Simple Homomorphisms of Cocks IBE and Applications

The Cocks Identity Based Encryption (IBE) scheme, proposed in 2001 by Clifford Cocks, has been the standard for Quadratic Residue-based IBE. It had been long believed that this IBE did not have enough structure to have homomorphic properties. In 2013, Clear, Hughes, and Tewari (Africacrypt 2013) created a Cocks scheme derivative where they viewed ciphertexts as polynomials modulo a quadratic. While the scheme was homomorphic, it required sending twice as much information per ciphertext as the original Cocks scheme. A recent result by Joye (PKC 2016) used complex algebraic structures to demonstrate the fact that Cocks IBE, on its own, is additively homomorphic. In this work, we build upon the results from CHT and Joye. We take the simple intuition from CHT, that ciphertexts can be seen as polynomials, but also demonstrate that we only need to send as much data as in the original Cocks scheme. This perspective leads to better intuition as to why these ciphertexts are homomorphic and to explicit efficient algorithms for computing this homomorphic addition. We believe that our approach will facilitate other extensions of Cocks IBE. As an example, we exhibit a two-way proxy re-encryption algorithm, which arises as a simple consequence of the structure we propose. That is, given a re-encryption key, we can securely convert a ciphertext under one key to a ciphertext under the other key and vice-versa (hence two-way)

A Note on Attribute-Based Group Homomorphic Encryption

Group Homomorphic Encryption (GHE), formally defined by Armknecht, Katzenbeisser and Peter, is a public-key encryption primitive where the decryption algorithm is a group homomorphism. Hence it supports homomorphic evaluation of a single algebraic operation such as modular addition or modular multiplication. Most classical homomorphic encryption schemes such as as Goldwasser-Micali and Paillier are instances of GHE. In this work, we extend GHE to the attribute-based setting. We introduce and formally define the notion of Attribute-Based GHE (ABGHE) and explore its properties. We then examine the algebraic structure on attributes induced by the group operation in an ABGHE. This algebraic stricture is a bounded semilattice. We consider some possible semilattices and how they can be realized by an ABGHE supporting inner product predicates. We then examine existing schemes from the literature and show that they meet our definition of ABGHE for either an additive or multiplicative homomorphism. Some of these schemes are in fact Identity-Based Group Homomorphic Encryption (IBGHE) schemes i.e. instances of ABGHE whose class of access policies are point functions. We then present a possibility result for IBGHE from indistinguishability obfuscation for any group for which a (public-key) GHE scheme exists

Anonymous Homomorphic IBE with Application to Anonymous Aggregation

All anonymous identity-based encryption (IBE) schemes that are group homomorphic (to the best of our knowledge) require knowledge of the identity to compute the homomorphic operation. This paper is motivated by this open problem, namely to construct an anonymous group-homomorphic IBE scheme that does not sacrifice anonymity to perform homomorphic operations. Note that even when strong assumptions such as indistinguishability obfuscation (iO) are permitted, no schemes are known. We succeed in solving this open problem by assuming iO and the hardness of the DBDH problem over rings (specifically, $Z_{N^2}$ for RSA modulus $N$). We then use the existence of such a scheme to construct an IBE scheme with re-randomizable anonymous encryption keys, which we prove to be IND-ID-RCCA secure. Finally, we use our results to construct identity-based anonymous aggregation protocols

Topology-Hiding Computation on all Graphs

A distributed computation in which nodes are connected by a partial communication graph is called topology-hiding if it does not reveal information about the graph beyond what is revealed by the output of the function. Previous results have shown that topology-hiding computation protocols exist for graphs of constant degree and logarithmic diameter in the number of nodes [Moran-Orlov-Richelson, TCC\u2715; Hirt \etal, Crypto\u2716] as well as for other graph families, such as cycles, trees, and low circumference graphs [Akavia-Moran, Eurocrypt\u2717], but the feasibility question for general graphs was open. In this work we positively resolve the above open problem: we prove that topology-hiding computation is feasible for all graphs under either the Decisional Diffie-Hellman or Quadratic-Residuosity assumption. Our techniques employ random-walks to generate paths covering the graph, upon which we apply the Akavia-Moran topology-hiding broadcast for chain-graphs (paths). To prevent topology information revealed by the random-walk, we design multiple random-walks that, together, are locally identical to receiving at each round a message from each neighbors and sending back processed messages in a randomly permuted order

Implementation of the Cocks cryptosystem

RESUMEN: Los sistemas de encriptación basados en la identidad (IBE, Identity-Based Cryptosystem) son criptosistemas donde la clave pública de cada usuario se calcula a partir de un «string» que lo identifique. En este trabajo, veremos la implementación del criptosistema de Cocks, un sistema de encriptación basado en la identidad cuya seguridad reside en la dificultad de resolver el problema de la residuidad cuadrática. Además del estudio teórico del sistema de Cocks y de su implementación en python, se presentarán los fundamentos de teoría de complejidad, la dificultad del problema de la residuosidad cuadrática, las reducciones de seguridad, así como aplicaciones del criptosistema de Cocks a la computación con texto encriptado.ABSTRACT: The Identity-Based Cryptosystem (IBE) are cryptosystems where the public key of each user is calculated from a «string» that identifies it. In this work, we will see the implementation of the Cocks cryptosystem, an identity-based encryption system whose security lies in the difficulty of solving the quadratic residuosity problem. In addition to the theoretical study of the Cocks system and its implementation in python, will be presented the fundamentals of complexity theory, the difficulty of the quadratic residual problem, the security reductions, as well as Cocks cryptosystem applications to encrypted text computing.Grado en Matemática

