The Odyssey of Entropy: Cryptography
Abstract
:1. Introduction
2. Entropy Measures and Related Concepts
2.1. Entropy Measures
2.2. Related Concepts
3. Entropy and Encryption
3.1. Modeling, Design and Implementation
3.1.1. Security Models Related to Entropy
- Entropic Security: Entropic security is a relaxed version of semantic security. In semantic security, only negligible information about the plaintext can be extractable (in any feasible way) from the ciphertext. More specifically, suppose that a probabilistic polynomial time algorithm (PPTA) knows the ciphertext c generated from a message m (regardless of the related distribution) and the length of m. The algorithm should still be unable to extract any partial information regarding m with a probability that is non-negligibly larger than all other PPTAs that know only the length of m (and not c).In entropic security, the cryptosystem needs to guarantee that the entropy of the message space is high from the point of view of the adversary [34]. A few research reports have worked on entropic security for high-entropy plaintexts [35]. Moreover, this model was used in honey encryption [36]. In honey encryption, decrypting the ciphertext using an incorrect key (guessed by the adversary) leads to a meaningful, but incorrect plaintext, which fools the adversary.
- Unconditional Security: A cryptosystem is said to have unconditional security (also called information-theoretic security) if the system is secure against adversaries with unlimited computational power, resources, memory space, and time. Information-theoretic methods and techniques have been utilized in studying unconditionally secure cryptosystems [37]. Some researchers have focused on this security model. For example, Renner and Wolf [38] investigated the possibility of asymmetric unconditional security, which corresponds to asymmetric-key cryptography in the computational security model.
- Provable Security: Some researchers have used entropy in provable security. For example, Kim et al. [39] argued that the assumption of uniform key distribution, which is made in traditional provable security, is far from reality. They modeled realistic key distributions by entropy sources. As another example, it was shown by Ruana [40] that the explicit authenticated key agreement protocol presented by Zheng [41] is vulnerable to impersonation attack due to the low entropy of the keys.
- Perfect Secrecy: Perfect secrecy (defined by Shannon) guarantees that , where P is the set of possible values for the plaintext and C denotes the set of possible values for the ciphertext. Put alternatively, a cryptosystem is perfectly secure if the adversary is unable to make any guesses about the plaintext, even in the case of full access to the channel (and, consequently, to the ciphertext). Several research works have focused on perfect secrecy. Gersho [42] argued that message quality degradation is inevitable for a perfectly secure cryptosystem that encrypts an analog message using a digital key with a finite size. He designed a perfectly secure analog signal encryption scheme that keeps the bandwidth of the encrypted signal from growing above that of the original analog signal without altering the key size or increasing the quality degradation incurred on the decrypted signal. The notion of “finite-state encryptability” for an individual plaintext sequence was introduced by Merhav [43] as the minimum asymptotic key rate required to guarantee perfect secrecy for that sequence. He demonstrated that the finite-state encryptability is equal to the finite-state compressibility (defined by Ziv and Lempel [44]) for every individual sequence. Perfect secrecy in radio signal encryption using DFT (discrete Fourier transform) was studied by Bi et al. [45]. They proved perfect secrecy to be asymptotically achievable for any baseband signaling method, provided that the signal block length approaches infinity. It is well known that the only real-world implementation of perfect secrecy (in its pure notion) is one-time pad (OTP), wherein the key is at least as long as the plaintext and needs to be updated with each new plaintext. However, some variants of perfect secrecy have received research focus. In the following, we review some well-studied variants of perfect secrecy as well as some related research works.
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- Perfect Forward Secrecy:Perfect forward secrecy depends on frequent changes in the encryption/decryption key (e.g., with each call or each message in a conversation, or each web page reload) in order to prevent the cryptosystem from being broken if a key is compromised. Several researchers have proposed encryption systems providing perfect forward secrecy to be used in different communication systems. For example, two email protocols with perfect forward secrecy were proposed by Sun et al. [46]. However, some flaws in the reasoning presented by Sun et al. [46] were reported by Dent [47], and two new robust email protocols with guaranteed perfect forward secrecy were introduced by Ziv and Lempel [48]. Later on, a method for cryptanalysis of the protocols proposed by Ziv and Lempel [48] was presented by Yoon and Yoo [49].In recent years, perfect forward secrecy has been considered in several other areas. For example, a lightweight transport layer security protocol with perfect secrecy was proposed by Pengkun et al. [50]. As another example, perfect secrecy is guaranteed to be provided by a high-performance key agreement protocol proposed by Yang et al. [51].
