Key Recovery Attacks against NTRU-based Somewhat Homomorphic Encryption Schemes

A key recovery attack allows an attacker to recover the private key of an underlying encryption scheme when given a number of decryption oracle accesses. Previous research has shown that most existing Somewhat Homomorphic Encryption (SHE) schemes suffer from this attack. In this paper, we propose efficient key recovery attacks against two NTRU-based SHE schemes, which have not gained much attention in the literature. One is published by Lopez-Alt et al. at STOC conference 2012 and the other is published by Bos et al. at the IMACC conference 2013. Parallel to our work, Dahab, Galbraith and Morais have also proposed similar attacks but only for specific parameter settings at ICITS conference 2015. In comparison, our attacks apply to all parameter settings and are more efficient than theirs

On the IND-CCA1 Security of FHE Schemes

Fully homomorphic encryption (FHE) is a powerful tool in cryptography that allows one to perform arbitrary computations on encrypted material without having to decrypt it first. There are numerous FHE schemes, all of which are expanded from somewhat homomorphic encryption (SHE) schemes, and some of which are considered viable in practice. However, while these FHE schemes are semantically (IND-CPA) secure, the question of their IND-CCA1 security is much less studied, and we therefore provide an overview of the IND-CCA1 security of all acknowledged FHE schemes in this paper. To give this overview, we grouped the SHE schemes into broad categories based on their similarities and underlying hardness problems. For each category, we show that the SHE schemes are susceptible to either known adaptive key recovery attacks, a natural extension of known attacks, or our proposed attacks. Finally, we discuss the known techniques to achieve IND-CCA1-secure FHE and SHE schemes. We concluded that none of the proposed schemes were IND-CCA1-secure and that the known general constructions all had their shortcomings.publishedVersio

Encriptação parcialmente homomórfica CCA1-segura

Orientadores: Ricardo Dahab, Diego de Freitas AranhaTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Nesta tese nosso tema de pesquisa é a encriptação homomórfica, com foco em uma solução prática e segura para encriptação parcialmente homomórfica (somewhat homomorphic encryption - SHE), considerando o modelo de segurança conhecido como ataque de texto encriptado escolhido (chosen ciphertext attack - CCA). Este modelo pode ser subdividido em duas categorias, a saber, CCA1 e CCA2, sendo CCA2 o mais forte. Sabe-se que é impossível construir métodos de encriptação homomórfica que sejam CCA2-seguros. Por outro lado, é possível obter segurança CCA1, mas apenas um esquema foi proposto até hoje na literatura; assim, seria interessante haver outras construções oferecendo este tipo de segurança. Resumimos os principais resultados desta tese de doutorado em duas contribuições. A primeira é mostrar que a família NTRU de esquemas SHE é vulnerável a ataques de recuperação de chave privada, e portanto não são CCA1-seguros. A segunda é a utilização de computação verificável para obter esquemas SHE que são CCA1-seguros e que podem ser usados para avaliar polinômios multivariáveis quadráticos. Atualmente, métodos de encriptação homomórfica são construídos usando como substrato dois problemas de difícil solução: o MDC aproximado (approximate GCD problem - AGCD) e o problema de aprendizado com erros (learning with errors - LWE). O problema AGCD leva, em geral, a construções mais simples mas com desempenho inferior, enquanto que os esquemas baseados no problema LWE correspondem ao estado da arte nesta área de pesquisa. Recentemente, Cheon e Stehlé demonstraram que ambos problemas estão relacionados, e é uma questão interessante investigar se esquemas baseados no problema AGCD podem ser tão eficientes quanto esquemas baseados no problema LWE. Nós respondemos afirmativamente a esta questão para um cenário específico: estendemos o esquema de computação verificável proposto por Fiore, Gennaro e Pastro, de forma que use a suposição de que o problema AGCD é difícil, juntamente com o esquema DGHV adaptado para uso do Teorema Chinês dos Restos (Chinese remainder theorem - CRT) de forma a evitar ataques de recuperação de chave privadaAbstract: In this thesis we study homomorphic encryption with focus on practical and secure somewhat homomorphic encryption (SHE), under the chosen ciphertext attack (CCA) security model. This model is classified into two different main categories: CCA1 and CCA2, with CCA2 being the strongest. It is known that it is impossible to construct CCA2-secure homomorphic encryption schemes. On the other hand, CCA1-security is possible, but only one scheme is known to achieve it. It would thus be interesting to have other CCA1-secure constructions. The main results of this thesis are summarized in two contributions. The first is to show that the NTRU-family of SHE schemes is vulnerable to key recovery attacks, hence not CCA1-secure. The second is the utilization of verifiable computation to obtain a CCA1-secure SHE scheme that can be used to evaluate quadratic multivariate polynomials. Homomorphic encryption schemes are usually constructed under the assumption that two distinct problems are hard, namely the Approximate GCD (AGCD) Problem and the Learning with Errors (LWE) Problem. The AGCD problem leads, in general, to simpler constructions, but with worse performance, wheras LWE-based schemes correspond to the state-of-the-art in this research area. Recently, Cheon and Stehlé proved that both problems are related, and thus it is an interesting problem to investigate if AGCD-based SHE schemes can be made as efficient as their LWE counterparts. We answer this question positively for a specific scenario, extending the verifiable computation scheme proposed by Fiore, Gennaro and Pastro to work under the AGCD assumption, and using it together with the Chinese Remainder Theorem (CRT)-version of the DGHV scheme, in order to avoid key recovery attacksDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação143484/2011-7CNPQCAPE

