Method and device for generating random numbers

Abstract

The invention relates to a random number sequence generator comprising measuring means for measuring a plurality of continuous observables of an electromagnetic field, digital conversion means operatively connected with the measuring means for obtaining the measurement of each observable of the plurality of random number sequences, characterized in that the measuring means comprise one or more differential photodetection blocks followed by electronic homodyne (EO) or electronic heterodyne (EE) type blocks, possibly in parallel, with single- frequency or multi-frequency optical / electrical local oscillators, followed by analog / digital converters and by a subsequent digital filtering.

Description

[0001] PATENT APPLICATION FOR THE INDUSTRIAL INVENTION ENTITLED:

[0002] “Method and device for generating random numbers” ********************

[0003] FIELD OF THE INVENTION

[0004] The present invention relates to the technical field of devices capable of generating sequences of unpredictable random numbers (Random Number Generators, RNG).

[0005] BACKGROUND ART

[0006] RNG devices form a fundamentally important service in the field of cybersecurity. Most of the encryption systems, authentication protocols, and cryptographic key exchanges (just to name a few) used today rely on such RNGs. Despite the centrality thereof, almost all of the generators adopted are not capable of satisfying the requirement of randomness (crucial for security purposes): the basic operating principles are deterministic (algorithms or classic physical phenomena), making the output of such devices predictable. By being inherently probabilistic, the quantum processes, on the other hand, allow systems based thereon (Quantum RNG, i.e. , QRNG) to generate truly random numbers.

[0007] However, for cybersecurity-oriented applications, in addition to randomness, other requirements must be considered:

[0008] • The string of numbers must be evenly distributed (statistically, each number must have the same probability of being generated);

[0009] • Each number generated must be independent of the others (observing the creations of the RNG, it is not possible to predict future achievements); • Each sequence of numbers must be independent from the outside (information of any kind possessed by a possible attacker is not useful to predict the output of the RNG).

[0010] In general, quantum phenomena are not capable, alone, of satisfying such properties. Therefore, QRNGs use particular algorithms capable of extracting, from the raw string of measurements, a (shorter) sequence of numbers which is compliant with the aforementioned requirements. The required compression (reduction of the raw string length) depends on a parameter called "quantum conditional min-entropy" (Hmin). The quantity Hmin represents the number of random bits which can be extracted by the generator for each measurement.

[0011] The hypotheses underlying the estimation of Hmin divides the QRNGs into three groups:

[0012] • Fully Trusted (FT): they require a complete characterization of the various systems used (source and quantum measurement apparatus) and of the quantum phenomenon exploited;

[0013] • Device Independent (DI): it is not required to characterize the devices used or even to be aware of the physical phenomenon exploited (relying solely on the fact that quantum mechanics is the correct theory for the description of the reality surrounding us);

[0014] • Semi-Device Independent (SDI): characterization of a part of the equipment used (source or receiver) or knowledge (even partial) of the physical phenomenon used. In terms of security (of the numbers generated), performance (generation speed) and implementation difficulties (physical creation of the device), FT, DI and

[0015] SDI type protocols have the following features:

[0016] • FT: limited security of the numbers generated for applications related to cybersecurity (the working hypotheses required for the correct operation are numerous and stringent), high generation speed and (relative) ease of implementation;

[0017] • DI: high security of the numbers generated (very general working hypotheses), low performance and great difficulties in the construction thereof (the current schemes require a loophole-free violation of Bell's inequality);

[0018] • SDI: good security levels (reduced number of hypotheses) and high performance. They represent an excellent compromise in terms of speed and security with respect to purely FT and DI solutions.

[0019] In addition to the FT, SDI and DI classification, QRNG systems are also distinguished based on the number of degrees of freedom of the quantum physical phenomena considered: finite (discrete case, known as "with discrete variables", DV) or infinite (continuous case, known as "with continuous variables", CV). While the FT, DI and SDI classification emphasizes the security of the numbers generated, the distinction between DV and CV focuses attention on the type of implementation (and performance) of a QRNG. The former usually require the use of single-photon detectors, while the latter use analog-to-digital converters (ADCs). Since the number of measurements per unit time of an ADC is orders of magnitude higher than single- photon detectors, CV-type QRNG systems are generally preferred for high- performance applications.

[0020] Homodyne and Heterodyne Measurements

[0021] In the CV field, the measurement of one or both of the quadratures, Q and P, of the electromagnetic field of an optical quantum state can be exploited for the creation of an RNG. The most common schemes capable of conducting such measurements are:

[0022] • Homodyne receiver: apparatus capable of measuring one of the two available quadratures;

[0023] • Double homodyne (or heterodyne) receiver: apparatus capable of simultaneously measuring both quadratures of the field.

[0024] As shown in Fig. 1 , the reference scheme of a homodyne receiver exploits a balanced Beam Splitter (BS) (50:50) to "combine" the quantum state to be measured (P1 ) with the intense light of a laser (P2), then distributing the output to the two outlets. BS forms a "2x2 optical group", with 2 input optical signals and 2 output optical signals. The 2 output optical signals are then converted into an electrical signal by a "differential photodetection" block consisting of two photodiodes (FD1 , FD2) followed by an (analog) subtraction of the photocurrents. The output electrical signal is combined with a sinusoidal signal at frequency fi and phase φ1 (block “OE= electronic homodyne”) to obtain an output signal which represents the measurement of a quadrature of the field. In some implementations the "OE" block is not present and the signal is analyzed directly at the output of the differential photodetection block. Therefore, the measurement operator associated with a homodyne system is of the projective type. Denoting the input quantum signal as and the local oscillator as of amplitude and phase (remember that this is intense enough so that it can be treated classically), the output of the measurement is given by: where Q is the projection operator on the rotated quadrature of the angle . In the above equation, \aL0) is a coherent state. The coherent states are particular Gaussian optical states representing the light emitted by a laser. Thermal states, vacuum and squeezed states are also included in the class of Gaussian states.

[0025] Fig. 2a) instead shows the reference scheme of a double homodyne device (also called heterodyne in the quantum communications community) consisting of two homodyne type receivers (H1 , H2) (with the OE block not present) each of which measures a "copy" of the quantum state.

