# True Random Number Generators

## Abstract

Random numbers are needed in many areas: cryptography, Monte Carlo computation and simulation, industrial testing and labeling, hazard games, gambling, etc. Our assumption has been that random numbers cannot be computed; because digital computers operate deterministically, they cannot produce random numbers. Instead, random numbers are best obtained using physical (true) random number generators (TRNG), which operate by measuring a well-controlled and specially prepared physical process. Randomness of a TRNG can be precisely, scientifically characterized and measured. Especially valuable are the information-theoretic provable random number generators (RNGs), which, at the state of the art, seem to be possible only by exploiting randomness inherent to certain quantum systems. On the other hand, current industry standards dictate the use of RNGs based on free-running oscillators (FRO) whose randomness is derived from electronic noise present in logic circuits and which cannot be strictly proven as uniformly random, but offer easier technological realization. The FRO approach is currently used in 3rd- and 4th-generation FPGA and ASIC hardware, unsuitable for realization of quantum RNGs. In this chapter we compare weak and strong aspects of the two approaches. Finally, we discuss several examples where use of a true RNG is critical and show how it can significantly improve security of cryptographic systems, and discuss industrial and research challenges that prevent widespread use of TRNGs.

## 1 Introduction

True random numbers and physical nondeterministic random number generators (RNGs) seem to be of an ever-increasing importance. Random numbers are essential in cryptography (mathematical, stochastic, and quantum), Monte Carlo calculations, numerical simulations, statistical research, randomized algorithms, lotteries, etc. Today, true random numbers are most critically required in cryptography and its numerous applications to our everyday life: mobile communications, e-mail access, online payments, cashless payments, ATMs, e-banking, Internet trade, point of sale, prepaid cards, wireless keys, general cybersecurity, distributed power grid security (SCADA), etc.

Without loss of generality in the rest of this chapter, we will assume that generators produce random bits.

In applications where provability is essential, randomness sources (if involved) must also be provably random; otherwise, the whole chain of proofs collapses. In cryptography, where due to Kerckhoffs’ principle all parts of protocols are publicly known except some secret (the key or other information) known only to the sender and the recipient, it is clear that the secret must not be calculable by an eavesdropper, i.e., it must be random. For example, the well-known BB84 quantum key distribution protocol [5] (described in Sect. 3.4) would be completely insecure if only an eavesdropper could calculate (or predict) either Alice’s random numbers or Bob’s random numbers or both. From analysis of the secret key rate presented therein, it is obvious that any predictability of random numbers by the eavesdropper would leak relevant information to him, thus diminishing the effective key rate. It is intriguing [79] that in the case that the eavesdropper could calculate the numbers exactly, the cryptographic potential of the BB84 protocol would be zero. Indeed one of the recent successful attacks on quantum cryptography exploits the possibility to control local quantum RNGs by exploiting a design flaw of two commercial quantum cryptographic systems and one practical scientific system. This example, discussed below, shows that the local RNGs assumed in BB84 are essential for its security and may not be exempt from the security proof.

Lotteries are yet another serious business where random numbers are essential. Due to the large sum of money involved (estimated six billion USD annually only online and only in the USA [36]), some countries have set explicit requirements for RNGs for use in online gambling and lottery machines and have set certificate issuing authorities. For example, the Lotteries and Gaming Authority (LGA) of Malta has prescribed a list of requirements for RNGs, stipulated in the Remote Gaming Regulations Act [45]. An RNG that does not conform to this act may not be legally used for gambling business. These rules have been put forward in order to ensure fair game by providers and to prevent possibility that gamers manipulate the system by foreseeing outcomes.

RNGs have been an occupation of scientists and inventors for a long time. Whole branches of mathematics have been invented out of a need to understand random numbers and ways to obtain them. At the dawn of the modern computing era, John von Neumann was one of the first to note that deterministic Turing computers are not able to produce true random numbers, as he put it in his well-known statement that “Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.”

RNGs are one of the hottest topics of research in recent years. There have been about 83 patents per year in the last decade, 1418 in total since 1970, and countless scientific articles published regarding true RNGs. Still, a sharp discrepancy between the number of publications and very modest number of products (only four quantum RNGs and a handful of Zener noise-based mostly phased-out RNGs) that ever made it to the market [33, 34, 57, 60] clearly indicates immaturity of most of the art. In our view, the main problems are lack of randomness proofs and poor reproducibility of the majority of solutions presented so far. The search for true randomness continues.

## 2 Pseudorandom Number Generators

Historically, there have been two approaches to random number generation: algorithmic (pseudorandom) and by a physical process (nondeterministic).

Pseudorandom number generators (PRNG) are well known in the art and we are not going to address them here in great detail. Surveys and individual examples of PRNGs can be found elsewhere [32, 40, 48, 49, 92]. In a nutshell, a PRNG is nothing more than a mathematical formula, which produces deterministic, periodic sequence of numbers, which is completely determined by the initial state called seed. By definition such generators are not provably random. In practice, PRNGs feature perfect balance between 0s and 1s (zero bias) but also strong long-range correlations, which undermine cryptographic strength and can show up as unexpected errors in Monte Carlo calculations and modeling.

While most modern PRNGs pass all known statistical tests, there are myths about some PRNGs being much better than the others. The truth is that every PRNG shows its weakness in some particular application. Indeed PRNGs are often found to be the cause of erroneous stochastic simulations and calculations [11, 12, 15, 21, 29, 40, 45, 55, 58, 70, 87]. As for cryptographic purposes, all major families of PRNGs have been cryptanalyzed so far [40, 61, 74], and use of PRNG where an RNG should be used will therefore present a big security risk for the protocol in question. We will revisit this point in more detail in Sect. 6.

In any case, due to strict determinism of PRNG algorithms, no PRNG is random by any reasonable definition of randomness. Let us illustrate this by a fictitious anecdote. Alice wanted to impress Bob, by a particular version of Mersenne Twister PRNG [49] for which she claimed that it produces true random numbers, by asking him to test them. Bob agreed but asked a minimum of 1 Giga bytes of random data to be sent to him via e-mail. Alice produced the huge file but her mailing program refused to send such a big file. Cutting a file into small pieces and sending multiple e-mails, etc. was an option but too big a nuisance for both of them. Finally, Bob received from Alice a 1 kilobyte e-mail containing the following short notice: “Dear Bob, Please find attached a program in C++. Compile it, use the following seed: 12345678 and stop the program after producing 1 Giga bytes of data. That is what I wanted to send you." Instead of reproducing the file and running on his computer very time-consuming tests, Bob shortly answered: “Dear Alice, if you think that 1 Giga bytes of truly random data can, under any circumstances, can be compressed without loss to just 1,000 bytes, than I have nothing more to say to you!”

Advantages of PRNGs are their low cost, ease of implementation, and user-friendliness, especially in a CPU-available environment such as a PC computer, but one has to be cautious when it comes to use of PR numbers for simulations, cryptography, and in fact any use.

## 3 True Random Number Generators

*a*

_{k}), defined, for example, in [40]:

*b*

_{1},

*b*

_{2},

*…*,

*b*

_{N}} is an

*N*bits long random string and

*k*is the lag or “order” of the coefficient. Both

*b*and

*a*

_{k}are normalized such that they can take on values in the interval [−1, 1] and that an ideal RNG exhibits

*b*= 0 and

*a*

_{k}= 0. True RNGs are generally constructed such that the correlation among bits is small—which is, namely, the idea of randomness. In some cases the physical system that is measured is being “reset” to an initial condition after production of each bit in order to reduce autocorrelation. Therefore in most cases only a few lowest-order autocorrelation coefficients are significant, ideally only the first one, which is named autocorrelation and denoted by

*a*.

