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**1 - 4**of**4**### Survey on Cryptographic Obfuscation

, 2015

"... The recent result of Garg et al. (FOCS 2013) changed the previously pessimistic attitude towards general purpose cryptographic obfuscation. Since their first candidate construction, several authors proposed newer and newer schemes with more persuasive security arguments and better efficiency. At th ..."

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The recent result of Garg et al. (FOCS 2013) changed the previously pessimistic attitude towards general purpose cryptographic obfuscation. Since their first candidate construction, several authors proposed newer and newer schemes with more persuasive security arguments and better efficiency. At the same time, indistinguishability obfuscation proved its extreme usefulness by becoming the basis of many solutions for long-standing open problems in cryptography e.g. functional or witness encryption and others. In this survey, we give an overview of recent research, focusing on the theoretical results on general purpose obfuscation, particularly, indistinguishability obfuscation.

### On Constructing One-Way Permutations from Indistinguishability

"... We prove that there is no black-box construction of a one-way permutation family from a one-way function and an indistinguishability obfuscator for the class of all oracle-aided circuits, where the construction is “domain invariant ” (i.e., where each permutation may have its own domain, but these d ..."

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We prove that there is no black-box construction of a one-way permutation family from a one-way function and an indistinguishability obfuscator for the class of all oracle-aided circuits, where the construction is “domain invariant ” (i.e., where each permutation may have its own domain, but these domains are independent of the underlying building blocks). Following the framework of Asharov and Segev (FOCS ’15), by considering indistinguisha-bility obfuscation for oracle-aided circuits we capture the common techniques that have been used so far in constructions based on indistinguishability obfuscation. These include, in particu-lar, non-black-box techniques such as the punctured programming approach of Sahai and Waters (STOC ’14) and its variants, as well as sub-exponential security assumptions. For example, we fully capture the construction of a trapdoor permutation family from a one-way function and an indistinguishability obfuscator due to Bitansky, Paneth and Wichs (TCC ’16). Their construc-tion is not domain invariant and our result shows that this, somewhat undesirable property, is unavoidable using the common techniques. In fact, we observe that constructions which are not domain invariant circumvent all known

### Hierarchical Functional Encryption *

"... Abstract Functional encryption provides fine-grained access control for encrypted data, allowing each user to learn only specific functions of the encrypted data. We study the notion of hierarchical functional encryption, which augments functional encryption with delegation capabilities, offering s ..."

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Abstract Functional encryption provides fine-grained access control for encrypted data, allowing each user to learn only specific functions of the encrypted data. We study the notion of hierarchical functional encryption, which augments functional encryption with delegation capabilities, offering significantly more expressive access control. We present a generic transformation that converts any general-purpose public-key functional encryption scheme into a hierarchical one without relying on any additional assumptions. This significantly refines our understanding of the power of functional encryption, showing that the existence of functional encryption is equivalent to that of its hierarchical generalization. Instantiating our transformation with the existing functional encryption schemes yields a variety of hierarchical schemes offering various trade-offs between their delegation capabilities (i.e., the depth and width of their hierarchical structures) and underlying assumptions. When starting with a scheme secure against an unbounded number of collusions, we can support arbitrary hierarchical structures. In addition, even when starting with schemes that are secure against a bounded number of collusions (which are known to exist under rather minimal assumptions such as the existence of public-key encryption and shallow pseudorandom generators), we can support hierarchical structures of bounded depth and width. * In this extended abstract we present results from

### Lower Bounds on Assumptions behind

"... Abstract. Since the seminal work of Garg et al. (FOCS’13) in which they proposed the first candidate construction for indistinguishability ob-fuscation (iO for short), iO has become a central cryptographic primitive with numerous applications. The security of the proposed construction of Garg et al. ..."

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Abstract. Since the seminal work of Garg et al. (FOCS’13) in which they proposed the first candidate construction for indistinguishability ob-fuscation (iO for short), iO has become a central cryptographic primitive with numerous applications. The security of the proposed construction of Garg et al. and its variants are proved based on multi-linear maps (Garg et al. Eurocrypt’13) and their idealized model called the graded encoding model (Brakerski and Rothblum TCC’14 and Barak et al. Eurocrypt’14). Whether or not iO could be based on standard and well-studied hard-ness assumptions has remain an elusive open question. In this work we prove lower bounds on the assumptions that imply iO in a black-box way, based on computational assumptions. Note that any lower bound for iO needs to somehow rely on computational assumptions, because if P = NP then statistically secure iO does exist. Our results are twofold: 1. There is no fully black-box construction of iO from (exponentially secure) collision-resistant hash functions unless the polynomial hier-archy collapses. Our lower bound extends to (separate iO from) any primitive implied by a random oracle in a black-box way. 2. Let P be any primitive that exists relative to random trapdoor per-mutations, the generic group model for any finite abelian group, or degree-O(1) graded encoding model for any finite ring. We show that