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Information-theoretic security
In 1949, Shannon introduced the principle of measuring the secrecy level of a communication system by a quantitative value [44]. He considered the model depicted in Figure 1.5 where a transmitter, Alice, wants to convey messages to a legitimate receiver, Bob, through a perfect channel, while an eavesdropper, Eve, is wiretapping the communication. He supposed that Eve also observes an error-free copy of all the transmitted messages. He dened the notion of perfect secrecy which is achieved if the eavesdropper’s equiv-ocation of the message remains the same with and without the knowledge of its received vector H(MjZ) = H(M): (1.10)
This is equivalent to having zero mutual information between the source message and the eavesdropper’s received vector I(M;Z) = 0: (1.11)
In his considered model, this perfect secrecy is achieved if the transmitted codeword is statistically independent of the message. It can be guaranteed by means of a one-time pad where the message is secured using a secret key known only by the transmitter and the legitimate receiver, and the key is used only once. For instance, the codeword can be computed as the binary addition (XOR) of the message and the secret key. However, the problem is that, since each key cannot be used more than once, the transmitter and the legitimate receiver have to store long sequences of random keys and share them over a secure channel. That caused information-theoretic security to be regarded as a theoretical concept unfeasible in practice.
In 1975, Wyner was the rst to introduce the concept of creating security by exploiting the randomness of the noise present in all communication systems [45], which is known today as physical layer security. He considered the channel model depicted in Figure 1.6 which he called the wiretap channel. In this model, Alice conveys messages to Bob through a discrete memoryless channel (DMS). Eve observes the communication through a second discrete memoryless channel. The channel between Alice and Eve is called the wiretap channel and is considered to be degraded with respect to the main channel.
Main results in physical layer security
After Wyner’s discovery, determining this secrecy capacity has been addressed extensively and has led to a plethora of information-theoretical results for many classes of channels such as erasure, Gaussian, MIMO, broadcast, interference and relay channels (see [47] and references therein). The Gaussian wiretap channel was introduced in [48] and was then investigated in dierent scenarios [49, 50, 51]. Secure communications Sarah Kamel 15 over fading channels were also investigated [52, 53, 54]. The point-to-point scenario was extended to multi-user systems for dierent channels, e.g. multiple access channel [55, 56, 57], broadcast channel [58, 59, 60, 61] and relay channel [62, 63, 64]. In order to achieve this secrecy capacity, practical wiretap codes were designed based on the application of classic and modern coding techniques [65, 66]. First approaches considered capacity-achieving graph-based codes such as LDPC codes [67] and extended their applications for secrecy [68]. Then, some researches examined secrecy rate using structured nested lattice codes for wiretap coding [69, 70, 71]. Polar codes were also considered as secrecy achieving codes for some wiretap channels [72, 73, 74]. Physical layer security issues were also studied based on network coding theory in [75]. Secrecy capacities of multiple antenna channels were extensively studied. The main challenge in securing multiple-input multiple-output (MIMO) channels is that the wiretap MIMO channel is not degraded in general. Secret communications in MIMO channels where multiple antennas are used for transmission and reception were rst studied by Hero in [76]. He introduced two constraints, called low probability of intercept and low probability of detection, and studied them in dierent scenarios where the transmitter, the legitimate receiver and the eavesdropper are either informed or uninformed about their channel states. The secrecy capacity of the single-input multiple-output (SIMO) wiretap channel where the eavesdropper has more than one antenna was analyzed in [77]. The secrecy capacity of the multiple-input single-output (MISO) wiretap channel was also investigated when the eavesdropper has a single antenna [78, 51] or multiple antennas [79, 80]. These studies were then generalized to the MIMO channel with single or multiple eavesdropper antennas [81, 82, 83, 84].
Original GGH scheme
The Goldreich-Goldwasser-Halevi cryptosystem stemmed from the lattice closest vector problem and was proposed originally in [1]. The authors took advantage of the fact that it is easy to generate a random vector close to a lattice point; however, it is hard to nd the lattice point which is the closest to this random vector. This is the main idea on which is based the GGH trapdoor function. Lattice-based GGH scheme uses also the lattice properties of having an innite number of bases and generating easily a lattice point using any basis of the lattice. However, after adding a noise to the lattice point and obtaining a \close-to-lattice » point, only a near orthogonal basis can be used to recover the initial lattice point. Thus, the GGH private key is chosen as a good orthogonal basis allowing simple recovery of the encrypted message. For the public key, a bad non-orthogonal basis is chosen rendering the decryption by any attacker nearly impossible. The badness of the basis is measured by its orthogonality defect, dened in (1.7).
