Practically Solving LPN in High Noise Regimes Faster Using Neural Networks

We conduct a systematic study of solving the learning parity with noise problem (LPN) using neural networks. Our main contribution is designing families of two-layer neural networks that practically outperform classical algorithms in high-noise, low-dimension regimes. We consider three settings where the numbers of LPN samples are abundant, very limited, and in between. In each setting we provide neural network models that solve LPN as fast as possible. For some settings we are also able to provide theories that explain the rationale of the design of our models. Comparing with the previous experiments of Esser, Kubler, and May (CRYPTO 2017), for dimension $n = 26$, noise rate $\tau = 0.498$, the ''Guess-then-Gaussian-elimination'' algorithm takes 3.12 days on 64 CPU cores, whereas our neural network algorithm takes 66 minutes on 8 GPUs. Our algorithm can also be plugged into the hybrid algorithms for solving middle or large dimension LPN instances.Comment: 37 page

Anonymous IBE, Leakage Resilience and Circular Security from New Assumptions

© International Association for Cryptologic Research 2018. In anonymous identity-based encryption (IBE), ciphertexts not only hide their corresponding messages, but also their target identity. We construct an anonymous IBE scheme based on the Computational Diffie-Hellman (CDH) assumption in general groups (and thus, as a special case, based on the hardness of factoring Blum integers). Our approach extends and refines the recent tree-based approach of Cho et al. (CRYPTO’17) and Döttling and Garg (CRYPTO’17). Whereas the tools underlying their approach do not seem to provide any form of anonymity, we introduce two new building blocks which we utilize for achieving anonymity: blind garbled circuits (which we construct based on any one-way function), and blind batch encryption (which we construct based on CDH). We then further demonstrate the applicability of our newly-developed tools by showing that batch encryption implies a public-key encryption scheme that is both resilient to leakage of a (1o(1)) -fraction of its secret key, and KDM secure (or circular secure) with respect to all linear functions of its secret key (which, in turn, is known to imply KDM security for bounded-size circuits). These yield the first high-rate leakage-resilient encryption scheme and the first KDM-secure encryption scheme based on the CDH or Factoring assumptions. Finally, relying on our techniques we also construct a batch encryption scheme based on the hardness of the Learning Parity with Noise (LPN) problem, albeit with very small noise rate Ω(log2(n)/n). Although this batch encryption scheme is not blind, we show that it still implies standard (i.e., non-anonymous) IBE, leakage resilience and KDM security. IBE and high-rate leakage resilience were not previously known from LPN, even with extremely low noise

Amortizing Rate-1 OT and Applications to PIR and PSI

Recent new constructions of rate-1 OT [Döttling, Garg, Ishai, Malavolta, Mour, and Ostrovsky, CRYPTO 2019] have brought this primitive under the spotlight and the techniques have led to new feasibility results for private-information retrieval, and homomorphic encryption for branching programs. The receiver communication of this construction consists of a quadratic (in the sender\u27s input size) number of group elements for a single instance of rate-1 OT. Recently [Garg, Hajiabadi, Ostrovsky, TCC 2020] improved the receiver communication to a linear number of group elements for a single string-OT. However, most applications of rate-1 OT require executing it multiple times, resulting in large communication costs for the receiver. In this work, we introduce a new technique for amortizing the cost of multiple rate-1 OTs. Specifically, based on standard pairing assumptions, we obtain a two-message rate-1 OT protocol for which the amortized cost per string-OT is asymptotically reduced to only four group elements. Our results lead to significant communication improvements in PSI and PIR, special cases of SFE for branching programs. - PIR: We obtain a rate-1 PIR scheme with client communication cost of $O(\lambda\cdot\log N)$ group elements for security parameter $\lambda$ and database size $N$. Notably, after a one-time setup (or one PIR instance), any following PIR instance only requires communication cost $O(\log N)$ number of group elements. - PSI with unbalanced inputs: We apply our techniques to private set intersection with unbalanced set sizes (where the receiver has a smaller set) and achieve receiver communication of $O((m+\lambda) \log N)$ group elements where $m, N$ are the sizes of the receiver and sender sets, respectively. Similarly, after a one-time setup (or one PSI instance), any following PSI instance only requires communication cost $O(m \cdot \log N)$ number of group elements. All previous sublinear-communication non-FHE based PSI protocols for the above unbalanced setting were also based on rate-1 OT, but incurred at least $O(\lambda^2 m \log N)$ group elements

Optimal Bounded-Collusion Secure Functional Encryption

We construct private-key and public-key functional encryption schemes secure against adversaries that corrupt an a-priori bounded number of users and obtain their functional keys, from minimal assumptions. For a collusion bound of $Q=Q(\lambda)$ (where $\lambda$ is the security parameter), our public-key (resp. private-key) functional encryption scheme (a) supports the class of all polynomial-size circuits; (b) can be built solely from a vanilla public-key (resp. private-key) encryption scheme; and (c) has ciphertexts that grow linearly with the collusion bound $Q$. Previous constructions were sub-optimal with respect to one or more of the above properties. The first two of these properties are the best possible and any improvement in the third property, namely the ciphertext size dependence on the collusion bound $Q$, can be used to realize an indistinguishability obfuscation scheme. In addition, our schemes are adaptively secure and make black-box use of the underlying cryptographic primitives

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)

