We consider protocols for secure multi-party computation (MPC) built from FHE under honest majority, i.e., for $n=2t 1$ players of which $t$ are corrupt, that is robust. Surprisingly there exists no robust threshold FHE scheme based on BFV to design an MPC protocol. Precisely, all existing methods for generating a common relinearization key can abort as soon as one player deviates. We solve this issue, with a new relinearization key (adapted from [CDKS19, CCS'19]) which we show how to...

Homomorphically multiplying a plaintext matrix with a ciphertext matrix (PC-MM) is a central task for the private evaluation of transformers, commonly used for large language models. We provide several RLWE-based algorithms for PC-MM that consist of multiplications of plaintext matrices (PC-MM) and comparatively cheap pre-processing and post-processing steps: for small and large dimensions compared to the RLWE ring degree, and with and without precomputation. For the algorithms with...

We propose a new NTRU-based Public-Key Encryption (PKE) scheme called $\mathsf{NTRU }\mathsf{PKE}$, which effectively incorporates the Fujisaki-Okamoto transformation for PKE (denoted as $\mathsf{FO}_{\mathsf{PKE}}$) to achieve chosen-ciphertext security in the Quantum Random Oracle Model (QROM). While $\mathsf{NTRUEncrypt}$, a first-round candidate in the NIST PQC standardization process, was proven to be chosen-ciphertext secure in the Random Oracle Model (ROM), it lacked corresponding...

Lattice cryptography schemes based on the learning with errors (LWE) hardness assumption have been standardized by NIST for use as post-quantum cryptosystems, and by HomomorphicEncryption.org for encrypted compute on sensitive data. Thus, understanding their concrete security is critical. Most work on LWE security focuses on theoretical estimates of attack performance, which is important but may overlook attack nuances arising in real-world implementations. The sole existing concrete...

Homomorphic encryption allows for computations on encrypted data without exposing the underlying plaintext, enabling secure and private data processing in various applications such as cloud computing and machine learning. This paper presents a comprehensive mathematical foundation for three prominent homomorphic encryption schemes: Brakerski-Gentry-Vaikuntanathan (BGV), Brakerski-Fan-Vercauteren (BFV), and Cheon-Kim-Kim-Song (CKKS), all based on the Ring Learning with Errors (RLWE) problem....

We construct an efficient proxy re-encryption (PRE) scheme secure against honest re-encryption attacks (HRA-secure) with precise concrete security estimates. To get these precise concrete security estimates, we introduce the tight, fine-grained noise-flooding techniques of Li et al. (CRYPTO'22) to RLWE-based (homomorphic) PRE schemes, as well as a mixed statistical-computational security to HRA security analysis. Our solution also supports homomorphic operations on the ciphertexts. Such...

Fully homomorphic encryption (FHE) allows for evaluating arbitrary functions over encrypted data. In Multi-party FHE applications, different parties encrypt their secret data and submit ciphertexts to a server, which, according to the application logic, performs homomorphic operations on them. For example, in a secret voting application, the tally is computed by summing up the ciphertexts encoding the votes. Valid encrypted votes are of the form $E(0)$ and $E(1)$. A malicious voter could...

Sparse binary LWE secrets are under consideration for standardization for Homomorphic Encryption and its applications to private computation. Known attacks on sparse binary LWE secrets include the sparse dual attack and the hybrid sparse dual-meet in the middle attack, which requires significant memory. In this paper, we provide a new statistical attack with low memory requirement. The attack relies on some initial parallelized lattice reduction. The key observation is that, after...

We present a novel circuit bootstrapping algorithm that outperforms the state-of-the-art TFHE method with 9.9× speedup and 15.6× key size reduction. These improvements can be attributed to two technical contributions. Firstly, we redesigned the circuit bootstrapping workflow to operate exclusively under the ring ciphertext type, which eliminates the need of conversion between LWE and RLWE ciphertexts. Secondly, we improve the LMKC blind rotation algorithm by reducing the number of...

Homomorphic encryption (HE) is in the spotlight as a solution for privacy-related issues in various real-world scenarios. However, the limited types of operations supported by each HE scheme have been a major drawback in applications. While HE schemes based on learning-with-error (LWE) problem provide efficient lookup table (LUT) evaluation in terms of latency, they have downsides in arithmetic operations and low throughput compared to HE schemes based on ring LWE (RLWE) problem. The use of...

The splitting field $F$ of the polynomial $Y^n-2$ is an extension over $\mathbb{Q}$ generated by $\zeta_n=\exp(2 \pi \sqrt{-1} /n)$ and $\sqrt[n]{2}$. In this paper, we lay the foundation for applying the Order-LWE in the integral ring $\mathcal{R}=\mathbb{Z}[\zeta_n, \sqrt[n]{2}]$ to cryptographic uses when $n$ is a power-of-two integer. We explicitly compute the Galois group $\text{Gal}\left(F/\mathbb{Q} \right)$ and the canonical embedding of $F$, based on which we study the properties of...

We extend the known pseudorandomness of Ring-LWE to be based on lattices that do not correspond to any ideal of any order in the underlying number field. In earlier works of Lyubashevsky et al (EUROCRYPT 2010) and Peikert et al (STOC 2017), the hardness of RLWE was based on ideal lattices of ring of integers of number fields, which are known to be Dedekind domains. While these works extended Regev's (STOC 2005) quantum polynomial-time reduction for LWE, thus allowing more efficient and more...