Noise-based RNGs

Free-running oscillator RNGs

Chaos RNGs

Quantum RNGs

Note that our definition of a true RNG is not to be confused with a pseudorandom number generator implemented in CMOS logic or similar hardware; such generator is still a PRNG, since it is just a hardware implementation of a mathematical method. Next, we are going to address each of the above families in some detail.

### 3.1 Noise-Based RNGs

Johnson’s effect [54] creates random voltage on terminals of any resistive material which is held at a temperature higher than absolute zero. Johnson’s noise is due to random thermal motion of the quantized electric charge (i.e., carriers). However, long-range carrier correlations in conductors cause correlations in movements of electric charges, and, therefore, the resulting voltage is not completely random [4].

Zener noise (in semiconductor Zener diodes) is caused by tunneling of carriers through quantum barriers of ideally constant height and width. If current is sufficiently low, individual “jumps” of carriers through barriers will be seen as voltage peaks across the diode, forming a pink noise of perfect randomness. An interesting property of this kind of noise is that at sufficiently high inverse voltage, the diode exhibits high internal avalanche gain. Such a gain mechanism leads to large amplitude of the noise and is highly insensitive to electromagnetic radiation from the environment. However, the Zener effect is never found well isolated in physical devices from other effects nor is the quantum barrier constant. Most of the aforementioned processes in resistors and Zener diodes have some memory effect. This means that an instantaneous voltage across the device depends on voltages in the (near) past and this in turn leads to a correlation among random numbers extracted therefrom.

Other popular sources of noise include: inverse base-emitter breakdown in bipolar transistors, laser phase noise [30], chaos noise [44], etc. The biggest problem with all kinds of noises is that randomness of noise sources cannot be well characterized, measured, or even controlled during fabrication of the device. Furthermore, some noise mechanisms (notably Johnson’s noise) produce rather tiny voltages that need to be strongly amplified before conversion to digital form. The strong amplification introduces further deviations from randomness due to the limited amplifier bandwidth and gain nonlinearity. Also, fast electrical switching of binary logic used in the RNG circuitry produces strong electromagnetic interference so that multiple nearby RNGs (especially if on chip) tend to mutually synchronize causing the dramatic drop of overall entropy. On top of that, highly sensitive amplifiers allow easy manipulation of noise-based RNGs by external electromagnetic fields which can be exploited for cryptographic attacks.

- 1.
Provability of randomness of the exploited noise source

- 2.
Effect of the sampling/digitizing procedure

- 3.
Eventual use of deterministic post-processing

In the Bagini–Bucci generator [2] shown in Fig. 3, the analog voltage from the free-running noise source is periodically sampled at frequency *f*_{Ck1} and compared to a threshold value at the comparator. Whenever the comparator produced logical “1” the T-type flip-flop (TFF) changes its state. If the sampled process is random and stationary, because of time symmetry of this process, the output of the TFF spends half of the time in the low state and the other half in the high state. There are a couple of problems with that design. First, the holding capacitor acts as a memory that remembers previous analog voltage. Due to finite impedances in the circuit when charged with the next voltage level, the voltage will be to some extent dependent on the previous one, thus creating the autocorrelation. The second problem is that if the TFF is interrogated at too high a rate, it will tend to give the same answer several times in a row, thus producing positively autocorrelated output, even when the basic random process is truly random! The only way to circumvent this problem is to use a bit sampling frequency *f*_{Ck2} much lower than the noise sampling frequency *f*_{Ck1}, for example, \(f_{\mathit{Ck}2} = f_{\mathit{Ck}1}/N\), thus arriving to asymptotically random sequence of bits in the limit of \(N \rightarrow \infty \).

*N*) goes to infinity. In practice however

*N*only needs to be sufficiently large to keep correlations at the desired level.

The bad side of this “sampling” principle, illustrated in Figs. 3 and 4, is that it required *N* random events to produce one random bit (low efficiency). The good side is that by letting *N* be large enough, one can obtain any desired level of randomness quality, at least theoretically. Practically however, small imperfections in logical circuits, such as uneven high-to-low and low-to-high transitions, will ultimately limit the achievable randomness. Regarding provability of randomness of this principle, technical imperfections of the individual components, unclear theory of operation of the “noise source,” and overall complexity of the circuit make it impossible to arrive to a credible proof of randomness.

All the above examples utilize electronic noise: a resource which is becoming less and less available because manufacturers of electronics components and chips make every possible effort to make it ever smaller. Therefore researchers have turned to sources capable of producing fluctuating voltage similar to electronics noise but whose origin is more fundamental and therefore has less sensitivity to technological advances. For example, very fast noise can be obtained by lasers. Lasers exhibit very fast fluctuations which can be detected by fast PIN or avalanche photodiodes (APDs), thus producing wide-band electrical noise.

This is an example of a white noise-based generator where a Gaussian-shaped distribution of analog electrical amplitudes has been obtained by optical rather than electrical means (e.g., such as discussed in the Bagini–Bucci generator [2] and some others described above). The noise source, shown in Fig. 7, is realized by use of a single mode VCSEL laser where the signal and its delayed copy have been brought to interference on an APD detector via a Michelson–Morley type interferometer, a system also known as “homodyne” detection. The electrical noise produced at a very high gain-bandwidth photodiode is the result of phase jitter of the laser. The noise voltage produced by the APD is then digitized by a fast (40 MHz) analog to digital converter (ADC) with 8-bit resolution and the numbers so obtained are further processed to obtain random bits at 20 Mbit/s. Authors show that if the delay is much longer than the laser coherence time of 1.6 ns, then the phase jitter is dominated by quantum effects which are separate from any construction detail and depend only on the laws of physics. In that regime, adding sufficient jitter leads to near-perfect Gaussian distribution via the central limit theorem, similar to the principle utilized in [77]. The authors further measure the autocorrelation function of the analog noise and show that after about 10 ns, all correlations die off. To be on the safe side, sampling of noise is made every 25 ns, and after further simple post-processing, one obtained 20 Mbit/s of random data that passed all relevant statistical tests (mentioned in Chap. 4). The similar phase self-interference principle is exploited in [59]. The advantage of the quantum phase noise over the electronic noise is that its amplitude is determined by fundamental laws and is therefore (in the ideal case) independent of technological details of the laser. In our finding though, the authors here were not considering two important points. First, the time delay introduces a “rolling” memory effect that necessarily leads to autocorrelation of the noise voltage generated by the APD, and, therefore, the bits obtained therefrom would not be random even if the phase jitter itself is random. Second, the bit generating algorithm, which most critically includes digitization of an analog quantum-random effect, is only approximate and good care has to be exercised in order to keep randomness at the desired level at all times. Even so, this is one of the very rare noise-based generators which are characterized by clean sequence of in-principle provable and well-understood physical and algorithmic processes.

More examples of noise-based true random number generators can be found in the scientific literature and in the free-access worldwide patent database Espacenet [20].

For all noise-based generators, some kind of post-processing is required. In some cases a simple *ad hoc* post-processing such as XORing several subsequent bits or von Neumann [94] de-biasing may be good enough. But if the raw bits exhibit strong correlations, simple procedures may not be sufficient to eliminate correlations among bits which can even be enhanced by simple de-biasing procedures or changed from short-range ones to long-range ones. A better approach is found in complex, often offline post-processing which however brings in its own problems (see Sect. 3.4).

