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Context: Secrecy has kept researchers spanning over centuries engaged in the creation of data protection techniques. With the growing rate of data breach and intervention of adversaries in confidential data storage and communication, efficient data protection has found to be a challenge. Homomorphic encryption is one such data protection technique in the cryptographic domain which can perform arbitrary computations on the enciphered data without disclosing the original plaintext or message. The first working fully homomorphic encryption scheme was proposed in the year 2009 and since then there has been a tremendous increase in the development of homomorphic encryption schemes such that they can be applied to a wide range of data services that demand security. All homomorphic encryption schemes can be categorized as partially homomorphic (PHE), somewhat homomorphic (SHE), leveled Homomorphic (LHE), and fully homomorphic encryption (FHE). Each encryption algorithm has its own importance and usage in different realms of security. DHGV, Paillier, HElib, and FHEW are the algorithms chosen in this study considering their wide usage and scope for further advancement in this subject area. A public-key algorithm named RSA is also chosen for comparison of the impact of HE and PKE (Public-key encryption) algorithm on the CPU and Memory. The utilization of various homomorphic schemes and concepts in the trending cloud storage systems is a prevailing field of research and can be expanded further by knowing the current state-of-the-art of homomorphic encryption. Hence, the necessity of comprehending the knowledge of homomorphic encryption schemes and their aspect in cloud security becomes vital. Objectives: The objective of this study is to analytically assess homomorphic encryption and various homomorphic encryption schemes. A comprehensive investigation on working and performance of the selected HE schemes is another objective of this research. Also, an experiment to run publicly available libraries of DGHV, Paillier, HElib, and FHEW is one of the main objectives. In addition to these, comprehending the impact of HE and PKE on CPU and Memory is also among the objectives of the study. The role and practice of homomorphic encryption in the cloud storage system are among the secondary objectives of this research in terms of securing confidential data. These objectives are set based on the research gap identified by conducting an exhaustive literature review. Methods: The objectives of this study are achieved by adopting the methods exhaustive literature review and experiment. Scientific databases such as IEEE Xplore, ACM Digital Library, Inspec, Springer Link etc. are used and literature is accordingly selected based on the relevance to the research topic. An exhaustive literature review is conducted and extensive bibliographic research is done to accomplish the objective of comprehending the working, applications, significance of homomorphic encryption. Apart from literature review, bibliographic research, an experiment is also conducted to run the publicly available homomorphic encryption libraries to evaluate, compare, and analyze the performance of DGHV, Paillier, HElib, and FHEW schemes. Experiment to run publicly available PKE algorithm is also conducted. Finally, the conclusion and outcome by adopting these research methods for accomplishing the objectives are theoretically presented in detail. Results: By conducting an exhaustive literature review, the importance, working, application of homomorphic encryption and its schemes is discerned. And by conducting an experiment, the impact of HE and PKE is also discerned. Apart from this, the limitations of HE and selected HE schemes along with the distinction between public and private key cryptography is understood by finding and mapping in connection with each other. From the experiment conducted, it is examined that despite the encryption libraries being publicly available for use, the possibility of running and employing few libraries successfully is remarkably low inferring that there is much improvement needed in this cryptographic discipline. Conclusions: From this research, it can be concluded that homomorphic encryption has a wide scope of extending towards efficiency and application in various fields concerned with data protection. It can also me concluded that the experimental assessment of state of the art of few HE schemes libraries that are available online are remarkably impractical for real-time practice. By analyzing the selected ii schemes, it can be concluded few HE schemes do not support any other operations on encrypted data other than addition and multiplication due to which chances of increasing noise for each encryption is relatively high. From the experiment conducted for Paillier encryption (HE) and RSA (PKE) encryption, it is concluded that both the schemes increase linearly with an increase in the input size when CPU and Memory utilization is measured. Apart from these conclusions, it can also be inferred that not all the homomorphic encryption algorithms are IND-CCA1 and IND-CCA2 secure. From this study, it can be deduced that more empirical validation and analysis of HE algorithms is required in terms of their performance and security. In order to address these problems, much research and improvement are required as it inferred from the results of this research that Homomorphic encryption is still in its early stage of development and enormous utility can be anticipated when enhanced correctly.