[0026] In order to simultaneously obtain the two electromagnetic field measurements, a BS on the quantum state P1 and one on the local oscillator P2 are used with a subsequent 90° phase shift of one of the two outlets. The optical signals obtained are combined by two successive BSs. Therefore, the "2x4 optical unit" (2 inlets and 4 outlets) has the effect of combining the two input optical signals P1 and P2 to obtain 2 pairs of output optical signals. Each pair of optical signals is measured by means of a "differential photodetection" block: the two electrical outlets correspond to the 2 quadratures of the field. In this case, the measurement operator is not of the projective type but it is equivalent to a Positive Operator Valued Measurement (POVM). Maintaining the same conventions for the inlet ports used in Fig. 1 , the measurement operator is where is a coherent state.

[0027] Subsequently, regardless of the scheme adopted (homodyne, double homodyne or heterodyne, see below), the measurements of the quadratures are digitized by means of a special ADC and then processed with the extraction algorithms mentioned above.

[0028] It should be noted that homodyne, double homodyne and heterodyne detection systems, in addition to QRNGs, are commonly used in all those applications where the measurement of quantum properties of light is required (for example tomography and quantum communication systems). The publication Brunner, H. et al. (2017). A low-complexity heterodyne CV-QKD architecture. International Conference on Transparent Optical Networks. https: / / doi.orq / 10.1109 / ICTQN.2017.8025030 provides an overview of the various applications used in classical optical communications and for Quantum Key Distribution (QKD).

[0029] In classical optical communities, homodyne is used to indicate schemes in which the local oscillator has a frequency equal to the central frequency of the signal, while in the heterodyne case the local oscillator has a different frequency with respect to that of the signal. For this reason, demodulators capable of extracting two quadratures of the signal directly in the baseband are defined as double homodyne while those which arrive at the same result by passing from a first demodulation at intermediate frequency fi are defined as heterodyne. A heterodyne measurement scheme is shown in Fig. 2b). The optical part is identical to a homodyne measurement, while the electrical signal obtained from the differential photodetection block is divided into two copies and combined with two sinusoidal electrical signals at frequency fi and offset by 90°.

[0030] Quantum nomenclature, on the other hand, does not differentiate double homodyne from heterodyne, as both configurations allow the quantum measurement of the same operator . Both configurations shown in

[0031] Fig. 2a) and 2b) have two electrical output signals corresponding to the two quadratures. Unless otherwise indicated, the notation of the classical optical community will be used in the present description: with homodyne the scheme in Fig.1 is indicated, with double homodyne the scheme in Fig. 2a), with heterodyne the scheme in Fig. 2b).

[0032] Homodyne-based QRNG

[0033] The most direct approach to making a homodyne receiver-based QRNG FT is to measure the state of the electromagnetic vacuum, as shown in Fig.. 3. This scheme is described in several works, including Gabriel, C., Wittmann, C., Sych, D. et al. A generator for unique quantum random numbers based on vacuum states. Nature Photon 4, 711-715 (2010). https: / / doi.org / 10.1038 / nphoton.2010.197 and Gehring, T., Lupo, C., Kordts, A. et al. Homodyne-based quantum random number generator at 2.9 Gbps secure against quantum side-information. Nat Commun 12, 605 (2021 ). https: / / doi.org / 10.1038 / s41467-020-20813 The approach is distinguished by the implementation simplicity thereof. However, the complete characterization of the apparatuses involved is required, in addition to taking note of the properties of the quantum phenomenon used.

[0034] With regard to the latter aspect, generalizations are possible; in the work by Gehring, T., Lupo, C., Kordts, A. et al. Homodyne-based quantum random number generator at 2.9 Gbps secure against quantum side-information. Nat Commun 12, 605 (2021 ). https: / / doi.Org / 10.1038 / s41467-020-20813-w, instead of considering a specific quantum phenomenon, the class of Gaussian states was examined (it should be noted that the scheme is still of the FT type).

[0035] Greater security of the numbers generated is achieved by adopting the scheme proposed in EP3040853A1 , derived from Marangon, D. G., Vallone, G., & Villoresi, P. (2017). Source-Device-Independent Ultrafast Quantum Random Number Generation. Phys. Rev. Lett., 118(6), 060503. https: / / doi.org / 10.1103 / PhysRevLett.118.060503. The protocol in question, shown in Fig. 4, is of the Source Independent type (SI, particular type of SDI QRNG) where the only homodyne receiver is characterized (the source is considered controlled by an attacker). The certification of the numbers occurs through Entropic Uncertainty Principle (EUP): in addition to the quadrature used to generate the random numbers (P), a second one (Q) is "defined" through which the purity of the measured state is verified. Through the measurements obtained, it is possible to lower the m in-entropy quantum conditional, regardless of the quantum state used:

[0036] It should be noted that K is related to the overlap factor between the two measurements (parameter estimated during receiver characterization), while is a corrective factor related to the measurements in the additional base (estimated while the device is in operation). Given the need to measure along two quadratures, despite the receiver being of the homodyne type (remember that it is capable of measuring along only one quadrature), the diagram makes use of a further device: a phase modulator (PM). By applying a voltage to the latter, it is possible to modify the phase of the laser, obtaining the effect of measuring along a different quadrature of the electromagnetic field. Since, as already mentioned, the system can only measure one quadrature at any time, in order to guarantee the security of the numbers generated, the decision to acquire data from the control or generation quadrature must occur randomly. Therefore, attention must be paid to the following aspects:

[0037] • To maximize performance, the probability of measuring generation quadrature must be greater than the probability of measuring along the control base;

[0038] • The asymmetry between the probability of measuring the control quadrature and that of generation must not be such as to overestimate the corrective factor.

[0039] Despite the higher level of security offered, the use of a phase modulator involves a complication of the protocol, adds a component to the receiver scheme and negatively impacts the cost of implementation. Furthermore, the active electronics necessary to drive the phase modulator PM must be considered:

[0040] • being active, the higher the required speeds, the higher the cost thereof,

[0041] • For technological reasons, this has limits in terms of speed, thus representing a potential bottleneck in terms of QRNG performance.

[0042] Finally, having to choose randomly when to measure in one base or the other, an initial random number string (seed) is necessary to be able to start the generation process and randomness is spent during the execution of the protocol for the choice of the measurement bases, negatively impacting the rate.

[0043] Another possible method for making an SI generator by means of homodyne detection is presented in Smith et al. (2019). Simple source device-independent continuous-variable quantum random number generator. Phys. Rev. A, 99(6), 062326. https: / / doi.orq / 10.1103 / PhysevA.99.062326. In this document, the laser used as a local oscillator is driven (by the gain-switching technique) obtaining optical pulses having, mutually, completely random phases, as shown in Fig. 5. With this method, it is possible, for each pulse generated by the laser, to measure the (unknown) input quantum state in a different base. Through the above procedure, the authors manage to limit the conditioned min-entropy: of the incoming quantum state, always being higher than the min-entropy calculated for the quantum vacuum state.

