Cryptanalysis of a random number generator based on continuous‐time chaos

IET Circuits, Devices & SystemsVolume 14, Issue 5 p. 569-575 Research ArticleFree Access Cryptanalysis of a random number generator based on continuous-time chaos Kaya Demir, Kaya Demir orcid.org/0000-0002-4915-6386 TÜBİTAK – Informatics and Information Security Research Center, PO Box 74, Gebze, 41470 Kocaeli, TurkeySearch for more papers by this authorSalih Ergün, Corresponding Author Salih Ergün salih.ergun@tubitak.gov.tr TÜBİTAK – Informatics and Information Security Research Center, PO Box 74, Gebze, 41470 Kocaeli, TurkeySearch for more papers by this author Kaya Demir, Kaya Demir orcid.org/0000-0002-4915-6386 TÜBİTAK – Informatics and Information Security Research Center, PO Box 74, Gebze, 41470 Kocaeli, TurkeySearch for more papers by this authorSalih Ergün, Corresponding Author Salih Ergün salih.ergun@tubitak.gov.tr TÜBİTAK – Informatics and Information Security Research Center, PO Box 74, Gebze, 41470 Kocaeli, TurkeySearch for more papers by this author First published: 30 April 2020 https://doi.org/10.1049/iet-cds.2019.0356Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study presents the algebraic cryptanalysis of a continuous-time chaotic random number generator (RNG) based on a Zhongtang chaotic system. The chaos-based RNG provides key values for logic encryption, round key formation and the substitution box (S-box) generation steps in AES (advanced encryption standard). In this study, an attack method is proposed to exploit the weaknesses of the chaos-based RNG used in this cryptographic system. By observing a chaotic state variable and knowing the structure of the target RNG, output bit sequence of the target RNG is reproduced by a clone RNG used in master–slave synchronisation scheme. The feasibility of the attack system is confirmed through simulation and numerical results. Through this attack method, the same S-box parameters and key values can be obtained which can be used for decryption of the proposed encryption algorithm. This study targets a specific continuous-time chaos-based RNG and the associated cryptographic system as a case study. However, the cryptanalysis method described in this study is applicable to any continuous-time or discrete-time chaos-based RNGs. Therefore, this study highlights the security vulnerabilities of chaos-based RNGs and emphasises that deterministic chaos phenomena itself should not be considered as the actual source of randomness. 1 Introduction With ever-increasing usage of electronic financial transactions, digital signature and encrypted messaging applications, information secrecy has become a major concern. Therefore, cryptographic systems turned into inevitable components responsible for protecting the confidentiality, the integrity and the authenticity of information. Almost all cryptographic systems rely on random number generators (RNGs) to produce key values for use in their associated specific encryption algorithms, one-time pad [1], cryptographic nonces, padding bytes and blinding values [2]. In order to ensure the information security, the RNG deployed in a cryptographic system must satisfy the following criteria: (i) the next bit of a RNG must be unpredictable [3], (ii) the output bit sequence of a RNG must be unreproducible even if all details about the RNG structure and random bit generation method are known and (iii) the output bit stream must qualify all statistical tests of randomness [4]. Therefore, it is of utmost importance to cryptanalyse a RNG to reveal its security weaknesses and its susceptibility to a variety of attacks before it is deployed in a cryptographic system. Basically, RNGs are made up of three elements: (i) a physical entropy source such as thermal noise from a resistor, jittered oscillator, charge on a capacitor or quantum phenomena [5], (ii) a sampler such as a comparator or a D flip-flop, (iii) a post-processing operation such as Von Neumann or XOR to alleviate statistical imperfections such as 1/0 balance in RNG output. Four fundamental methods are frequently used for building RNGs: (i) amplification of a noise source [6, 7], (ii) dual-oscillator architecture with a jittered clock [8], (iii) discrete-time chaotic maps [9] and (iv) continuous-time chaotic oscillators [10-12]. Compared to other methods, utilisation of continuous-time chaotic oscillators is shown to achieve higher throughput without need for amplification and post-processing [13, 14]. Therefore, RNG applications based on continuous-time chaotic systems have received great attention. A cryptographic system is secure as long as the key values produced by the RNG cannot be accurately predicted by an attacker [15]. However, chaos is defined by deterministic equations and if purely deterministic chaos is used for generation of random numbers, the output of the RNG can be obtained through synchronisation of chaotic oscillators, thus information security can be compromised [16]. In chaos-based RNGs described in [17, 18], the effect of circuit noise is analysed to address this security issue. Contrarily, the RNG proposed in [19] uses solely the deterministic chaos as the source of randomness. In [20], a novel hybrid encryption algorithm based on the chaotic RNG proposed in [19] and S-AES algorithm is presented where the continuous-time chaotic signals corresponding to chaotic state variables are used to generate the crypto keys and S-box values. Then a sample image is encrypted and decrypted in order to demonstrate the feasibility of the proposed algorithm. This paper presents the cryptanalysis of this aforementioned encryption algorithm utilising a