, 41, Edificio INCIBE 24005, Le´on, Spain. EI Telecomunicaci´on, Department of Signal Theory and Communications, University of Vigo, Campus Universitario Lagoas-Marcosende, E-36310 Vigo, Spain. Juan Carlos Garcia-Escartin† Universidad de Valladolid, Dpto. Teor´ıa de la Se˜nal e Ing. Telem´atica, Paseo Bel´en no 15, 47011 Valladolid, Spain. (Dated: October 24, 2016) Random numbers are a fundamental resource in science and engineering with impor- tant applications in simulation and cryptography. The inherent randomness at the core of quantum mechanics makes quantum systems a perfect source of entropy. Quantum random number generation is one of the most mature quantum technologies with many alternative generation methods. We discuss the different technologies in quantum ran- dom number generation from the early devices based on radioactive decay to the multiple ways to use the quantum states of light to gather entropy from a quantum origin. We also discuss randomness extraction and amplification and the notable possibility of gen- erating trusted random numbers even with untrusted hardware using device independent generation protocols. CONTENTS G. Generators based on the phase noise of lasers 21 H. Generators based on amplified spontaneous I. Motivation 2 emission 23 I. Generators based on Raman scattering 24 II. Random numbers and their applications 2 J. Generators based on optical parametric oscillators 27 A. Pseudorandom number generators and true random number generators 3 B. Random numbers in simulation 5 VIII. Non-optical Quantum Random Number Generators 28 C. Random numbers in cryptography 5 D. Random numbers in fundamental science 7 IX. Random numbers certified by quantum mechanics 29 A. Self-testing in quantum random number generators 30 III. Block description 8 B. Device independent quantum random number IV. Entropy estimation 9 generators 31 C. Other forms of quantum certification 33 V. Quantum Random Number Generators based on radioactive decay 10 X. Postprocessing 34 arXiv:1604.03304v2 [quant-ph] 21 Oct 2016 A. The first quantum random number generators 10 A. Randomness extractors 35 B. Evolution 12 C. Limitations 12 1. Deterministic extractors 35 2. Seeded extractors 36 VI. Random Number Generators based on noise 13 XI. Quantum randomness extractors: randomness expansion VII. Optical Quantum Random Number Generators 14 and randomness amplification 37 A. Quantum optics in random number generators 14 B. Branching path generators 15 A. Quantum randomness expansion 38 C. Time of arrival generators 17 B. Quantum randomness amplification 38 D. Photon counting generators 19 E. Attenuated pulse generators 20 XII. Randomness testing 39 F. Generators based on quantum vacuum fluctuations 20 XIII. Discussion 40 Acknowledgments 40 ∗ [email protected] † [email protected] References 40 2 I. MOTIVATION when each method is more appropriate. Due to their im- portance, we concentrate on applications to simulation Quantum mechanics offers interesting new protocols in and cryptography. the intersection between computer science, telecommu- Section III describes the main functional elements of nications, information theory and physics. Results like quantum random number generators and their roles. In the protocols for quantum key distribution (Bennett and Section IV, we present some mathematical measures of Brassard, 1984; Ekert, 1991) and efficient algorithms for randomness which are particularly useful to analyse the problems that are thought or known to be hard for clas- amount of available random bits and the security of quan- sical computers (Childs and van Dam, 2010; Ekert and tum random number generators. Jozsa, 1996) show quantum physics can have a profound Section V discusses QRNGs based on radioactive de- impact in the way we think about security, cryptography cay, which were the first proposed QRNGs and are still and computation. in use today. Section VI introduces random number gen- Despite the impressive experimental achievements of erators based on electronic noise and analyses when they the last decades, the current state of technology is still can be said to be quantum. not advanced enough for a full-scale universal quan- Section VII discusses how optics has modernized tum computer. Quantum key distribution, on the other QRNGs. Most present-day QRNGs are based on quan- hand, has already become an established technology and tum optics and we review the multiple implementations the first commercial systems have been demonstrated in that work with the quantum states of light. practical scenarios (Peev et al., 2009; Sasaki et al., 2011). Section VIII covers alternative QRNGs based on non- Another important well-established quantum technol- optical quantum phenomena and Section IX is centered