Pairing Cryptography Meets Isogeny: A New Framework of Isogenous Pairing Groups

We put forth a new mathematical framework called Isogenous Pairing Groups (IPG) and new intractable assumptions in the framework, the Isogenous DBDH (Isog-DBDH) assumption and its variants. Three operations, i.e., exponentiation, pairing and isogeny on elliptic curves are treated under a unified notion of trapdoor homomorphisms, and combinations of the operations have potential new cryptographic applications, in which the compatibility of pairing and isogeny is a main ingredient in IPG. As an example, we present constructions of (small and large universe) key-policy attribute-based encryption (KP-ABE) schemes secure against pre-challenge quantum adversaries in the quantum random oracle model (QROM). Note that our small universe KP-ABE has asymptotically the same efficiency as Goyal et al.\u27s small universe KP-ABE, which has only classical security. As a by-product, we also propose practical (hierarchical) identity-based encryption ((H)IBE) schemes secure against pre-challenge quantum adversaries in the QROM from isogenies, which are based on the Boneh-Franklin IBE and the Gentry-Silverberg HIBE, respectively

Pairings in Cryptology: efficiency, security and applications

Abstract The study of pairings can be considered in so many di�erent ways that it may not be useless to state in a few words the plan which has been adopted, and the chief objects at which it has aimed. This is not an attempt to write the whole history of the pairings in cryptology, or to detail every discovery, but rather a general presentation motivated by the two main requirements in cryptology; e�ciency and security. Starting from the basic underlying mathematics, pairing maps are con- structed and a major security issue related to the question of the minimal embedding �eld [12]1 is resolved. This is followed by an exposition on how to compute e�ciently the �nal exponentiation occurring in the calculation of a pairing [124]2 and a thorough survey on the security of the discrete log- arithm problem from both theoretical and implementational perspectives. These two crucial cryptologic requirements being ful�lled an identity based encryption scheme taking advantage of pairings [24]3 is introduced. Then, perceiving the need to hash identities to points on a pairing-friendly elliptic curve in the more general context of identity based cryptography, a new technique to efficiently solve this practical issue is exhibited. Unveiling pairings in cryptology involves a good understanding of both mathematical and cryptologic principles. Therefore, although �rst pre- sented from an abstract mathematical viewpoint, pairings are then studied from a more practical perspective, slowly drifting away toward cryptologic applications

Cryptographic Pairings: Efficiency and DLP security

This thesis studies two important aspects of the use of pairings in cryptography, efficient algorithms and security. Pairings are very useful tools in cryptography, originally used for the cryptanalysis of elliptic curve cryptography, they are now used in key exchange protocols, signature schemes and Identity-based cryptography. This thesis comprises of two parts: Security and Efficient Algorithms. In Part I: Security, the security of pairing-based protocols is considered, with a thorough examination of the Discrete Logarithm Problem (DLP) as it occurs in PBC. Results on the relationship between the two instances of the DLP will be presented along with a discussion about the appropriate selection of parameters to ensure particular security level. In Part II: Efficient Algorithms, some of the computational issues which arise when using pairings in cryptography are addressed. Pairings can be computationally expensive, so the Pairing-Based Cryptography (PBC) research community is constantly striving to find computational improvements for all aspects of protocols using pairings. The improvements given in this section contribute towards more efficient methods for the computation of pairings, and increase the efficiency of operations necessary in some pairing-based protocol

Minicrypt Primitives with Algebraic Structure and Applications

Algebraic structure lies at the heart of much of Cryptomania as we know it. An interesting question is the following: instead of building (Cryptomania) primitives from concrete assumptions, can we build them from simple Minicrypt primitives endowed with additional algebraic structure? In this work, we affirmatively answer this question by adding algebraic structure to the following Minicrypt primitives: • One-Way Function (OWF) • Weak Unpredictable Function (wUF) • Weak Pseudorandom Function (wPRF) The algebraic structure that we consider is group homomorphism over the input/output spaces of these primitives. We also consider a “bounded” notion of homomorphism where the primitive only supports an a priori bounded number of homomorphic operations in order to capture lattice-based and other “noisy” assumptions. We show that these structured primitives can be used to construct many cryptographic protocols. In particular, we prove that: • (Bounded) Homomorphic OWFs (HOWFs) imply collision-resistant hash functions, Schnorr-style signatures, and chameleon hash functions. • (Bounded) Input-Homomorphic weak UFs (IHwUFs) imply CPA-secure PKE, non-interactive key exchange, trapdoor functions, blind batch encryption (which implies anonymous IBE, KDM-secure and leakage-resilient PKE), CCA2 deterministic PKE, and hinting PRGs (which in turn imply transformation of CPA to CCA security for ABE/1-sided PE). • (Bounded) Input-Homomorphic weak PRFs (IHwPRFs) imply PIR, lossy trapdoor functions, OT and MPC (in the plain model). In addition, we show how to realize any CDH/DDH-based protocol with certain properties in a generic manner using IHwUFs/IHwPRFs, and how to instantiate such a protocol from many concrete assumptions. We also consider primitives with substantially richer structure, namely Ring IHwPRFs and L-composable IHwPRFs. In particular, we show the following: • Ring IHwPRFs with certain properties imply FHE. • 2-composable IHwPRFs imply (black-box) IBE, and $L$-composable IHwPRFs imply non-interactive $(L + 1)$-party key exchange. Our framework allows us to categorize many cryptographic protocols based on which structured Minicrypt primitive implies them. In addition, it potentially makes showing the existence of many cryptosystems from novel assumptions substantially easier in the future