Digital signatures and encryption schemes constitute arguably an integral part of cryptographic schemes with the goal to meet the security needs of present and future private and business applications. However, almost all public key cryptosystems applied in practice are put at risk due to its vulnerability to quantum attacks as a result of Shor's quantum algorithm. The magnitude of economic and social impact is tremendous inherently asking for alternatives replacing classical schemes in case large-scale quantum computers are built. Lattice-based cryptography emerged as a powerful candidate attracting lots of attention not only due to its conjectured resistance against quantum attacks, but also because of its unique security guarantee to provide worst-case hardness of average-case instances. Hence, the requirement of imposing further assumptions on the hardness of randomly chosen instances disappears, resulting in more efficient instantiations of cryptographic schemes. The best known lattice attack algorithms run in exponential time. \nIn this thesis we contribute to a smooth transition into a world with practically efficient lattice-based cryptographic schemes. This is indeed accomplished by designing new algorithms and cryptographic schemes as well as improving existing ones. Our contributions are threefold. \nFirst, we construct new encryption schemes that fully exploit the error term in LWE instances. To this end, we introduce a novel computational problem that we call Augmented LWE (A-LWE), differing from the original LWE problem only in the way the error term is produced. In fact, we embed arbitrary data into the error term without changing the target distributions. Following this, we prove that A-LWE instances are indistinguishable from LWE samples. This allows to build powerful encryption schemes on top of the A-LWE problem that are simple in its representations and efficient in practice while encrypting huge amounts of data realizing message expansion factors close to 1. This improves, to our knowledge, upon all existing encryption schemes. Due to the versatility of the error term, we further add various security features such as CCA and RCCA security or even plug lattice-based signatures into parts of the error term, thus providing an additional mechanism to authenticate encrypted data. Based on the methodology to embed arbitrary data into the error term while keeping the target distributions, we realize a novel CDT-like discrete Gaussian sampler that beats the best known samplers such as Knuth-Yao or the standard CDT sampler in terms of running time. At run time the table size amounting to 44 elements is constant for every discrete Gaussian parameter and the total space requirements are exactly as large as for the standard CDT sampler. Further results include a very efficient inversion algorithm for ring elements in special classes of cyclotomic rings. In fact, by use of the NTT it is possible to efficiently check for invertibility and deduce a representation of the corresponding unit group. Moreover, we generalize the LWE inversion algorithm for the trapdoor candidate of Micciancio and Peikert from power of two moduli to arbitrary composed integers using a different approach. \nIn the second part of this thesis, we present an efficient trapdoor construction for ideal lattices and an associated description of the GPV signature scheme. Furthermore, we improve the signing step using a different representation of the involved perturbation matrix leading to enhanced memory usage and running times. Subsequently, we introduce an advanced compression algorithm for GPV signatures, which previously suffered from huge signature sizes as a result of the construction or due to the requirement of the security proof. We circumvent this problem by introducing the notion of public and secret randomness for signatures. In particular, we generate the public portion of a signature from a short uniform random seed without violating the previous conditions. This concept is subsequently transferred to the multi-signer setting which increases the efficiency of the compression scheme in presence of multiple signers. Finally in this part, we propose the first lattice-based sequential aggregate signature scheme that enables a group of signers to sequentially generate an aggregate signature of reduced storage size such that the verifier is still able to check that each signer indeed signed a message. This approach is realized based on lattice-based trapdoor functions and has many application areas such as wireless sensor networks. \nIn the final part of this thesis, we extend the theoretical foundations of lattices and propose new representations of lattice problems by use of Cauchy integrals. Considering lattice points as simple poles of some complex functions allows to operate on lattice points via Cauchy integrals and its generalizations. For instance, we can deduce for the one-dimensional and two-dimensional case simple expressions for the number of lattice points inside a domain using trigonometric or elliptic functions.