chaos-based RNG and S-AES as described in [19, 20]. The chaotic RNG is targeted in order to illustrate the possible security issues and vulnerabilities when deterministic chaos is used as the sole entropy source for generation of random numbers. An attack system is proposed in order to synchronise to the target RNG by means of master–slave synchronisation scheme and accurately predict the output of the target RNG as described in [21]. The mathematical analysis for achieving synchronisation of attack and target systems is also revisited. Although this paper targets the specific chaos-based RNG and the cryptograhic system described in [19, 20], this study comprehensively presents a cryptanalysis method which can be applied to any chaos-based RNG. Therefore, this study highlights the vital importance of the RNG security in a cryptographic system on information security. In Section 2, target RNG system is explained in detail. Section 3 describes the theory behind the attack method to cryptanalyse the target RNG by using three clone systems corresponding to each observed chaotic state variable. Section 4 presents numerical results depicting the convergence of target and attack RNGs, followed by conclusions. 2 Target system In this study, the target system is a RNG based on a Zhongtang continuous-time chaotic system as described in [19, 20]. Chaotic systems are defined by deterministic equations. However, their non-linearity, extreme sensitivity to initial conditions and divergent aperiodicity due to positive Lyapunov exponents make chaotic signals unpredictable in the long term, thus attractive to use in RNGs. Chaotic systems can be divided into two broad categories namely discrete-time and continuous-time depending on their change over time. In discrete-time chaotic systems, switched capacitors are often used with multipliers and op-amps, leading to complicated circuits. Conversely, continuous-time chaotic systems can be realised as simpler electronic circuits as they do not require sample-and-hold and multiplier blocks. Therefore, continuous-time chaotic systems are advantageous over discrete-time chaotic systems in terms of their ease of implementation. In [19], a continuous-time chaotic system named as scaled Zhongtang system is chosen as the core of RNG due to its high chaotic features and wide-band spectrum. The equations defining the continuous-time chaotic system are given as (1) The initial conditions and parameters in [20] are given as (2) Using the initial conditions and parameters in (2), (1) can be numerically solved using a fourth-order Runge–Kutta algorithm and a fixed step size as also used in [19]. The phase portraits of chaotic state variables are shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Phase portraits of the Zhongtang chaotic system(a) , (b) , (c) , (d) For the given parameters, Lyapunov exponents corresponding to this chaotic system can be calculated approximately as (3.87, 0, −37.95) using Wolf's algorithm [22]. As there is only one positive Lyapunov exponent, the system exhibits chaotic behaviour. Having seven parameters enabling wider key space and depicting a wider frequency range are the reasons to use this chaotic system according to [19]. The bit generation method is based on numerically calculating the float values of x - y - z chaotic state variables with a fixed step size of and with a precision of 15 digits after the comma. Then mod 2 operation is applied on each of 15 digits. Therefore, 15 bits are created from each calculated x - y - z value. Then xy, xz and yz bit series are formed by putting each digit through bit-wise xor operation. Following this method 1 MBit length series is generated for all three combinations (xy, xz and yz) and it is shown that all the bit series satisfy the NIST-800-22 statistical tests of randomness. The yz series is used in logic encryption and the xy series are used in round key generation parts of the AES (advanced encryption standard) algorithm. In [20], generation of the S-box is also accomplished similarly by using the x - y series generated directly from the solutions of the chaotic system equations. The least significant 8 digits after the comma belonging to x and z phases are put through a bit-wise xor operation. Then the decimal value in the range 0–255 is calculated and input to the S-box unless the S-box contains that value. The process is continued until all 256 components of S-box are filled with distinct numbers. Then as a demonstration of the hybrid algorithm based on chaos, an image is encrypted and decrypted using this S-box and AES algorithm. In this chaos-based encryption scheme, initial conditions and parameters of the chaotic system are suggested as encryption keys. Due to the sensitivity of chaotic systems to initial conditions, it is claimed that the chaos-based hybrid encryption algorithm is impossible to decrypt [20]. 