Lattice-based cryptography has generated considerable interest in the last two decades due toattractive features, including conjectured security against quantum attacks, strong securityguarantees from worst-case hardness assumptions and constructions of fully homomorphicencryption schemes. On the other hand, even though it is a crucial part of many lattice-basedschemes, Gaussian sampling is still lagging and continues to limit the effectiveness of this newcryptography. The first goal of this thesis is to improve the efficiency of Gaussian sampling forlattice-based hash-and-sign signature schemes. We propose a non-centered algorithm, with aflexible time-memory tradeoff, as fast as its centered variant for practicable size of precomputedtables. We also use the Rényi divergence to bound the precision requirement to the standarddouble precision. Our second objective is to construct Falcon, a new hash-and-sign signaturescheme, based on the theoretical framework of Gentry, Peikert and Vaikuntanathan for latticebasedsignatures. We instantiate that framework over NTRU lattices with a new trapdoor sampler.

" In the last decade, cloud computing became popular among companies for outsourcing some of their services. Companies use cloud services to store crucial information such as financial and client data. Cloud services are not only cost effective but also easier to manage since the companies avoid maintenance of servers. Although cloud has its advantages, maintaining the security is a big concern. Cloud services might not have any malicious intent, but attacks targeting cloud systems could easily steal vital data belong to the companies. The only protection that assures the security of the information is a strong encryption. However, these schemes only protects the information but prevent you to do any computation on the data. This was an open problem for more than 30 years and it has been solved recently by the introduction of the first fully homomorphic encryption (FHE) scheme by Gentry. The FHE schemes allow you to do arbitrary computation on an encrypted data by still preserving the encryption. Namely, the message is not revealed (decrypted) at any given time while computing the arbitrary circuit. However, the first FHE scheme is not practical for any practical application. Later, numerous research work has been published aiming at making fully homomorphic encryption practical for daily use, but still they were too inefficient to be used in everyday practical applications. In this dissertation we tackle the efficiency problems of fully homomorphic encryption (FHE) schemes. We propose two new FHE schemes that improve the storage requirement and runtime performance. The first scheme (Doröz, Hu and Sunar) reduces the size of the evaluation keys in existing NTRU based FHE schemes. In the second scheme (F-NTRU) we designed an NTRU based FHE scheme which is not only free of costly evaluation keys but also competitive in runtime performance. We further proposed two hardware accelerators to increase the performance of arithmetic operations underlying the schemes. The first accelerator is a custom hardware architecture for realizing the Gentry-Halevi fully homomorphic encryption scheme. This contribution presents the first full realization of FHE in hardware. The architecture features an optimized multi-million bit multiplier based on the Schönhage-Strassen multiplication algorithm. Moreover, a number of optimizations including spectral techniques as well as a precomputation strategy is used to significantly improve the performance of the overall design. The other accelerator is optimized for a class of reconfigurable logic for somewhat homomorphic encryption (SWHE) based schemes. Our design works as a co-processor: the most compute-heavy operations are offloaded to this specialized hardware. The core of our design is an efficient polynomial multiplier as it is the most compute-heavy operation of our target scheme. The presented architecture can compute the product of very-large polynomials more efficiently than software implementations on CPUs. Finally, to assess the performance of proposed schemes and hardware accelerators we homomorphically evaluate the AES and the Prince block ciphers. We introduce various optimizations including a storage-runtime trade-off. Our benchmarking results show significant speedups over other existing instantiations. Also, we present a private information retrieval (PIR) scheme based on a modified version of Doröz, Hu and Sunar’s homomorphic scheme. The scheme is capable of privately retrieving data from a database containing 4 billion entries. We achieve asymptotically lower bandwidth cost compared to other PIR schemes which makes it more practical. "

