Security analysis of encrypted audio based on elliptic curve and hybrid chaotic maps within GFDM modulator in 5G networks

ABSTRACT


INTRODUCTION
The development of 5G wireless communications has resulted in a significant increase in data transfer and massive growth in the number and types of mobile programs.When data are transmitted in the public wireless channel, they are exposed to negative attacks that affect the security of the information represented by the authenticity, confidentiality, and integrity of data, which is a major concern.Consequently, data must be encrypted on many systems before being sent to provide security [1].
Communications security based on chaos theory has emerged as a new subject in wireless communication research in recent years due to it providing the strongest approaches to protect data that travel through insecure channels.The chaos signal is deterministic, aperiodic, nonlinear, and long-term prediction.Since it is highly sensitive to initial conditions and control parameters, no two chaotic systems will develop identically.The chaotic system is classified into one-dimensional and multidimensional based on the number of positive Lyapunov exponents in the chaos system.Multidimensional chaotic systems are more complex and unexpected than one-dimension because of their numerous control parameters and initial conditions [2].The production of unpredictable amounts, large enough and random enough, is necessary for the security of the majority of known cryptographic systems.Pseudorandom number generators (PRNG) can be generated using elliptic curves (EC).Elliptic curve cryptography (ECC) is a public-key algorithm that is significant in cryptography.Because of the fact of ECC has a small key space and offers a similar level of protection to other public-key algorithms that make use of larger key spaces.Additionally, it provides higher security levels while utilizing less computational power, memory, and bandwidth [3], [4].
Audio-based communication is increasing in the administrative, military, and various aspects of life.Audio has distinct characteristics, structure, and larger file sizes than images and texts.The encryption process guards the data against being accessed or destroyed by unofficial parties.The silent portion of the audio is filled with noise signals during encryption so that only a legitimate receiver can determine the audio content [5].
Massive multiple-input-multiple-output (MIMO) is an interesting wireless technique for meeting 5G requirements by maximizing capacity, throughput, and reliability.In order to improve the spectral efficiency of the transmission system, massive MIMO allows attaching thousands of device antennas to one base station.Also, spatial modulation (SM) scheme is employed with a 5G network to exploit some information resources in transmission [6].The parallel spatial modulation (PSM) divides the transmitted antennas into subgroups and SM is then carried out independently in each group using similar signal constellations [2].
Furthermore, a massive MIMO must integrate with a multicarrier scheme to deal with the frequency-selective channels in 5G wireless networks.Generalized frequency division multiplexing (GFDM) is a promising new waveform for multicarrier design and is regarded as a generalized form of orthogonal frequency division multiplexing (OFDM) because it is more adaptable to the parameter selection process.Arranging the data in a two-dimensional time-frequency block minimizes the number of cyclic prefixes (CP) compared to valuable information [7].The combination of massive MIMO-GFDM is very attractive to meet the ever-increasing needs for higher link readability and spectrum efficiency in 5G wireless communication networks [8].
Several audio encryption techniques have been proposed in past studies.Alwahbani and Bashier [9] presented audio encryption using two chaotic maps; a logistic map generates a one-time pad (OTP) for the diffusion phase and a circle map does the confusion phase.Alazawi and Kadhim [10] presented a speech encryption technique based on chaotic Lorenz fractional order.Saad and Hashim [11] introduced a voice encryption system using wavelet transforms and chaotic sequences to perform two-stage of key permutations.A voice encryption technique based on the principles of substitution and permutation by employing 2D logistic, Henon, and Baker maps was suggested in [12].Mahdi and Hreshee [13] proposed an encryption algorithm to encrypt audio files based on XORed between the voice signal and the binary sequences generated from the chaotic Henon signal.Ismael and Sadkhan [14] presented an audio encryption algorithm using three chaotic maps named Chen, Lorenz, and Henon that mix with audio using a linear and nonlinear function.Kordov [15] suggested a voice encryption scheme based on a circle map and rotation equation to perform permutation and substitution for voice samples.Raheema et al. [16] carried out an audio encryption model using Henon and logistic maps for the OFDM technique through the additive white Gaussian noise (AWGN) channel.Raheema et al. [17] suggested a voice encryption algorithm for the OFDM waveform using triple chaotic signals named (random logistic map, random Lorenz system, and random Chen system) through the AWGN channel.Abdelfatah et al. [18] presented an audio encryption algorithm based on ECC and utilizing a 2D logistic-Lorenz chaotic sequence and hash function.Khoirom et al. [19] introduced a voice encryption model using ElGamal encryption algorithm over a finite field.
Through the preceding literature, the classical and contemporary voice encryption techniques do not provide adequate protection, and some modern systems do not yet have protection.In addition, some cryptosystems take a long time to process and leave residue noise in the decrypted audio.In order to overcome these difficulties, in this work, a novel and powerful audio encryption has been proposed, which provides a high level of security to withstand attacks.The main contributions of the proposed cryptosystem in this paper are: -The encryption process performs inside communication components by securing the GFDM modulator.
-Minimizing residual intelligibility (R.I.) and high reconstructed audio signal quality.
-The subcarriers and subsymbols of GFDM are permuted and substituted using chaotic maps and EC-LCG sequence.-The Ikeda and Tent map sequences are mixed with the EC-LCG sequence based on the modulo operator and then combined with real and imaginary parts of each subsymbol inside GFDM.