There is a strong tendency among researchers to name noise-based RNGs “quantum RNG” because noise is ultimately caused by small particles governed by laws of quantum mechanics. But noise is also a collective effect, a summation of many individual motions, and therefore its quantum property is “blurred” by a collective behavior which is somewhere between quantum and classical worlds. Furthermore, motion of particles which generate noise (e.g., electrons in a wire) is usually intercorrelated by action of forces among them to such extent that the noise may not be completely random [4]. Note that an autocorrelation of the order of a percent may not be important when motion of electrons is considered but if so generated random numbers with the serial autocorrelation of the same order (0.01) are used in numerical simulations, results may be completely wrong. Finally noise cannot be “restarted” in order to interrupt correlations between successive measurement/bit production.

### 3.2 Chaos RNG

- 1.
Conceptual mixing of chaos and randomness

- 2.
(Mis)Belief that hard-to-describe systems necessarily behave in random fashion

- 3.
Robustness of certain chaotic systems to produce macroscopic levels of “noise” easily utilizable to generate random numbers essentially via noise RNG methods (as described in Sect. 3.1)

At present state-of-the-art most convenient chaotic systems for fast generation of random numbers are optical, electrical, or opto-electrical, although mechanical constructions have also been demonstrated, for example, in [52]. In this section we present several typical designs.

Lasers can be brought to chaotic fluctuation of output power by many different mechanisms. Well known are chaotic constructions involving distributed feedback lasers [44]. One very simple but extremely fast self-feedback chaotic laser system [38] is shown in Fig. 8. Again, the light of chaotically fluctuating amplitude is detected by a fast photodiode (PD) whose amplitude is sampled by a fast 8-bit ADC and further processed by performing a high-order differentiation, to yield a world record bit production speed of 300 Gigabit/s.

Lasers offer means for realizing very fast chaotic systems and are frequently used for random number generation. Due to the possibility to build tiny lasers, resonators, and various passive and active optical elements on a chip, such generators can be completely integrated and can feature a low power consumption.

A RNG shown in Fig. 9 [44] consists of an ultra-wide-band (UWB) chaotic laser (a), amplitude sampler (b), and comparator (c). Its principle of operation is a copy-paste of the Bagini–Bucci noise generator described earlier (Fig. 3) with the difference that instead of electrical noise here the light intensity of a chaotic laser is used as a source of randomness. The interesting distinguishing characteristic of this RNG is that it is “all optical,” meaning that all signals and signal processing are done at the optical level, even the output numbers are in fact digital levels of light intensity: low light intensity signifies “0,” while high intensity signifies “1.” This is interesting for use in all-optical processing chips, and furthermore, if so needed, the output can be easily converted into an electrical signal by use of a fast photodiode and a suitable amplifier.

In a body of research related to chaotic RNGs, some authors claim to use system(s) in chaos without actually providing any direct evidence that the system in use for random number generation is indeed in chaos [85], some are able to demonstrate chaotic behavior, for example, by studying ballistic maps or Lyapunov exponents [62], and some even go so far as to model the chaotic behavior of the system and confirm it experimentally [44, 97, 98]. But whichever the case, chaotic RNGs have a theoretic base common to those PRNGs which operate by simulating a deterministic chaotic system, for example, and therefore in the long run became short-breathed in producing new entropy, inevitably ending in producing not more than a small fraction of 1 bit of entropy per each new generated random bit.

A general objection to the very idea of the generation of random numbers by chaos is that chaotic behavior is defined as a specific type of solution of the differential equation which, supplemented by initial conditions, describes the system. Because any such equation and data contain only a limited (small) amount of information, once that much information is extracted from the system by measurements there is no new information that can be extracted from it, and consequently all further measurements contain (asymptotically) zero new information. In particular it means that a chaotic system, in theory, can only produce a limited set of random bits and that all the rest must be perfectly or near perfectly correlated to that set. Having said that, we understand that a realistic chaotic system never behaves exactly as it would by obeying the “equation of motion” that models it because of random quantum or statistical effects which randomize the system’s phase-space trajectory all the time. However, these additional effects are not the basis for a chaotic RNG (and therefore not accounted for in its definition) and also are usually too tiny or ineffective to make any significant difference in a system whose behavior is mostly determined by a macroscopically observable chaos. On the other hand, fundamental quantum randomness alone can be harnessed for the production of provable random numbers, as we will discuss in Sect. 3.4.

### 3.3 Free-Running Oscillator RNGs

An inverting gate is in practice a very high-gain inverting amplifier. Connecting its output to the input creates the Zeno paradox: if the output is in logical HIGH state, then the input will be as well and the NOT action will drive the output to go LOW. Once the output goes LOW, the NOT action will drive it to HIGH and so forth. Theoretical Boolean logic analysis will yield that the output is undetermined, but in practice due to the finite propagation delay of the NOT element, the circuit will oscillate. The peculiarity of this oscillation is that it appears in a circuit with negative feedback (180^{∘} phase shift), while in electronics theory negative feedback leads to “stabilization” rather than to oscillation. The reason for that is that by analyzing logical states we assumed infinite gain. However, since in practice gain is never infinite, it may happen that the circuit locks (stabilizes) into some voltage state between zero and one without any or with very small amplitude oscillations which are not capable of driving further logic circuits. To help oscillations, one may intentionally add some reactance in the feedback loop so as to produce phase shift different from ± 180^{∘}. The same function may be provided with stray reactances. In that case, the Barkhausen criterion may be satisfied for some high-frequency pole and oscillations will appear. Due to the complex mechanism of free oscillations, their frequency is typically quite sensitive to variation of power supply voltage and temperature but these changes are slow compared to the oscillation frequency. On the other hand, the electronic noise present at the input adds to the signal fed back from the output and after being strongly amplified causes very fast, random jitter of frequency and phase of oscillations. In that sense, FRO RNG can be regarded as a special case of a noise-based generator. Since the noise of each such circuit is individual, it is reasonable to assume that the multiple oscillators even when on the same chip have different frequencies and that their mutual phases walk off randomly in time. But when multiple such oscillators are close to each other (e.g., on a single chip), they tend to synchronize through electromagnetic interaction facilitated by the high gain of FRO amplifiers. In effect, the immense gain of NOT gates required to amplify tiny electronic noise to a noticeable level also helps to pick up any other nearby interference. This effect known as “phased interlock” [54] may adversely affect the performance of the design and is a major problem inherent with FROs. Interlocked rings have waveforms that share (nearly) the same phase and this will lead to (near) pseudorandom operation. The same effect of high gain makes FROs vulnerable to attacks with external electromagnetic radiation.