The main objective of this project is to find the original symmetric presentations of some very important finite groups and to give our constructions of some of these groups. We have found the Mathieu sporadic group M11, HS × D5, where HS is the sporadic group Higman-Sim group, the projective special unitary group U(3; 5) and the projective special linear group L2(149) as homomorphic images of the monomial progenitors 11*4 :m (5 :4), 5*6 :m S5 and 149*2 :m D37. We have also discovered 24 : S3 × C2, 24 : A5, (25 : S4), 25 : S3 × S3, 33 : S4 × C2, S6, 29: PGL(2,7), 22 • (S6 : S6), PGL(2,19), ((A5 : A5 × A5) : D6), 6 • (U4(3): 2), 2 • PGL(2,13), S7, PGL (2,8), PSL(2,19), 2 × PGL(2,81), 25 : (S6 × A5), 26 : S4 × D3, U(4,3), 34 : S4, 32 :D6, 2 • (PGL(2,7) :PSL(2,7), 22 : (S5 : S5) and 23 : (PSL3(4) : 2) as homomorphic images of the permutation progenitors 2*8 : (2 × 4 : 2), 2*16: (2 × 4 :C2 × C2), 2*9: (S3 × S3), 2*9: (S3 × A3), 2*9: (32 × 23) and 2*9: (33 × A3). We have also constructed 24: S3 × C2, 24 : A5, (25: S4), 25 : S3 × S3,: 33: S4 × C2, S6, M11 and U (3,5) by using the technique of double coset enumeration. We have determined the isomorphism types of the most of the images mentioned in this thesis. We demonstrate our work for the following examples: 34 : (32 * 23) × 2, 29 : PGL(2,7), 2•S6, (54 : (D4 × S3)), and 3: •PSL(2,19) ×2.

The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J1, the Mathieu group M12, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2*60 : (2 x A5), 2*60 : A5, 2*56 : (23 : 7), and 2*28 : (PGL(2,7):2), respectively. We have also discovered the groups 24 : A5, 34 : S5, PSL(2,31), PSL(2,11), PSL(2,19), PSL(2,41), A8, 34 : S5, A52, 2• A52, 2 : A62, PSL(2,49), 28 : A5, PGL(2,19), PSL(2,71), 24 : A5, 24 : A6, PSL(2,7), 3 x PSL(3,4), 2• PSL(3,4), PSL(3,4), 2• (M12 : 2), 37:S7, 35 : S5, S6, 25 : S6, 35 : S6, 25 : S5, 24 : S6, and M12 as homomorphic images of the permutation progenitors 2*60 : (2 x A5), 2*60 : A5, 2*21 : (7: 3), 2*60 : (2 x A5), 2*120 : S5, and 2*144 : (32 : 24). We have given original proof of the 2*n Symmetric Presentation Theorem. In addition, we have also provided original proof for the Extension of the Factoring Lemma (involutory and non-involutory progenitors). We have constructed S5, PSL(2,7), and U(3,5):2 using the technique of double coset enumeration and by way of linear fractional mappings. Furthermore, we have given proofs of isomorphism types for 7 x 22, U(3,5):2, 2•(M12 : 2), and (4 x 2) :• 22.

In this thesis, we provide a summary of fully homomorphic encryption, and in particular, look at the BGV encryption scheme by Brakerski, Gentry, and Vaikuntanathan; as well the DGHV encryption scheme by van Dijk, Gentry, Halevi, and Vaikuntanathan. We explain the mechanisms developed by Gentry in his breakthrough work, and show examples of how they are used. While looking at the BGV encryption scheme, we make improvements to the underlying lemmas dealing with modulus switching and noise management, and show that the lemmas as currently stated are false. We then examine a lower bound on the hardness of the Learning With Errors lattice problem, and use this to develop specific parameters for the BGV encryption scheme at a variety of security levels. We then study the DGHV encryption scheme, and show how the somewhat homomorphic encryption scheme can be implemented as both a fully homomorphic encryption scheme with bootstrapping, as well as a leveled fully homomorphic encryption scheme using the techniques from the BGV encryption scheme. We then extend the parameters from the optimized version of this scheme to higher security levels, and describe a more straightforward way of arriving at these parameters.

