Naor-Yung paradigm with shared randomness and applications

The Naor-Yung paradigm (Naor and Yung, STOC’90) allows to generically boost security under chosen-plaintext attacks (CPA) to security against chosen-ciphertext attacks (CCA) for public-key encryption (PKE) schemes. The main idea is to encrypt the plaintext twice (under independent public keys), and to append a non-interactive zero-knowledge (NIZK) proof that the two ciphertexts indeed encrypt the same message. Later work by Camenisch, Chandran, and Shoup (Eurocrypt’09) and Naor and Segev (Crypto’09 and SIAM J. Comput.’12) established that the very same techniques can also be used in the settings of key-dependent message (KDM) and key-leakage attacks (respectively). In this paper we study the conditions under which the two ciphertexts in the Naor-Yung construction can share the same random coins. We find that this is possible, provided that the underlying PKE scheme meets an additional simple property. The motivation for re-using the same random coins is that this allows to design much more efficient NIZK proofs. We showcase such an improvement in the random oracle model, under standard complexity assumptions including Decisional Diffie-Hellman, Quadratic Residuosity, and Subset Sum. The length of the resulting ciphertexts is reduced by 50%, yielding truly efficient PKE schemes achieving CCA security under KDM and key-leakage attacks. As an additional contribution, we design the first PKE scheme whose CPA security under KDM attacks can be directly reduced to (low-density instances of) the Subset Sum assumption. The scheme supports keydependent messages computed via any affine function of the secret ke

On the Non-malleability of the Fiat-Shamir Transform

The Fiat-Shamir transform is a well studied paradigm for removing interaction from public-coin protocols. We investigate whether the resulting non-interactive zero-knowledge (NIZK) proof systems also exhibit non-malleability properties that have up to now only been studied for NIZK proof systems in the common reference string model: first, we formally define simulation soundness and a weak form of simulation extraction in the random oracle model (ROM). Second, we show that in the ROM the Fiat-Shamir transform meets these properties under lenient conditions. A consequence of our result is that, in the ROM, we obtain truly efficient non malleable NIZK proof systems essentially for free. Our definitions are sufficient for instantiating the Naor-Yung paradigm for CCA2-secure encryption, as well as a generic construction for signature schemes from hard relations and simulation-extractable NIZK proof systems. These two constructions are interesting as the former preserves both the leakage resilience and key-dependent message security of the underlying CPA-secure encryption scheme, while the latter lifts the leakage resilience of the hard relation to the leakage resilience of the resulting signature scheme

Circular chosen-ciphertext security with compact ciphertexts

A key-dependent message (KDM) secure encryption scheme is secure even if an adversary obtains encryptions of messages that depend on the secret key. Such key-dependent encryptions naturally occur in scenarios such as harddisk encryption, formal cryptography, or in specific protocols. However, there are not many provably secure constructions of KDM-secure encryption schemes. Moreover, only one construction, due to Camenisch, Chandran, and Shoup (Eurocrypt 2009) is known to be secure against active (i.e., CCA) attacks. In this work, we construct the first public-key encryption scheme that is KDM-secure against active adversaries and has compact ciphertexts. As usual, we allow only circular key dependencies, meaning that encryptions of arbitrary *entire* secret keys under arbitrary public keys are considered in a multi-user setting. Technically, we follow the approach of Boneh, Halevi, Hamburg, and Ostrovsky (Crypto 2008) to KDM security, which however only achieves security against passive adversaries. We explain an inherent problem in adapting their techniques to active security, and resolve this problem using a new technical tool called ``lossy algebraic filters\u27\u27 (LAFs). We stress that we significantly deviate from the approach of Camenisch, Chandran, and Shoup to obtain KDM security against active adversaries. This allows us to develop a scheme with compact ciphertexts that consist only of a constant number of group elements