In spite of all these problems, current security standards [65] practically dictate the use of RNGs based on free FROs. The NIST standard FIPS140-2 [19] says: “There are no FIPS Approved nondeterministic random number generators.” Consequently, the FRO approach currently is used in 3rd- and 4th-generation FPGA, CPLD, and ASIC hardware for various cryptographic purposes. One real-life example that illustrates well the combinatorial cuisine typically needed to obtain a decent RNG is the entropy source for PadLock “quantum” RNG implemented in VIA C3 processors [88, 89, 90, 91]. It consists of four FROs, 3 fast (450–810 MHz) and 1 slow (20–68 MHz). Wide tolerance on the frequencies already shows problems that we mentioned before: it is very hard to control the parameters of FROs during fabrication. In this topology fast FRO (A) is sampled by a slow FRO (D) as discovered in the patent application [78]. At least one of the two FROs must be of good randomness, and since it is easier to achieve with the slower one, VIA went for that option. The slow generator is made of FROs B, C, and D. First, B and C are slowed down by 1/8 dividers and their XORed outputs are used to disturb slow FRO D (which is the only one featuring digital input). Resulting bits appear at the output Q of the D-type flip-flop in synchronization with pulses from the FRO D. Optionally, the output is filtered through a von Neumann corrector [94] which cuts the bit production rate roughly by a factor of 4 (see description in Chap. 4). Looking at this schematic, it is clear that it is impossible to arrive to a proof of its randomness. According to VIA [88], the analog bias voltage injected into this otherwise digital circuitry “may (or may not!) improve the statistical characteristics of the random bits.” The bottom line is that the random numbers are still of low quality and in order to pass tests must be corrected (Sect. 3) by a full-blown secure hash algorithm SHA-1 which is hardwired into the logic circuits on the same chip [88].

Because the digital logic chip infrastructure is unsuitable for realization of a quantum RNG (Sect. 3.4), an FRO approach seems to be a reasonable viable alternative. However, a caveat with FROs is that the semiconductor industry is making an enormous effort to make the electronics noise as small as possible and it generally goes down with newer versions of a chip. Consequently the effect of jitter can be very small and cause the FRO-based RNG to operate in nearly PRNG regime. Therefore implementation details of an FRO-based RNG most often have to be tailored for each specific type or generation, and technology of a programmable/reconfigurable or ASIC chip and uniformity of operation cannot be guaranteed from batch to batch.

Still, even with this interesting and innovative principle, the problem is cross-platform non-portability of the design and the requirement of sufficiently large noise for the scheme to work in a reasonably random (far from pseudorandom) regime. Furthermore, the authors warn that the design must be done most carefully in order to minimize interlocking with the system clock and other logic circuits in the chip, including nearby FROs. Therefore they experimented with spatial placement of FROs in the chip. They also conclude that randomness of either of the two generator families by itself is not perfect and could be “enhanced” by XORing two independent generators, most favorably one GARO and one FIRO.

More examples on FRO pre- and post-processing gymnastics, including XORing multiple generators, combinations with LFSRs, etc., can be found in [81]. The complexity of post-processing procedures required to pass the statistical tests with FRO-based RNGs is often such that any randomness proof is impossible, but even more interestingly the authors almost never seem to be aware that a proof is needed. A rare exemption in that respect is the work of Sunar et al. [82] where a theoretical model of an FRO-based RNG has been presented, analyzed, and proven but later criticized in [101] as nonrealistic. We however found the whole proof unsatisfactory because it is based on McNeill’s model of FRO which simply postulates that free oscillations occur as a nonstationary random process without actually linking the postulate to reality, for example, by means of laws of physics. An excellent further reading and summary of problems and cuisine used to minimize them is found in [101]. Further reading on FRO-based RNGs is given in [81].

In conclusion, FRO-based RNGs are low-cost, low-entropy solutions whose only good side is the fact that they can be easily implemented in conventional programmable or reconfigurable logical chips which are used in various cybersecurity solutions, but they do not offer either very good or provable randomness.

### 3.4 Quantum RNGs

What is a quantum random number generator? Since we live in a world governed by the laws of quantum physics, any true random number generator (e.g., a rolling dice, or a flipping coin) may be named “quantum.” However, we want to reserve this name for only those generators which utilize a single intrinsically random quantum effect (realized as close as possible to its theoretical idealization) measured over and over again in order to produce random bits in such a way that between any two sets of measurements used to deduce random bits, the system is reset to the same initial conditions. It may seem strange that such a physical setup (generator) is even possible, namely, that starting from exactly the same initial conditions and measured in exactly the same way, it gives different results, but quantum physics allows it. In this section we describe and explain multiple examples described in scientific articles and patents.

*qubit*. A single quantum of light (photon) can be used as a carrier of one qubit, but there are many other examples and they are not limited only to elementary particles. Qubit can be thought of as a linear combination of two bit values: 0 and 1. When a certain type of measurement is performed on a qubit, it will “project” to either pure 0 or pure 1 state in the basis in which the measurement has been carried out. Very often photons are used in QRNGs because they are easy to create, manipulate, and detect. To illustrate this let us consider circularly polarized photon entering a polarizing beam splitter (PBS) (Fig. 14). The PBS decomposes polarization of incident light and sends the linear horizontal component to one output port and the linear vertical component to the other port.

Thus, a circularly polarized photon has equal content of both linear polarizations, but since it cannot be split in half, it has exactly 50 % chance to exit either port. If now we label one of the ports as “0” and the other as “1,” we immediately get a theoretically perfect RNG whose randomness is guaranteed by the laws of quantum physics. Note that the system being “measured” is always the same yet it always gives a new random result. This is completely different from chaotic and noisy generators where in order to get a different result systems must change.

Quantum RNGs based on this (or other principles) can be made pretty good, and the imperfections of any type (multiphoton emission, non-perfect circular polarization, beam splitter port axis misalignment, detector dead time, afterpulsing and memory effects, etc.) can be measured independently of the bit generation process so their effect on random numbers can be estimated with precision and dealt with in post-processing (see Sect. 3.5). This method is a basis for a commercial generator [33].

The beam splitter RNG is an example of “spatial principle” in which the value of the random bit, 0 or 1, is determined by the place at which photon ends up. A complementary “temporal principle” uses time information of random photon emission, for example, in direct atomic (or quantum dot) relaxation, from well-saturated lasers, etc.

A simple time interval method shown in Fig. 16, which is particularly immune to hardware imperfections, has been proposed in [80]. It uses time rather than space information contained in a random event generator (REG). In [80] photon emission and detection processes are used for the first time instead of much slower (and more dangerous!) radioactive decay [22, 26]. The bit production principle is as follows. Time intervals *t*_{1} and *t*_{2} spanned by three subsequent photon detections are compared: if \(t_{1} > t_{2}\) then produce “0,” if \(t_{2} > t_{1}\) then produce “1,” and if *t*_{1} = *t*_{2} then produce nothing (skip).

The problem with this method is how the time intervals are measured. The crucial improvement made in [80] is the notion that clock measuring time intervals (*t*_{i}) must be started in synchronization with beginning of each interval; otherwise, the method would produce correlated bits even if fed by perfectly random events. This was not understood in previous works and patents [22] which consequently must have yielded correlated output, but this was not detected at the time because clock frequency ( ≈ 10 MHz) was much higher than the source mean frequency ( ≈ 10 kHz) in which case correlations are small. It can be shown that this method not only performs well at low ratio between clock and RPG frequencies but that it also cancels out almost all imperfections: intensity change of the source, efficiency change, dead time, and afterpulsing of the photon detector. It is also highly immune to actual distribution of random interval times, as long as events are independent of each other. Furthermore, random bit production is self-clocked so if either source or detector dies, there will be no bits at the output. This generator was the first one found to be passing all known tests including “usual” *t*-statistical tests [47, 67, 68, 96] and some undisclosed algorithmic randomness tests [31].

^{∘}), detectors D1 and D2 should produce random, mutually uncorrelated pulses of equal frequency (however adjustment of the polarizer angle is an insurmountable task, as explained for RNG in Fig. 3). While pulses from D1 reset (input R) the RS-type flip-flop setting the output to LOW state, the D2 set (input S) the flip-flop to HIGH state. The output of the said flip-flop is sampled at periodic times in order to generate random bits (Fig. 18).