Homomorphic cryptography provides a third party with the ability to perform simple computations on encrypted data without revealing any information about the data itself. Typically, a third party can calculate one of the encrypted sum or the encrypted product of two encrypted messages. This is possible due to the fact that the encryption function is a group homomorphism, and thus preserves group operations. This makes homomorphic cryptosystems useful in a wide variety of privacy preserving protocols. A comprehensive survey of known homomorphic cryptosystems is provided, including formal definitions, security assumptions, and outlines of security proofs for each cryptosystem presented. Threshold variants of several homomorphic cryptosystems are also considered, with the first construction of a threshold Boneh-Goh-Nissim cryptosystem given, along with a complete proof of security under the threshold semantic security game of Fouque, Poupard, and Stern. This approach is based on Shoup's approach to threshold RSA signatures, which has been previously applied to the Paillier and Damg\aa rd-Jurik cryptosystems. The question of whether or not this approach is suitable for other homomorphic cryptosystems is investigated, with results suggesting that a different approach is required when decryption requires a reduction modulo a secret value. The wide variety of protocols utilizing homomorphic cryptography makes it difficult to provide a comprehensive survey, and while an overview of applications is given, it is limited in scope and intended to provide an introduction to the various ways in which homomorphic cryptography is used beyond simple addition or multiplication of encrypted messages. In the case of strong conditional oblivious tranfser, a new protocol implementing the greater than predicate is presented, utilizing some special properties of the Boneh-Goh-Nissim cryptosystem to achieve security against a malicious receiver.

Homomorphic cryptography provides a third party with the ability to perform simple computations on encrypted data without revealing any information about the data itself. Typically, a third party can calculate one of the encrypted sum or the encrypted product of two encrypted messages. This is possible due to the fact that the encryption function is a group homomorphism, and thus preserves group operations. This makes homomorphic cryptosystems useful in a wide variety of privacy preserving protocols. A comprehensive survey of known homomorphic cryptosystems is provided, including formal definitions, security assumptions, and outlines of security proofs for each cryptosystem presented. Threshold variants of several homomorphic cryptosystems are also considered, with the first construction of a threshold Boneh-Goh-Nissim cryptosystem given, along with a complete proof of security under the threshold semantic security game of Fouque, Poupard, and Stern. This approach is based on Shoup's approach to threshold RSA signatures, which has been previously applied to the Paillier and Damg\aa rd-Jurik cryptosystems. The question of whether or not this approach is suitable for other homomorphic cryptosystems is investigated, with results suggesting that a different approach is required when decryption requires a reduction modulo a secret value. The wide variety of protocols utilizing homomorphic cryptography makes it difficult to provide a comprehensive survey, and while an overview of applications is given, it is limited in scope and intended to provide an introduction to the various ways in which homomorphic cryptography is used beyond simple addition or multiplication of encrypted messages. In the case of strong conditional oblivious tranfser, a new protocol implementing the greater than predicate is presented, utilizing some special properties of the Boneh-Goh-Nissim cryptosystem to achieve security against a malicious receiver.

In this thesis, we provide a summary of fully homomorphic encryption, and in particular, look at the BGV encryption scheme by Brakerski, Gentry, and Vaikuntanathan; as well the DGHV encryption scheme by van Dijk, Gentry, Halevi, and Vaikuntanathan. We explain the mechanisms developed by Gentry in his breakthrough work, and show examples of how they are used. While looking at the BGV encryption scheme, we make improvements to the underlying lemmas dealing with modulus switching and noise management, and show that the lemmas as currently stated are false. We then examine a lower bound on the hardness of the Learning With Errors lattice problem, and use this to develop specific parameters for the BGV encryption scheme at a variety of security levels. We then study the DGHV encryption scheme, and show how the somewhat homomorphic encryption scheme can be implemented as both a fully homomorphic encryption scheme with bootstrapping, as well as a leveled fully homomorphic encryption scheme using the techniques from the BGV encryption scheme. We then extend the parameters from the optimized version of this scheme to higher security levels, and describe a more straightforward way of arriving at these parameters.