Revisiting Variable Output Length XOR Pseudorandom Function

Let σ be some positive integer and C ⊆ {(i, j) : 1 ≤ i < j ≤ σ}. The theory behind finding a lower bound on the number of distinct blocks P1, . . . , Pσ ∈ {0, 1}n satisfying a set of linear equations {Pi ⊕Pj = ci,j : (i, j) ∈ C} for some ci,j ∈ {0, 1}n, is called mirror theory. Patarin introduced the mirror theory and provided a proof for this. However, the proof, even for a special class of equations, is complex and contains several non-trivial gaps. As an application of mirror theory, XORP[w] (known as XOR construction) returning (w−1) block output, is a pseudorandom function (PRF) for some parameter w, called width. The XOR construction can be seen as a basic structure of some encryption algorithms, e.g., the CENC encryption and the CHM authenticated encryption, proposed by Iwata in 2006. Due to potential application of XORP[w] and the nontrivial gaps in the proof of mirror theory, an alternative simpler analysis of PRF-security of XORP[w] would be much desired. Recently (in Crypto 2017) Dai et al. introduced a tool, called the χ2 method, for analyzing PRF-security. Using this tool, the authors have provided a proof of PRF-security of XORP[2] without relying on the mirror theory. In this paper, we resolve the general case; we apply the χ2 method to obtain a simpler security proof of XORP[w] for any w ≥ 2. For w = 2, we obtain a tighter bound for a wider range of parameters than that of Dai et al.. Moreover, we consider variable width construction XORP[∗] (in which the widths are chosen by adversaries adaptively), and also provide variable output length pseudorandom function (VOLPRF) security analysis for it. As an application of VOLPRF, we propose an authenticated encryption which is a simple variant of CHM or AES-GCM and provides much higher security than those at the cost of one extra blockcipher call for every message

A Standalone FPGA-based Miner for Lyra2REv2 Cryptocurrencies

Lyra2REv2 is a hashing algorithm that consists of a chain of individual hashing algorithms, and it is used as a proof-of-work function in several cryptocurrencies. The most crucial and exotic hashing algorithm in the Lyra2REv2 chain is a specific instance of the general Lyra2 algorithm. This work presents the first hardware implementation of the specific instance of Lyra2 that is used in Lyra2REv2. Several properties of the aforementioned algorithm are exploited in order to optimize the design. In addition, an FPGA-based hardware implementation of a standalone miner for Lyra2REv2 on a Xilinx Multi-Processor System on Chip is presented. The proposed Lyra2REv2 miner is shown to be significantly more energy efficient than both a GPU and a commercially available FPGA-based miner. Finally, we also explain how the simplified Lyra2 and Lyra2REv2 architectures can be modified with minimal effort to also support the recent Lyra2REv3 chained hashing algorithm.Comment: 13 pages, accepted for publication in IEEE Trans. Circuits Syst. I. arXiv admin note: substantial text overlap with arXiv:1807.0576

Revisiting Shared Data Protection Against Key Exposure

This paper puts a new light on secure data storage inside distributed systems. Specifically, it revisits computational secret sharing in a situation where the encryption key is exposed to an attacker. It comes with several contributions: First, it defines a security model for encryption schemes, where we ask for additional resilience against exposure of the encryption key. Precisely we ask for (1) indistinguishability of plaintexts under full ciphertext knowledge, (2) indistinguishability for an adversary who learns: the encryption key, plus all but one share of the ciphertext. (2) relaxes the "all-or-nothing" property to a more realistic setting, where the ciphertext is transformed into a number of shares, such that the adversary can't access one of them. (1) asks that, unless the user's key is disclosed, noone else than the user can retrieve information about the plaintext. Second, it introduces a new computationally secure encryption-then-sharing scheme, that protects the data in the previously defined attacker model. It consists in data encryption followed by a linear transformation of the ciphertext, then its fragmentation into shares, along with secret sharing of the randomness used for encryption. The computational overhead in addition to data encryption is reduced by half with respect to state of the art. Third, it provides for the first time cryptographic proofs in this context of key exposure. It emphasizes that the security of our scheme relies only on a simple cryptanalysis resilience assumption for blockciphers in public key mode: indistinguishability from random, of the sequence of diferentials of a random value. Fourth, it provides an alternative scheme relying on the more theoretical random permutation model. It consists in encrypting with sponge functions in duplex mode then, as before, secret-sharing the randomness