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The remainder of the paper is formatted as follows.Section 2 briefly describes the techniques used, including PSM, GFDM, and massive MIMO.Section 3 introduces the principle of chaos theory in cryptography.Section 4 summarizes EC arithmetic over finite field.Section 5 presents the proposed audio encryption system using secure GFDM.Section 6 presents the performance evaluation and security tests for encrypted audio.Section 7 gives simulation results of the proposed scheme.Section 8 presents a security performance analysis and a comparison study with previous works.Finally, conclusions is in section 9.

BASIC PRINCIPLES 2.1. Parallel spatial modulation
The working principle of PSM can be summarized in the steps [7]: a.The transmitter antennas are divided into equal groups, each group contains g=Nt/P antenna, where 2≤g≤Nt, and only one antenna is activated (sends data) per group individually.When P=1, this means that g=Nt, and therefore we get the conventional SM. b.The information bits are divided into (P+1) parts, as seen in Figure 1, the first one part has log 2 () bits and the remaining P parts comprise log 2 () bits of each.c.The first part of information bits is used to modulate signal while the remaining parts is exploited to activate one antenna in each group d.The PSM spectrum efficiency in terms of bps/Hz can be expressed as ( 1):

Generalized frequency division multiplexing
GFDM is a digital and generic multicarrier modulation technique with pulse shaping, and it satisfies the different requirements of 5G networks.GFDM represents a non-orthogonal modulation method carried out on separate time-frequency blocks, each of which contains a collection of subcarriers in frequency and subsymbols in time.By circularly shifting in the time and frequency domain, the subcarriers on each subsymbol are filtered with applications that require a prototype filter.This approach will eliminate unwanted out-of-band (OOB) radiation and pave the way for spectrum distribution to succeed.
Also, GFDM provides additional properties in terms of reducing inter-carrier and inter-symbol interferences by using CP and pulse shaping.The time-frequency grid layout necessitates highly adaptable sophisticated methods in the receiver to achieve demodulating activities [20].A single CP is utilized for the whole block in GFDM, even if it contains multiple subsymbols, whereas one CP should be used for every subsymbol in OFDM, as shown in Figure 2(a).OFDM has a considerably high peak-to-average power ratio (PAPR).It could be reduced by raising the subcarrier's bandwidth and reducing the number of subcarriers.When designing GFDM and OFDM with the same length, the subcarriers become wider, and the block has lower subcarriers, lowering the PAPR in the GFDM block, as shown in Figure 2(b) [21].The time-frequency resource grid of GFDM consists of K and M, which denote the numbers of subcarriers and subsymbols, respectively.There are KM sample locations in each resource block.As a result, if KM=N is met, the quantity of information transmission by GFDM will equal the amount of transmission data for OFDM across the same symbol time and bandwidth.A pulse-shaping filter is used to identify the position of each resource block.The mathematical formula for the GFDM signal is as (2): The convolution functions are represented by where ( * ).In (2),  ̃[ −  ] ≜ [( − )  ] is the pulse shaping filter with  time-shifting, the mod operator is equal to the tail-biting procedure that makes a circular convolution filter.In (2) can simplify to OFDM when M=1 and the rectangular pulse shaping filter is used.Similarly, in (2) can convert to the single-carrier transmission in the case of K=1; therefore, this method is known as GFDM.The GFDM modulator is as described in Figure 3