The evolution of quantum algorithms threatens to break public key cryptography in polynomial time. The development of quantum-resistant algorithms for the post-quantum era has seen a significant growth in the field of post quantum cryptography (PQC). Polynomial multiplication is the core of Ring Learning with Error (RLWE) lattice based cryptography (LBC) which is one of the most promising PQC candidates. In this work, we present the design of fast and energy-efficient pipelined Number...

We present two new constructions for private information retrieval (PIR) in the classical setting where the clients do not need to do any preprocessing or store any database dependent information, and the server does not need to store any client-dependent information. Our first construction (HintlessPIR) eliminates the client preprocessing step from the recent LWE-based SimplePIR (Henzinger et. al., USENIX Security 2023) by outsourcing the "hint" related computation to the server,...

Homomorphic Encryption (HE) is one of the mainstream cryptographic tools used to enable secure outsourced computation. A typical task is secure matrix computation. Popular HE schemes are all based on the problem of Ring Learning with Errors (RLWE), where the messages are encrypted in a ring. In general, the ring dimension should be large to ensure security, which is often larger than the matrix size. Hence, exploiting the ring structure to make fast homomorphic matrix computation has been an...

Blind rotation is one of the key techniques to construct fully homomorphic encryptions with the best known bootstrapping algorithms running in less than one second. Currently, the two main approaches, namely, AP and GINX, for realizing blind rotation are first introduced by Alperin-Sheriff and Peikert (CRYPTO 2014) and Gama, Izabachene, Nguyen and Xie (EUROCRYPT 2016), respectively. \qquad In this paper, we propose a new blind rotation algorithm based on a GSW-like encryption from the...

Key-policy attribute-based encryption scheme (KP-ABE) uses a set of attributes as public keys for encryption. It allows homomorphic evaluation of ciphertext into another ciphertext of the same message, which can be decrypted if a certain access policy based on the attributes is satisfied. A lattice-based KP-ABE scheme is reported in several works in the literature, and its software implementation is available in an open-source library called PALISADE. However, as the cryptographic primitives...

In this paper, we present NEV -- a faster and smaller NTRU Encryption using Vector decoding, which is provably IND-CPA secure in the standard model under the decisional NTRU and RLWE assumptions over the cyclotomic ring $R_q = \mathbb{Z}_q[X]/(X^n 1)$. Our main technique is a novel and non-trivial way to integrate a previously known plaintext encoding and decoding mechanism into the provably IND-CPA secure NTRU variant by Stehl\'e and Steinfeld (Eurocrypt 2011). Unlike the original NTRU...

Most of the current fully homomorphic encryption (FHE) schemes are based on either the learning-with-errors (LWE) problem or on its ring variant (RLWE) for storing plaintexts. During the homomorphic computation of FHE schemes, RLWE formats provide high throughput when considering several messages, and LWE formats provide a low latency when there are only a few messages. Efficient conversion can bridge the advantages of each format. However, converting LWE formats into RLWE format, which is...

The post-quantum digital signature scheme CRYSTALS-Dilithium has been recently selected by the NIST for standardization. Implementing CRYSTALS-Dilithium, and other post-quantum cryptography schemes, on embedded devices raises a new set of challenges, including ones related to performance in terms of speed and memory requirements, but also related to side-channel and fault injection attacks security. In this work, we investigated the latter and describe a differential fault attack on the...

In this paper we provide the implementation details and performance analysis of the lattice-based post-quantum commitment scheme introduced by Martínez and Morillo in their work titled «RLWE-Based Zero-Knowledge Proofs for Linear and Multiplicative Relations» together with the corresponding Zero-Knowledge Proofs of Knowledge (ZKPoK) of valid openings, linear and multiplicative relations among committed elements. We bridge the gap between the existing theoretical proposals and practical...

Fully Homomorphic Encryption over the Torus (TFHE) is a homomorphic encryption scheme which supports efficient Boolean operations over encrypted bits. TFHE has a unique feature in that the evaluation of each binary gate is followed by a bootstrapping procedure to refresh the noise of a ciphertext. In particular, this gate bootstrapping involves two algorithms called the blind rotation and key-switching. In this work, we introduce several optimization techniques for the TFHE bootstrapping....

In recent years, there has been much focus on developing core cryptographic primitives based on lattice assumptions, driven by the NIST cal for post-quantum key encapsulation and digital signature algorithms. However, more work must be conducted on efficient privacy-preserving protocols with post-quantum security. Electronic voting is one such privacy-preserving protocol whose adoption is increasing across the democratic world. E-voting offers both a fast and convenient alternative to...

The promising field of homomorphic encryption enables functions to be evaluated on encrypted data and produce results that mimic the same computations done on plaintexts. It, therefore, comes as no surprise that many ventures at constructing homomorphic encryption schemes have come into the limelight in recent years. Most popular are those that rely on the hard lattice problem, called the Ring Learning with Errors problem (RLWE). One major limitation of these homomorphic encryption schemes...