3 Attack system A continuous-time chaotic signal exhibits unpredictable irregular behaviour and wide-band spectrum similar to noise. Therefore, chaotic signals appear convenient for generation of random numbers. However, short-term predictability of chaotic systems introduce concerns about security [23, 24]. A variety of cryptanalysis methods such as return-map attacks [25], prediction attacks [26], synchronisation attacks [27] and parameter identification [28] attacks have been reported regarding the security of continuous-time chaos-based systems. In this study, the attack system is based on the synchronisation of chaotic systems as described in the pioneering study by Pecora and Carroll [29, 30]. Convergence of target and attack systems is achieved using the master–slave synchronisation by linear continuous feedback method. Corresponding to each observed chaotic state variable, three clone systems are proposed in order to provide an algebraic cryptanalysis of the target RNG system. Synchronisation between clone and target systems is accomplished by adjusting the feedback coupling coefficient such that the largest conditional Lyapunov exponent (CLE) becomes negative. The continuous-time chaotic system given in (1) is targeted by three clone systems while the information available is the structure and the scalar time series of the corresponding chaotic state variable. In this paper, all parameters a, b, c, d, f and g of the target RNG system are assumed to be known. However, there are also mathematical methods to extract these parameters [28] which is beyond the scope of this paper and will be analysed in a future study. 3.1 Clone system for is observable Assuming out of the three chaotic state variables only is observable and applying continuous linear coupling, the equations defining the clone system for cryptanalysis of the target RNG is given as (3) where k is the strength of coupling between the target (master) and clone (slave) systems for continuous linear feedback. In this study, the parameters of the clone RNG system are assumed equal to the parameters of the target RNG. The error signals are defined as , , . To achieve synchronisation, the purpose is to adjust k such that as . To find the range of k values where the master–slave synchronisation is stable, the CLEs are calculated by defining the Jacobian matrix of the difference system and calculating its eigenvalues by using QR decomposition method and fourth-order Runge–Kutta algorithm. The equations defining the difference system can be obtained by subtracting (4) from (1). Assuming the parameters of target and clone systems are identical (4) The derivatives of with respect to , and can be found as (5) The derivatives of with respect to , and is given by (6) While calculating , is assumed, whereas while calculating , is assumed as described in [31]. The derivatives of with respect to , and can be found by (7) Similarly, during calculation of , is assumed, whereas while calculating , is assumed as in [31]. Then the Jacobian of the difference system can be defined as (8) The CLE values are the Lyapunov exponents of the matrix (8). As there are three chaotic state variables, the CLE spectrum consists of three components and if the largest CLE is negative, then synchronisation is achievable and stable. Fig. 2 illustrates the variation of largest CLE with respect to coupling strength k. It is observed that largest CLE does not become negative despite changing coupling strength k values. Therefore, when the observable scalar time series is , the synchronisation between the target and the clone RNG is not possible. Fig. 2Open in figure viewerPowerPoint Change of largest CLE with coupling strength k for observable 3.2 Clone system for is observable Similarly, assuming the available scalar time series is , then the clone system for cryptanalysis of the target RNG is given as (9) Following the similar methodology, the Jacobian of the difference system can be written as (10) CLE values can be calculated using (10). Fig. 3 displays the relation between coupling strength k and CLE when is observable. It is seen that for , the largest CLE is negative, thus the synchronisation between target and attack RNGs is possible and stable. Fig. 3Open in figure viewerPowerPoint Change of largest CLE with coupling strength k for observable 3.3 Clone system for is observable If the observable chaotic state variable is and continuous linear coupling is assumed, the clone system for cryptanalysis of the target RNG is given as (11) Then the Jacobian of the difference system can be defined as (12) Similarly, using (12) and Wolf's algorithm [22], CLE values can be calculated. The change of largest CLE with coupling coefficient k for is shown in Fig. 4. It can be observed that the largest CLE is positive for a wide range of coupling strength k values. Hence, the synchronisation of target and clone systems is not possible by observing only chaotic state variable. Fig. 4Open in figure viewerPowerPoint Change of largest CLE with coupling strength k for observable Fig. 5Open in figure viewerPowerPoint Unsynchronised behaviour of and while is observable and Fig. 6Open in figure viewerPowerPoint Unsynchronised behaviour of and while is observable and Fig. 7Open in figure viewerPowerPoint illustrating the synchronisation of target and clone systems while is observable and Fig. 8Open in figure viewerPowerPoint illustrating the synchronisation of target and clone systems while is observable and Therefore, by adjusting the value of coupling strength k, observing the chaotic state variable and applying linear continuous coupling, it is possible to obtain stable synchronisation between the target and clone RNGs hence, predict the next bit and reproduce the bit stream of the target RNG simultaneously. 