Configuration of massive multiple-input-multiple-output system
This paper's point-to-point massive MIMO transceiver system comprises the number of Nt and Nr antennas at the transmitter and receiver, respectively.The received signal Y=[y1, y2, …, yNr] at the base station can be estimated using the formula Y=Hx+n and the received signal in matrix form as (3): Where

CRYPTOGRAPHY SYSTEM BASED ON CHAOS THEORY
Chaos theory is a mathematical field that studies nonlinear, dynamical, and complex systems that are apparently random but deterministic.Chaos theory deals with the study of deterministic differencedifferential mathematical equations that exhibit sensitive dependence on initial conditions (SDIC) by producing time pathways that appear random.This means that even a tiny variation in measuring the system's state can cause a significant change in its future behavior.This makes it impossible to predict the system's long-term behavior with any accuracy.
Chaos theory has been applied to cryptography to create innovative encryption techniques for audio, image, video, text, data, and watermarking encryption.Cryptography can use a pseudo-chaotic system that is based on chaos theory.Chaos initial conditions and parameters can be utilized as a cryptographic key.Cryptography use diffusion, representing the sensitivity of chaotic parameters to the initial condition.The initial state of the chaos sequence can be mixed with the plaintext to produce keys in cryptography.The final state of the chaos sequence represents ciphertext in cryptography.Asymptotic independence of initial and final states in chaos can be employed as confusion in cryptography.
The chaotic systems used in cryptography can be classified into two categories: maps and flows.Maps are represented by difference equations and are often referred to as discrete systems.Flows are represented by differential equations and are often referred to as continuous systems.Chaotic system dynamics behavior is described using the time domain called time series or in phase space called a strange attractor.
Within this taxonomy, chaotic maps are graded from one-dimensional to multidimensional depending on the number of variables in the equations of the chaotic maps.When mixing two or more types of chaotic maps, a chaotic hybrid map can be created with a more complicated chaotic property than the majority of single chaotic maps and more SDIC.This is because the hybrid approach takes advantage of all the strengths of the combined chaotic maps and attempts to reduce the weakness of one of the weak chaotic maps.Table 1 outlines discrete maps which are used in this paper [7], [25]- [27].

CURVE ARITHMETIC OVER FINITE FIELD
For a given prime number p, suppose Fp represents the finite field of p, so the EC over Fp can be defined by the relation in (4): where the coefficients a, b ∈ Fp and should be satisfied by (5): Scalar multiplication is necessary to perform point multiplication on EC.The fundamental operations of EC are point addition and point doubling.By assuming there are two points, P (x1, y1) and Q (x2, y2), belonging to Fp, the coordinate addition P+Q yields the third point, R (x3, y3), satisfying the EC equation as in ( 6)-( 8): In the subtraction case, the sign inversion of the y-coordinate of the second point is needed and solved as (9): Point multiplication calculations can be improved by effectively using point addition and doubling operations.Point multiplication can be calculated as many points are added together.For instance, when one needs to estimate nP, where n is a positive integer, then: The PRNG can be generated based on the group of points of an EC defined over a prime finite field.This work uses a linear congruential generator on elliptic curve (EC-LCG), a type of PRNG sequence.The EC-LCG sequence for a given Uo and G ∈ Fp, where G represents the generating point, and Uo represents the initial value (seed), is defined as (11): The initial value U0=(x0, y0), and the constants G, a, and b can be taken as secret keys in the cryptographic algorithm [19], [27], [28].In this work, the parameter values used are p=4,093, a=9, b=7, G=(4,1110), and U0 =(332,1395).Therefore, the keyspace of the EC-LCG sequence is 2 60 by assuming a and b are constant.