Digital signatures and encryption schemes constitute arguably an integral part of cryptographic schemes with the goal to meet the security needs of present and future private and business applications. However, almost all public key cryptosystems applied in practice are put at risk due to its vulnerability to quantum attacks as a result of Shor's quantum algorithm. The magnitude of economic and social impact is tremendous inherently asking for alternatives replacing classical schemes in case large-scale quantum computers are built. Lattice-based cryptography emerged as a powerful candidate attracting lots of attention not only due to its conjectured resistance against quantum attacks, but also because of its unique security guarantee to provide worst-case hardness of average-case instances. Hence, the requirement of imposing further assumptions on the hardness of randomly chosen instances disappears, resulting in more efficient instantiations of cryptographic schemes. The best known lattice attack algorithms run in exponential time. \nIn this thesis we contribute to a smooth transition into a world with practically efficient lattice-based cryptographic schemes. This is indeed accomplished by designing new algorithms and cryptographic schemes as well as improving existing ones. Our contributions are threefold. \nFirst, we construct new encryption schemes that fully exploit the error term in LWE instances. To this end, we introduce a novel computational problem that we call Augmented LWE (A-LWE), differing from the original LWE problem only in the way the error term is produced. In fact, we embed arbitrary data into the error term without changing the target distributions. Following this, we prove that A-LWE instances are indistinguishable from LWE samples. This allows to build powerful encryption schemes on top of the A-LWE problem that are simple in its representations and efficient in practice while encrypting huge amounts of data realizing message expansion factors close to 1. This improves, to our knowledge, upon all existing encryption schemes. Due to the versatility of the error term, we further add various security features such as CCA and RCCA security or even plug lattice-based signatures into parts of the error term, thus providing an additional mechanism to authenticate encrypted data. Based on the methodology to embed arbitrary data into the error term while keeping the target distributions, we realize a novel CDT-like discrete Gaussian sampler that beats the best known samplers such as Knuth-Yao or the standard CDT sampler in terms of running time. At run time the table size amounting to 44 elements is constant for every discrete Gaussian parameter and the total space requirements are exactly as large as for the standard CDT sampler. Further results include a very efficient inversion algorithm for ring elements in special classes of cyclotomic rings. In fact, by use of the NTT it is possible to efficiently check for invertibility and deduce a representation of the corresponding unit group. Moreover, we generalize the LWE inversion algorithm for the trapdoor candidate of Micciancio and Peikert from power of two moduli to arbitrary composed integers using a different approach. \nIn the second part of this thesis, we present an efficient trapdoor construction for ideal lattices and an associated description of the GPV signature scheme. Furthermore, we improve the signing step using a different representation of the involved perturbation matrix leading to enhanced memory usage and running times. Subsequently, we introduce an advanced compression algorithm for GPV signatures, which previously suffered from huge signature sizes as a result of the construction or due to the requirement of the security proof. We circumvent this problem by introducing the notion of public and secret randomness for signatures. In particular, we generate the public portion of a signature from a short uniform random seed without violating the previous conditions. This concept is subsequently transferred to the multi-signer setting which increases the efficiency of the compression scheme in presence of multiple signers. Finally in this part, we propose the first lattice-based sequential aggregate signature scheme that enables a group of signers to sequentially generate an aggregate signature of reduced storage size such that the verifier is still able to check that each signer indeed signed a message. This approach is realized based on lattice-based trapdoor functions and has many application areas such as wireless sensor networks. \nIn the final part of this thesis, we extend the theoretical foundations of lattices and propose new representations of lattice problems by use of Cauchy integrals. Considering lattice points as simple poles of some complex functions allows to operate on lattice points via Cauchy integrals and its generalizations. For instance, we can deduce for the one-dimensional and two-dimensional case simple expressions for the number of lattice points inside a domain using trigonometric or elliptic functions.
Building cryptographic schemes upon as many fundamentally different hard problems as possible, seems to be the best way to hedge against future threats such as quantum computers. Being mainly based on the hardness of factoring and computing discrete logarithms, the present security landscape is at risk. In contrast, problems in lattices, such as finding short non-zero vectors, seem to withstand quantum computer attacks and the best known algorithms run in exponential time. In sharp contrast to other fields of cryptography, lattices admit a worst-case to average-case reduction (Ajtai 1996). Instead of assuming that a problem is hard for randomly chosen instances, lattice-based cryptosystems merely require the existence of a single hard instance, i.e., hardness in the worst case. With such an additional "trust anchor", the resulting security guarantees are much more plausible. Quite recently, we have seen an increased interest in lattice-based cryptography with many striking results. In this thesis, we are particularly interested in signature schemes, which provide a supporting pillar for today's economy. While we have seen basic signature schemes from lattices, e.g., (Gentry, Peikert, Vaikuntanathan 2008), (Lyubashevsky, Micciancio 2008), (Lyubashevsky 2009), or (Cash, Hofheinz, Kiltz, Peikert 2009), there are hardly any results dealing with the specific needs of applications, where ordinary signatures often fall too short. In this thesis, we build upon the above results and equip them with additional features, motivated by an exemplary selection of application scenarios. Hence, we demonstrate the great versatility of lattices in cryptography. In particular, we facilitate privacy-friendly electronic elections, fair online contract signing, signature compression, secure signatures in the strongest sense, as well as identity-based primitives. As far as possible, we avoid simplifying assumptions, such as the random oracle model. We believe that our techniques can be transferred to other application scenarios as well. Independently of the these results, we discuss the practical hardness of lattice problems and provide a framework for estimating the security of essentially all modern lattice-based cryptography.