[0044] Double homodyne-based QRNG

[0045] QRNG systems based on double homodyne optical receivers allow the implementation of SDI protocols without the use of additional components, as shown in the work of Avesani et al. “Source-device-independent heterodyne-based quantum random number generator at 17 Gbps”. Nat Commun 9, 5365 (2018). https: / / doi.orq / 10.1038 / s41467-018-07585-0

[0046] In the work of Avesani et al. (see Fig. 6), regardless of the source used (SI type system), the properties of the measurement apparatus (after characterization) are exploited so as to have a conservative estimate of Hmin. However, such a system turns out to be more complex with respect to a homodyne apparatus: since a double homodyne type measurement is exploited, it is necessary to use an optical component called 90-hybrid, the behavior of which is equivalent to two separate homodynes with two lasers mutually offset by 90 degrees. It should be noted that the 90-hybrid is a passive optical device, and as such does not require control electronics, therefore allowing both costs and speed limitations to be limited (please, compare with the SDI homodyne by Marangon et al. discussed above). However, from the technological point of view, the creation of the 90-hybrid presents critical issues attributable to the need to simultaneously measure two quadratures of the electromagnetic field. Since any deviation from the working hypothesis can compromise the safety of a QRNG, the tolerances of the production processes required for the implementation thereof are tight, with a possible impact in terms of construction costs on the optical components depending on the platform adopted.

[0047] In addition to SI schemes, the use of double homodyne receivers in other SDI protocols has been demonstrated in Avesani et al. (2021 ). Semi-Device- Independent Heterodyne-Based Quantum Random-Number Generator. Phys. Rev. Appl., 75(3), 034034. https: / / doi.orq / 10.1103 / PhysRevApplied.15. In this document, the transmitter provides two states (considered controlled by an attacker) to be sent to a receiver (pessimistically assumed always owned by the attacker). It should be noted that:

[0048] • it is assumed that the attacker must submit to an energy constraint (average number of photons) on the possible states which can be made available by the transmitter side;

[0049] • An external source independent of the QRNG chooses which status to send to the receiver;

[0050] • On the receiver side, there are two possible outcomes coded as ‘0 ’and ‘1 ’(they also represent the raw string of the QRNG itself).

[0051] Although for the purposes of the protocol the use of a double homodyne is not necessary (given the independence of the scheme from the nature of both transmitter and receiver), the work has demonstrated the use thereof in SDI schemes other than the SI framework. Furthermore, the ability of the double homodyne to simultaneously measure the quadratures of the field guarantees a certain resilience of the receiver with respect to the phase fluctuations of the laser used as a local oscillator; this property has proven useful in the post-processing phase. It should be noted that, although it is not necessary to characterize the devices used, with respect to SI-SDI devices,

[0052] • The speeds achievable are lower (also considering a Fully Trusted implementation thereof);

[0053] • The performance is sensitive to the losses and inefficiencies introduced by the peripheral devices (BS, photodiodes, etc.) used (the greater the losses, the lower the rates obtainable);

[0054] • The request from the external source independent of the QRNG (and therefore from the attacker) means that the maximum achievable speed is subordinated to the speed of the latter.

[0055] In summary, for the purposes of creating a CV-QRNG based on the measurement of the quadratures of the electromagnetic field, the existing protocols exploit homodyne and double homodyne type receivers.

[0056] The first stands out for the implementation simplicity thereof, but going beyond FT implementations requires the use of additional components which negatively impact in terms of costs, complexity of the protocol itself and performance. The SI- SDI scheme based on a phase modulator such as that described in Marangon et al. requires the use, being an active optical component, of a fast electronic control system so as to achieve high generation speeds. Creating such a high-performance controller is complex (as well as expensive). Furthermore, the higher the generation speed, the more difficult and onerous the design of the aforesaid control system will be. The scheme of Smith et al. instead requires a high-performance electronic driver to modulate the laser which acts as a local oscillator so as to obtain high generation speeds (similar problems are faced by Marangon et al., also in terms of scaling, design difficulties and costs). In addition, Smith's proposal must consider a (transient) relaxation time for the photodetectors once the pulse is received; time within which there can be no generation of random numbers (an effect which tends to reduce the performance of the QRNG).

[0057] QRNGs based on a dual-homodyne receiver lend themselves, without the addition of additional components, to the creation of SDI protocols, however the creation thereof, with respect to homodyne receivers, is more complex and expensive given the presence of the 90-hybrid. Furthermore, the latter component must be made respecting strict constructive tolerances so as to ensure low losses and a phase difference of the output signals of 90 degrees.

[0058] FILTERS

[0059] In CV-type devices, the analog electrical signal associated with quantum measurements (usually a voltage) must be amplified so as to adapt it to the range (usually fixed) of the Analog-Digital Converter (ADC). Such amplification (see Fig.

[0060] 3) makes it possible to adequately exploit the resources of the ADC, thus optimizing performance. The operation requires the use of a system with a flat frequency response in the band of interest; a requirement generally not met by commercially available amplifiers (especially if intended for broadband applications). Such a deviation from the flat pattern of the frequency response introduces unwanted correlations in the signal, which increase in magnitude as the deviation becomes more pronounced.

[0061] The presence of correlations negatively impacts the performance and safety of a QRNG. However, it is possible to eliminate them by designing a filter with a response function Hprj(ƒ) to be added in cascade to the amplification chain. If the frequency response of the amplifier is Hamp(ƒ), the filter response function must be the inverse of Hamp(ƒ). In fact, for cascade systems the overall frequency response is given by the product of the individual response functions: where the unit, in the frequency domain, indicates flat response.