The Brakerski-Gentry-Vaikuntanathan (BGV) scheme is a Fully Homomorphic Encryption (FHE) cryptosystem based on the Ring Learning With Error (RLWE) problem. Ciphertexts in this scheme contain an error term that grows with operations and causes decryption failure when it surpasses a certain threshold. For this reason, the parameters of BGV need to be estimated carefully, with a trade-off between security and error margin. The ciphertext space of BGV is the ring $\mathcal R_q=\mathbb...

With the increased use of data and communication through the internet and the abundant misuse of personal data by many organizations, people are more sensitive about their privacy. Privacy-preserving computation is becoming increasingly important in this era. Functional encryption allows a user to evaluate a function on encrypted data without revealing sensitive information. Most implementations of functional encryption schemes are too time-consuming for practical use. Mera et al. first...

A large part of current research in homomorphic encryption (HE) aims towards making HE practical for real-world applications. In any practical HE, an important issue is to convert the application data (type) to the data type suitable for the HE. The main purpose of this work is to investigate an efficient HE-compatible encoding method that is generic, and can be easily adapted to apply to the HE schemes over integers or polynomials. $p$-adic number theory provides a way to transform...

In the last decade, zero-knowledge proof of knowledge protocols have been extensively studied to achieve active security of various cryptographic protocols. However, the existing solutions simply seek zero-knowledge for both message and randomness, which is an overkill in many applications since protocols may remain secure even if some information about randomness is leaked to the adversary. We develop this idea to improve the state-of-the-art proof of knowledge protocols for RLWE-based...

This note introduces a public-key variant of TFHE. The output ciphertexts are of LWE type. Interestingly, the public key is shorter and the resulting ciphertexts are less noisy. The security of the scheme holds under the standard RLWE assumption. Several variations and extensions are also described.

The Brakerski/Fan-Vercauteren (BFV) scheme is a state-of-the-art scheme in Fully Homomorphic Encryption based on the Ring Learning with Errors (RLWE) problem. Thus, ciphertexts contain an error that increases with each homomorphic operation and has to stay below a certain threshold for correctness. This can be achieved by setting the ciphertext modulus big enough. On the other hand, a larger ciphertext modulus decreases the level of security and computational efficiency, making parameters...

Ring-Learning-with-Errors (RLWE) has emerged as the foundation of many important techniques for improving security and privacy, including homomorphic encryption and post-quantum cryptography. While promising, these techniques have received limited use due to their extreme overheads of running on general-purpose machines. In this paper, we present a novel vector Instruction Set Architecture (ISA) and microarchitecture for accelerating the ring-based computations of RLWE. The ISA, named B512,...

In this work, we introduce a lightweight communication-efficient multi-key approach suitable for the Federated Averaging rule. By combining secret-key RLWE-based HE, additive secret sharing and PRFs, we reduce approximately by a half the communication cost per party when compared to the usual public-key instantiations, while keeping practical homomorphic aggregation performances. Additionally, for LWE-based instantiations, our approach reduces the communication cost per party from quadratic...

NTRU was the first practical public key encryption scheme constructed on a lattice over a polynomial-based ring and has been considered secure against significant cryptanalytic attacks over the past few decades. However, NTRU and its variants suffer from several drawbacks, including difficulties in achieving worst-case correctness error in a moderate modulus, inconvenient sampling distributions for messages, and relatively slower algorithms compared to other lattice-based schemes. In...

Primal attack, BKW attack, and dual attack are three well-known attacks to LWE. To build efficient post-quantum cryptosystems in practice, the structured variants of LWE (i.e. MLWE/RLWE) are often used. Some efforts have been spent on addressing concerns about additional vulnerabilities introduced by algebraic structures and no effective attack method based on ideal lattices or module lattices has been proposed so far; these include refining primal attack and BKW attack to MLWE/RLWE. It is...

The quantum resistance Key Encapsulation Mechanism (PQC-KEM) design aims to replace cryptography in legacy security protocols. It would be nice if PQC-KEM were faster and lighter than ECDH or DH for easy migration to legacy security protocols. However, it seems impossible due to the temperament of the secure underlying problems in a quantum environment. Therefore, it makes reason to determine the threshold of the scheme by analyzing the maximum bandwidth the legacy security protocol can...

In this work we extend the known pseudorandomness of Ring-LWE (RLWE) to be based on ideal lattices of non Dedekind domains. In earlier works of Lyubashevsky et al (EUROCRYPT 2010) and Peikert et al (STOC 2017), the hardness of RLWE was based on ideal lattices of ring of integers of number fields, which are known to be Dedekind domains. While these works extended Regev's (STOC 2005) quantum polynomial-time reduction for LWE, thus allowing more efficient and more structured cryptosystems, the...

Quantum computing threatens classical cryptography, leading to the search for stronger alternatives. The cryptographic approach based on lattices is considered as a viable option. Schemes with that approach use Gaussian sampling, a design which brings along two concerns: efficiency and information leakage. This work addresses those concerns in the RLWE formulation, for digital signatures. Efficiency mitigation uses the central limit theorem, and the Walsh–Hadamard transform, whereas the...

Secure aggregation enables a server to learn the sum of client-held vectors in a privacy-preserving way, and has been successfully applied to distributed statistical analysis and machine learning. In this paper, we both introduce a more efficient secure aggregation construction and extend secure aggregation by enabling input validation, in which the server can check that clients' inputs satisfy required constraints such as $L_0$, $L_2$, and $L_\infty$ bounds. This prevents malicious clients...