- 1.
Bias is unstable (sensitive to temperature variations) and only mechanically adjustable.

- 2.
Correlations due to the finite sampling period as discussed in noise generators, and all that even if a perfectly random source of photons is assumed.

^{−8}which suppresses any eventual photon correlations far beyond the detectable level. The bit extraction method is implemented in an FPGA reconfigurable chip and is as follows. The number of detected photons is counted in intervals of a constant time yielding a Poissonian statistics. An even number of events within an interval is interpreted as “1” and odd as “0.” The authors note that due to the nonsymmetric shape of the Poissonian distribution, the probability of ones is not equal to the probability of zeros. However, due to the two imperfections in the photon detector (nonzero dead time and dependence of dead time with the detection frequency), the resulting distribution is not Poissonian but more bell shaped, thus favorably leading to a bias that quickly tends to zero as the counting interval length rises. The authors show and compare experimental and theoretical results for modeled bias; however they do not model or prove anything about correlations. Instead, correlations are simply evaluated from generated bits using a linear autocorrelation coefficient. Theoretically, bias tends to zero as detection frequency goes to infinity. Empirically, the preferred operating condition is close to as high as possible a detection frequency but a bit smaller due to rising problems in the photon detector. But in the same limit, it is to be expected that fluctuating bias produces an increasing level of complex short-range correlations among bits—which however has not been mathematically modeled and/or brought into connection with the imperfections of the setup. The problem with this approach is that it fails to describe a theoretical model of an RNG that gives perfect random numbers based on a (nearly) ideally random quantum effect (e.g., low-intensity emission from LED) and assuming ideal apparatus. Consequently it fails to clearly prove randomness and to model deviation from perfect randomness introduced by implementation-related imperfections. Nevertheless, this generator has a practical value because it apparently passes all relevant statistical tests. It is however to be understood that an acceptable randomness proof cannot be obtained by passing any number of randomness tests (as will be discussed in Sect. 3.5).

Yet another commercial quantum RNG which utilizes photon arrival time information has been presented by Picoquant [57, 95]. Here the complete chain of reasoning required for a convincing randomness proof has been at least attempted and, according to the authors, successfully established. As in the previous example, a random event source of the type shown in Fig. 17 is made utilizing essentially the same technique as in [23] (LED + photomultiplier tube). The specific difference of this construction with respect to previously described ones which use high-speed photon detection and produce ≤ 1 bit per detection [23, 35, 75, 80] is that the random detections are made at a relatively low mean frequency of 12.5 MHz, thus operating in a regime far from dead time and pileup effects producing a highly precise exponential distribution of time intervals (Fig. 20 left). The time intervals *t*_{1}, *t*_{2}, *t*_{3}, *…* are measured by a nanosecond precision, and quasi-exponentially distributed integer numbers so obtained are used to generate on average ≈ 14 random bits per each detected photon yielding ≈ 160 million raw random bits per second. The imperfections both in the extraction method and in hardware (timers, detectors, light source) are modeled resulting in a convincing lower bound on the average per-bit entropy of the raw bits. The average entropy is then improved by compression of the raw stream by resilient functions (see Sect. 3.5) to the level theoretically indistinguishable from true randomness even for bit strings of unrealistic length. The weakest link, in our opinion, is this last post-processing step because it is not clearly proven that resilient functions are effective against the specific type of imperfections present in raw bits, that is, that bounds on post-processed bits hold. However, raw bits are already very close to randomness, and further post-processing by resilient functions clearly improves the pass rate of statistical tests indicating that the resulting bits are very close to true randomness. Indeed, post-processed bits at an average speed of 150 Mbit/s pass all relevant statistical tests as well as some undisclosed statistical and algorithmic tests performed by the University of Twente research group [31]. Still, caution is maybe in order when resilient functions are used because some researchers [81] point out that resilient functions appear to be limited in their ability to eliminate the effects of active adversaries on the output bits. A similar principle but by digitization of an analog quantum amplitude is described in [1].

*t*

_{0}is the pulse length) and practical (pulse shape is achieved by an analog, only partially precise circuit, thus not allowing us to properly conduct proof of randomness. The authors also note that this circuit produces strong electrical disturbances in nearby circuitry which, in our view, makes it unsuitable for miniaturization to a chip level. And finally, the theoretical basis for exponential time-arrival distribution is drawn out of a steady field assumption, whereas here the strength of the light electromagnetic field is wildly varying so even theoretical grounds for this generator are not clean (Figs. 21 and 22).

This generator belongs to a broad niche of RNG constructs whose general philosophy is to produce partially random data and then filter it through a pseudorandom hash function (such as SHA-256 used in this example) in hope of improving the randomness (see Sect. 3.5). We believe this is a very problematic approach and here is our reasoning. Proof of randomness in this case relies on estimating the entropy of the source of raw bits and on the process of randomness amplification by hashing. The hashing procedure is generally not foolproof [3] and does not allow just blind application of the hash function to a badly constructed generator. Let us imagine, for example, that raw RNG source produces some sequences more often than the others (which indeed is the case if it is nonrandom). Then the hash of these sequences (the hash function being deterministic) would also produce some sequences more often than the others meaning that even the “corrected” bits would not be random. A nice confirmation of this comes from this very example: even after hashing, the produced bits are not completely random and fail some statistical tests.

We saw that photon emission and photon detection techniques are often used in quantum random number generators. The photon detection rate of current single photon detectors is a limiting factor in achievable random bit product rate especially for semiconductor APDs. APDs are small and convenient for single photon detection on a chip scale; however they suffer from imperfections that are especially bad for random number generation and consequently rarely used for that purpose. The biggest problems are relatively long dead time (induced by requirement to quench avalanche between subsequent detections) and high afterpulsing rate (usually in the range 1–10 %). In order to advance on this, Toshiba has developed a special, so-called “self-differencing” approach [102] to readout of semiconductor avalanche photodiodes which promises significantly higher detection rates (lower dead time) than the usual active quenching method while suppressing afterpulses by effectively squaring the afterpulsing probability. This new technique has been used for random number generation by the same group of authors [18]. Namely, even though this method does not offer spectacular improvements in general because it inherently prefers operating the APD at low detection efficiency, it is very well suited for use in random number generation because of its high gating speed and complete irrelevance of the photon detection efficiency for that application.

A distributed feedback (DFB) laser in continuous wave (cw) mode is used as the light source. The power of the DFB laser (1,550 nm) is adjusted (by means of variable attenuator) such that the strength of the electromagnetic field reaching the surface of the APD causes roughly 0.004 avalanche detections per gate. When detection occurs, a new random bit is generated and its value is “0” if it occurred on an even gate or “1” if on an odd gate. Taking into account the detection efficiency of 0.004, this method yields 4.01 MHz of random bits. This bit generating process is intrinsically biasless (probabilities of zeros and ones are equal) but (what is not noted by the authors of this article) there is an intrinsic negative autocorrelation which rises with detection efficiency. Namely, in the limiting case of efficiency 1 (one detection per gate), there would always be a “1” after “0” and vice versa, thus producing a completely deterministic sequence \(01010101\cdots \) which has autocorrelation equal to − 1. Even though the authors claim that this method of generating random numbers could, in principle, be extended to much higher rates by using a higher laser power and a detection rate of up to 100 MHz (efficiency of 0.100), it is clear that at that point the autocorrelation would amount to approximately − 0. 1 and the bits would not pass any randomness test.