Digital signatures are indispensable for security on the Internet, because they guarantee authenticity, integrity, and non-repudiation, of namely e-mails, software updates, and in the Transport Layer Security (TLS) protocol which is used for secure data transfer, for example. Most signature schemes that are currently in use such as the RSA signature scheme, are considered secure as long as the integer factorization problem or the discrete logarithm (DL) problem are computationally hard. At present, no algorithms have yet been found to solve these problems on conventional computers in polynomial time. However, in 1997, Shor published a polynomial-time algorithm that uses quantum computation to solve the integer factorization and the DL problem. In particular, this means that RSA signatures are considered broken as soon as large-scale quantum computers exist. Due to significant advances in the area of quantum computing, it is reasonable to assume that within 20 years, quantum computers that are able to break the RSA scheme, could exist. In order to maintain authenticity, integrity, and non-repudiation of data, cryptographic schemes that cannot be broken by quantum attacks are required. In addition, these so-called post-quantum secure schemes should be sufficiently efficient to be suitable for all established applications. Furthermore, solutions enabling a timely and secure transition from classical to post-quantum schemes are needed. This thesis contributes to the above-mentioned transition. In this thesis, we present the two lattice-based digital signature schemes TESLA and qTESLA, whereby lattice-based cryptography is one of five approaches to construct post-quantum secure schemes. Furthermore, we prove that our signature schemes are secure as long as the so-called Learning With Errors (LWE) problem is computationally hard to solve. It is presumed that even quantum computers cannot solve the LWE problem in polynomial time. The security of our schemes is proven using security reductions. Since our reductions are tight and explicit, efficient instantiations are possible that provably guarantee a selected security level, as long as the corresponding LWE instance provides a certain hardness level. Since both our reductions (as proven in the quantum random oracle model) and instantiations, take into account quantum attackers, TESLA and qTESLA are considered post-quantum secure. Concurrently, the run-times for generating and verifying signatures of qTESLA are similar (or faster) than those of the RSA scheme. However, key and signature sizes of RSA are smaller than those of qTESLA. In order to protect both the theoretical signature schemes and their implementations against attacks, we analyze possible vulnerabilities against implementation attacks. In particular, cache-side-channel attacks resulting from observing the cache behavior and fault attacks, which recover secret information by actively disrupting the execution of an algorithm are focused. We present effective countermeasures for each implementation attack we found. Our analyses and countermeasures also influence the design and implementation of qTESLA. Although our schemes are considered (post-quantum) secure according to state-of-the-art LWE attacks, cryptanalysis of lattice-based schemes is still a relatively new field of research in comparison to RSA schemes. Hence, there is a lack of confidence in the concrete instantiations and their promised security levels. However, due to developments within the field of quantum computers, a transition to post-quantum secure solutions seems to be more urgently required than ever. To solve this dilemma, we present an approach to combine two schemes, e.g., qTESLA and the RSA signature scheme, so that the combination is secure as long as one of the two combined schemes is secure. We present several of such combiners to construct hybrid signature schemes and hybrid key encapsulation mechanisms to ensure both authenticity and confidentiality in our Public-Key Infrastructure (PKI). Lastly, we also demonstrate how to apply the resulting hybrid schemes in standards such as X.509 or TLS. To summarize, this work presents post-quantum secure candidates which can, using our hybrid schemes, add post-quantum security to the current classical security in our PKI.