THE PROPOSED SECURE GFDM SYSTEM
The proposed GFDM modulator is described in detail in this section, as shown in Figure 4.The proposed secure model depends on substitution and permutation principles.Three chaotic and EC-LCG sequences are mixed separately with the data inside GFDM.Substitution is performed inside the GFDM system by separating subsymbols into real and imaginary parts.Then the result of the modulo operation between the Ikeda map and the x-coordinate of EC-LCG is combined with the real part of the GFDM subsymbol.Similarly, the result of the modulo operator between the Tent map and the y-coordinate of EC-LCG is combined with the imaginary part of the GFDM subsymbol.

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After that, the real and imaginary parts are swapped and reconstructed data again to complex-valued.For permutation operation, the data index is randomized using the Duffing map.Now, the proposed GFDM is being implemented onto 5G wireless communication networks architecture that consists of massive MIMO and PSM; the details of each part are illustrated in Table 2 and Figure 5. On the transmitter side, the original audio is sampled at a frequency of 8 kHz and saved in WAV audio format.Then, an analogue-to-digital converter (ADC) is used to convert the audio into a binary system.In order to satisfy the PSM requirement, the incoming bitstream is partitioned into six groups, each with 4 bits, one group used for 16 quadrature amplitude modulation (QAM) modulation, and the remaining groups used for antenna index.The outputs of each PSM group are modulated using the proposed GFDM and then transmitted using multiple antennas through the fading channel.On the receiver side, the decrypting process can be completed successfully by performing procedures similar to the transmitter but in the opposite direction and generating the same secret key and its positions.

SECURITY ASSESSMENT KEYS
The encryption algorithm must effectively protect audio transmitted on the public wireless channel and be impervious to eavesdropping techniques.A valuable indicator for considering and establishing a system's security needs is R.I.If the audio's R.I. is low, the audio is indistinct (high-security level) [2].The proposed cryptosystem algorithm's R.I. was evaluated using the following tests.

Signal to noise ratio
SNR can be used to assess the audio encryption/decryption waveform quality.When the SNR is low, this means a good encryption algorithm.The encrypted audio SNR values are calculated as (12): where   and   are original and encrypted audio samples, respectively.

Peak signal to noise ratio
The mean square error of the original (  ) and encrypted (  ) audio can be estimated as (13): Then, peak signal to noise ratio (PSNR) can be determined using (14): Where MAX denotes the maximum value of the encrypted audio.Lower PSNR implies that the audio has a high amount of noise, indicating that the encryption technique is good and resistant to attacks [7].

Spectral segment signal to noise ratio
The SSSNR is abbreviated as in (15): where xi and yi are DFT of clean and encrypted audio samples respectively.

Cepstral distance
The cepstral distance (dCD) is written as (16): Where   () and   () are the cepstral coefficient of original and encrypted audio respectively.Also, the linear predictive coding (LPC) coefficient can be used to calculate the dCD [13].

Frequency weighted log spectral distance (dFWLOG)
It is possible to understand audio with a frequency band of 300-500 Hz.In other words, the audio R.I. components are found within this frequency band.As a result, distance measures more appropriate for cryptanalysis of transform domain audio encryption would prefer accurate coefficient reposition in this frequency band.One method is to mask elements of the original and cryptanalysis audio signal spectrum to zero, restricting distance measurements to the 300-500 Hz region.A masking window can be applied to the spectrum of the two frames to compare these frequencies as (17): where a and b are the index of upper and lower frequency spectral coefficients, and n=b-a+1 [7].

SECURITY ASSESSMENT KEYS 7.1. Histogram analysis
The quality of scrambled audio can be determined using histogram analysis, which is a simple and practical approach.The original audio converts to random-like noise with an approximately flat sample value distribution to be a more secure encryption method.Figures 6(a)-(c) show the histogram of the original, encrypted, and decrypted audio, respectively.