There are numerous other variations of space and time principles that can be found in the scientific and patent literature.

In conclusion, the most distinctive characteristic of a quantum approach to random number generation is that, at least in principle, it makes it possible to establish a simple relation between the randomness of numbers, the exploited physical process, and the implementation imperfections, thus offering a possibility for scientific proof of randomness. Careful practical realizations come sufficiently close to theoretical idealization and allow for an independent assessment of implementation imperfections, the effects of which can, if required, be dealt with by information-theoretic post-processing (see Sect. 3.5). On top of that, quantum random detection processes exist that are inherently highly insensitive to electromagnetic radiation (e.g., avalanche amplification in semiconductor photodiodes), thus offering immunity to side-channel manipulation by external fields. Because of all said, quantum RNGs are the best choice for true random number generation for cryptography and other applications which critically require true random numbers. The most significant drawback of the present solutions is that they make use of bulky physical objects and therefore cannot be miniaturized to the chip level using present technologies. Furthermore, due to the frequent use of photon detectors, QRNGs are typically very expensive and much slower than software PRNGs. Fortunately, the nascent science and technology of optical chips offers a promising avenue for fast, miniature, and affordable quantum RNGs, and significant advances can be expected in this exciting field in the near future.

### 3.5 Post-processing

Although there is a “gray zone” of what part of random number production belongs to bit extraction method and which to post-processing, the bit extraction is usually a first and very simple step which converts physical measurement of an analog or digital signal into the “raw” digital random binary number (such as digitizing analog noise via a threshold comparator shown in Fig. 2), whereas post-processing is a more complex process designed to reduce or completely remove imperfections that are necessarily present after the first step. While bit extraction is always made in hardware, post-processing algorithms are usually so complicated that they can only be executed by a computer (or a microcontroller or FPGA) although the most valuable post-processing techniques are those simple enough to be suitable for direct implementation in hardware.

Generally, post-processing takes a lot of resources and blurs. In our opinion, a good true RNG should be post-processing-free or use minimal ad hoc post-processing. Most popular post-processing techniques can be categorized in four families as described in the following.

#### 3.5.1 Ad Hoc Simple Correctors

Ad hoc corrector examples are XORing two or more neighboring bits from the same RNG [72], omitting bits (decimator), feeding an LFSR with imperfect random numbers [84], Latin square bit reshuffling [47], von Neumann [94] and Peres [56] de-biasing, XORing two or more RNGs that work in parallel [14, 44], etc.

It is important to note that ad hoc, naïve processing can lead to unexpected problems. For example, it is usually considered a good idea to apply von Neumann de-biasing scheme [94] in order to completely remove any bias from the sequence of bits. The scheme works as follows. The biased bit sequence is cut into a sequence of non-overlapping pairs of bits. Pairs 11 and 00 are discarded, 01 is converted to “0,” and 10 is converted into “1.” While it is tempting to think that the probability of occurrence of 10 is equal to the probability of 01 (and therefore the resulting sequence has no bias), it is often overlooked that this is true only if the bits are completely independent (no correlations). The following extreme example illustrates how miserably von Neumann’s procedure can fail. Let us consider the sequence: \(101010101010\cdots \). It obviously readily has no bias. After application of von Neumann de-biasing, the sequence reads: \(111111\cdots \) which is a maximally biased and maximally correlated sequence. The reason for this unexpected result is that the original sequence is maximally anticorrelated and therefore quite far from the assumption of complete statistical independence. Generally, if the raw string is correlated, naïve de-biasing procedure may even increase the bias or create other unexpected statistical deficiencies. On the other hand, simple and easy-to-understand ad hoc correctors have the advantage over more complex procedures, that they are easier to include in a randomness proof.

#### 3.5.2 Cryptographic One-Way (Hash) Functions

*N*bits, where

*N*usually is in the range from 128 to 512. Hash functions are characterized by two requirements:

Given an output value, there is no faster way to find a corresponding input than by random guessing (i.e., a hash function is “one way”).

The probability of two different inputs yielding the same output is less or equal to 1∕2

*N*.

One of the most popular post-processing techniques is “whitening” of output of a TRNG by means of a cryptographic hash function, such as MD5, SHA-1, SHA-2, SHA-256, and SHA-512. Many authors believe that a bad RNG that does not pass statistical tests, and runs through a “cryptographic” hash compression procedure would magically become very good, without actually demonstrating any theoretical understanding on why this should be the case. Indeed a very interesting example given in [99] demonstrates that hashing a bad generator can fail to enhance randomness enough to pass statistical tests.

From a performance perspective, implementing a hash function in hardware chips is pretty resource demanding, so in most cases hashing is done on a computer; two exemptions to this rule are the aforementioned Intel’s RNG in Fig. 5 and VIA C3 in Fig. 12, which make use of SHA-1 hardwired right next to the RNG on the same chip. Regarding the provability of randomness of the hashed output, even though interesting results on privacy (and randomness) amplification have been theoretically exercised for Wegman’s Universal Hash Function(s) [7], in the case of real-life, black-box hash functions (which probably contain unknown statistical or security weaknesses), it is hard to perform a convincing proof of randomness. For example, a hash function may contain statistical problems like some output strings being more probable than others which would then be inherited by the output bits even if the function is fed by perfect random numbers. On top of that, hash functions are usually used at the end of the post-processing leaving a bitter aftertaste in the mouth that physically generated random numbers actually exit out of a deterministic, complex, black-box piece of software which has not been specifically designed for the purpose.

#### 3.5.3 Extractor Functions

A more scrutinized approach to randomness healing is offered by the young theory of extractors [71]. A randomness extractor is an algorithm that converts a long weakly random sequence into a shorter sequence with almost perfect randomness. For some randomness sources, provable extractors exist but no single randomness extractor currently exists that has been proven to work when applied blindly to any type of a high-entropy source. The problem with extractor algorithms is that they require a memory buffer and a lot of CPU which complicates the hardware and slows down the overall output bit rate.

Extractor functions for post-processing of true random number generators were proposed by Barak et al. [3]. The initial purpose was to achieve designs robust against changes in the physical generators due to, for example, aging, temperature changes, or attacks. Extractor functions are stateless functions with quantifiable properties originally developed as a tool for complexity theory. The aforementioned group of authors has developed a mathematical model to capture an adversary’s influence on the randomness source and give an explicit construction based on universal hash functions which is proven for its output properties even if nonlocal correlations exist in the input source.

More on the theory and practice of extractors can be found in [72].

#### 3.5.4 Resilient Functions

Yet another approach to enhancing randomness by filtering through some deterministic process is the use of resilient functions that were introduced by Sunar et al. [82] as the post-processing step for an FRO RNG design. The idea is, according to the authors, to “filter out any deterministic bits” from the raw string in the environment where some bits may be under the control of an attacker and that bits are then considered “deterministic.” The authors of [82] study the degree of resilience of the procedure against active adversaries (therefrom comes the name of these functions). In short, an (*n*, *m*, *k*)-resilient function is a function \(f: F^{n} \rightarrow F_{m}\) such that every possible output *m*-tuple is equally likely to occur when the values of *k* input bits are fixed and the remaining *n* − *k* bits are each chosen at random. The elements of *F* are binary values 0 and 1. The important distinguishing characteristic of resilient functions is that they have been constructed specifically to nullify the attacks on a (certain percentage of) random bits—a point of high importance in cryptographic applications of random numbers (see Sects. 4–6).