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- One-Time Pad (OTP):OTP is the only perfectly secure cryptographic scheme used in real-world applications. Although some researchers believe that OTP is more of a key safeguarding scheme than a cryptosystem [52], a vast number of research works have considered OTP as (part of) the security solution in a broad spectrum of applications. The feasibility of perfectly secure cryptography using imperfect random sources was studied by Dodis and Spencer [53]. Liu et al. [54] proposed an OTP cryptosystem in which the receiver does not need the OTP to decrypt the ciphertext, while the OTP fully affects the plaintext from the adversaries point of view. The application of OTP in scenarios where the receiver may not be trustworthy was studied by Matt and Maurer [55].An OTP-based cryptosystem was proposed by Büsching and Wolf [56] for BANs (body area networks), wherein messages are short, and large volumes of NVM (non-volatile memory) are available. This cryptosystem stores pre-calculated OTPs in the NVM for future use. Moreover, OTPs have been used in a spectrum of environments, such as multi-user one-hop wireless networks [57], IMDs (implantable medical devices) [58], UAVs (ynmanned aerial vehicles) [59], medical images [60], mobile instant messaging [61], coded networks [62] and credit cards [63]. OTP has been also used in quantum computing, especially in QKD (quantum key distribution) [64,65,66,67,68,69,70].
3.1.2. Cryptosystem Elements
- Encryption and Decryption Algorithms: Entropy has played a role in several research reports focusing on the design of encryption and decryption algorithms. Some of these research works are discussed below.
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- Encryption:The role of image block entropy in image encryption was studied by researchers [72] just like the case of image steganography (reviewed in Section 4.3.1). A multimedia encryption scheme based on entropy coding with low computational overhead was proposed by Xie and Kuo [73]. A method for encrypting entropy-coded compressed video streams without the need for decoding was introduced by Almasalha et al. [74]. Moreover, the encryption of entropy-coded videos was studied in some other research works. To mention a few, one may refer to Refs. [75,76,77,78]. The impact of key entropy on the security of an image encryption scheme was studied by Ye et al. [79]. Külekci [80] investigated the security of high-entropy volumes, where the most typical sources are entropy-encoded multimedia files or compressed text sequences. Min entropy was used by Saeb [81] to reduce the size of the key search space of an encryption scheme to a value lower than that of a brute-force or birthday attack. A chaotic encryption scheme for low-entropy images was proposed by Yavuz et al. [82]. This method uses confusion and diffusion techniques to make it difficult for the adversary to perform statistical analysis on adjacent pixels, which are likely to have close values.
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- Decryption:There are few works focusing on the security of the encryption algorithm. For example, the multiple decryption problem was introduced by Domaszewicz and Vaishampayan [83] as a generalization of the problem of source coding subject to a fidelity criterion. They used entropy to evaluate the security of a multiple-channel system in this scenario.
- Key Generation and Management Module: Several research works have focused on the role of entropy in key generation and management. Some of these works are briefly reviewed in the following.The entropy of the key has been of interest to researchers as a measure of security for decades [84]. Golic and Baltatu [85] used Shannon entropy analysis to evaluate the security of their proposed biometric key generation scheme. Wang et al. [86] tried to alleviate the quantization discrepancy problem in quantization-based key generation methods using an ECQS (entropy-constrained-like quantization scheme).It was highlighted by Shikata [87] that the existing bounds on the key entropy for retaining information-theoretical security are not tight enough. The reason is that existing random number generators do not create truly random sequences. More realistic bounds for the key entropy were derived in this research. "Personal Entropy" was introduced by Ellison et al. [88] as a means for remembering personal passphrases based on which secret keys are generated. Personal entropy is created via asking the user several personal questions.
- Key Agreement and Exchange Protocol: There are a few research works focusing on the applications of entropy in key exchange protocols. Among these works, we can refer to a framework designed by Luo et al. [89] for fingerprinting key exchange protocols using their impact on high-entropy data blocks. Another example is the key transmission method presented by Boyer and Delpha [90] for MISO (multiple-input single-output) flat-fading channels. This method tries to increase relative entropy using an artificial noise in order to minimize the BER (bit error rate) for the key receiver, while keeping it close to unity (maximum) for the eavesdropper.