One essential quest in cryptography is the search for hard instances of a given computational problem that is known to be hard in the worst-case. In lattice cryptography we are in the unique situation that we have found a way of picking random instances which are at least as hard as well-studied lattice problems in the worst-case. At the same time, no attack running in subexponential time is known to break these problems, even for an adversary using quantum computers. Virtually all public-key schemes in use today are subject to such attacks, and the development of quantum computers is actively pursued, so it is prudent to investigate lattice-based alternatives. There are two fundamental open problems in lattice cryptography today and this thesis contributes to solving them. First, there exists a widely used efficiency improvement that allows for trapdoors whose asymptotic keysize and evaluation time are both quasilinear in the dimension of an associated lattice. This is accomplished by restricting oneself to lattices with special structure, so-called ideal lattices. This entails the use of newer security assumptions, but these have not been analyzed thoroughly so far. We start this work by comparing the class of ideal lattice problems with its general counterpart in terms of size. Affirming folklore, we find the number of restricted instances among all instances to be asymptotically negligible for those classes of lattices suggested for practical use. The second open problem is that while the connection to worst-case problems is well understood, the practical hardness of the related average-case problems is not. Specifically, there have been parameters suggested for practical usage, where current lattice basis reduction algorithms can solve these worst-case problems, but the related average-case problems, which these are reduced to and which represent the basis of practical security of the cryptosystems, are completely infeasible. In most cases, this lack of understanding has lead either to a very conservative choice of parameters, or none at all. This in turn makes it impossible for the resulting lattice schemes to compete with their counterparts based on other paradigms. We further the understanding of this practical security and at the same time improve the efficiency of several common related cryptographic schemes. Among other things, we find that for the SWIFFT compression function, solving certain problems closely related to finding collisions is easier than previously thought and we suggest efficient replacement parameters. We propose a novel zero-knowledge identification scheme that, to our knowledge, beats all competing post-quantum schemes, even those based on other paradigms. Possibly most important, we help to tighten the efficiency gap between lattice encryption schemes that are provably secure and the acclaimed ad-hoc encryption scheme NTRU. This is done by unifying many recent developments into a new provably secure design and providing a comprehensive analysis of practical security, which together results in a great leap of efficiency.
Lattice-based cryptography is a branch of cryptography exploiting the presumed hardness of some well-known problems on lattices. Its main advantages are its simplicity, efficiency, and apparent security against quantum computers. The principle of the security proofs in lattice-based cryptography is to show that attacking a given scheme is at least as hard as solving a particular problem, as the Learning with Errors problem (LWE) or the Small Integer Solution problem (SIS). Then, by showing that those two problems are at least as hard to solve than a hard problem on lattices, presumed polynomial time intractable, we conclude that the constructed scheme is secure.In this thesis, we improve the foundation of the security proofs and build new cryptographic schemes. We study the hardness of the SIS and LWE problems, and of some of their variants on integer rings of cyclotomic fields and on modules on those rings. We show that there is a classical hardness proof for the LWE problem (Regev's prior reduction was quantum), and that the module variants of SIS and LWE are also hard to solve. We also give two new lattice-based group signature schemes, with security based on SIS and LWE. One is the first lattice-based group signature with logarithmic signature size in the number of users. And the other construction allows another functionality, verifier-local revocation. Finally, we improve the size of some parameters in the work on cryptographic multilinear maps of Garg, Gentry and Halevi in 2013.
Encryption and signature schemes based on worst-case lattice problems are promising candidates for the post-quantum era, where classic number-theoretic assumptions are rendered false. Although there have been many important results and breakthroughs in lattice cryptography, the questions of how to systematically evaluate their security in practice and how to choose secure parameters are still open. This is mainly due to the fact that most security proofs are essentially asymptotic statements. In addition, the hardness of the underlying complexity assumption is controlled by several interdependent parameters rather than just a simple bit length as in many classic schemes. With our work, we close this gap by providing a framework that (1) distills a hardness estimate out of a given parameter set and (2) relates the complexity of practical lattice-based attacks to symmetric “bit security” for the first time. Our approach takes various security levels, or attacker types, into account. Moreover, we use it to predict long-term security in a similar fashion as the results that are collected on www.keylength.com. In contrast to the experiments by Gama and Nguyen (Eurocrypt 2008), our estimates are based on precisely the family of lattices that is relevant in modern lattice-based cryptography. Our framework can be applied in two ways: Firstly, to assess the hardness of the (few) proposed parameter sets so far and secondly, to propose secure parameters in the first place. Our methodology is applicable to essentially all lattice-based schemes that are based on the learning with errors problem (LWE) or the small integer solution problem (SIS) and it allows us to compare efficiency and security across different schemes and even across different types of cryptographic primitives.
Abstract. Lattice-based encryption schemes still suffer from a low message throughput per ciphertext. This is mainly due to the fact that the underlying schemes do not tap the full potentials of LWE. Many constructions still follow the one-time-pad approach considering LWE instances as random vectors added to a message, most often encoded bit vectors. Recently, at Financial Crypto 2015 El Bansarkhani et al. proposed a novel encryption scheme based on the A-LWE assumption (Augmented LWE), where data is embedded into the error term without changing its target distributions. By this novelty it is possible to encrypt much more data as compared to the traditional one-time-pad approach and it is even possible to combine both concepts. In this paper we revisit this approach and propose amongst others several algebraic techniques in order to significantly improve the message throughput per ciphertext. Furthermore, we give a thorough security analysis as well as an efficient implementation of the CCA1-secure encryption scheme instantiated with the most efficient trapdoor construction. In particular, we attest that it even outperforms the CPA-secure encryption scheme from Lindner and Peikert presented at CT-RSA 2011.