More on the theory and practice resilient functions can be found in [10, 42, 43, 69, 82].

## 4 Randomness Evaluation (Testing)

The most important notion about statistical testing is the following: if a generator passes all known statistical tests, this does not prove that it is random—it only means that it passes all currently known randomness tests. Tomorrow it can fail some new test or it already fails in the way known only to its constructors.

Most randomness tests check one or more statistical properties of long sequences of random numbers, for example, bias, serial autocorrelation, etc. Some compilations of tests are more oriented toward problems in PRNGs (e.g., DIEHARD [47]), some more to true RNGs (e.g., ENT [96]), while some are of general nature (e.g., Universal Test [50], NIST STS [68]). The unfortunate fact is that there is an infinite number of statistical properties which truly random numbers must satisfy. Tests themselves are not perfect: some contain errors discovered later [44, 65] or constants of questionable precision obtained by simulation using “trusted” RNGs such as combination of white noise and “black noise” [47].

Running a comprehensive set of tests takes many CPU hours: to test 1E9 bits with NIST STS, it takes about 6 h on the fastest single core CPU, while to produce that many bits with a commercial QRNG, it takes between 7 and 250 s.

Randomness tests are very time consuming—it takes much shorter time to generate numbers than to test them. Nevertheless, randomness testing is important for constructors of RNGs. Therefore in some cases where one can reasonably expect only certain type of imperfections (especially for quantum RNGs), one will tend to use only special tests sensitive to these particular imperfections in order to arrive at more efficient testing.

## 5 Random Numbers in Quantum Cryptography

Quantum cryptography is a protocol of public agreement of a symmetric cryptographic key, meaning if two parties A and B possess a small common secret key, then using this protocol they will be able to establish a common secret key of any length. This cryptographic function is also known as “secret key growing.” The ultimate goal of establishing a long secret key is to use it as a one-time pad and thus obtain transfer of data in absolute secrecy. There are several mathematically identical QC protocols. The first one, named after its creators as BB84, appeared in 1984 and was experimentally realized in 1991 [5].

In the BB84 scenario, Alice and Bob are connected via two different channels: the quantum channel (usually well-shielded optic fiber) capable of conducting single photons of light and an unsecured “classical” channel such as a telephone line, radio link, or the Internet.

Here is the simplified schematic of how the protocol works: Alice can prepare photons in different polarization states. In order to establish a secret key, Alice sends to Bob a sequence of random numbers encoded in photon polarizations as follows: “1” is equiprobably encoded either as linear vertical (LV) or left circular (LC) polarization, while “0” is equiprobably encoded either as linear horizontal (LH) or right circular (RC). Bob has two polarization analyzers: one which can correctly measure linear polarizations (L) and the other which can correctly measure circular polarizations (C). Alice chooses one polarization at random, prepares the photon, and sends it to Bob. Bob chooses one of the two analyzers at random and measures with it the photon received from Alice. If, by chance, Bob has chosen right polarizer, he will receive 0 or 1 as sent by Alice. If Bob has chosen wrong polarizer, he will receive 0 or 1 with equal probability regardless of what Alice has sent. So after receiving a photon from Alice, Bob announces (over an authenticated but not secret public channel) which polarizer he has just used (L or C). Note that this says nothing to a potential eavesdropper about the value of the bit Bob has got. Alice responds with “Keep it” or “Trash it.” So bit by bit the two of them are building their secret key. The laws of quantum mechanics prevent qubits from being faithfully copied, so an eavesdropper can obtain only limited information about Alice’s and Bob’s string and furthermore eavesdropping can be detected by Alice and Bob.

It is straightforward to see that the whole protocol would be completely insecure if only the eavesdropper could calculate (or predict) either Alice’s random numbers or Bob’s random numbers or both. From the analysis of the secret key rate presented in [6], it is obvious that any predictability of random numbers by the eavesdropper would leak relevant information to him, thus diminishing the effective key rate. It is intriguing (and obvious) that in the case that the eavesdropper could calculate the numbers exactly, the cryptographic potential of the BB84 protocol would be zero. This example shows that the local RNGs assumed in BB84 are essential for its security and should not be taken for granted.

Apart from what has been described above, the BB84 protocol has two more subprotocols. Namely, due to the quantum incoherence, losses in the quantum channel or eavesdropping Alice and Bob will not have the exact same strings of bits after the first phase, although the two strings will have a lot of common information. Therefore the second subprotocol, the “Data Reconciliation,” is used to equalize the two strings, albeit at a cost of leaking some small information to an eavesdropper. Fortunately, Alice and Bob can calculate a lower limit of their mutual information after the two initial phases and then perform the privacy amplification phase in order to arrive to a shorter but much more private key. These two subprotocols require further random numbers.

The protocol BB84 is considered information theoretically proven [28, 73] meaning that an attacker simply has not enough information to calculate the plaintext even given infinite computing resources. This is in strong contrast with the widely used “deterministic cryptography” where an attacker has enough information to calculate the key except that it would probably require insurmountably large computation resources and/or time. The caveat with QC is that the security proof holds only against the family of attacks considered in the proof. Unfortunately, with time, it became evident that unexpected attacks on QC which utilize various quantum effects are feasible which makes QC much less “untouchable.”

For example, in 2007 an MIT group presented attack that gave Eve as much as 100 % of information about the key albeit at an expense of elevated BER [39], but the attack was reassuringly classified as “simulation only” because it assumed that Eve has a specific information about Bob’s receiver that she apparently could not get.

As with any other cryptographic procedure, some problems in real-world implementation of the protocol, especially of the quantum channel and real photon detectors, could be used to weaken the cryptographic security of the protocol and open pathways for attacks.

A beautiful demonstration of serious weakening and even 100 % breaking of the key without any notice to legitimate parties has been made by Makarov et al. in 2010 [24, 46]. The demonstration has been made on the commercial QC systems from Swiss company ID Quantique, based in Geneva, Switzerland, and one by MagiQ Technologies, based in Boston, Massachusetts. Improvements that would make QC resistant to those attacks are possible and have been proposed [46], but the lesson learned from this is that even protocols whose theoretical base is proven secure in some scenario are not to be automatically assumed immune to all practical attacks. The attack was made possible because the authors have found a way to manipulate RNG at the receiving station by exploiting weaknesses of single photon detectors. To make things even worse, this strategy made the previously mentioned MIT attack truly viable (not anymore just a simulation). This is yet another example of the importance of (local) RNGs to the security of a cryptographic scheme.

In conclusion, quantum cryptographic protocol BB84 requires that both Alice and Bob possess their private (local) provable RNGs. This is a highly critical requirement. Note that a public server of random numbers cannot substitute for local generators because the random numbers would have to be delivered to Alice and Bob in perfect secrecy in the first place, and the server would have to be trusted.

## 6 Random Numbers in Statistical Cryptography

Statistical cryptography was invented by Ueli Maurer in 1991. The so-called SKAPD protocol [51] resembles quantum cryptography and likewise consists of three subprotocols. In fact the last two subprotocols (the Data Reconciliation and the Privacy Amplification) are the same as in QC. However, the first subprotocol, named “Advantage Distillation” (AD), is completely different and it does not involve the quantum channel which potentially makes it much more practical. Instead, it requires something called “binary channel with noise” which is theoretically a classical communication channel complemented with a provable RNG.