3.1.3. Cryptographic Primitives
- Random Number Generation Algorithm: Entropy has been considered by researchers as a measure for randomness for decades [91]. One well-known issue with pseudo-random number generators is that the entropy of their output depends on the entropy of the seed. This issue was reported by Kim et al. [92] to exist in entropy sources of random number generators used in real-world cryptographic protocols, such as SSL (secure socket layer). Several research works have proposed methods for increasing the entropy of the seed via harvesting entropy from execution times of programs [93] or chaotic functions [94]. In recent years, entropy has been used as an objective for improving different chaos-based random number generators [95,96] as well as randomness tests for image encryption [97,98]. The criticality of entropy in research on true random number generators is due to the fact that their susceptibility to process variations as well as intrusion attacks, degrades the generated entropy. This makes it necessary to include an on-the-fly mechanism for the detection and correction of bias variations [99].
- Hashing Functions and Algorithms: A hash function with maximized conditional entropy was used by Lin et al. [100] as part of the solution to the ANN (approximate nearest neighbor) problem. Later on, Wang et al. [101] suggested LSH (locality sensitive hashing) as a promising solution to the ANN problem. However, they argued that in LSH, points are often mapped to poor distributions. They proposed a number of novel hash map functions based on entropy to alleviate this problem. Maximum-entropy hash functions were used in some other applications, such as packet classification [102]. The role of graph entropy in perfect hashing was studied by Newman et al. [103]. Later on, a graph entropy bound was calculated by Arikan [104] for the size of perfect hash function families. A fuzzy hash method based on quantum entropy distribution was used to construct a biometric authentication algorithm by Cao and Song [105]. Entropy measurement and improvement techniques were used by Zhang et al. [106] along with perceptual hashing for key frame extraction in content-based video retrieval. Moreover, entropy reduction on layout data combined with lossless compression and cryptographic hashing was used by Koranne et al. [107] to manage IP (intellectual property) via tracking geometrical layout from design through manufacturing and into production. Inaccessible entropy was used in the design of one-way hash functions by Haitner et al. [108]. A generator G is said to have inaccessible entropy if the total accessible entropy (calculated over all blocks blocks) is considerably smaller than the real entropy of G’s output. The possibility of designing a hash function with a hash-bit-rate equal to the conditional entropy was investigated by Li et al. [109].
3.1.4. Cryptographic Hardware
- Hardware Random Number Generators: The metal oxide semi-conductor (CMOS) implementation of full-entropy true random number generators was investigated by Mathew et al. [110,110]. CMOS is a fabrication process used in integrated circuits with high noise immunity and low static power consumption. Cicek et al. [111] proposed architectures for the CMOS implementation of true random number generators with dual entropy cores. In these implementations, different entropy sources, such as MRAMs [112], beta radioisotopes [113], the jitter of event propagation in self-timed rings [114] or thermal phenomena [115], were examined by researchers. Other hardware implementations depend on field programmable gate arrays (FPGAs) [116,117] or system-on-chip (SoC) devices [115]. An FPGA is a programmable semiconductor device consisting of a matrix of configurable logic blocks connected via networks of bistate connections. Furthermore, an SoC is a single integrated circuit containing (almost) all components of a computer such as a central processing unit, secondary storage, input/output ports, memory, etc. In the hardware implementation of true random number generators, objectives, such as power consumption [118], were considered by researchers.
- Physically Unclonable Functions (PUFs): In recent years, it was shown that some unclonable properties in some elements such as devices, waves or materials can vary randomly in different experiments or uniquely between similar elements. PUFs use these properties to create random and/or unique signals. They are used in cryptographic primitives, such as random number generation as well as message/device authentication. PUFs have been of interest to researchers in recent years [119,120]. The architecture of a PUF is shown in Figure 3.As shown in Figure 3, the core of a PUF is an unclonable element to which we simply refer as the element for short. The element can be a material, such as paper, carbon nanotube, etc. It can even be a wave, such as an optical or magnetic wave. However, most commonly, it is a device. It varies from sensors to microprocessors. The element along with its unique/random property (property for short) build the source of uniqueness/randomness (source for short). The property varies from eye-opening oscillation in humans to the geometry of the substrate in CMOS devices. As shown in Figure 3, an extraction circuit extracts this randomness, and (possibly) some post-processing improves the performance of the resulting signal to create the final output signal.Entropy analysis has appeared in several research reports focusing on the implementation of PUFs. For example, a connection between the min entropy and the randomness of PUFs was established by Gu et al. [121]. Gu et al. [122] and Schaub et al. [123] used entropy to evaluate the randomness of PUFs. Similarly, Koyily et al. [124] used entropy to evaluate the non-linearity of PUFS. Upper bounds on the entropy of some types of PUFs were calculated by Delvaux et al. [21]. Some bounds on the conditional min entropy of PUFs were presented by Wilde et al. [125]. Liu et al. [119] argued that some previously calculated upper/lower bounds on the entropy of PUFs are too loose or too conservative. They proposed a method for calculating a new bound via predicting the expectation of the point where min entropy bounds obtained from different experiments will converge. The loss of entropy in key generation using PUFs was studied by Koeberl et al. [126]. Other research works used PUFs as pumps of entropy [120].