New Code-Based Privacy-Preserving Cryptographic Constructions

Code-based cryptography has a long history but did suffer from periods of slow development. The field has recently attracted a lot of attention as one of the major branches of post-quantum cryptography. However, its subfield of privacy-preserving cryptographic constructions is still rather underdeveloped, e.g., important building blocks such as zero-knowledge range proofs and set membership proofs, and even proofs of knowledge of a hash preimage, have not been known under code-based assumptions. Moreover, almost no substantial technical development has been introduced in the last several years. This work introduces several new code-based privacy-preserving cryptographic constructions that considerably advance the state-of-the-art in code-based cryptography. Specifically, we present $3$ major contributions, each of which potentially yields various other applications. Our first contribution is a code-based statistically hiding and computationally binding commitment scheme with companion zero-knowledge (ZK) argument of knowledge of a valid opening that can be easily extended to prove that the committed bits satisfy other relations. Our second contribution is the first code-based zero-knowledge range argument for committed values, with communication cost logarithmic in the size of the range. A special feature of our range argument is that, while previous works on range proofs/arguments (in all branches of cryptography) only address ranges of non-negative integers, our protocol can handle signed fractional numbers, and hence, can potentially find a larger scope of applications. Our third contribution is the first code-based Merkle-tree accumulator supported by ZK argument of membership, which has been known to enable various interesting applications. In particular, it allows us to obtain the first code-based ring signatures and group signatures with logarithmic signature sizes

Leakage-Resilient IBE/ABE with Optimal Leakage Rates from Lattices

We derive the first adaptively secure IBE and ABE for t-CNF, and selectively secure ABE for general circuits from lattices, with $1-o(1)$ leakage rates, in the both relative leakage model and bounded retrieval model (BRM). To achieve this, we first identify a new fine-grained security notion for ABE -- partially adaptive/selective security, and instantiate this notion from LWE. Then, by using this notion, we design a new key compressing mechanism for identity-based/attributed-based weak hash proof system (IB/AB-wHPS) for various policy classes, achieving (1) succinct secret keys and (2) adaptive/selective security matching the existing non-leakage resilient lattice-based designs. Using the existing connection between weak hash proof system and leakage resilient encryption, the succinct-key IB/AB-wHPS can yield the desired leakage resilient IBE/ABE schemes with the optimal leakage rates in the relative leakage model. Finally, by further improving the prior analysis of the compatible locally computable extractors, we can achieve the optimal leakage rates in the BRM

Non-Interactive Batch Arguments for NP from Standard Assumptions

We study the problem of designing *non-interactive batch arguments* for $\mathsf{NP}$. Such an argument system allows an efficient prover to prove multiple $\mathsf{NP}$ statements, with size smaller than the combined witness length. We provide the first construction of such an argument system for $\mathsf{NP}$ in the common reference string model based on standard cryptographic assumptions. Prior works either require non-standard assumptions (or the random oracle model) or can only support private verification. At the heart of our result is a new *dual mode* interactive batch argument system for $\mathsf{NP}$. We show how to apply the correlation-intractability framework for Fiat-Shamir -- that has primarily been applied to proof systems -- to such interactive arguments

Rate-1 Key-Dependent Message Security via Reusable Homomorphic Extractor against Correlated-Source Attacks

In this work, we first present general methods to construct information rate-1 PKE that is \KDM^{(n)}-secure with respect to \emph{block-affine} functions for any unbounded polynomial $n$. To achieve this, we propose a new notion of extractor that satisfies \emph{reusability}, \emph{homomorphic}, and \emph{security against correlated-source attacks}, and show how to use this extractor to improve the information rate of the \KDM-secure PKE of Brakerski et al.~(Eurocrypt 18). Then, we show how to amplify \KDM~security from block-affine function class into general bounded size circuits via a variant of the technique of Applebaum (Eurocrypt 11), achieving better efficiency. Furthermore, we show how to generalize these approaches to the IBE setting. Additionally, our PKE and IBE schemes are also leakage resilient, with leakage rates $1-o(1)$ against a slightly smaller yet still general class -- block leakage functions. We can instantiate the required building blocks from \LWE or \DDH

Adaptively Secure and Succinct Functional Encryption: Improving Security and Efficiency, Simultaneously

Functional encryption (FE) is advanced encryption that enables us to issue functional decryption keys where functions are hardwired. When we decrypt a ciphertext of a message $m$ by a functional decryption key where a function $f$ is hardwired, we can obtain $f(m)$ and nothing else. We say FE is selectively or adaptively secure when target messages are chosen at the beginning or after function queries are sent, respectively. In the weakly-selective setting, function queries are also chosen at the beginning. We say FE is single-key/collusion-resistant when it is secure against adversaries that are given only-one/polynomially-many functional decryption keys, respectively. We say FE is sublinearly-succinct/succinct when the running time of an encryption algorithm is sublinear/poly-logarithmic in the function description size, respectively. In this study, we propose a generic transformation from weakly-selectively secure, single-key, and sublinearly-succinct (we call ``building block\u27\u27) PKFE for circuits into adaptively secure, collusion-resistant, and succinct (we call ``fully-equipped\u27\u27) one for circuits. We assume only the existence of the building block PKFE for circuits. That is, our transformation relies on neither concrete assumptions such as learning with errors nor indistinguishability obfuscation (IO). This is the first generic construction of fully-equipped PKFE that does not rely on IO. As side-benefits of our results, we obtain the following primitives from the building block PKFE for circuits: (1) laconic oblivious transfer (2) succinct garbling scheme for Turing machines (3) selectively secure, collusion-resistant, and succinct PKFE for Turing machines (4) low-overhead adaptively secure traitor tracing (5) key-dependent-message secure and leakage-resilient public-key encryption. We also obtain a generic transformation from simulation-based adaptively secure garbling schemes that satisfy a natural decomposability property into adaptively indistinguishable garbling schemes whose online complexity does not depend on the output length