Elementary Remarks on Some Quadratic Based Identity Based Encryption Schemes

In the design of an identity-based encryption (IBE) scheme, the primary security assumptions center around quadratic residues, bilinear mappings, and lattices. Among these approaches, one of the most intriguing is introduced by Clifford Cocks and is based on quadratic residues. However, this scheme has a significant drawback: a large ciphertext to plaintext ratio. A different approach is taken by Zhao et al., who design an IBE still based on quadratic residues, but with an encryption process reminiscent of the Goldwasser-Micali cryptosystem. In the following pages, we will introduce an elementary method to accelerate Cocks\u27 encryption process and adapt a space-efficient encryption technique for both Cocks\u27 and Zhao et al.\u27s cryptosystems

Anonymous IBE From Quadratic Residuosity With Fast Encryption

We develop two variants of Cocks\u27 identity-based encryption. One variant has faster encryption, where the most time-consuming part only requires several modular multiplications. The other variant makes the first variant anonymous under suitable complexity assumptions, while its decryption efficiency is about twice lower than the first one. Both the variants have ciphertext expansion twice more extensive than the original Cocks\u27 identity-based encryption. To alleviate the issue of the second variant\u27s large ciphertext expansion, we consider using it to construct a public-key encryption with keyword search scheme with a fast encryption algorithm

Generalized Galbraith\u27s Test: Characterization and Applications to Anonymous IBE Schemes

The main approaches currently used to construct identity based encryption (IBE) schemes are based on bilinear mappings, quadratic residues and lattices. Among them, the most attractive approach is the one based on quadratic residues, due to the fact that the underlying security assumption is a well understood hard problem. The first such IBE scheme was constructed by Cocks and some of its deficiencies were addressed in subsequent works. In this paper, we will focus on two constructions that address the anonymity problem inherent in Cocks\u27 scheme and we will tackle some of their incomplete theoretical claims. More precisely, we rigorously study Clear et. al and Zhao et. al\u27s schemes and give accurate probabilities of successful decryption and identity detection in the non-anonymized version of the schemes. Also, in the case of Zhao \emph{et. al}\u27s scheme, we give a proper description of the underlying security assumptions

The Jacobi Symbol Problem for Quadratic Congruences and Applications to Cryptography

The hardness of solving the quadratic residuosity problem is the basis for establishing the security of many cryptographic schemes. Two of these are the public key encryption scheme and the identity-based encryption scheme proposed by Cocks. In this paper, we introduce a new computational problem: the problem of distinguishing between the Jacobi symbols of the solutions of a quadratic congruence modulo an RSA integer. We show that the security of the two encryption schemes is equivalent to the hardness of this problem, while the quadratic residuosity problem reduces to this new problem. We then specialize the problem to roots of quadratic residues and establish several computational indistinguishability relationships

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

On the Distribution of Quadratic Residues and Non-residues Modulo Composite Integers and Applications to Cryptography

We develop exact formulas for the distribution of quadratic residues and non-residues in sets of the form $a+X=\{(a+x)\bmod n\mid x\in X\}$, where $n$ is a prime or the product of two primes and $X$ is a subset of integers with given Jacobi symbols modulo prime factors of $n$. We then present applications of these formulas to Cocks\u27 identity-based encryption scheme and statistical indistinguishability

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