3.1.5. Modification and Use of Existing Cryptosystems
3.2. Analysis and Evaluation
3.2.1. Entropy as a Security Measure
- As an Independent Measure: Entropy is a widely used security measure. Among the research works that have used entropy as an independent measure for evaluating cryptographic schemes, one may refer to the following. A multichannel system was introduced by Voronych et al. [135] for the purpose of structuring and transmitting entropy-manipulated encrypted signals. Schulman [136] argued that entropy makes a cryptographic pseudo-random number generator indistinguishable from a truly random number generator. He studied different ways of creating and increasing entropy. A method was introduced by Wua et al. [97] to measure the entropy of small blocks in an encrypted image. The average of the entropy over the blocks of an image was suggested as an efficient measure for evaluating the security of an image encryption scheme.
- Relation with Other Cryptographic Measures: In the following, we study the research works that have established connections between entropy and other security measures, such as unicity distance, malleability, guesswork, confusion, diffusion and indistinguishability.
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- Unicity Distance:The unicity distance of a cryptosystem is defined as the minimum number of ciphertext bits needed for an adversary with unlimited computational power to recover the key. The connection between entropy and unicity distance has been of interest to some researchers. For example, an entropy analysis presented by AlJabri [137] highlighted the unicity distance as an upper bound on the probability of the key being guessed by an eavesdropper.
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- Malleability:Consider a cryptosystem and a function f. Let us assume that encrypts a plaintext p to a ciphertext c, and encrypts to . If there is a transform g that guarantees , then is called a malleable cryptosystem with respect to the function f. The notion of non-malleable extractors was introduced by Dodis and Wichs [138] (inspired by the notion of malleability) for the purpose of symmetric-key cryptography from low-entropy keys. Later on, a widely believed conjecture on the distribution of prime numbers in arithmetic progressions was used by Dodis et al. [139] along with an estimate for character sums in order to build some new non-malleable extractors. Moreover, entropy analysis was used by Cohen et al. [140] to present an unconditional construction for non-malleable extractors with short seeds. Recently, some researchers worked on entropy lower bounds for non-malleable extractors [141].
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- Guesswork:There is a clear relation between entropy and guesswork. While entropy can be interpreted as the average number of guesses required by an optimal binary search attack to break a cryptosystem, guesswork is defined as the average number of guesses required in an optimal linear search attack scenario [142]. It was shown by Christiansen and Duffy [143] that if appropriately scaled, when the key is long enough, the expectation of the logarithm of the guesswork approaches the Shannon entropy of the key selection process. A similar research work studied the relation between guesswork and Rényi entropy [144]. Pliam [145] demonstrated that there cannot be any general inequality between Shannon entropy and the logarithm of the minimum search space size necessary to guarantee a certain level of guesswork. Another research reported by Malone and Sullivan [146] showed that entropy and guesswork cannot be interchangeably used in normal conditions. The LDP (large deviation principle) was used by Malone and Sullivan [147] to derive the moments of the guesswork for a source of information determined by a Markov chain. It was shown by Lundin [148] how entropy and guesswork can be simultaneously used to evaluate the security of selectively encrypted information.
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- Confusion and Diffusion:Confusion and diffusion are two properties suggested by Shannon [1] in order to make the statistical analysis of a cryptosystem as difficult as possible. Confusion states that the ciphertext is a complex function of several portions of the key, and this function cannot be simplified to an easily analyzable function. On the other hand, diffusion requires that each plaintext symbol affects several symbols in the ciphertext and each ciphertext symbol is a function of several symbols in the plaintext. This property diffuses the statistical structures of the plaintext over the symbols of ciphertext. The relation between entropy and the mentioned two properties were studied in several research works. For example, entropy was used to evaluate the security of chaotic confusion–diffusion image encryption schemes [149,150]. Moreover, Wu et al. [151] used entropy improvement techniques in combination with confusion and diffusion mechanisms in their proposed cryptographic schemes.