Updatable Public Key Encryption in the Standard Model

Forward security (FS) ensures that corrupting the current secret key in the system preserves the privacy or integrity of the prior usages of the system. Achieving forward security is especially hard in the setting of public-key encryption (PKE), where time is divided into periods, and in each period the receiver derives the next-period secret key from their current secret key, while the public key stays constant. Indeed, all current constructions of FS-PKE are built from hierarchical identity-based encryption (HIBE) and are rather complicated. Motivated by applications to secure messaging, recent works of Jost et al. (Eurocrypt’19) and Alwen et al. (CRYPTO’20) consider a natural relaxation of FS-PKE, which they term updatable PKE (UPKE). In this setting, the transition to the next period can be initiated by any sender, who can compute a special update ciphertext. This ciphertext directly produces the next-period public key and can be processed by the receiver to compute the next-period secret key. If done honestly, future (regular) ciphertexts produced with the new public key can be decrypted with the new secret key, but past such ciphertexts cannot be decrypted with the new secret key. Moreover, this is true even if all other previous-period updates were initiated by untrusted senders. Both papers also constructed a very simple UPKE scheme based on the CDH assumption in the random oracle model. However, they left open the question of building such schemes in the standard model, or based on other (e.g., post-quantum) assumptions, without using the heavy HIBE techniques. In this work, we construct two efficient UPKE schemes in the standard model, based on the DDH and LWE assumptions, respectively. Somewhat interestingly, our constructions gain their efficiency (compared to prior FS-PKE schemes) by using tools from the area of circular-secure and leakage resilient public-key encryption schemes (rather than HIBE)

On Generic Constructions of Circularly-Secure, Leakage-Resilient Public-Key Encryption Schemes

Abstract. We propose generic constructions of public-key encryption schemes, satisfying key- dependent message (KDM) security for projections and different forms of key-leakage resilience, from CPA-secure private key encryption schemes with two main abstract properties: (1) additive homomorphism with respect to both messages and randomness, and (2) reproducibility, providing a means for reusing encryption randomness across independent secret keys. More precisely, our construction transforms a private-key scheme with the stated properties (and one more mild condition) into a public-key one, providing: - n-KDM-projection security, an extension of circular security, where the adversary may also ask for encryptions of negated secret key bits; – a (1-o(1)) resilience rate in the bounded-memory leakage model of Akavia et al. (TCC 2009); and – Auxiliary-input security against subexponentially-hard functions. We introduce homomorphic weak pseudorandom functions, a homomorphic version of the weak PRFs proposed by Naor and Reingold (FOCS ’95) and use them to realize our base encryption scheme. We obtain homomorphic weak PRFs under assumptions including subgroup indistinguishability (implied, in particular, by QR and DCR) and homomorphic hash-proof systems (HHPS). As corollaries of our results, we obtain (1) a projection-secure encryption scheme (as well as a scheme with a (1-o(1)) resilience rate) based solely on the HHPS assumption, and (2) a unifying approach explaining the results of Boneh et al (CRYPTO ’08) and Brakerski and Goldwasser (CRYPTO ’10). Finally, by observing that Applebaum’s KDM amplification method (EUROCRYPT ’11) preserves both types of leakage resilience, we obtain schemes providing at the same time high leakage resilience and KDM security against any fixed polynomial-sized circuit family

Compactly Hiding Linear Spans: Tightly Secure Constant-Size Simulation-Sound QA-NIZK Proofs and Applications

International audienceQuasi-adaptive non-interactive zero-knowledge (QA-NIZK) proofs is a powerful paradigm, suggested recently by Jutla and Roy (Asiacrypt '13), which is motivated by the Groth-Sahai seminal techniques for efficient non-interactive zero-knowledge (NIZK) proofs. In this paradigm, the common reference string may depend on specific language parameters, a fact that allows much shorter proofs in important cases. It even makes certain standard model applications competitive with the Fiat-Shamir heuristic in the Random Oracle idealization (such QA-NIZK proofs were recently optimized to constant size by Jutla and Roy (Crypto '14) and Libert et al. (Eurocrypt '14) for the important case of proving that a vector of group elements belongs to a linear subspace). While, e.g., the QA-NIZK arguments of Libert et al. provide unbounded simulation-soundness and constant proof length, their simulation-soundness is only loosely related to the underlying assumption (with a gap proportional to the number of adversarial queries) and it is unknown how to alleviate this limitation without sacrificing efficiency. Here, we deal with the basic question of whether and to what extent we can simultaneously optimize the proof size and the tightness of security reductions, allowing for important applications with tight security (which are typically to date quite lengthy) to be of shorter size. In this paper, we resolve this question by describing a novel simulation-sound QA-NIZK argument showing that a vector v ∈ G n belongs to a subspace of rank t < n using a constant number of group elements. Unlike previous constant-size QA-NIZK proofs of such statements, the unbounded simulation-soundness of our system is nearly tightly related (i.e., the reduction only loses a factor proportional to the security parameter) to the standard Decision Linear assumption. To show simulation-soundness in the constrained context of tight reductions, we employ a number of techniques, and explicitly point at a technique – which may be of independent interest – of hiding the linear span of a structure-preserving homomorphic signature (which is part of an OR proof). As an application, we design a public-key cryptosystem with almost tight CCA2-security in the multi-challenge, multiuser setting with improved length (asymptotically optimal for long messages). We also adapt our scheme to provide CCA security in the key-dependent message scenario (KDM-CCA2) with ciphertext length reduced by 75% when compared to the best known tightly secure KDM-CCA2 system so far