[0062] In Cedric Bruynsteen et al. (Bruynsteen, C., Gehring, T., Lupo, C., Bauwelinck, J., & Yin, X. (2023). 100-Gbit / s Integrated Quantum Random Number Generator Based on Vacuum Fluctuations. PRX Quantum, 4(1 ), 010330. https: / / doi.Org / 10.1103 / PRXQuantum.4.010330) a receiver frequency response equalization system was used. In the work, a Fully Trusted QRNG of the homodyne type is considered, demonstrating (in addition with further hypotheses) that the temporal correlations introduced by the reception system (especially the RF amplification chain) have a direct (negative) impact with regard to the estimation of the min-entropy. However, by equalizing (removing correlations) the receiver response by means of a special filter, the generator performance is improved. Many details of the equalizer adopted are not provided, but the following is presented:

[0063] • The equalizer (filter with reverse frequency response with respect to that of the homodyne receiver) is referred to as Zero-Forcing Equalizer (ZFL);

[0064] • The filter was made digitally using the FIR (Finite Impulse Response) approach. In order to demonstrate the achievable increase in performance, the number of coefficients (taps, in jargon) used to make the filter was varied: reference is made to 9 taps (first increase in performance) and 201 taps (maximum increase in performance). The work declares that the proposed system is implementable (although it has not been done) also analogically (use of physical, non-digital components);

[0065] • the filter has a normalization coefficient in order to ensure that the variance of the equalized signal is equal to the variance of the unequalized signal;

[0066] • it is stated that the filter coefficients are estimated by performing a Least Square Fit of the inverse of the response of the homodyne receiver.

[0067] In the digital field, once the specifications of a filter have been defined, it can be made according to the HR (Infinite Impulse Response) or FIR (Finite Impulse Response) method. For real-time applications, the FIR approach is usually preferred, as it can guarantee greater numerical stability. On the other hand, with the same specifications, the number of coefficients / taps (filter length, in jargon) required is greater with respect to the IIRs. Such a request can potentially run counter to the limited resources made available by real-time hardware platforms.

[0068] In the field of decorrelation filters (whitening filters, in jargon), this limit is particularly felt. Standard synthesis techniques are based on least squares regression procedures, which lead to systems of considerable length (often greater than the available resources). However, it is common to truncate the number thereof without significantly affecting the properties of the filter. Nevertheless, the procedure is often sub-optimal (in terms of performance and / or resources saved).

[0069] Alternatively, regularization techniques can be adopted to address the problem more rigorously. By adding additional (very general) assumptions regarding the required frequency response, the length can be compressed while preserving the required decorrelating properties.

[0070] The object of the present invention is to provide a simple and practical device, capable of generating secure random numbers at high speed, through coherent detection schemes which simplify the optical implementation.

[0071] BRIEF DESCRIPTION OF THE INVENTION

[0072] The invention achieves the object with a random number sequence generator comprising measuring means for measuring a plurality of continuous observables of an electromagnetic field, digital conversion means operatively connected with the measuring means for obtaining sequences of random numbers from the measurement of each observable of the plurality. The measuring means comprise one or more differential photodetection blocks followed by electronic homodyne OE or electronic heterodyne EE blocks (possibly in parallel) with single-frequency or multi-frequency optical / electrical local oscillators, followed by analog / digital converters and, preferably, by a subsequent digital filtering.

[0073] Electronic homodyne / heterodyne circuits in combination with optical homodynes have never been used to generate random numbers, in particular to obtain a plurality of signals in parallel to be used to increase the random number generation rate.

[0074] In the most general form, the system comprises: a) N generators of optical signals P1, P2, ... PN. Each generator Pjis obtained by combining a number Kj of Gaussian states at different frequencies through an optical combiner. b) An Nx2M optical group consisting of a linear optical system which combines the N input optical signals to obtain 2M output optical signals. c) M differential photodetection blocks, which transform the 2M optical signals into M electrical signals. Each differential photodetection block comprises: i. a pair of photodetectors for converting two optical signals exiting the optical group into electrical signals; ii. a subtractor device which receives the electrical signals from the two photodetectors as input and provides a difference electrical signal on an output port; d) M electronic demodulators demux1, demux2,..., demuxM. Each electronic demodulator demuxj comprises a plurality Djof type OE or EE components connected in parallel with the same input electrical signal. Each OE or EE type component combines the input electrical signal with sinusoidal electrical signals of known frequency; e) a plurality of analog-digital converters connected to the electrical outlets for converting all the electrical signals exiting the M electronic demodulators into number sequences. Such converters can include electrical amplifiers to adapt the signal to the operating range of the ADCs; f) a plurality of digital filters for eliminating any correlations present in the signal.

[0075] In an embodiment, one of the optical signals Pjcorresponds to a quantum state, while the others are classic signals of the coherent state, vacuum state or thermal state type.

[0076] In another embodiment, the generator comprises: a) a 2x2 optical unit comprising: i. a beam splitter BS capable of combining an optical quantum state P1 entering a first port with at least one known frequency-optical signal P2 entering a second port and outputting the combined signal divided onto two outlet ports; b) a differential photodetection block which transforms the two optical signals exiting the BS into an electrical signal; c) at least one electronic demodulator provided with at least one channel in which the input difference electrical signal is mixed with an input electrical signal with known frequency so as to obtain as output at least one electrical signal demodulated in baseband or at an intermediate frequency, combination of the frequency of the input optical signal and the frequency of the input electrical signal; d) at least one analog-digital converter connected to the at least one demodulator for converting the at least one output demodulated electrical signal into number sequences.

[0077] The at least one electronic demodulator is advantageously a phase and quadrature demodulator provided with two channels in which the input difference electrical signal is mixed with the same input electrical signal at a known frequency but offset by 90° so as to output a pair of quadrature signals demodulated in baseband or at an intermediate frequency, as a combination of the frequency of the input optical signal and the frequency of the input electrical signal.

[0078] In an embodiment, the generator comprises a plurality N of electronic demodulators which use sinusoidal electrical signals of known frequency connected in parallel with the same electrical signal as input difference and with the equal or different input electrical signals of known frequency.

[0079] In particular, the N electronic demodulators have input electrical signals with different known frequencies, in particular, called fsthe center frequency of the quantum optical signal and the frequency of the optical signal with known input frequency, the N demodulators have, respectively, input electrical signals with frequency f1, f2,..., fN so that the central frequency of the demodulated electrical signal, or of the demodulated electrical signals in the event of in-phase and quadrature demodulators, exiting from each demodulator is

[0080] In another embodiment, the input optical signal comprises a plurality of different known frequencies the electronic demodulator comprising a plurality of electronic demodulators connected in parallel with the same input difference electrical signal and with known frequency-input electrical signals f1, f2,..., fNso that the central frequency of the demodulated electrical signal, or of the demodulated electrical signals in the event of in-phase and quadrature demodulators, exiting from each demodulator, depends on a frequency of the plurality of frequencies of the known frequency-input optical signal. In another embodiment, the N optical inlets consist of a quantum state, a local oscillator consisting of a combination of K Gaussian states (for example, coherent states or thermal states), at known frequencies and of N-2 vacuum states and the optical group NxN consists of a unitary matrix U.