Studies on the Security of Selected Advanced Asymmetric Cryptographic Primitives

The main goal of asymmetric cryptography is to provide confidential communication, which allows two parties to communicate securely even in the presence of adversaries. Ever since its invention in the seventies, asymmetric cryptography has been improved and developed further, and a formal security framework has been established around it. This framework includes different security goals, attack models, and security notions. As progress was made in the field, more advanced asymmetric cryptographic primitives were proposed, with other properties in addition to confidentiality. These new primitives also have their own definitions and notions of security. This thesis consists of two parts, where the first relates to the security of fully homomorphic encryption and related primitives. The second part presents a novel cryptographic primitive, and defines what security goals the primitive should achieve. The first part of the thesis consists of Article I, II, and III, which all pertain to the security of homomorphic encryption schemes in one respect or another. Article I demonstrates that a particular fully homomorphic encryption scheme is insecure in the sense that an adversary with access only to the public material can recover the secret key. It is also shown that this insecurity mainly stems from the operations necessary to make the scheme fully homomorphic. Article II presents an adaptive key recovery attack on a leveled homomorphic encryption scheme. The scheme in question claimed to withstand precisely such attacks, and was the only scheme of its kind to do so at the time. This part of the thesis culminates with Article III, which is an overview article on the IND-CCA1 security of all acknowledged homomorphic encryption schemes. The second part of the thesis consists of Article IV, which presents Vetted Encryption (VE), a novel asymmetric cryptographic primitive. The primitive is designed to allow a recipient to vet who may send them messages, by setting up a public filter with a public verification key, and providing each vetted sender with their own encryption key. There are three different variants of VE, based on whether the sender is identifiable to the filter and/or the recipient. Security definitions, general constructions and comparisons to already existing cryptographic primitives are provided for all three variants.Doktorgradsavhandlin

Adaptive key recovery attacks on NTRU-based somewhat homomorphic encryption schemes

In this paper we present adaptive key recovery attacks on NTRU-based somewhat homomorphic encryption schemes. Among such schemes, we study the proposal by Bos et al [BLLN13] in 2013. Given access to a decryption oracle, the attack allows us to compute the private key for all parameter choices. Such attacks show that one must be very careful about the use of homomorphic encryption in practice. The existence of a key recovery attack means that the scheme is not CCA1-secure. Indeed, almost every somewhat homomorphic construction proposed till now in the literature is vulnerable to an attack of this type. Hence our result adds to a body of literature that shows that building CCA1-secure homomorphic schemes is not trivial