The condition for successful key agreement is that prior to the AD protocol, the common information shared by Alice and Bob is greater than the common information shared by either Alice and Eve or Bob and Eve.

The practical problem with SKAPD is that it contains an unspoken “zeroth phase” in which Alice and Bob obtain their partially correlated initial strings of bits which satisfy the above condition. There is no known plausible way to make the zeroth phase possible although some scenarios have been proposed (scanning the surface of the Moon, listening to noise from faraway galaxies, taking big chunks of Internet data, etc.).

## 7 Random Numbers in Deterministic Cryptography

What we call here “deterministic cryptography” in this chapter is what is widely known as just “cryptography.” Some authors use the name “mathematical cryptography.” It is the contemporary cryptography based on the difficulty of computing discrete logarithms in Galois groups and elliptic curve groups, and also the factorization of composite numbers into primes. It also needs and uses random data; an excellent short survey is given in [8]. Since all such security protocols are by definition deterministic and therefore reversible, the only true security resource is that nondeterministic part: a key or one-time data which is supposed to be “random.” The quality and provability of randomness are therefore crucial for the security of the whole system.

- 1.
The Diffie–Hellman key establishment protocol [17] enables the same functionality as the above-mentioned BB84 and SKAPD protocols and is used, for example, in the “https” protocol in order to establish a session key. The protocol requires both parties (Alice and Bob) to generate private random data and after some operations send them to each other. A more resistant version of DH requires further random data used for digital signatures. A vulnerability of the PRNG built in an early version of the Netscape Internet browser led to a complete compromise of the subsequent cryptographic protocol. An example is the attack on Netscape’s 40-bit RC4-40 [64] challenge data and encryption keys, which was able to break the https protocol in a minute or so, as described in [25]. The authors of this article stipulate that 128-bit version RC4-128 would not be much harder to break either if seeding is done in a similar fashion.

- 2.
The RSA public key protocol relies on the generation of public and private keys separately by Alice and Bob. In order to create a private/public pair of keys, it is necessary to generate two unique, large prime numbers. Already calculating prime number candidates involves random numbers. After that, candidates need to be tested for primality using the Miller–Rabin algorithm which requires random numbers as the bases in order to properly test for primality. Additional one-time random numbers may be used in the process of actual communication. Where high-entropy physical random bits are not available or are time expensive (like on a typical PC computer), there is a tendency to “expand” a short random string to a long one by pseudorandom methods. This approach can create serious cryptographic weaknesses because an attacker must guess a much smaller number of bits than he or she would in the case of use of truly random numbers.

- 3.
Similarly, research into a cryptographic attack on the partially pseudorandom number generator of an AES-based commercial cryptographic system is described in [66].

To conclude, in deterministic cryptography, random numbers are the only part of the protocol which is different from point to point, and furthermore their true randomness is sometimes a prerogative for correct calculations. Therefore, even though most deterministic cryptography primitives are not secure, using true random numbers ensures the highest achievable security with these methods.

## 8 Open Problems and Outlook

In this survey chapter we attempt to show the importance of random numbers for the strength of cryptographic protocols not only for quantum and stochastic cryptographies where random numbers are an essential part of the data exchanged between communicating parties but also for contemporary deterministic cryptography where the unpredictability and maximal entropy of the random numbers used therein maximize the overall cryptographic strength.

- 1.
the lack of generator designs whose proofs of randomness would be at the same time correct, convincing, and demonstrated to be resilient to expected imperfections in hardware

- 2.
the (widespread) lack of understanding that pseudorandom numbers cannot be used as a substitute for true random numbers in so many applications, notably cryptography, both classical and quantum, computer security, Monte Carlo simulations, lotteries, testing of products and their functionality, and many more

- 3.
the high price of true random number generators

- 4.
the lack of support of true random number generators in various popular software that requires random numbers which makes them hard to use

## 9 Additional Comments and References

A distinctive difference between PRNG and TRNG is the provability of the latter. While mathematical proof of randomness is impossible, for TRNGs we do not rely solely on mathematics but also on sets of physical postulates which lie outside of mathematics. Indeed, the only provable feature of a PRNG is that it is not random because all numbers produced thereby can be calculated from a single initial number: the seed. On the other hand, TRNGs seem to be inevitably plagued with “small imperfections” in hardware which turn into measurable deviations from randomness which calls for post-processing. But complex post-processing blurs or weakens our belief in the eventual randomness proof. Furthermore, the practice of withholding information on the operating of a TRNG as well as a scientific proof of its randomness seems to be almost a rule when it comes to commercial TRNGs. Manufacturers justify this by the need to protect their intellectual property and technology. While such justification is fine when it regards common products (e.g., a dishwasher), it is exactly what ruins the purpose of a TRNG because without a clear insight into the technology and randomness proof of a TRNG, one falls back to the unprovability situation of a PRNG. On the other side of the coin, in scientific publications proofs of randomness are offered very rarely too, probably because the proof is the hard part of the research while it does not seem to be required by the editors of scientific journals. Most researchers therefore fall back to the minimum-action strategy: make a TRNG, obtain at least one random number sequence that passes the chosen set of randomness tests, and publish. However, without a detailed investigation of the sensitivity of the extracted randomness on small variations in hardware and without a randomness proof, a scientific design cannot proceed toward a product. In our view this situation has been improving and will continue to improve very slowly over time, thus ensuring a longevity and freshness of the research on TRNGs.

Even though there is a large collection of publications which document the fact that PRNGs may fail their purpose as RNGs [11, 12, 15, 19, 21, 29, 36, 40, 45, 55, 58, 70, 78, 87], we see that PRNGs are still in much more widespread use than TRNGs even in most critical applications. Among the reasons is the fact that PRNGs are so much more convenient, simpler, and cheaper to use than TRNGs, and also there is a ubiquitous lack of understanding of what randomness is and what it isn’t, supported by the nonexistence of a widely accepted definition of randomness [40]. Clearly, further research on that subject is needed.

All commercial TRNGs whose speed is at least 1 Mbit/s are bulky and the price is in the range $5–25k, which is more expensive than most of the software that would use a TRNG. It therefore generally does not pay a software manufacturer to make its product much more expensive by requiring a third-party TRNG for generating random numbers. In extreme rare cases though, a software has been married to a selected TRNG: for example, Mathematica and Quantis (by a third party) [53].

Commercial TRNGs typically come with drivers that support transfer of random numbers to programming languages such as Pascal or C++ on selected operating systems, e.g., [33], using a product specific subroutine or program library function. This is probably the maximum that a manufacturer can reasonably do to support its product. On the other hand, most commercial or free software that uses or needs random numbers does not come with support for any TRNG. This means that having a precompiled software, there is practically no way to connect it to any TRNG. The only viable solution to include a TRNG in a software package would be to write it from scratch and include in it specific function calls associated with the specific chosen TRNG. Since there is no industry standard for access to a TRNG from within a computer program (unlike, e.g., to access printers or other common peripherals), one could support only a specific TRNG per programming effort. In our view it is clear that as long as there is no standardized way to access TRNGs, or, better yet, until TRNGs are physically integrated in computers and are accessible in major programming languages, their popularity will remain minimal.

While it is clear that true randomness cannot be generated by deterministic operations and that therefore it must rely on physical phenomena, the problem of generating good enough randomness and the provability of randomness remain the main open problems with physical RNGs. New directions in the development of physical RNGs will probably concentrate on self-calibrating [41, 86] or no-calibration devices [80] with fundamentally random quantum phenomena as a source of randomness.

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