[0081] In another embodiment, the 2x2 optical unit consists of a BS, one optical input consists of a quantum state p, the other optical input consists of a thermal state at known frequencies. The two output optical signals are converted into an electrical signal by a differential photodetection block and then demodulated with an EE or OE type block.

[0082] According to another aspect, the invention relates to a method for generating random numbers comprising: a) mixing an optical quantum state with an optical signal at predefined frequency; b) extracting an electrical quadrature of said quantum state by means of one or more differential photodetection blocks; c) subjecting the electrical quadrature to a homodyne or heterodyne demodulation by means of mixing with an electrical signal at predefined frequency so as to obtain at least one demodulated electrical signal in baseband or at an intermediate frequency as combination of the optical signal frequency and the electrical signal frequency; d) convert into digital the demodulated signal(s) to obtain a random number sequence. According to another aspect, the invention further relates to a method for filtering signals in which it is envisaged to subject the demodulated digital signal(s) to a decorrelation filter, also subject matter of the invention, having a frequency response equal to assuming that the frequency response of the receiver system is that of an auto- regressive model of the type where N and a; are, respectively, the length and the parameters of the model, xnthe inlets and ynthe outlets, where, with N fixed, the coefficients a; are determined by solving the matrix system of linear equations aR = p where being ry(n) the autocorrelation coefficients of y(ri).

[0083] Further features and improvements are the subject of the dependent claims.

[0084] BRIEF DESCRIPTION OF THE FIGURES

[0085] Further features and advantages of the invention will become apparent from reading the following detailed description, given by way of a non-limiting example, with the aid of the figures shown in the accompanying drawings, in which: Fig. 1 shows the basic diagram of a homodyne optical receiver.

[0086] Fig. 2a) shows the basic diagram of a double homodyne optical receiver. Fig. 2b) shows the basic diagram of a heterodyne optical receiver.

[0087] Fig. 3 shows the diagram of a homodyne optical receiver for QRNG applications according to the background art.

[0088] Fig. 4 shows another diagram of a homodyne optical receiver for QRNG applications according to the background art.

[0089] Fig. 5 shows a further diagram of a homodyne optical receiver for QRNG applications according to the background art.

[0090] Fig. 6 shows the functional diagram of a generic source-device-independent QRNG in CV.

[0091] Fig. 7 shows the general diagram of a QRNG according to the invention.

[0092] Fig. 8 shows the conceptual diagram of a receiver for QRNG applications according to a first embodiment of the invention.

[0093] Fig. 9 shows the conceptual diagram of a multi-chromatic heterodyne followed by an electronic homodyne (left) or heterodyne (right) according to another embodiment.

[0094] Fig. 10 shows the conceptual diagram of a further embodiment comprising an optical homodyne with a single local oscillator and demodulation of the electrical signal produced with N multi-frequency RF local oscillators in both the homodyne and electronic heterodyne variants.

[0095] Fig. 1 1 shows the conceptual diagram of a multi-port homodyne receiver according to a further embodiment.

[0096] Fig. 12 shows the conceptual diagram of a homodyne receiver with thermal oscillator according to another embodiment. Fig. 13 shows the spectrum of the raw measurements (on the top left), the frequency response of the decorrelation filter (on the top right) and the spectrum of the raw measurements after the decorrelation filter (at the bottom).

[0097] Fig. 14 shows the coefficients of the decorrelation filter in the case of least squares minimization (LS) and regularized minimization (ReLS).

[0098] The following description of exemplary embodiments relates to the accompanying drawings. The same reference numbers in the various drawings identify the same elements or similar elements. The following detailed description does not limit the invention. The scope of the invention is defined by the appended claims.

[0099] DETAILED DESCRIPTION OF THE INVENTION

[0100] With reference to Fig. 7, a continuous variable quantum random number generation (CV-QRNG) system according to the most general form of the invention comprises: a) N generators 100 of optical signals P1, P2, ... PN. Each optical signal Pjis obtained by combining a number Kj of Gaussian states at different frequencies through an optical combiner 101. Such Gaussian states may be coherent states, vacuum states, squeezed states, thermal states. One or more optical signals may be general quantum states (not necessarily Gaussian ones); b) An Nx2M optical group 200 consisting of a linear optical system which combines the N input optical signals to obtain 2M output optical signals; c) M differential photodetection blocks 300, which allow transforming the 2M optical signals into M electrical signals. Each differential photodetection block 300 comprises: i. a pair of photodetectors 301 for converting a pair of optical signals exiting the optical group into two electrical signals; ii. a subtractor device 302 receiving the electrical signals from the two photodetectors 301 as input and provides a difference electrical signal on an output port; d) M electronic demodulators 400 demux1, demux2,..., demux.MEach electronic demodulator demuxj comprises a plurality Dj of type OE or EE components connected in parallel with the same input electrical signal. Each OE or EE type component combines the input electrical signal with sinusoidal electrical signals of known frequency. In some embodiments, some EE or OE type components can be removed and therefore the electrical signal is directly connected to the next analog-to-digital converter; e) a plurality of analog-digital converters 500 connected to the electrical outlets for converting all the electrical signals exiting the M electronic demodulators into number sequences. Such converters can include electrical amplifiers to adapt the signal to the operating range of the ADCs; f) a plurality of digital filters 600 for eliminating any correlations present in the signal.

[0101] The diagram depicted in Fig. 8 derives from the more general one of Fig. 7 where N=2 and M=1. In this simplified embodiment, an optical / electronic hybrid receiver capable of simultaneously measuring the two orthogonal quadratures of the electromagnetic field (in a manner equivalent to a purely optical double homodyne receiver) is used. Specifically, the optical part implements an optical homodyne receiver the photo-current (electrical signal) of which is processed by an electronic heterodyne, obtaining the measurement of the two quadratures of the electromagnetic field required.

[0102] In detail, the optical part 1 is a homodyne receiver comprising: i. a beam splitter 11 capable of combining an optical quantum state 21 entering a first port 31 with at least one known frequency-optical signal 41 entering a second port 51 and outputting the combined signal divided onto two outlet ports 61 ; ii. a pair of photodetectors 71 coupled with the outlet ports 61 of the beam splitter 11 to convert the output optical signals into electrical signals; iii. a subtractor device 81 which receives the electrical signals from the two photodetectors 71 as input and provides a difference electrical signal 3 at an intermediate frequency on an output port 91 .