Efficient KDM-CCA Secure Public-Key Encryption for Polynomial Functions

KDM$[\mathcal{F}]$-CCA secure public-key encryption (PKE) protects the security of message $f(sk)$, with $f \in \mathcal{F}$, that is computed directly from the secret key, even if the adversary has access to a decryption oracle. An efficient KDM$[\mathcal{F}_{\text{aff}}]$-CCA secure PKE scheme for affine functions was proposed by Lu, Li and Jia (LLJ, EuroCrypt2015). We point out that their security proof cannot go through based on the DDH assumption. In this paper, we introduce a new concept _Authenticated Encryption with Auxiliary-Input_ $\mathsf{AIAE}$ and define for it new security notions dealing with related-key attacks, namely _IND-RKA security_ and _weak INT-RKA security_. We also construct such an $\mathsf{AIAE}$ w.r.t. a set of restricted affine functions from the DDH assumption. With our $\mathsf{AIAE}$, -- we construct the first efficient KDM$[\mathcal{F}_{\text{aff}}]$-CCA secure PKE w.r.t. affine functions with compact ciphertexts, which consist only of a constant number of group elements; -- we construct the first efficient KDM$[\mathcal{F}_{\text{poly}}^d]$-CCA secure PKE w.r.t. polynomial functions of bounded degree $d$ with almost compact ciphertexts, and the number of group elements in a ciphertext is polynomial in $d$, independent of the security parameter. Our PKEs are both based on the DDH & DCR assumptions, free of NIZK and free of pairing

DDH-Like Assumptions Based on Extension Rings

Abstract. We introduce and study a new type of DDH-like assumptions based on groups of prime order q. Whereas standard DDH is based on encoding elements of Fq “in the exponent ” of elements in the group, we ask what happens if instead we put in the exponent elements of the extension ring Rf = Fq[X]/(f) where f is a degree-d polynomial. The decision problem that follows naturally reduces to the case where f is irreducible. This variant is called the d-DDH problem, where 1-DDH is standard DDH. We show in the generic group model that d-DDH is harder than DDH for d&gt; 1 and that we obtain, in fact, an infinite hierarchy of progressively weaker assumptions whose complexities lie “between” DDH and CDH. This leads to a large number of new schemes because virtually all known DDH-based constructions can very easily be upgraded to be based on d-DDH. We use the same construction and security proof but get better security and moreover, the amortized complexity (e.g, computation per encrypted bit) is the same as when using DDH. We also show that d-DDH, just like DDH, is easy in bilinear groups. We therefore suggest a different type of assumption, the d-vector DDH problems (d-VDDH), which are based on f(X) = X d, but with a twist to avoid problems with reducible polynomials. We show in the generic group model that d-VDDH is hard in bilinear groups and that the problems become harder with increasing d. We show that hardness of d-VDDH implies CCA-secure encryption, efficient Naor-Reingold style pseudorandom functions, and auxiliary input secure encryption. This can be seen as an alternative to the known family of k-LIN assumptions.

Reproducible Circularly-Secure Bit Encryption: Applications and Realizations

We give generic constructions of several fundamental cryptographic primitives based on a new encryption primitive that combines circular security for bit encryption with the so-called reproducibility property (Bellare et al. PKC 2003). At the heart of our constructions is a novel technique which gives a way of de-randomizing reproducible public-key bit-encryption schemes and also a way of reducing one-wayness conditions of a constructed trapdoor-function family (TDF) to circular security of the base scheme. The main primitives that we build from our encryption primitive include k-wise one- way TDFs (Rosen and Segev TCC 2009), CCA2-secure encryption and deterministic encryption. Our results demonstrate a new set of applications of circularly-secure encryption beyond fully-homomorphic encryption and symbolic soundness. Finally, we show the plausibility of our assumptions by showing that the DDH-based circularly-secure scheme of Boneh et al. (Crypto 2008) and the subgroup indistinguishability based scheme of Brakerski and Goldwasser (Crypto 2010) are both reproducible