Revisiting the Concrete Security of Goldreich's Pseudorandom Generator

Local pseudorandom generators are a class of fundamental cryptographic primitives having very broad applications in theoretical cryptography. Following Couteau et al.'s work in ASIACRYPT 2018, this paper further studies the concrete security of one important class of local pseudorandom generators, i.e., Goldreich's pseudorandom generators. Our first attack is of the guess-and-determine type. Our result significantly improves the state-of-the-art algorithm proposed by Couteau et al., in terms of both asymptotic and concrete complexity, and breaks all the challenge parameters they proposed. For instance, for a parameter set suggested for 128 bits of security, we could solve the instance faster by a factor of about $2^{61}$, thereby destroying the claimed security completely. Our second attack further exploits the extremely sparse structure of the predicate $P_5$ and combines ideas from iterative decoding. This novel attack, named guess-and-decode, substantially improves the guess-and-determine approaches for cryptographic-relevant parameters. All the challenge parameter sets proposed in Couteau et al.'s work in ASIACRYPT 2018 aiming for 80-bit (128-bit) security levels can be solved in about $2^{58}$ ($2^{78}$) operations. We suggest new parameters for achieving 80-bit (128-bit) security with respect to our attacks. We also extend the attack to other promising predicates and investigate their resistance.Comment: 20 pages, 9 figure

Pseudorandom generators and the BQP vs. PH problem

It is a longstanding open problem to devise an oracle relative to which BQP does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural conjecture about the capacity of the Nisan-Wigderson pseudorandom generator [NW94] to fool AC_0, with MAJORITY as its hard function. Our conjecture is essentially that the loss due to the hybrid argument (which is a component of the standard proof from [NW94]) can be avoided in this setting. This is a question that has been asked previously in the pseudorandomness literature [BSW03]. We then make three main contributions: (1) We show that our conjecture implies the existence of an oracle relative to which BQP is not in the PH. This entails giving an explicit construction of unitary matrices, realizable by small quantum circuits, whose row-supports are "nearly-disjoint." (2) We give a simple framework (generalizing the setting of Aaronson [A10]) in which any efficiently quantumly computable unitary gives rise to a distribution that can be distinguished from the uniform distribution by an efficient quantum algorithm. When applied to the unitaries we construct, this framework yields a problem that can be solved quantumly, and which forms the basis for the desired oracle. (3) We prove that Aaronson's "GLN conjecture" [A10] implies our conjecture; our conjecture is thus formally easier to prove. The GLN conjecture was recently proved false for depth greater than 2 [A10a], but it remains open for depth 2. If true, the depth-2 version of either conjecture would imply an oracle relative to which BQP is not in AM, which is itself an outstanding open problem. Taken together, our results have the following interesting interpretation: they give an instantiation of the Nisan-Wigderson generator that can be broken by quantum computers, but not by the relevant modes of classical computation, if our conjecture is true.Comment: Updated in light of counterexample to the GLN conjectur

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

Contributions to Confidentiality and Integrity Algorithms for 5G

The confidentiality and integrity algorithms in cellular networks protect the transmission of user and signaling data over the air between users and the network, e.g., the base stations. There are three standardised cryptographic suites for confidentiality and integrity protection in 4G, which are based on the AES, SNOW 3G, and ZUC primitives, respectively. These primitives are used for providing a 128-bit security level and are usually implemented in hardware, e.g., using IP (intellectual property) cores, thus can be quite efficient. When we come to 5G, the innovative network architecture and high-performance demands pose new challenges to security. For the confidentiality and integrity protection, there are some new requirements on the underlying cryptographic algorithms. Specifically, these algorithms should: 1) provide 256 bits of security to protect against attackers equipped with quantum computing capabilities; and 2) provide at least 20 Gbps (Gigabits per second) speed in pure software environments, which is the downlink peak data rate in 5G. The reason for considering software environments is that the encryption in 5G will likely be moved to the cloud and implemented in software. Therefore, it is crucial to investigate existing algorithms in 4G, checking if they can satisfy the 5G requirements in terms of security and speed, and possibly propose new dedicated algorithms targeting these goals. This is the motivation of this thesis, which focuses on the confidentiality and integrity algorithms for 5G. The results can be summarised as follows.1. We investigate the security of SNOW 3G under 256-bit keys and propose two linear attacks against it with complexities 2172 and 2177, respectively. These cryptanalysis results indicate that SNOW 3G cannot provide the full 256-bit security level. 2. We design some spectral tools for linear cryptanalysis and apply these tools to investigate the security of ZUC-256, the 256-bit version of ZUC. We propose a distinguishing attack against ZUC-256 with complexity 2236, which is 220 faster than exhaustive key search. 3. We design a new stream cipher called SNOW-V in response to the new requirements for 5G confidentiality and integrity protection, in terms of security and speed. SNOW-V can provide a 256-bit security level and achieve a speed as high as 58 Gbps in software based on our extensive evaluation. The cipher is currently under evaluation in ETSI SAGE (Security Algorithms Group of Experts) as a promising candidate for 5G confidentiality and integrity algorithms. 4. We perform deeper cryptanalysis of SNOW-V to ensure that two common cryptanalysis techniques, guess-and-determine attacks and linear cryptanalysis, do not apply to SNOW-V faster than exhaustive key search. 5. We introduce two minor modifications in SNOW-V and propose an extreme performance variant, called SNOW-Vi, in response to the feedback about SNOW-V that some use cases are not fully covered. SNOW-Vi covers more use cases, especially some platforms with less capabilities. The speeds in software are increased by 50% in average over SNOW-V and can be up to 92 Gbps.Besides these works on 5G confidentiality and integrity algorithms, the thesis is also devoted to local pseudorandom generators (PRGs). 6. We investigate the security of local PRGs and propose two attacks against some constructions instantiated on the P5 predicate. The attacks improve existing results with a large gap and narrow down the secure parameter regime. We also extend the attacks to other local PRGs instantiated on general XOR-AND and XOR-MAJ predicates and provide some insight in the choice of safe parameters