[0103] The electrical part 2 is a heterodyne receiver comprising an electronic demodulator provided with two channels in which the input difference electrical signal 3 is mixed in the mixers 32, 42 with input electrical signals 12, 22 of known frequency and offset by 90° so as to obtain a pair of quadrature electrical signals X, P demodulated in baseband or at an intermediate frequency. The circuit is completed by a digital analog converter 4 for digitizing the signals output to the electronic demodulator and subsequent digital filtering. The scheme allows preserving the advantages of a purely optical double homodyne while avoiding the complexity thereof in terms of construction (the problem of the 90-optical hybrid is avoided):

[0104] • With regard to the optical part, a homodyne optical receiver (simpler to implement) is relied on;

[0105] • An electronic heterodyne is used for the extraction of the measurement of the two quadratures of the electromagnetic field of the quantum state.

[0106] This approach, in constructive terms, makes it possible to simplify the system, limit construction costs and increase the generation rate.

[0107] In each experimental embodiment, the POVM associated with the quantum measurement is discretized (due to the discretization of the ADC in the receiver) with parameter δ. The discretized version of the measurement POVM is indicated as the experimental imperfections (optical and electrical) of the receiver are also included in the explicit form of . Regarding the quantification of the conditional quantum min-entropy it is proposed to use a generalization of what was obtained in Avesani, M., Marangon, D.G., Vallone, G. et al. Source-device- independent heterodyne-based quantum random number generator at 17 Gbps. Nat Commun 9, 5365 (2018). where e represents the maximum eigenvalue of

[0108] Fig. 9 shows another embodiment of the invention. In this case, the QRNG is based on an optical receiver of the homodyne type described above, but with a multi-chromatic coherent local oscillator, (N>1 laser with different frequency followed by N single (Fig. 9a) or double electronic homodynes (Fig. 9b), i.e., N single-channel or double-channel demodulators in phase and quadrature with

[0109] RF oscillator at frequency ƒRF. With respect to a standard homodyne or heterodyne (single local oscillator), the multi-chromatic system allows, with specific quantum sources, using photodetectors with a lower band (less expensive), appropriately exploiting the distance (in the frequency domain) of the lasers used as a local oscillator.

[0110] For each intermediate frequency, there is the equivalent of an optical homodyne QRNG system (Fig. 9a) or optical homodyne plus independent electronic heterodyne (FIG. 9b). Therefore, having multiple generation channels available in parallel results in a higher generation speed. The advantage of the schemes in Fig. 9 with respect to the scheme with a single local oscillator is linked to the photo- detector band. Suppose the photodetectors have band Δƒ. The systems in Fig. 9 with N lasers are equivalent to a system with a single laser but with an N* band photodetector Δ .

[0111] A variant of the previous scheme corresponds to the use of a single optical oscillator at frequency and the use of RF electrical signals at frequencies - a homodyne or multiple electrical heterodyne) and combined with determined phases (even different from 90°) with the electrical signal downstream of the balanced receiver. Fig. 10a shows the case of a multiple electric homodyne, i.e., with N single-channel demodulators, while Fig. 10b shows the case of a multiple electric heterodyne, i.e., with N electric heterodyne demodulators. A further embodiment corresponds to having a single frequency optical oscillator ƒ0and linear optics system which combines the quantum signal and the local oscillator to obtain 2N optical output modes, followed by N balanced receivers, as shown in Fig. 11 . Thereby, N electrical signals are obtained which generalize the double homodyne detector. Such a scheme can be used to generate random numbers, with the same technique shown above: the conditional quantum min- entropy is linked to the maximum eigenvalue among all the POVM operators describing the quantum measurement. The N electrical signals obtained are combined with N local electrical oscillators with frequencies (not necessarily different from each other) in the form of a single (OE type device) or double homodyne (EE type device).

[0112] The scheme can be generalized by assuming a multi-chromatic oscillator or / and using multiple local oscillators as input to the different input modes.

[0113] The multi-port system described above is nothing more than an advantageous generalization of the homodyne and heterodyne system. The first can be seen as a system with two optical inlets (local oscillator plus quantum signal to be measured), two optical outlets from which an electrical output is obtained (the quadrature of the electromagnetic field). Instead, the second, as a device with four optical inlets (of which only two are actively used, i.e., the one related to the quantum signal to be measured and the local oscillator), four optical outlets from which two electrical outputs are obtained (the two quadratures mutually offset by 90 degrees). It should be noted that although only two inlets are used, the other two must still contribute to the final result (the principles of quantum mechanics require it). Now the 2x1 (homodyne, with 2 optical inlets and 1 electrical output) and 4x2 (double homodyne, with 4 optical inlets and 2 electrical outputs) system can be generalized to NxM: N are the inlets, 2M the optical outlets and M the electrical outlets. Of these N inlets, in the embodiment shown in Figure 11 only two inlets are different from the vacuum. In the most general form, all the inlets can be different from the vacuum: having several channels available in parallel allows having a higher generation speed.

[0114] A variant of the protocol presented by Smith, et al. (2019). Simple source device- independent continuous-variable quantum random number generator. Phys. Rev. A, 99(6), 062326. https: / / doi.Org / 10.1103 / PhvsRevA.99.062326, is shown in Fig. 12 where, instead of using a pulsed laser as a local oscillator, an optical thermal source is used, for example an LED (Light Emitting Diode) or EDFA (Erbium Doped Fiber Amplifier) optical amplifier: the associated light (thermal state) has an uniformly distributed phase similarly to the laser pulses produced by Smith. However, unlike Schmit et al.:

[0115] • the laser drive electronics are not required, leading to a simplification of the system;

[0116] • Downtime (within which there is no generation of random numbers) due to the relaxation time of the photodiodes is avoided, consequently improving the performance of the QRNG.

[0117] However, for the scheme to work properly, the system sampling rate must be slow enough to avoid correlations.

[0118] Smith's scheme is more complex, as it requires a pulsed laser to generate a local oscillator with randomized phase. Furthermore, for each pulse, it must wait for the system to stabilize before it can acquire the signals from which it can then extract the random numbers. Hence the idea of using, instead, a thermal source to randomize the phase. This works continuously, avoiding the waiting times necessary for stabilization (leading to a possible increase in the generation speed) and the circuit diagram necessary to drive the heat source is simpler than a pulsed system. Furthermore, it can make use of a heterodyne system for measuring quadratures. This implementation is in fact a particular case of the general case, considering a 2x2 optical group, one inlet consisting of a quantum state and the other consisting of a thermal state.