Key-Recovery Attacks Against Somewhat Homomorphic Encryption Schemes

In 1978, Rivest, Adleman and Dertouzos introduced the concept of privacy homomorphism and asked whether it is possible to perform arbitrary operations on encrypted ciphertexts. Thirty years later, Gentry gave a positive answer in his seminal paper at STOC 2009, by proposing an ingenious approach to construct fully homomorphic encryption (FHE) schemes. With this approach, one starts with a somewhat homomorphic encryption (SHE) scheme that can perform only limited number of operations on ciphertexts (i.e. it can evaluate only low-degree polynomials). Then, through the so-called bootstrapping step, it is possible to turn this SHE scheme into an FHE scheme. After Gentry's work, many SHE and FHE schemes have been proposed; in total, they can be divided into four categories, according to the hardness assumptions underlying each SHE (and hence, FHE) scheme: hard problems on lattices, the approximate common divisor problem, the (ring) learning with errors problem, and the NTRU encryption scheme. Even though SHE schemes are less powerful than FHE schemes, they can already be used in many useful real-world applications, such as medical and financial applications. It is therefore of primary concern to understand what level of security these SHE schemes provide. By default, all the SHE schemes developed so far offer IND-CPA security - i.e. resistant against a chosen-plaintext attack - but nothing is said about their IND-CCA1 security - i.e. secure against an adversary who is able to perform a non-adaptive chosen-ciphertext attack. Considering such an adversary is in fact a more realistic scenario. Gentry emphasized it as a future work to investigate SHE schemes with IND-CCA1 security, and the task to make some clarity about it was initiated by Loftus, May, Smart and Vercauteren: at SAC 2011 they showed how one family of SHE schemes is not IND-CCA1 secure, opening the doors to an interesting investigation on the IND-CCA1 security of the existing schemes in the other three families of schemes. In this work we therefore continue this line of research and show that most existing somewhat homomorphic encryption schemes are not IND-CCA1 secure. In fact, we show that these schemes suffer from key recovery attacks (stronger than a typical IND-CCA1 attack), which allow an adversary to completely recover the private keys through a number of decryption oracle queries. As a result, this dissertation shows that all known SHE schemes fail to provide IND-CCA1 security. While it is true that IND-CPA security may be enough to construct cryptographic protocols in presence of semi-honest attackers, key recovery attacks will pose serious threats for practical usage of SHE and FHE schemes: if a malicious attacker (or a compromised honest party) submits manipulated ciphertexts and observes the behavior (side channel leakage) of the decryptor, then it may be able to recover all plaintexts in the system. Therefore, it is very desirable to design SHE and FHE with IND-CCA1 security, or at least design them to prevent key recovery attacks. This raises the interesting question whether it is possible or not to develop such IND-CCA1 secure SHE scheme. Up to date, the only positive result in this direction is a SHE scheme proposed by Loftus et al. at SAC 2011 (in fact, a modification of an existing SHE scheme and IND-CCA1 insecure). However, this IND-CCA1 secure SHE scheme makes use of a non standard knowledge assumption, while it would be more interesting to only rely on standard assumptions. We propose then a variant of the SHE scheme proposed by Lopez-Alt, Tromer, and Vaikuntanathan at STOC 2012, which offers good indicators about its possible IND-CCA1 security