Fully Homomorphic encryption allows the evaluation of any circuits over encrypted data while preserving the privacy of the data. However, without any additional properties, no guarantee is provided for the privacy of the circuits which are evaluated. A sanitization algorithm allows to destroy all previous information about how a ciphertext was obtained, ensuring that the circuit which was evaluated remains secret. In this paper, we present two techniques to randomize RLWE ciphertexts, and...

In most homomorphic encryption schemes based on the RLWE, the native plaintexts are represented as polynomials in a ring $Z_t[x]/x^N 1$ where $t$ is a plaintext modulus and $x^N 1$ is a cyclotomic polynomial with degree power of two. An encoding scheme should be used to transform some natural data types(such as integers and rational numbers) into polynomials in the ring. After a homomorphic computation on the polynomial is finished, the decoding procedure is invoked to obtain the result....

This paper introduces Ibex, an advertising system that reduces the amount of data that is collected on users while still allowing advertisers to bid on real-time ad auctions and measure the effectiveness of their ad campaigns. Specifically, Ibex addresses an issue in recent proposals such as Google’s Privacy Sandbox Topics API in which browsers send information about topics that are of interest to a user to advertisers and demand-side platforms (DSPs). DSPs use this information to (1)...

As for any cryptographic algorithm, the deployment of post-quantum CCA-secure public-key encryption schemes may come with the need to be protected against side-channel attacks. For existing post-quantum schemes that have not been developed with leakage in mind, recent results showed that the cost of these protections can make their implementations more expensive by orders of magnitude. In this paper, we describe a new design, coined \(\texttt{POLKA}\), that is specifically tailored for this...

We propose and implement a multiparty homomorphic encryption (MHE) scheme with a $t$-out-of-$N$-threshold access-structure that is efficient and does not require a trusted dealer in the common random-string model. We construct this scheme from the ring-learning-with-error (RLWE) assumptions, and as an extension of the MHE scheme of Mouchet et al. (PETS 21). By means of a specially adapted share re-sharing procedure, this extension can be used to relax the $N$-out-of-$N$-threshold access...

Fully homomorphic encryption~(FHE) has flourished since it was first constructed by Gentry~(STOC 2009). Single instruction multiple data~(SIMD) gave rise to efficient homomorphic operations on vectors in \((\mathbb{F}_{t^d})^\ell\), for prime \(t\). RLWE instantiated with cyclotomic polynomials of the form \(X^{2^N} 1\) dominate implementations of FHE due to highly efficient fast Fourier transformations. However, this choice yields very short SIMD plaintext vectors and high degree extension...

Introducing small errors in the lattice-based key exchange protocols, although it is resistant to quantum computing attacks, will cause both parties to only get roughly equal secret values, which brings uncertainty to the negotiation of the key agreement. The role of the error reconciliation mechanism is to eliminate this uncertainty and ensure that both parties can reach a consensus. This paper designs a new error reconciliation mechanism: Speedy Error Reconciliation (SER), which can...

Fully Homomorphic Encryption (FHE) is a groundbreaking technology that allows for arbitrary computations to be performed on encrypted data. State-of-the-art schemes such as Brakerski Gentry Vaikuntanathan (BGV) are based on the Learning with Errors over rings (RLWE) assumption, and each ciphertext has an associated error that grows with each homomorphic operation. For correctness, the error needs to stay below a certain threshold, requiring a trade-off between security and error margin for...

A multi-key fully homomorphic encryption (MKFHE) scheme allows a public server to evaluate arbitrary circuits over ciphertexts encrypted under different keys. One of the main drawbacks of MKFHE schemes is the need for a ciphertext expansion procedure prior to evaluation, which combines ciphertexts encrypted under different keys to a (much larger) ciphertext encrypted under a concatenated key. In this paper, we present a new (leveled) RLWE-based MKFHE scheme without ciphertext expansion.

Ever since the appearance of quantum computers, prime factoring and discrete logarithm based cryptography has been put in question, giving birth to the so called post-quantum cryptography. The most prominent field in post-quantum cryptography is lattice-based cryptography, protocols that are proved to be as difficult to break as certain difficult lattice problems like Learning With Errors (LWE) or Ring Learning With Errors (RLWE). Furthermore, the application of cryptographic techniques to...

We present fully homomorphic encryption schemes for fixed-point arithmetic with fixed precision. Our scheme achieves $\mathsf{IND}$-$\mathsf{CPA^D}$ security and uses $\mathsf{RLWE}$ ring with dimension ${2^{13}}$ or less. Our techniques could also be extended to construct fully homomorphic encryption schemes for approximate numbers with $\mathsf{IND}$-$\mathsf{CPA}$ security. The bootstrapping process of our $\mathsf{IND}$-$\mathsf{CPA}$ scheme preserves about 39-bit precision with ring...

Recently, the application of multi-party secure computing schemes based on homomorphic encryption in the field of machine learning attracts attentions across the research fields. Previous studies have demonstrated that secure protocols adopting packed additive homomorphic encryption (PAHE) schemes based on the ring learning with errors (RLWE) problem exhibit significant practical merits, and are particularly promising in enabling efficient secure inference in machine-learning-as-a-service...