[0119] We consider now a pre-processing system (filter / equalizer) of the raw measurements of CV-QRNG systems based on homodyne / heterodyne systems according to the invention, so as to remove the correlations introduced by the hardware of the receiver itself (photodiodes, RF amplifiers, etc.). The presence of such correlations negatively impacts the m in-entropy quantum conditional, and consequently the performance of the device. The use of equalization techniques can alleviate the problem. Typically, works concerning QRNG schemes do not take this aspect into account, implicitly assuming de facto a frequency response of the flat reception system.

[0120] In the improvement described below, we start from the estimation of a model of the receiver so that the frequency response of the first fits that of the second. Since in the digital domain the frequency response is a function with support in [0,2π], assuming that the frequency response modulus is at least continuous, then it is possible to approximate such a frequency response with an AR (Auto-Regressive) model: where N and the coefficients are, respectively, the length and parameters of the model. The correlations of the system are represented by the dependence with respect to the past (memory effect), by means of the y , of the output at the n-th instant. Since the following inequality applies in general: the memory effects introduced by the receiver (represented as classic side- information E2to be added to the quantum one can lead to a decrease in the entropy and therefore in the performance of the QRNG.

[0121] However, since the model is invertible, it is possible, starting from this, to synthesize a filter capable of removing the aforementioned memory effect. As already outlined in the background art ("Filters" section), a flat (constant) frequency response indicates the absence of correlations. In the case of the AR model, the associated frequency response is: therefore, the decorrelation filter response must be:

[0122] Given N, the coefficients aiare determined by solving the Yule-Walker equations

[0123] (or Wiener-Hopf, except for a sign) (system of linear equations) aR = p (represented in matrix form), where: with the autocorrelation coefficients of . The latter and N are estimated starting from the raw measurements using cross-validation algorithms and (un- biased estimators, respectively. For the purposes of estimating the coefficients at, the above problem can be reformulated as a least squares regression (LS): where Y is the vector of the experimental data (raw measurements) and the matrix

[0124] It should be noted that every increasing n (number of raw measurements).

[0125] By way of example, Fig. 13 shows the spectrum of the raw measurements (first left), the frequency response of the decorrelation filter (second right) and the spectrum of the data processed with said filter (below). The constant trend of the spectrum of the latter should be noted, proving that the correlations introduced by the receiver have been successfully removed.

[0126] The decorrelation filter, being of finite length, can be implemented as a digital FIR. This leads to further advantages from the technological / implementation point of view:

[0127] • being digital, they are more reliable than an entirely analog implementation;

[0128] • they ensure greater implementation flexibility with respect to HR solutions; • they are numerically more stable with respect to HR implementations,

[0129] • for real-time applications, many of the hardware platforms designed for the purpose already implement dedicated modules so as to ensure an efficient implementation thereof.

[0130] However, it should be borne in mind that at the same length N, all the coefficients tend to be relevant, even if the underlying model may require a smaller number of coefficients. This is due to the fact that the solution of Yule-Walker (or Wiener- Hopf) equations, as already mentioned above, in practice, can be reconducted to a problem of least squares minimization (minimization norm L2):

[0131] To overcome the problem, it is possible to adopt regularization techniques for the LS solution: with which to promote solutions, in our case, with a more compact number of relevant coefficients. To this end, the matrix P can be placed: with constant c and k. It should be noted that linear transformations of P can also be adopted so as to consider the relationship between coefficients. Fig. 14 shows an application example of such a procedure: while in the LS solution all the coefficients participate in the solution, in the regularized case (with P defined as above) only the first 20s have a role (the others are clearly negligible). References

[0132] 1. Brunner, H. H., Comandar, L. C., Karinou, F., Bettelli, S., Hillerkuss, D., Fung, F., Wang, D., Mikroulis, S., Yi, Q., Kuschnerov, M., Poppe, A., Xie, C., & Peev, M. (2017). A low-complexity heterodyne CV-QKD architecture. International Conference on Transparent Optical Networks. https: / / doi.orq / 10.1109 / ICTON.2017.8025030

[0133] 2. Gabriel, C., Wittmann, C., Sych, D. et al. A generator for unique quantum random numbers based on vacuum states. Nature Photon 4, 711-715 (2010). https: / / doi.orq / 10.1038 / nphoton.2010.197

[0134] 3. Gehring, T., Lupo, C., Kordts, A. et al. Homodyne-based quantum random number generator at 2.9 Gbps secure against quantum side-information. Nat Commun 12, 605 (2021 ). https: / / doi.Org / 10.1038 / s41467-020-20813-w

[0135] 4. Marangon, D. G., Vallone, G., & Villoresi, P. (2017). Source-Device-Independent Ultrafast Quantum Random Number Generation. Phys. Rev. Lett., 118(6), 060503. https: / / doi.orq / 10.1103 / PhvsRevLett.118.060503

[0136] 5. Smith, P. R., Marangon, D. G., Lucamarini, M., Yuan, Z. L., & Shields, A. J.

[0137] (2019). Simple source device-independent continuous-variable quantum random number generator. Phys. Rev. A, 99(6), 062326. https: / / doi.orq / 10.1103 / PhvsRevA.99.062326

[0138] 6. Avesani, M., Marangon, D.G., Vallone, G. et al. Source-device-independent heterodyne-based quantum random number generator at 17 Gbps. Nat Commun 9, 5365 (2018). https: / / doi.Org / 10.1038 / s41467-018-07585-0

[0139] 7. Avesani, M., Tebyanian, H., Villoresi, P., & Vallone, G. (2021 ). Semi-Device- Independent Heterodyne-Based Quantum Random-Number Generator. Phys. Rev. Appl., 75(3), 034034. https: / / doi.Org / 10.1103 / PhysRevApplied.15.034034

[0140] 8. Marino, A. M., C. R. Stroud, Jr., Wong, V., Bennink, R. S., & Boyd, R. W. (2007). Bichromatic local oscillator for detection of two-mode squeezed states of light. J. Opt. Soc. Am. B, 24(2), 335-339. https: / / doi.Org / 10.1364 / JQSAB.24.000335 9. Fan, H., He, D., & Feng, S. (2015). Experimental study of a phase-sensitive heterodyne detector. J. Opt. Soc. Am. B, 32(10), 2172-2177. https: / / doi.Org / 10.1364 / JQSAB.32.002172

[0141] 10. Bruynsteen, C., Gehring, T., Lupo, C., Bauwelinck, J., & Yin, X. (2023). 100- Gbit / s Integrated Quantum Random Number Generator Based on Vacuum Fluctuations. PRX Quantum, 4(1 ), 010330. https: / / doi.Org / 10.1103 / PRXQuantum.4.010330.