Table of contents :
Acknowledgements
Abstract
Table of contents
List of Figures
List of Tables
List of Abbreviations
Resume Detaille de la These
Introduction
1 Security in Communication Systems
1.1 Cryptography
1.1.1 Code-based cryptography
1.1.2 Lattice-based cryptography
1.2 Information-theoretic security
1.2.1 Main results in physical layer security
1.2.2 Secrecy in caching scenario
1.3 Conclusion
2 GLD Lattice-Based Cryptosystem
2.1 Original GGH scheme
2.2 GGH improvements
2.2.1 Micciancio’s scheme
2.2.2 LDLC scheme
2.2.3 Other GGH improvements
2.3 GLD lattice-based cryptosystem
2.3.1 GLD lattices
2.3.2 Proposed public-key scheme
2.4 Security analysis
2.4.1 Brute-force attack
2.4.2 Decoding attacks
2.4.3 Dual code attacks
2.5 Complexity analysis
2.5.1 Public key size
2.5.2 Key generation
2.5.3 Decryption
2.6 Conclusion
3 Individual Secrecy in Caching Scenario
3.1 Problem denition
3.1.1 Channel model
3.1.2 Message library and receiver demands
3.1.3 Caching phase
3.1.4 Delivery phase
3.1.5 Secure capacity-memory tradeo
3.2 Secure joint cache-channel coding
3.2.1 Message splitting
3.2.2 Codebook generation
3.2.3 Caching phase
3.2.4 Delivery phase
3.2.5 Decoding at receiver 1
3.2.6 Decoding at receiver 2
3.2.7 Analysis of the error probability
3.2.8 Analysis of the information leakage
3.2.9 Securely achievable rate-memory tuples
3.3 Upper bound under one-sided cache assignment
3.3.1 Proof of the upper bound
3.4 Lower bound under symmetric cache assignment
3.4.1 Message splitting
3.4.2 Caching phase
3.4.3 Delivery phase
3.4.4 Analysis of the error probability
3.4.5 Analysis of the information leakage
3.5 Upper bound under symmetric cache assignment
3.6 Separate cache-channel coding
3.6.1 Message splitting
3.6.2 Caching phase
3.6.3 Delivery phase
3.6.4 Analysis of the probability of error
3.6.5 Analysis of the information leakage
3.7 Discussion and numerical results
3.7.1 Impact of the secrecy constraint
3.7.2 Impact of cache assignment
3.7.3 Impact of joint cache-channel coding
3.8 General lower bound
3.8.1 Scheme achieving rate-memory pair (R(K)
3.8.2 Scheme achieving rate-memory pair (R(K)
3.8.3 Scheme achieving rate-memory pair (R(K)
3.9 General upper bound
3.10 Examples
3.11 Conclusion
4 Joint Secrecy in Caching Scenario
4.1 Problem denition
4.2 Lower bound under one-sided cache assignment
4.2.1 Scheme achieving rate-memory pair (R1;M1)
4.2.2 Scheme achieving rate-memory pair (R2;M2)
4.2.3 Scheme achieving rate-memory pair (R3;M3)
4.2.4 Scheme achieving rate-memory pair (R4;M4)
4.3 Upper bound under one-sided cache assignment
4.3.1 Proof of the upper bound
4.4 Lower bound under symmetric cache assignment
4.4.1 Scheme achieving rate-memory pair (R1;Sym;M1;Sym)
4.4.2 Scheme achieving rate-memory pair (R2;Sym;M2;Sym)
4.5 Upper bound under symmetric cache assignment
4.6 Lower bound under asymmetric cache assignment
4.6.1 Scheme achieving rate-memory pair (R2;Asym;M2;Asym)
4.6.2 Scheme achieving rate-memory pair (R3;Asym;M3;Asym)
4.6.3 Scheme achieving rate-memory pair (R4;Asym;M4;Asym)
4.7 Upper bound under asymmetric cache assignment
4.8 Discussion and numerical results
4.8.1 Discussion on the obtained bounds
4.8.2 Impact of the joint secrecy constraint
4.8.3 Impact of the cache assignment
4.9 General lower bound
4.9.1 Scheme achieving rate-memory pair (R(K)
4.9.2 Scheme achieving rate-memory pair (R(K)
4.9.3 Scheme achieving rate-memory pair (R(K)
4.9.4 Scheme achieving rate-memory pair (R(K)
4.10 General upper bound
4.10.1 Proof of the general upper bound
4.11 Examples
4.12 Conclusion
Conclusion