We propose a new bootstrapping approach that works for all three Brakerski-Gentry-Vaikuntanathan (BGV), Brakerski/Fan-Vercauteren (BFV), and Cheon-Kim-Kim-Song (CKKS) schemes. This approach adopts a blind rotation technique from FHEW-type schemes. For BGV and BFV, our bootstrapping does not have any restrictions on plaintext modulus unlike typical cases of the previous methods. For CKKS, our approach introduces an error comparable to a rescaling error which enables more than 70 bits of...

Cryptographic constructions based on $\textit{ring learning with errors}$ (RLWE) have emerged as one of the front runners for the standardization of post quantum public key cryptography. As the standardization process continues, optimizing specific parts of proposed schemes becomes a worthwhile endeavor. In this work we focus on using error correcting codes to alleviate a natural trade-off present in most schemes; namely, we would like a wider error distribution to increase security, but a...

Several recent proposals of efficient public-key encryption are based on variants of the polynomial learning with errors problem ($\mathsf{PLWE}^f$) in which the underlying polynomial ring $\mathbb{Z}_q[x]/f$ \ is replaced with the (related) modular integer ring $\mathbb{Z}_{f(q)}$; the corresponding problem is known as Integer Polynomial Learning with Errors ($\mathsf{I-PLWE}^f$). Cryptosystems based on $\mathsf{I-PLWE}^f$ and its variants can exploit optimised big-integer arithmetic to...

In the recent years, many research lines on Functional Encryption (FE) have been suggested and studied regarding the functionality, security, or efficiency. Nevertheless, an open problem on a basic functionality, the single-input inner-product (IPFE), remains: can IPFE be instantiated based on the Ring Learning With Errors (RLWE) assumption? The RLWE assumption provides quantum-resistance security while in comparison with LWE assumption gives significant performance and compactness...

Private Stream Aggregation (PSA) protocols allow for the secure aggregation of time-series data, affording security and privacy to users' private data, with significantly better efficiency than general secure computation such as homomorphic encryption, multiparty computation, and secure hardware based approaches. Earlier PSA protocols face limitations including needless complexity, a lack of post-quantum security, or other practical issues. In this work, we present SLAP, a Simple...

We present a two-message oblivious transfer protocol achieving statistical sender privacy and computational receiver privacy based on the RLWE assumption for cyclotomic number fields. This work improves upon prior lattice-based statistically sender-private oblivious transfer protocols by reducing the total communication between parties by a factor $O(n\log q)$ for transfer of length $O(n)$ messages. Prior work of Brakerski and Döttling uses transference theorems to show that either a...

One of the NIST Post-Quantum Cryptography Standardization Process Round 2 candidates is the NewHope cryptosystem, which is a suite of two RLWE based key encapsulation mechanisms. Recently, four key reuse attacks were proposed against NewHope by Bauer et al., Qin et al., Bhasin et al. and Okada et al. In these attacks, the adversary has access to the key mismatch oracle which tells her if a given ciphertext decrypts to a given message under the targeted secret key. Previous attacks either...

In this paper we give efficient statistical zero-knowledge proofs (SNARKs) for Module/Ring LWE and Module/Ring SIS relations, providing the remaining ingredient for building efficient cryptographic protocols from lattice-based hardness assumptions. We achieve our results by exploiting the linear-algebraic nature of the statements supported by the Aurora proof system (Ben-Sasson et al.), which allows us to easily and efficiently encode the linear-algebraic statements that arise in lattice...

The hardness of the Ring Learning with Errors problem (RLWE) is a central building block for efficiency-oriented lattice-based cryptography. Many applications use an ``entropic'' variant of the problem where the so-called ``secret'' is not distributed uniformly as prescribed but instead comes from some distribution with sufficient min-entropy. However, the hardness of the entropic variant has not been substantiated thus far. For standard LWE (not over rings) entropic results are known,...

Anonymous veto networks (AV-nets), originally proposed by Hao and Zielinski (2006), are particularly lightweight protocols for evaluating a veto function in a peer-to-peer network such that anonymity of all protocol participants is preserved. Prior to this work, anonymity in all AV-nets from the literature relied on the decisional Diffie-Hellman (DDH) assumption and can thus be broken by (scalable) quantum computers. In order to defend against this threat, we propose two practical and...

In 2019, Gu Chunsheng introduced Integer-RLWE, a variant of RLWE devoid of some of its efficiency flaws. Most notably, he proposes a setting where $n$ can be an arbitrary positive integer, contrarily to the typical construction $n = 2^k$. In this paper, we analyze the new problem and implement the classical meet-in-the-middle and lattice-based attacks. We then use the peculiarity of the construction of $n$ to build an improved lattice-based attack in cases where $n$ is composite with an odd...

An oblivious linear function evaluation protocol, or OLE, is a two-party protocol for the function $f(x) = ax b$, where a sender inputs the field elements $a,b$, and a receiver inputs $x$ and learns $f(x)$. OLE can be used to build secret-shared multiplication, and is an essential component of many secure computation applications including general-purpose multi-party computation, private set intersection and more. In this work, we present several efficient OLE protocols from the ring...

