We describe the theoretical concept of multilinear maps and its practical real- ization using new construction - Garg-Gentry-Halevi (GGH) Graded Encoding Scheme. In this construction, which is based on ideal lattices, we justify its assumptions and clarify some algebraic inaccuracies, especially the inversibility of the randomly chosen z from commutative ring Rq. We also present applica- tion of theoretical concept and its practical realization GGH to one-round N-way Diffie-Hellman key exchange.
We describe the theoretical concept of multilinear maps and its practical real- ization using new construction - Garg-Gentry-Halevi (GGH) Graded Encoding Scheme. In this construction, which is based on ideal lattices, we justify its assumptions and clarify some algebraic inaccuracies, especially the inversibility of the randomly chosen z from commutative ring Rq. We also present applica- tion of theoretical concept and its practical realization GGH to one-round N-way Diffie-Hellman key exchange.
Constructing multilinear maps has been long-standing open problem, before recently the first construction based on ideal lattices has been proposed by Garg et al. After this breakthrough, various new cryptographic systems have been proposed. They introduce the concept of level into the encodings, and the system has a function that extracts a deterministic value at only a specific level, and the encodings are unable to downgrade to the lower levels. These properties are useful for cryptography. We study how this graded encoding system be applied to cryptosystems, and we propose two protocols, group key exchange and witness encryption. In our group key exchange, we achieve the communication size and the computation costs per party are both O(1) with respect to the number of parties by piling the encodings of passed parties in one encoding. A witness encryption is a new type cryptosystem using NP-complete problem. The first construction is based on EXACT-COVER problem. We construct it based on another NP complete Hamilton Cycle problem, and prove its security under the Generic Cyclic Colored Matrix Model.
Multi-recipient encryption is an important public key cryptosystem, which can be applied for a variety of purposes, such as broadcasting data. In order to design an secure multi-recipient public key encryption (MRPKE) in post-quantum era, in this paper, we construct a novel MRPKE scheme base on Garg-Gentry-Halevi (GGH) framework which is a graded algebras analogue of multilinear maps from ideal lattice. Under the grade decisional Diffie-Hellman (GDDH) assumption of GGH, the proposed scheme has semantically safety against chosen plaintext attack (CPA). At the same time, each recipient, without first decrypting, can verify whether the message to be received is from a legitimate sender. Furthermore, the encryption and decryption only involves the polynomial modular addition and multiplication in polynomial ring, so the efficiency of the proposed scheme is higher.
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.
Today, lattice-based cryptography is a thriving scientific field. Its swift expansion is due, among others, to the attractiveness of fully homomorphic encryption and cryptographic multilinear maps. Lattice-based cryptography has also been recognized for its thrilling properties: a security that can be reduced to worst-case instances of problems over lattices, a quasi-optimal asymptotic efficiency and an alleged resistance to quantum computers. However, its practical use in real-world products leaves a lot to be desired. This thesis accomplishes a step towards this goal by narrowing the gap between theoretical research and practical implementation of recent public key cryptosystems. In this thesis, we design and implement a lattice-based digital signature, two fully homomorphic encryption schemes and cryptographic multilinear maps. Our highly efficient signature scheme, BLISS, opened the way to implementing lattice-based cryptography on constrained devices and remains as of today a promising primitive for post-quantum cryptography. Our fully homomorphic encryption schemes enjoy competitive homomorphic evaluations of nontrivial circuits. Finally, we describe the first implementation of cryptographic multilinear maps. Based on our implementation, a non interactive key exchange between more than three parties has been realized for the first time, and amounts to a few seconds per party.
In recent years, lattice-based cryptography has been recognized for its many attractive properties, such as strong provable security guarantees and apparent resistance to quantum attacks, flexibility for realizing powerful tools like fully homomorphic encryption, and high asymptotic efficiency. Indeed, several works have demonstrated that for basic tasks like encryption and authentication, lattice-based primitives can have performance competitive with (or even surpassing) those based on classical mechanisms like RSA or Diffie-Hellman. However, there still has been relatively little work on developing lattice cryptography for deployment in real-world cryptosystems and protocols. In this work we take a step toward that goal, by giving efficient and practical lattice-based protocols for key transport, encryption, and authenticated key exchange that are suitable as “drop-in ” components for proposed Internet standards and other open protocols. The security of all our proposals is provably based (sometimes in the random-oracle model) on the well-studied “learning with errors over rings” problem, and hence on the conjectured worst-case hardness of problems on ideal lattices (against quantum algorithms). One of our main technical innovations (which may be of independent interest) is a simple, low-bandwidth reconciliation technique that allows two parties who “approximately agree ” on a secret value to reach exact agreement, a setting common to essentially all lattice-based encryption schemes. Our technique reduces the ciphertext length of prior (already compact) encryption schemes nearly twofold, at essentially no cost. 1
In this dissertation, we explore the frontiers of theory of cryptography along two lines. In the first direction, we explore Lattice Cryptography, which is the primary sub-area of post-quantum cryptographic research. Our first contribution is the construction of a deniable attribute-based encryption scheme from lattices. A deniable encryption scheme is secure against not only eavesdropping attacks as required by semantic security, but also stronger coercion attacks performed after the fact. An attribute-based encryption scheme allows ``fine-grained'' access to ciphertexts, allowing for a decryption access policy to be embedded in ciphertexts and keys. We achieve both properties simultaneously for the first time from lattices. Our second contribution is the construction of a digital signature scheme that enjoys both short signatures and a completely tight security reduction from lattices. As a matter of independent interest, we give an improved method of randomized inversion of the G gadget matrix, which reduces the noise growth rate in homomorphic evaluations performed in a large number of lattice-based cryptographic schemes, without incurring the high cost of sampling discrete Gaussians. In the second direction, we explore Cryptographic Program Obfuscation. A program obfuscator is a type of cryptographic software compiler that outputs executable code with the guarantee that ``whatever can be hidden about the internal workings of program code, is hidden.'' Indeed, program obfuscation can be viewed as a ``universal and cryptographically-complete'' tool. Our third contribution is the first, full-scale implementation of secure program obfuscation in software. Our toolchain takes code written in a C-like programming language, specialized for cryptography, and produces secure, obfuscated software. Our fourth contribution is a new cryptanalytic attack against a variety of ``early'' program obfuscation candidates. We provide a general, efficiently-testable property for any two branching programs, called partial inequivalence, which we show is sufficient for launching an ``annihilation attack'' against several obfuscation candidates based on Garg-Gentry-Halevi multilinear maps.