Claims

CLAIMS1. A random number sequence generator comprising measuring means for measuring a plurality of continuous observables of an electromagnetic field, digital conversion means operatively connected with the measuring means for obtaining the measurement of each observable of the plurality of random number sequences, characterized in that the measuring means comprise one or more differential photodetection blocks (300) followed by electronic homodyne (EO) or electronic heterodyne (EE) type blocks, possibly in parallel, with single-frequency or multi-frequency optical / electrical local oscillators, followed by analog / digital converters and by a subsequent digital filtering.

2. A generator according to claim 1 comprising: a) N generators (100) of optical signals Pi, P2, ... PN, wherein each optical signal Pjis obtained by combining a number Kj of Gaussian states at different frequencies through an optical combiner (101 ); b) An Nx2M optical group (200) comprising a linear optical system which combines the N input optical signals to obtain 2M output optical signals; c) M differential photodetection blocks (300) adapted to transform the 2M optical signals into M electrical signals, wherein each differential photodetection block comprises: i. a pair of photodetectors (301 ) for converting a pair of optical signals exiting the optical group into two electrical signals;ii. a subtractor device (302) which receives the electrical signals from the two photodetectors (301 ) as input and provides an electrical difference signal on an output port; d) M electronic demodulators (400) demux-i, demux 2, demuxM, wherein each electronic demodulator demuxj comprises, at least in part, a plurality Dj of electronic homodyne (OE) or electronic heterodyne (EE) type components (400) connected in parallel with the same input electrical signal, each electronic homodyne or electronic heterodyne type component combining the input electrical signal with sinusoidal electrical signals of known frequency; e) a plurality of analog-digital converters (500) connected to the electrical outlets for converting the electrical signals exiting the M electronic demodulators (400) into number sequences; f) a plurality of digital filters (600) for eliminating any correlations present in the signal.

3. A generator according to claim 2, wherein at least one of the optical signals Pjcorresponds to a general quantum state, not necessarily Gaussian, while the other optical signals are Gaussian signals comprising coherent states, vacuum states, squeezed states or thermal states.

4. A generator according to one or more of the preceding claims, comprising: a) a 2x2 optical group (200) comprising a beam splitter (11 ) capable of combining an optical quantum state Pi entering a first port (31 ) with at least one known frequency-optical signal P2 entering a second port (51 ) and outputting the combined signal divided onto two outlet ports (61 );b) a differential photodetection block (71 , 81 ) which transforms the two optical signals exiting the beam splitter (11 ) into an electrical signal; c) at least one electronic demodulator (2) provided with at least one channel in which the input difference electrical signal is mixed with an input electrical signal with known frequency so as to obtain as output at least one electrical signal demodulated in baseband or at an intermediate frequency combination of the frequency of the input optical signal and the frequency of the input electrical signal; d) at least one analog-digital converter (4) connected to the at least one demodulator for converting the at least one demodulated electrical signal into output number sequences.

5. A generator according to claim 4, wherein the at least one electronic demodulator (2) is a phase and quadrature demodulator provided with two channels in which the input difference electrical signal (3) is mixed with the same input electrical signal (12, 22) at a known frequency but offset by 90° so as to output a pair of quadrature signals (X, P) demodulated in baseband or at an intermediate frequency as a combination of the frequency of the input optical signal (41 ) and the frequency of the input electrical signal (12, 22).

6. A generator according to one or more of the preceding claims, characterized in that it comprises a plurality N of electronic demodulators (2) that use sinusoidal electrical signals of known frequency connected in parallel with the same electrical signal as input difference (3) and with equal or different input electrical signals of known frequency (12, 22).

7. A generator according to claim 3, wherein the N electronic demodulators (2) have input electrical signals (12, 22) with different known frequencies, in particular, said fsthe center frequency of the quantum optical signal andthe frequency of the optical signal with known input frequency, the N demodulators (2) have, respectively, input electrical signals with frequency so that the central frequency of the demodulated electrical signal,or of the demodulated electrical signals in the event of in-phase and quadrature demodulators, exiting from each demodulator is8. A generator according to one or more of the preceding claims, wherein the known frequency- input optical signal (41 ) comprises a plurality of different known frequenciesthe electronic demodulator (2) comprising a plurality of electronic demodulators connected in parallel with the same input difference electrical signal (3) and with known frequency-input electrical signalsso that the central frequency of the demodulated electrical signal, or of the demodulated electrical signals in the event of in- phase and quadrature demodulators, exiting from each demodulator depends on a frequency of the plurality of frequencies of the known frequency-input optical signal.

9. A generator according to one or more of the preceding claims, wherein the N optical inlets consist of a quantum state, a local oscillator consisting of a combination of K Gaussian states, for example coherent states or thermal states, at known frequencies and of N-2 vacuum states and the optical group NxN consists of a unitary matrix U.

10. A generator according to one or more of the preceding claims, wherein the 2x2 optical group consists of a beam splitter, an optical inlet consists of a quantum state p, the other optical inlet consists of a thermal state at known frequencies, wherein the two output optical signals are converted into an electrical signal by a differential photodetection block and then demodulated with an electronic heterodyne or electronic homodyne type block.

11. A method for generating random numbers comprising: e) mixing an optical quantum state with an optical signal at predefined frequency: f) extracting an electrical quadrature of said quantum state by means of one or more differential photodetection blocks; g) subjecting the electrical quadrature to a homodyne or heterodyne demodulation by means of mixing with an electrical signal at predefined frequency so as to obtain at least one demodulated electrical signal in baseband or at an intermediate frequency as combination of the optical signal frequency and the electrical signal frequency; h) convert into digital the demodulated signal(s) to obtain a random number sequence.

12. A method according to one or more of claims 11 , wherein it is provided the step of subjecting the demodulated digital signal(s) to a decorrelation filter having a frequency response equal to:assum ing that the frequency response of the receiver system is the one of an auto-regressive model of the type:where N and a; are, respectively, the length and the parameters of the model, xnthe inlets and ynthe outlets, wherein, with N fixed, the coefficients a; are determined by solving the matrix system of linear equations αR = p where:being the autocorrelation coefficients of y(n).

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