Private function evaluation (PFE) allows to obliviously evaluate a private function on private inputs. PFE has several applications such as privacy-preserving credit checking and user-specific insurance tariffs. Recently, PFE protocols based on universal circuits (UCs), that have an inevitable superlinear overhead, have been investigated thoroughly. Specialized public key-based protocols with linear complexity were believed to be less efficient than UC-based approaches. In this paper, we...

Privacy protection and scalability are significant issues with blockchain. We propose an anonymous probabilistic payment under the general functionality for solving them. We consider the situation that several payers pay several payees through a tumbler. We have mediated the tumbler of the payment channel hub between payers and payees. Unlinkability means that the link, which payer pays which payee via the tumbler, is broken. A cryptographic puzzle plays a role in controlling the...

Oblivious linear evaluation (OLE) is a fundamental building block in multi-party computation protocols. In OLE, a sender holds a description of an affine function $f_{\alpha,\beta}(z)=\alpha z \beta$, the receiver holds an input $x$, and gets $\alpha x \beta$ (where all computations are done over some field, or more generally, a ring). Vector OLE (VOLE) is a generalization where the sender has many affine functions and the receiver learns the evaluation of all of these functions on a single...

Periodic key rotation is a common practice designed to limit the long-term power of cryptographic keys. Key rotation refers to the process of re-encrypting encrypted content under a fresh key, and overwriting the old ciphertext with the new one. When encrypted data is stored in the cloud, key rotation can be very costly: it may require downloading the entire encrypted content from the cloud, re-encrypting it on the client's machine, and uploading the new ciphertext back to the cloud. An...

Though Fully Homomorphic Encryption (FHE) has been realized, most practical implementations utilize leveled Somewhat Homomorphic Encryption (SHE) schemes, which have limits on the multiplicative depth of the circuits they can evaluate and avoid computationally intensive bootstrapping. Many SHE schemes exist, among which those based on Ring Learning With Error (RLWE) with operations on large polynomial rings are popular. Of these, variants allowing operations to occur fully in Residue Number...

A remarkable breakthrough in mathematics in recent years is the proof of the long-standing conjecture: sphere packing (i.e., packing unit balls) in the $E_8$ lattice is optimal in the sense of the best density \cite{V17} for sphere packing in $\mathbb{R}^8$. In this work, based on the $E_8$ lattice code, we design a mechanism for asymmetric key consensus from noise (AKCN), referred to as AKCN-E8, for error correction and key consensus. As a direct application of the AKCN-E8...

In this paper, we propose a novel lattice-based group key exchange protocol with dynamic membership. Our protocol is constructed by generalizing Dutta-Barua protocol to RLWE setting, inspired by Apon et al.’s recent paper in PQCrypto 2019. We describe our (static) group key exchange protocol from Apon et al.’s paper by modifying its third round and computation step. Then, we present both authenticated and dynamic group key exchange protocol with Join and Leave algorithms. The number of...

In the past few years, significant progresses on homomorphic encryption (HE) have been made toward both theory and practice. The most promising HE schemes are based on the hardness of the Learning With Errors (LWE) problem or its ring variant (RLWE). In this work, we present new conversion algorithms which switch between different (R)LWE-based HE schemes to take advantages of them. Specifically, we present and combine three ideas to improve the key-switching procedure between LWE...

We present efficient Zero-Knowledge Proofs of Knowledge (ZKPoK) for linear and multiplicative relations among secret messages hidden as Ring Learning With Errors (RLWE) samples. Messages are polynomials in $\mathbb{Z}_q[x]/\left<x^{n} 1\right>$ and our proposed protocols for a ZKPoK are based on the celebrated paper by Stern on identification schemes using coding problems (Crypto'93). Our $5$-move protocol achieves a soundness error slightly above $1/2$ and perfect Zero-Knowledge. As an...

We study the computation and communication costs and their possible trade-offs in various constructions for private information retrieval (PIR), including schemes based on homomorphic encryption and the Gentry--Ramzan PIR (ICALP'05). We improve over the construction of SealPIR (S&P'18) using compression techniques and a new oblivious expansion, which reduce the communication bandwidth by 60% while preserving essentially the same computation cost. We then present MulPIR, a PIR protocol...

Learning with Errors (LWE) and Ring-LWE (RLWE) problems allow the construction of efficient key exchange and public-key encryption schemes. However, while improving the security through the use of error distributions with large standard deviations, the decryption failure rate increases as well. Currently, the independence of individual coefficient failures is assumed to estimate the overall decryption failure rate of many LWE/RLWE schemes. However, previous work has shown that this...

Three-party key exchange, where two clients aim to agree a session key with the help of a trusted server, is prevalent in present-day systems. In this paper, we present a practical and secure three-party password-based authenticated key exchange protocol over ideal lattices. Aside from hash functions our protocol does not rely on external primitives in the construction and the security of our protocol is directly relied on the Ring Learning with Errors (RLWE) assumption. Our protocol...

Quantum computers threaten to compromise public-key cryptography schemes such as DSA and ECDSA in polynomial time, which poses an imminent threat to secure signal processing. The cryptography community has responded with the development and standardization of post-quantum cryptography (PQC) algorithms, a class of public-key algorithms based on hard problems that no known quantum algorithms can solve in polynomial time. Ring learning with error (RLWE) lattice- based cryptographic (LBC)...