Key space and sensitivity analysis
Keyspace and key sensitivity are major factors that affect audio encryption efficiency.Keyspace refers to the collection of secret keys used in audio encryption.When there is a small change in the encryption key, audio decryption is impossible, which means key sensitivity.Powerful audio encryption must have a large keyspace and great sensitivity to be secure against attacks [29].Table 3 shows the keyspace of each chaotic map employed in the proposed system.
Assuming that the precision of the calculation in MATLAB R2020a is about (10 -15 ), meaning that there are (10 15 ≈2 50 ) values for each secret key that can choose, then the overall secret keys space of the proposed model is 2,510.In order to examine the key sensitivity of the proposed system, one key is changed by a very small value, while the remaining keys are unaltered during implementation [5]

Time analysis
The speed with which a powerful encryption technique executes is an important quality factor.The encryption/decryption time is the duration required to complete the encryption/decryption technique procedures.This time is proportional to the length of the audio [30].Accordingly, the proposed algorithm is implemented in MATLAB R2020a under Windows 10, using a PC with Intel(R) Core (TM) i7-7500U @ 2.70 GHz 2.9 GHz, 8 GB RAM, and 64-bit operating system.Two audio files of different lengths were used.The computational time and speed can be summarized in Table 5.The model waveforms data are exhibited in Figures 7 and 8.The approach turns the original audio into practically noise-like encrypted audio.

Noise effect
In this part of the simulation results, the impact of noise on the recovered audio at the receiver was examined.This was done by calculating several metrics, including SSSNR, PSNR, dFWLOG, dCD, and MSE, which measure the quality of the audio.These calculations were performed between the original audio and the recovered audio of the proposed secure GFDM system at different SNR values.The results are presented in Figures 9-13, providing a comprehensive analysis of the system's performance under varying noise conditions.

SECURITY PERFORMANCE ANALYSIS
The investigation of the security performance in terms of the bit error rate (BER) was achieved, as shown in Figure 14.The BER of the authorized receiver and the eavesdropping receiver without knowing the secret keys for chaotic and EC-LCG sequences were tested.The plot shows that the eavesdropper could not retrieve the information without the secret keys due to the high BER of around 0.5.However, the legitimate receiver could obtain the information with an acceptable BER.In addition, to evaluate the weakness and strong points of the proposed cryptosystem, the suggested audio encryption technique's performance is compared to other related techniques.Keyspace, SNR, PSNR, SSSNR, P.Diff., dCD, dFWLOG, MSE, and speed are the security metrics considered for this comparison, as shown in Table 6.This comparison shows that the proposed method provides better results than existing works and is compatible with the requirements of 5G networks.

CONCLUSION
This paper proposed a novel audio encryption approach by employing several chaotic maps and EC arithmetic to encrypt audio in the massive MIMO-GFDM transmission system.This approach is based on chaotic maps of Ikeda, Tent, Duffing, and EC-LCG for the permutation and substitution processes inside

Figure 4 .
Figure 4. Block diagram of the proposed GFDM

Figure 5 .
Figure 5. Block diagram of the proposed cryptosystem

Figure 9 . 3477 Figure 13 .
Figure 9. SSSNR variation of the recovered audio Figure 10.PSNR variation of the recovered audio

Figure 14 .
Figure 14.BER of authorized and eavesdropper receiver

Table 2 .
Simulation parameters Security analysis of encrypted audio based on elliptic curve … (Mohammed Jabbar Mohammed Ameen) 3475 measurements to show the sensitivity key is the percentage of difference (P.Diff.),dCD,dFWLOG, MSE, SNR, PSNR, and SSSNR, as illustrated in Table4.When the values of P. Diff.MSE and dCD are high, and the values of SNR, PSNR, and SSSNR are small is means a large key sensitivity of the proposed system.The proposed system's secret keys are a, b, Xd (0), Yd (0), u, Xi (0), Yi (0), Xt (0), and µ for chaotic maps and p, Xo, Yo, Gx, and Gy for EC-LCG.The following factors should be considered when evaluating the proposed system from a statistical point of view: i) minimizing SNR, SSSNR, and PSNR values implies the reduction of R.I. in encrypted audio (high-security level) and ii) the increased dFWLOG, dCD, and MSE values mean a low R.I. of audio.

Table 3 .
Key space of proposed cryptosystem

Table 4 .
Key sensitivity examination of the proposed cryptosystem using audio-2 with a duration of 3 second

Table 6 .
Security metric comparison with previous techniques