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- Indistinguishability:Indistinguishability states that given the ciphertext corresponding to a plaintext randomly chosen from a plaintext space with only two elements (determined by the adversary), the adversary will not be able to identify the encrypted message with a probability significantly greater than that of random guessing (). Indistinguishability plays a significant role in provable security. Some research works have investigated the relation between indistinguishability and entropy. As an example, one may refer the research reported by Hayashi [152]. In this research, smoothed Rényi entropy and min entropy were used to evaluate the indistinguishability of universal hash functions. Universal hash functions have many important applications in QKD (quantum key distribution), cryptography, privacy amplification (leftover hash lemma), error-correcting codes, parallel computing, complexity theory, pseudorandomness, randomness extractors, randomized algorithms, data structures, etc. (see [153,154,155,156,157,158] and the references therein).
3.2.2. Applications in Security Proof
- Zero-Knowledge Proof:Zero-knowledge proof is about proving the possession of some information by one party (the prover) to the other party (the verifier) without revealing the information itself. Zero-knowledge proofs are widely studied in cryptography. Goldreich et al. [159] further developed the notion of non-interactive statistical zero-knowledge proof introduces by De Santis et al. [160]. They used entropy measures to highlight some conditions under which every statistical zero-knowledge proof can be made non-interactive. Lovett and Zhang [161] studied some black box algorithms in order to be used in zero-knowledge proofs. These algorithms can reverse the entropy of a function. It was shown in this report that a black box function of this type incurs an exponential loss of parameters, which makes it impossible for such an algorithm to be implemented in an efficient way. A new hard problem related to lattices, named ILP (isometric lattice problem) was introduced by Crépeau and Kazmi [162], who used entropy to show that there is an efficient zero-knowledge proof for this problem.
- Random Oracle:Random oracles are widely used in security proofs in order to model perfect hash algorithms. A random oracle is a hypothetical black box that responds to each query by producing a truly random number uniformly chosen from a predefined domain. There are a few research works that use entropy-related concepts in the analysis of random oracles. For example, it was demonstrated by Muchnik and Romashchenko [163] that random oracles cannot help the extraction of mutual information.
3.2.3. Applications in Adversarial Analysis
- Cryptanalysis: Entropy measures have been frequently used in research works focusing on cryptanalysis [166]. In particular, chaotic image encryption methods were cryptanalyzed using entropy calculations [167]. Moreover, some researchers used different methods for the cryptanalysis of chaotic image encryption schemes that use entropy improvement techniques [168].
- Eavesdropping: Measures of mutual information in quantum key distribution and their applications in eavesdropping were investigated by Rastegin [30].
- Encrypted Data Analysis: The analysis of encrypted data is another relevant area of application for entropy. For example, entropy analysis was used for identifying encrypted malware [169], detecting encrypted executable files [170], and correcting noisy encrypted images [171]. Moreover, some researchers focused on entropy analysis of encrypted strings [172].
- Covert Channel: Entropy has played role in research on adversarial analysis of cryptosystems via covert channels. For example, entropy was used by Chen et al. [173] to analyze the capacity of a covert channel as well as the factors affecting it.
- Attacks: Entropy analysis was used as part of several kinds of attack scenarios. To mention a few, we can refer to the following.
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- CPA (Chosen Plaintext Attack):Kiltz et al. [174] used entropy measures in their analysis of instantiability of RSA and optimal asymmetric encryption padding (OAEP) under a chosen plaintext attack. OAEP is a padding scheme proposed by Bellare and Rogaway [175], which is often used along with RSA encryption. In another research reported by Bard [176], entropy was used in a CPA against SSL. Moreover, Bard [177] tested several modes of operation for resistance against a blockwise adaptive chosen plaintext attack.
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- CCA (Chosen Ciphertext Attack):Like the case of chosen plain text attack, entropy analysis has played role in chosen ciphertext attack adversarial analysis. For example, a public-key cryptosystem featuring resistance against CCA was introduced by Zhao et al. [178]. Entropy assessment was used in order to prove the security of this cryptosystem against after-the-fact leakage without non-interactive zero-knowledge proof. Similarly, Sun et al. [179] presented a CCA-secure identity-based encryption system and used entropy to show its resistance against key leakage attacks. Another research study on CCA-resistant and leakage-resistant cryptosystems was reported by Zhou et al. [180] in which entropy was used in the security proof.