The RLWE family algorithms submitted to the NIST post-quantum cryptography standardization process have each merit in terms of security, correctness, performance, and bandwidth. However, there is no splendid algorithm in all respects. Besides, various recent studies have been published that affect security and correctness, such as side-channel attacks and error dependencies. To date, though, no algorithm has fully considered all the aspects. We propose a novel Key Encapsulation Mechanism...

Developing machine learning models from federated training data, containing many independent samples, is an important task that can significantly enhance the potential applicability and prediction power of learned models. Since single users, like hospitals or individual labs, typically collect data-sets that do not support accurate learning with high confidence, it is desirable to combine data from several users without compromising data privacy. In this paper, we develop a...

Authenticated encryption (AE) is very suitable for a resources constrained environment for it needs less computational costs and AE has become one of the important technologies of modern communication security. Identity concealment is one of research focuses in design and analysis of current secure transport protocols (such as TLS1.3 and Google's QUIC). In this paper, we present a provably secure identity-concealed authenticated encryption in the public-key setting over ideal lattices,...

The "Multivariate Ring Learning with Errors" problem was presented as a generalization of Ring Learning with Errors (RLWE), introducing efficiency improvements with respect to the RLWE counterpart thanks to its multivariate structure. Nevertheless, the recent attack presented by Bootland, Castryck and Vercauteren has some important consequences on the security of the multivariate RLWE problem with "non-coprime" cyclotomics; this attack transforms instances of $m$-RLWE with power-of-two...

The Number Theoretic Transform (NTT) technique is widely used in the implementation of the cryptographic schemes based on the Ring Learning With Errors problem(RLWE), since it provides efficient algorithm for multiplication of polynomials over the finite field. However, to employ NTT, the finite field is required to have some special root of unity, such as $n$-th root, which makes the module $q$ in RLWE big since we need $q\equiv 1\mod 2n$ to ensure such root exits. At Inscrypt 2018, Zhou...

A hash function family is called correlation intractable if for all sparse relations, it hard to find, given a random function from the family, an input output pair that satisfies the relation. Correlation intractability (CI) captures a strong Random Oracle like property of hash functions. In particular, when security holds for all sparse relations, CI suffices for guaranteeing the soundness of the Fiat-Shamir transformation from any constant round, statistically sound interactive proof to a...

Till now, the only reduction from the module learning with errors problem (MLWE) to the ring learning with errors problem (RLWE) is given by Albrecht $et\ al.$ in ASIACRYPT $2017$. Reductions from search MLWE to search RLWE were satisfactory over power-of-$2$ cyclotomic fields with relative small increase of errors. However, a direct reduction from decision MLWE to decision RLWE leads to a super-polynomial increase of errors and does not work even in the most special cases-\ -power-of-$2$...

Oblivious transfer is one of the main pillars of modern cryptography and plays a major role as a building block for other more complex cryptographic primitives. In this work, we present an efficient and versatile framework for oblivious transfer (OT) using one-round key-exchange (ORKE), a special class of key exchange (KE) where only one message is sent from each party to the other. Our contributions can be summarized as follows: i) We carefully analyze ORKE schemes and introduce new...

Key Exchange (KE) is, undoubtedly, one of the most used cryptographic primitives in practice. Its authenticated version, Authenticated Key Exchange (AKE), avoids man-in-the-middle-based attacks by providing authentication for both parties involved. It is widely used on the Internet, in protocols such as TLS or SSH. In this work, we provide new constructions for KE and AKE based on ideal lattices in the Random Oracle Model (ROM). The contributions of this work can be summarized as...

An important characteristic of recent MPC protocols is an input-independent setup phase in which most computations are offloaded, which greatly reduces the execution overhead of the online phase where parties provide their inputs. For a very efficient evaluation of arithmetic circuits in an information-theoretic online phase, the MPC protocols consume Beaver multiplication triples generated in the setup phase. Triple generation is generally the most expensive part of the protocol, and...

We present a fully homomorphic encryption scheme continuing the line of works of Ducas and Micciancio (2015, [DM15]), Chillotti et al. (2016, [CGGI16a]; 2017, [CGGI17]; 2018, [CGGI18a]), and Gao (2018,[Gao18]). Ducas and Micciancio (2015) show that homomorphic computation of one bit operation on LWE ciphers can be done in less than a second, which is then reduced by Chillotti et al. (2016, 2017, 2018) to 13ms. According to Chillotti et al. (2018, [CGGI18b]), the cipher expansion for TFHE is...

In this paper we present the first fully post-quantum proof of a shuffle for RLWE encryption schemes. Shuffles are commonly used to construct mixing networks (mix-nets), a key element to ensure anonymity in many applications such as electronic voting systems. They should preserve anonymity even against an attack using quantum computers in order to guarantee long-term privacy. The proof presented in this paper is built over RLWE commitments which are perfectly binding and computationally...

In this paper, we present a simple attack on LWE and Ring LWE encryption schemes used directly as Key Encapsulation Mechanisms (KEMs). This attack could work due to the fact that a key mismatch in a KEM is accessible to an adversary. Our method clearly indicates that any LWE or RLWE (or any similar type of construction) encryption directly used as KEM can be broken by modifying our attack method according to the respective cases.

