In this thesis, I explore the current multilinear map candidates and attacks against them. This involves analyzing the three proposals for multilinear maps which are based on ideal lattices, integers, and standard lattices. The attacks exploit the geometry of the lattices and linearity of the integers to break security. I also compare the applications of these schemes with what is required for attacks. Key agreement seems to need certain features of multilinear maps which expose vulnerabilities while other applications like indistinguishability obfuscation. I analyze the attacks against these maps and show why they are not able to break the program obfuscation application of multilinear maps.
The two-volume set of LNCS 11239 and LNCS 11240 constitutes the revised proceedings of the 16th International Conference on Theory of Cryptography, TCC 2018, held in Panaji, India, in November 2018. The total of 50 revised full papers presented in the proceedings were carefully reviewed and selected from 168 submissions. The Theory of Cryptography Conference deals with the paradigms, approaches, and techniques used to conceptualize natural cryptographic problems and provide algorithmic solutions to them and much more.
In this thesis, we describe plausible lattice-based constructions with properties that approximate the sought-after multilinear maps in hard-discrete-logarithm groups. The security of our constructions relies on seemingly hard problems in ideal lattices, which can be viewed as extensions of the assumed hardness of the NTRU function. These new constructions radically enhance our tool set and open a floodgate of applications.
Program Obfuscation is the art of making computer programs ``unintelligible" while preserving its functionality. There have been many attempts to formalize this notion and one such formalization, termed as indistinguishability obfuscation (iO), has led to several powerful implications: game theoretic hardness results, watermarking of programs, feasibility of time-lock puzzles, advanced encryption systems, leakage resilient circuit compilers, succinct randomized encodings and so on. On the construction side, in spite of intense research, the problem of basing iO on standard falsifiable cryptographic assumptions remains open. All the current known constructions of iO are based on the tool of degree-$d$ multilinear maps. The candidates for degree-$d$ multilinear maps, for arbitrary $d$, were only recently studied and its associated assumptions have been subject to several devastating cryptanalytic attacks. On the other hand, there are no known attacks on the candidates of degree-2 multilinear maps (also known bilinear maps), even after a decade of cryptanalytic research. The original construction of iO proposed by Garg, Gentry, Halevi, Raykova, Sahai in 2013 required degree-$d$ multilinear maps, where $d$ was a large polynomial in the security parameter. Although the works that followed improved the original work in different aspects, they still relied on $d$ to be a large polynomial in the security parameter. In this thesis, we (jointly with Jain, CRYPTO 2015) first propose a new template to construct iO starting from functional encryption, a primitive that has been explored for over a decade. Subsequently, several works used this template to construct iO. Notably, Lin (EUROCRYPT 2016) and subsequently Lin and Vaikuntanathan (FOCS 2016) showed how to construct iO relying upon degree-$d$ multilinear maps and other relatively mild assumptions, where $d$ is a constant ($> 30$). We (jointly with Sahai, EUROCRYPT 2017) improve upon these works and show how to base iO relying upon degree-5 multilinear maps and other relatively mild assumptions. This brings us tantalizingly close to basing iO on bilinear maps.
The aim of cryptography is to design primitives and protocols that withstand adversarial behavior. Information theoretic cryptography, how-so-ever desirable, is extremely restrictive and most non-trivial cryptographic tasks are known to be information theoretically impossible. In order to realize sophisticated cryptographic primitives, we forgo information theoretic security and assume limitations on what can be efficiently computed. In other words we attempt to build secure systems conditioned on some computational intractability assumption such as factoring, discrete log, decisional Diffie-Hellman, learning with errors, and many more. In this work, based on the 2013 ACM Doctoral Dissertation Award-winning thesis, we put forth new plausible lattice-based constructions with properties that approximate the sought after multilinear maps. The multilinear analog of the decision Diffie-Hellman problem appears to be hard in our construction, and this allows for their use in cryptography. These constructions open doors to providing solutions to a number of important open problems.
The two-volume set LNCS 9985 and LNCS 9986 constitutes the refereed proceedings of the 14th International Conference on Theory of Cryptography, TCC 2016-B, held in Beijing, China, in November 2016. The total of 45 revised full papers presented in the proceedings were carefully reviewed and selected from 113 submissions. The papers were organized in topical sections named: TCC test-of-time award; foundations; unconditional security; foundations of multi-party protocols; round complexity and efficiency of multi-party computation; differential privacy; delegation and IP; public-key encryption; obfuscation and multilinear maps; attribute-based encryption; functional encryption; secret sharing; new models.
The three volume-set LNCS 12105, 12106, and 12107 constitute the thoroughly refereed proceedings of the 39th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2020, which was due to be held in Zagreb, Croatia, in May 2020. The conference was held virtually due to the COVID-19 pandemic. The 81 full papers presented were carefully reviewed and selected from 375 submissions. The papers are organized into the following topical sections: invited talk; best paper awards; obfuscation and functional encryption; symmetric cryptanalysis; randomness extraction; symmetric cryptography I; secret sharing; fault-attack security; succinct proofs; generic models; secure computation I; quantum I; foundations; isogeny-based cryptography; lattice-based cryptography; symmetric cryptography II; secure computation II; asymmetric cryptanalysis; verifiable delay functions; signatures; attribute-based encryption; side-channel security; non-interactive zero-knowledge; public-key encryption; zero-knowledge; quantum II.
The three-volume set of LNCS 11921,11922, and 11923 constitutes the refereed proceedings of the 25th International Conference on the Theory and Applications of Cryptology and Information Security, ASIACRYPT 2019, held in Kobe, Japan, in December 2019. The 71 revised full papers presented were carefully reviewed and selected from 307 submissions. They are organized in topical sections on Lattices; Symmetric Cryptography; Isogenies; Obfuscation; Multiparty Computation; Quantum; E-cash and Blockchain; Codes; Authenticated Encryption; Multilinear Maps; Homomorphic Encryption; Combinatorial Cryptography; Signatures; Public Key Encryption; Side Channels; Functional Encryption; Zero Knowledge.
This book constitutes the refereed proceedings of the 19th International Conference on Cryptology in India, INDOCRYPT 2018, held in New Delhi, India, in December 2018. The 20 revised full papers presented in this book were carefully reviewed and selected from 60 submissions. The focus of the conference includes works on outsourced computation and searchable encryption; symmetric key cryptography and format preserving encryption; fault attacks and Hash functions; post quantum cryptography; asymmetric key cryptography and cryptanalysis; symmetric key cryptanalysis; theory; and secure computations and protocols.
The three-volume set LNCS 10624, 10625, 10626 constitutes the refereed proceedings of the 23rd International Conference on the Theory and Applications of Cryptology and Information Security, ASIACRYPT 2017, held in Hong Kong, China, in December 2017.The 65 revised full papers were carefully selected from 243 submissions. They are organized in topical sections on Post-Quantum Cryptography; Symmetric Key Cryptanalysis; Lattices; Homomorphic Encryptions; Access Control; Oblivious Protocols; Side Channel Analysis; Pairing-based Protocols; Quantum Algorithms; Elliptic Curves; Block Chains; Multi-Party Protocols; Operating Modes Security Proofs; Cryptographic Protocols; Foundations; Zero-Knowledge Proofs; and Symmetric Key Designs.