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- Side Channel Attack:Mutual information measure is frequently used in side channel attacks. The reason is that mutual information is capable of detecting any kind of statistical dependency, and many side channel analysis scenarios depend on a linear correlation coefficient as a wrong-key distinguisher [181]. Moreover, some research works have used entropy analyses to make cryptosystems more secure against side channel attacks. For example, a method for decreasing the entropy of the information leaked from side channels was introduced by Dhavlle et al. [182]. As another example, an information-theoretical model for side channel attacks was derived by Köpf and Basin [183]. The impact of the entropy of the masks in masking-based countermeasures against side channel attacks was studied by Nassar et al. [184]. This study shows that while these countermeasures are usually studied with the maximal possible entropy for the masks, some particular mask subsets may leak remarkably more as the entropy increases.
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- Replay Attack:Entropy analysis has been used in the detection of replay attacks. As an example, we can mention the research reported by Liu et al. [185], wherein a novel feature based on spectral entropy was introduced for detecting replay attacks.
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- Key Negotiation Attack:
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- Backdoor Attack:As an example of the applications of entropy in backdoor attacks, we can mention the research reported by Young and Yung [188]. They argued that some backdoor attacks, such as Monkey, require the attacker to obtain a large number of ciphertext blocks all encrypted by the same symmetric key, each containing one known plaintext bit. They proposed a new backdoor that eliminates the need for known plaintext while leaking a bound on the plaintext entropy to the reverse engineer.
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- Dictionary Attack:Some researchers have worked on the role of entropy in dictionary attacks. For example, it was shown by Nam et al. [189] that low-entropy keys make some PAKE (password-authenticated key exchange) protocols, such as the one presented by Abdalla and Pointcheval [190], vulnerable to dictionary attacks.
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- Algebraic Attack:In addition to dictionary attacks, low-entropy keys make cryptosystems vulnerable to algebraic attacks. For example, the complexity of finding low-entropy keys using SAT (Boolean satisfactory problem) solvers was studied by Hromada et al. [191].
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- Collision Attack:It was demonstrated by Rock [192] that replacing random permutations by random functions for the update of a stream cipher causes entropy loss, which makes the cipher vulnerable to collision attacks.
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- Correlation Attack:Wiemers and Klein [193] argued that the correlation-enhanced power analysis collision attack against AES proposed by Moradi et al. [194] usually yields a set of keys (instead of one) due to noise-related problems. To alleviate this problem, they proposed a practical search algorithm based on a theoretical analysis on how to quantify the remaining entropy.
3.2.4. Analysis of Well-Known Cryptographic Schemes
- Analysis of Emerging Cryptographic Paradigms:In addition to traditional cryptographic schemes, entropy has been used in research on cutting edge cryptographic schemes and paradigms, such as quantum cryptography, homomorphic encryption, white-box cryptography and attribute-based encryption. Some related research works are briefly reviewed in the following.
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- Quantum Cryptography:Entropy was used by Bienfang et al. [200] and Bienfang et al. [201] in order to evaluate OTP video stream encryption that use quantum-generated secret keys. Arnon-Friedman et al. [202] used entropy to analyze the security of a device-independent quantum cryptography scheme. Moreover, entropy was used in several research works for the purpose of evaluating QKD (quantum key distribution) protocols [203,204].
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- Attribute-Based Encryption:A technique aimed at increasing the entropy available for proving the security of dual system encryption schemes under decisional linear assumption was presented by Kowalczyk and Lewko [205]. They showed the efficiency of their method in an attribute-based encryption scheme as a case study.
3.2.5. Analysis of Cryptographic Problems and Functions
3.3. Application
4. Entropy and Other Cryptographic Areas
4.1. Obfuscation
4.2. Message Authentication Codes
4.3. Cryptography-Based Privacy
4.3.1. Steganography and Steganalysis
4.4. User/Device Authentication
4.5. Digital Signature
4.6. Secret Sharing
5. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Zolfaghari, B.; Bibak, K.; Koshiba, T. The Odyssey of Entropy: Cryptography. Entropy 2022, 24, 266. https://doi.org/10.3390/e24020266
Zolfaghari B, Bibak K, Koshiba T. The Odyssey of Entropy: Cryptography. Entropy. 2022; 24(2):266. https://doi.org/10.3390/e24020266
Chicago/Turabian StyleZolfaghari, Behrouz, Khodakhast Bibak, and Takeshi Koshiba. 2022. "The Odyssey of Entropy: Cryptography" Entropy 24, no. 2: 266. https://doi.org/10.3390/e24020266
APA StyleZolfaghari, B., Bibak, K., & Koshiba, T. (2022). The Odyssey of Entropy: Cryptography. Entropy, 24(2), 266. https://doi.org/10.3390/e24020266