4 Numerical results Using the proposed attack method, clone systems defined by (4), (9) and (11) are numerically modelled, and coupling strength is applied while the observable scalar time series are , and , respectively. As the largest CLE is positive for any value of coupling strength k when either or is observable, synchronisation between target and clone RNG is not possible. The unsynchronised behaviour is illustrated in Figs. 5 and 6 while the available scalar time series is from and , respectively. On the other hand, if is observable, then the CLE is negative for the coupling strength . Therefore, the synchronisation between the target and attack RNG is possible and stable. Figs. 7-9 illustrate the transition from unsynchronised behaviour shown in black to synchronisation shown in red corresponding to each chaotic state variable. Fig. 9Open in figure viewerPowerPoint illustrating the synchronisation of target and clone systems while is observable and Figs. 10-12 demonstrate the evolution of error signals defined as , and for the case of synchronisation. As observed, the error values approach to zero as the corresponding CLE value is negative. Fig. 10Open in figure viewerPowerPoint Error function values while is observable and Fig. 11Open in figure viewerPowerPoint Error function values while is observable and Fig. 12Open in figure viewerPowerPoint Error function (red) while is observable and Additionally, Fig. 13 shows a comparison between time to synchronisation of the clone system to target system for increasing coupling strength values of 15, 20, 30 and 40. It is observed that the time needed for convergence of clone RNG to attack RNG decreases with increasing coupling strength while the magnitude of largest negative CLE increases as shown in Fig. 3. Fig. 13Open in figure viewerPowerPoint Synchronisation errors of clone and target in time domain for varying coupling strength k values while is observable Based on previous discussions, despite starting from different initial conditions, identical synchronisation of clone and target systems can be accomplished and synchronisation is stable as long as the largest CLE is negative for coupling coefficient and observable scalar time series is from the chaotic state variable. Therefore, the next bit of the target RNG can be exactly predicted and the identical bit stream can be reproduced. This demonstrates that despite passing NIST statistical test suite of randomness, the target RNG in [19] is not cryptographically secure as it does not fulfil the first and second criteria of a secure RNG which state that it must be impossible to predict the next bit of the RNG and it must be impossible to regenerate the identical RNG output even though all details about the RNG are known, respectively. The security of any cryptographic system is jeopardised once the RNG output becomes predictable. In [20], as a demonstration of the use of RNG in a cryptographic system, the outputs of the chaotic RNG are used to provide the key values and S-box (substitution box) values for encryption of an image using AES algorithm. The AES algorithm is a commonly used standard encryption method. However, the strength of an encryption algorithm is directly related to the difficulty of predicting the crypto key values. This study showed that the output of the target RNG is obtained from the continuously coupled clone RNG by master–slave synchronisation. Furthermore, the deterministic steps for generating the encryption keys and the S-box values are also known as they are described in detail in [19] and also summarised in this paper. Therefore, the logic encryption key, round keys and the S-box values used in implementing the AES algorithm can be obtained following the same procedure as described in [19]. Additionally, deterministic steps of encryption and decryption processes are also known as they are presented in [20]. The strength of a cryptographic system relies on the secrecy of the key values rather than the type or complexity of the deterministic steps of encryption/decryption algorithm. Thus, using the key values obtained through synchronisation and following the deterministic steps of the algorithm, the encrypted image can be decrypted as illustrated in Fig. 14. Therefore, this study demonstrates that the chaos-based RNG in [19, 20] is not secure, despite the RNG output satisfies NIST 800-22 statistical tests of randomness. Fig. 14Open in figure viewerPowerPoint The original, encrypted and decrypted images of a sample image(a) Original image, (b) Encrypted image using key values from chaotic system, (c) Decrypted image 5 Conclusions The cryptanalysis of a continuous-time chaotic RNG is presented. As chaos is a noise-like but deterministic phenomena, it cannot be considered as the actual entropy source in a RNG. Instead, the non-deterministic drift in initial conditions is the actual source of randomness. The described method in this study is applicable to cryptanalyse any continuous-time or discrete-time chaos-based RNGs; however, as an example in this paper, the target RNG is the continuous-time chaos-based RNG utilising a Zhongtang chaotic system. As the target RNG wrongly uses deterministic chaos as the sole entropy source, it is subjected to cryptanalysis study. An attack method is proposed to reveal the security vulnerabilities of the chaos-based RNG. The theoretical background behind the synchronisation of chaotic systems is presented. Using master–slave synchronisation scheme, stable synchronisation between target and clone RNGs is numerically verified. By knowing the target RNG structure and observing scalar time series from a chaotic state variable of the target RNG, the output of bit stream of target and clone RNGs are synchronised by adjusting the magnitude of coupling. Through master–slave synchronisation scheme, it is possible to obtain the crypto-keys and S-box values generated by the chaotic RNG and used in the hybrid AES encryption algorithm, thus an image encrypted with the proposed algorithm can be decrypted. It is demonstrated that information security becomes compromised as the strength of any encryption method depends on the unpredictability of the key values generated by the associated RNG rather. 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