In this paper, we propose a Multi-Key Homomorphic Encryption (MKHE) scheme by generalizing the low-latency homomorphic encryption by Chillotti et al. (ASIACRYPT 2016). Our scheme can evaluate a binary gate on ciphertexts encrypted under different keys followed by a bootstrapping. The biggest challenge to meeting the goal is to design a multiplication between a bootstrapping key of a single party and a multi-key RLWE ciphertext. We propose two different algorithms for this hybrid product....

In 2017, a practical attack, referred to as the signal leakage attack, against reconciliation-based RLWE key exchange protocols was proposed. In particular, this attack can recover a long-term private key if a key pair is reused. Directly motivated by this attack, recently, Ding et al. proposed two countermeasures against the attack. One is the RLWE key exchange protocol with reusable keys (KERK), which is included in the Ding Key Exchange, a NIST submission. The idea for this construction...

Group signatures allow users of a group to sign messages anonymously in the name of the group, while incorporating a tracing mechanism to revoke anonymity and identify the signer of any message. Since its introduction by Chaum and van Heyst (EUROCRYPT 1991), numerous proposals have been put forward, yielding various improvements on security, efficiency and functionality. However, a drawback of traditional group signatures is that the opening authority is given too much power, i.e., he can...

We use an RLWE-based key exchange scheme to construct a simple and efficient post-quantum oblivious transfer based on the Ring Learning with Errors assumption. We prove that our protocol is secure in the Universal Composability framework against static malicious adversaries in the random oracle model. The main idea of the protocol is that the receiver and the sender interact using the RLWE-based key exchange in such a way that the sender computes two keys, one of them shared with the...

We initiate the study of partial key exposure in ring-LWE-based cryptosystems. Specifically, we - Introduce the search and decision Leaky-RLWE assumptions (Leaky-SRLWE, Leaky-DRLWE), to formalize the hardness of search/decision RLWE under leakage of some fraction of coordinates of the NTT transform of the RLWE secret and/or error. - Present and implement an efficient key exposure attack that, given certain $1/4$-fraction of the coordinates of the NTT transform of the RLWE secret, along...

The Number Theoretic Transform (NTT) provides efficient algorithm for multiplying large degree polynomials. It is commonly used in cryptographic schemes that are based on the hardness of the Ring Learning With Errors problem (RLWE), which is a popular basis for post-quantum key exchange, encryption and digital signature. To apply NTT, modulus q should satisfy that q = 1 mod 2n, RLWE-based schemes have to choose an oversized modulus, which leads to excessive bandwidth. In this work, we...

The Multivariate Ring Learning with Errors ($m$-RLWE) problem was introduced in 2015 by Pedrouzo-Ulloa, Troncoso-Pastoriza and Pérez-González. Instead of working over a polynomial residue ring with one variable as in RLWE, it works over a polynomial residue ring in several variables. However, care must be taken when choosing the multivariate rings for use in cryptographic applications as they can be either weak or simply equivalent to univariate RLWE. For example, Pedrouzo-Ulloa et al.\...

The Ring Learning with Errors (RLWE) problem over a cyclotomic ring has been the most widely used hardness assumption for the construction of practical homomorphic encryption schemes. However, this restricted choice of a base ring may cause a waste in terms of plaintext space usage. For example, an approximate homomorphic encryption scheme of Cheon et al. (ASIACRYPT 2017) is able to store a complex number in each of the plaintext slots since its canonical embedding of a cyclotomic field has...

Can Bob give Alice his decryption secret and be convinced that she will not give it to someone else? This is achieved by a proxy re-encryption scheme where Alice does not have Bob’s secret but instead she can transform ciphertexts in order to decrypt them with her own key. In this article, we answer this question in a different perspective, relying on a property that can be found in the well-known modified NTRU encryption scheme. We show how parties can collaborate to one-way-glue their...

Round5 is a Public Key Encryption and Key Encapsulation Mechanism (KEM) based on General Learning with Rounding (GLWR), a lattice problem. We argue that the ring variant of GLWR is better suited for embedded targets than the more common RLWE (Ring Learning With Errors) due to significantly shorter keys and messages. Round5 incorporates GLWR with error correction, building on design features from NIST Post-Quantum Standardization candidates Round2 and Hila5. The proposal avoids Number...

Since Gentry discovered in 2009 the first fully homomorphic encryption scheme, the last few years have witnessed dramatic progress on designing more efficient homomorphic encryption schemes, and some of them have been implemented for applications. The main bottlenecks are in bootstrapping and large cipher expansion (the ratio of the size of ciphertexts to that of messages). Ducas and Micciancio (2015) show that homomorphic computation of one bit operation on LWE ciphers can be done...

Constructing indistinguishability obfuscation (iO) [BGI 01] is a central open question in cryptography. We provide new methods to make progress towards this goal. Our contributions may be summarized as follows: 1. {\textbf Bootstrapping}. In a recent work, Lin and Tessaro [LT17] (LT) show that iO may be constructed using i) Functional Encryption (FE) for polynomials of degree $L$ , ii) Pseudorandom Generators (PRG) with blockwise locality $L$ and polynomial expansion, and iii) Learning With...