ogy is quantum random number generation. Quantum on those QRNGs whose randomness is backed by quan- random number generators, QRNGs, are devices that tum mechanics. use quantum mechanical effects to produce random num- Section X gives a brief tour on the available classi- bers and have applications that range from simulation cal randomness extraction methods and Section XI in- to cryptography. They are usually simpler than other troduces the quantum protocols for randomness expan- quantum devices and are mature enough to be applied. sion and amplification that allow to produce good-quality QRNGs using different quantum phenomena have gone random outputs from weak randomness sources. from the lab to the shelves with at least eight exist- Section XII is an introduction to the statistical tests ing commercial products (ComScire, 2014; Hughes and that are usually employed to assess the quality of the Nordholt, 2016; ID Quantique, 2014; MPD, 2014; Pico- final random bit stream. Quant, 2014; QRB121, 2014; Quintessence Labs, 2014; Finally, in Section XIII, we give an overview on the Qutools, 2014) and online servers that provide quantum current state of quantum random number generation and random numbers on demand (ANU, 2016; Humboldt- the challenges and opportunities for the next generation Universit¨at, 2016; Stevanovi´c et al., 2008; University of of quantum devices in the field of randomness. Geneva, 2004; Walker, 1996), as well as many patents (Beausoleil et al., 2008; Dultz et al., 2002; Dultz and Hidlebrandt, 2002; Kim and Klass, 2001; Klass, 2003, II. RANDOM NUMBERS AND THEIR APPLICATIONS 2005; Lutkenhaus et al., 2007; Ribordy et al., 2009; Sartor and Zimmermann, 2015; Trifonov and Vig, 2007; Vartsky Random numbers are an essential resource in science, et al., 2011). In the last few years there has also been technology and many aspects of everyday life (Hayes, a large number of proposals, experiments, improvements 2001). Randomness is required to different extents in and exciting theoretical results in randomness extraction applications like cryptography, simulation, coordination and randomness certification. in computer networks or lotteries. Some applications The aim of this review is to collect the most important require a small amount of random numbers and still proposals for quantum random number generation and use manual and mechanical methods to generate ran- give an introduction to the new advanced protocols that domness, like tossing a coin, throwing a die, spinning a use quantum physics to process, certify or otherwise deal roulette wheel or a drawing a ball from a lottery machine. with random strings. This paper complements previous Here, we will concern ourselves with the generation of surveys on the topics of physical and quantum random random numbers for computers. number generation (Stipˇcevi´c, 2012; Stipˇcevi´cand Ko¸c, Defining randomness is a deep philosophical problem 2014) with a focus on QRNGs based on quantum optics. and we will not attempt to solve it here. In this Section, Section II gives a brief description of the most impor- we give common operational definitions of randomness tant applications of randomness in science and comput- that fit the different purposes the random numbers must ers. We review the differences between algorithmic meth- fulfil. For instance, in simulation, a method that gen- ods to produce random looking numbers and physical erates numbers simulating the statistics of the desired methods to produce true random numbers and discuss distribution can be considered to be “random enough”, 3 even if it produces a predictable sequence. that the repetition does not appear during the intended operation time. Apart from congruential linear generators, there is an- A. Pseudorandom number generators and true random other large family of PRNGs based on linear shift feed- number generators back registers, LFSRs, and their generalizations. The most notable generator in this class is the Mersenne In computing, it is important to distinguish between Twister (Matsumoto and Nishimura, 1998), which be- algorithmically generated numbers that mimic the statis- longs to the family of twisted generalized linear shift feed- tics of random distributions and random numbers gener- back registers. The Mersenne Twister has a period which ated from unpredictable physical events. is a Mersenne prime of the form 2n 1, for an integer Generating random numbers directly from a computer n. The most widely used pseudorandom− number gen- seems a particularly attractive idea. Methods that pro- erator is the MT19937, the standard implementation of duce random numbers from a deterministic algorithm are the Mersenne Twister with
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