Full Indifferentiable Security of the Xor of Two or More Random Permutations Using the χ2\chi^2 Method

The construction $\mathsf{XORP}$ (bitwise-xor of outputs of two independent $n$-bit random permutations) has gained broad attention over the last two decades due to its high security. Very recently, Dai \textit{et al.} (CRYPTO\u2717), by using a method which they term the {\em Chi-squared method} ($\chi^2$ method), have shown $n$-bit security of $\mathsf{XORP}$ when the underlying random permutations are kept secret to the adversary. In this work, we consider the case where the underlying random permutations are publicly available to the adversary. The best known security of $\mathsf{XORP}$ in this security game (also known as {\em indifferentiable security}) is $\frac{2n}{3}$-bit, due to Mennink \textit{et al.} (ACNS\u2715). Later, Lee (IEEE-IT\u2717) proved a better $\frac{(k-1)n}{k}$-bit security for the general construction $\mathsf{XORP}[k]$ which returns the xor of $k$ ($\geq 2$) independent random permutations. However, the security was shown only for the cases where $k$ is an even integer. In this paper, we improve all these known bounds and prove full, {\em i.e.,} $n$-bit (indifferentiable) security of $\mathsf{XORP}$ as well as $\mathsf{XORP}[k]$ for any $k$. Our main result is $n$-bit security of $\mathsf{XORP}$, and we use the $\chi^2$ method to prove it

Adiantum: length-preserving encryption for entry-level processors

We present HBSH, a simple construction for tweakable length-preserving encryption which supports the fastest options for hashing and stream encryption for processors without AES or other crypto instructions, with a provable quadratic advantage bound. Our composition Adiantum uses NH, Poly1305, XChaCha12, and a single AES invocation. On an ARM Cortex-A7 processor, Adiantum decrypts 4096-byte messages at 10.6 cycles per byte, over five times faster than AES-256-XTS, with a constant-time implementation. We also define HPolyC which is simpler and has excellent key agility at 13.6 cycles per byte

The Summation-Truncation Hybrid: Reusing Discarded Bits for Free

A well-established PRP-to-PRF conversion design is truncation: one evaluates an $n$-bit pseudorandom permutation on a certain input, and truncates the result to $a$ bits. The construction is known to achieve tight $2^{n-a/2}$ security. Truncation has gained popularity due to its appearance in the GCM-SIV key derivation function (ACM CCS 2015). This key derivation function makes four evaluations of AES, truncates the outputs to $n/2$ bits, and concatenates these to get a $2n$-bit subkey. In this work, we demonstrate that truncation is wasteful. In more detail, we present the Summation-Truncation Hybrid (STH). At a high level, the construction consists of two parallel evaluations of truncation, where the truncated $(n-a)$-bit chunks are not discarded but rather summed together and appended to the output. We prove that STH achieves a similar security level as truncation, and thus that the $n-a$ bits of extra output is rendered for free. In the application of GCM-SIV, the current key derivation can be used to output $3n$ bits of random material, or it can be reduced to three primitive evaluations. Both changes come with no security loss