잡음키를 가지는 신원기반 동형암호에 관한 연구

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2020. 2. 천정희.클라우드 상의 데이터 분석 위임 시나리오는 동형암호의 가장 효과적인 응용 시나리오 중 하나이다. 그러나, 다양한 데이터 제공자와 분석결과 요구자가 존재하는 실제 현실의 모델에서는 기본적인 암복호화와 동형 연산 외에도 여전히 해결해야 할 과제들이 남아있는 실정이다. 본 학위논문에서는 이러한 모델에서 필요한 여러 요구사항들을 포착하고, 이에 대한 해결방안을 논하였다. 먼저, 기존의 알려진 동형 데이터 분석 솔루션들은 데이터 간의 층위나 수준을 고려하지 못한다는 점에 착안하여, 신원기반 암호와 동형암호를 결합하여 데이터 사이에 접근 권한을 설정하여 해당 데이터 사이의 연산을 허용하는 모델을 생각하였다. 또한 이 모델의 효율적인 동작을 위해서 동형암호 친화적인 신원기반 암호에 대하여 연구하였고, 기존에 알려진 NTRU 기반의 암호를 확장하여 module-NTRU 문제를 정의하고 이를 기반으로 한 신원기반 암호를 제안하였다. 둘째로, 동형암호의 복호화 과정에는 여전히 비밀키가 관여하고 있고, 따라서 비밀키 관리 문제가 남아있다는 점을 포착하였다. 이러한 점에서 생체정보를 활용할 수 있는 복호화 과정을 개발하여 해당 과정을 동형암호 복호화에 적용하였고, 이를 통해 암복호화와 동형 연산의 전 과정을 어느 곳에도 키가 저장되지 않은 상태로 수행할 수 있는 암호시스템을 제안하였다. 마지막으로, 동형암호의 구체적인 안전성 평가 방법을 고려하였다. 이를 위해 동형암호가 기반하고 있는 이른바 Learning With Errors (LWE) 문제의 실제적인 난해성을 면밀히 분석하였고, 그 결과 기존의 공격 알고리즘보다 평균적으로 1000배 이상 빠른 공격 알고리즘들을 개발하였다. 이를 통해 현재 사용하고 있는 동형암호 파라미터가 안전하지 않음을 보였고, 새로운 공격 알고리즘을 통한 파라미터 설정 방법에 대해서 논하였다.Secure data analysis delegation on cloud is one of the most powerful application that homomorphic encryption (HE) can bring. As the technical level of HE arrive at practical regime, this model is also being considered to be a more serious and realistic paradigm. In this regard, this increasing attention requires more versatile and secure model to deal with much complicated real world problems. First, as real world modeling involves a number of data owners and clients, an authorized control to data access is still required even for HE scenario. Second, we note that although homomorphic operation requires no secret key, the decryption requires the secret key. That is, the secret key management concern still remains even for HE. Last, in a rather fundamental view, we thoroughly analyze the concrete hardness of the base problem of HE, so-called Learning With Errors (LWE). In fact, for the sake of efficiency, HE exploits a weaker variant of LWE whose security is believed not fully understood. For the data encryption phase efficiency, we improve the previously suggested NTRU-lattice ID-based encryption by generalizing the NTRU concept into module-NTRU lattice. Moreover, we design a novel method that decrypts the resulting ciphertext with a noisy key. This enables the decryptor to use its own noisy source, in particular biometric, and hence fundamentally solves the key management problem. Finally, by considering further improvement on existing LWE solving algorithms, we propose new algorithms that shows much faster performance. Consequently, we argue that the HE parameter choice should be updated regarding our attacks in order to maintain the currently claimed security level.1 Introduction 1 1.1 Access Control based on Identity 2 1.2 Biometric Key Management 3 1.3 Concrete Security of HE 3 1.4 List of Papers 4 2 Background 6 2.1 Notation 6 2.2 Lattices 7 2.2.1 Lattice Reduction Algorithm 7 2.2.2 BKZ cost model 8 2.2.3 Geometric Series Assumption (GSA) 8 2.2.4 The Nearest Plane Algorithm 9 2.3 Gaussian Measures 9 2.3.1 Kullback-Leibler Divergence 11 2.4 Lattice-based Hard Problems 12 2.4.1 The Learning With Errors Problem 12 2.4.2 NTRU Problem 13 2.5 One-way and Pseudo-random Functions 14 3 ID-based Data Access Control 16 3.1 Module-NTRU Lattices 16 3.1.1 Construction of MNTRU lattice and trapdoor 17 3.1.2 Minimize the Gram-Schmidt norm 22 3.2 IBE-Scheme from Module-NTRU 24 3.2.1 Scheme Construction 24 3.2.2 Security Analysis by Attack Algorithms 29 3.2.3 Parameter Selections 31 3.3 Application to Signature 33 4 Noisy Key Cryptosystem 36 4.1 Reusable Fuzzy Extractors 37 4.2 Local Functions 40 4.2.1 Hardness over Non-uniform Sources 40 4.2.2 Flipping local functions 43 4.2.3 Noise stability of predicate functions: Xor-Maj 44 4.3 From Pseudorandom Local Functions 47 4.3.1 Basic Construction: One-bit Fuzzy Extractor 48 4.3.2 Expansion to multi-bit Fuzzy Extractor 50 4.3.3 Indistinguishable Reusability 52 4.3.4 One-way Reusability 56 4.4 From Local One-way Functions 59 5 Concrete Security of Homomorphic Encryption 63 5.1 Albrecht's Improved Dual Attack 64 5.1.1 Simple Dual Lattice Attack 64 5.1.2 Improved Dual Attack 66 5.2 Meet-in-the-Middle Attack on LWE 69 5.2.1 Noisy Collision Search 70 5.2.2 Noisy Meet-in-the-middle Attack on LWE 74 5.3 The Hybrid-Dual Attack 76 5.3.1 Dimension-error Trade-o of LWE 77 5.3.2 Our Hybrid Attack 79 5.4 The Hybrid-Primal Attack 82 5.4.1 The Primal Attack on LWE 83 5.4.2 The Hybrid Attack for SVP 86 5.4.3 The Hybrid-Primal attack for LWE 93 5.4.4 Complexity Analysis 96 5.5 Bit-security estimation 102 5.5.1 Estimations 104 5.5.2 Application to PKE 105 6 Conclusion 108 Abstract (in Korean) 120Docto

A key recovery attack to the scale-invariant NTRU-based somewhat homomorphic encryption scheme

In this paper we present a key recovery attack to the scale-invariant NTRU-based somewhat homomorphic encryption scheme proposed by Bos et al~\cite{NTRUbasedFHE} in 2013. The attack allows us to compute the private key for $t>2$ and when the private key is chosen with coefficients in $\{-1,0,1\}$. The efficiency of the attack is optimal since it requires just one decryption oracle query, showing that if we don\u27t look for this kind of vulnerabilities in homomorphic encryption constructions we are likely to choose insecure parameters. The existence of a key recovery attack means that the scheme is not CCA1-secure. Indeed, almost every somewhat homomorphic construction proposed till now in the literature is vulnerable to this kind of attack, hence our result indicates that building CCA1-secure homomorphic schemes is not trivial. We also provide tables showing how the multiplicative depth is affected when the critical parameter \Bkey is chosen in order to mitigatte the attack

A Subfield Lattice Attack on Overstretched NTRU Assumptions:Cryptanalysis of Some FHE and Graded Encoding Schemes

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