Recently, there has been a lot of interest in improving the understanding of the practical hardness of the 3-Tensor Isomorphism (3-TI) problem, which, given two 3-tensors, asks for an isometry between the two. The current state-of-the-art for solving this problem is the algebraic algorithm of Ran et al. '23 and the graph-theoretic algorithm of Narayanan et al. '24 that have both slightly reduced the security of the signature schemes MEDS and ALTEQ, based on variants of the 3-TI problem...

Side-channel-analysis (SCA) resistance with cost optimization in AES hardware implementations remains a significant challenge. While traditional masking-based schemes offer provable security, they often incur substantial resource overheads (latency, area, randomness, performance, power consumption). Alternatively, the RAMBAM scheme introduced a redundancy-based approach to control the signal-to-noise ratio, and achieves exponential leakage reduction as redundancy increases. This method...

Many lattice-based crypstosystems employ ideal lattices for high efficiency. However, the additional algebraic structure of ideal lattices usually makes us worry about the security, and it is widely believed that the algebraic structure will help us solve the hard problems in ideal lattices more efficiently. In this paper, we study the additional algebraic structure of ideal lattices further and find that a given ideal lattice in a polynomial ring can be embedded as an ideal into infinitely...

Secure multi-party computation (MPC) in a three-party, honest majority scenario is currently the state-of-the-art for running machine learning algorithms in a privacy-preserving manner. For efficiency reasons, fixed-point arithmetic is widely used to approximate computation over decimal numbers. After multiplication in fixed-point arithmetic, truncation is required to keep the result's precision. In this paper, we present an efficient three-party truncation protocol secure in the presence of...

In this work we study the efficiency of Zero-Knowledge (ZK) arguments of knowledge, particularly exploring Multi-Verifier ZK (MVZK) protocols as a midway point between Non-Interactive ZK and Designated-Verifier ZK, offering versatile applications across various domains. We introduce a new MVZK protocol designed for the preprocessing model, allowing any constant fraction of verifiers to be corrupted, potentially colluding with the prover. Our contributions include the first MVZK over rings....

Ring signatures are one of the crucial cryptographic primitives used in the design of privacy-preserving systems. Such a signature scheme allows a signer to anonymously sign a message on behalf of a spontaneously formed group. It not only ensures the authenticity of the message but also conceals the true signer within the group. An important extension of ring signatures is linkable ring signatures, which prevent a signer from signing twice without being detected (under some constraints)....

The cryptosystem RSA is a very popular cryptosystem in the study of Cryptography. In this article, we explore how the idea of a primitive m th root of unity in a ring can be integrated into the Discrete Fourier Transform, leading to the development of new cryptosystems known as RSA-DFT and RSA-HGR.

Ring signatures, a cryptographic primitive introduced by Rivest, Shamir and Tauman (ASIACRYPT 2001), offer signer anonymity within dynamically formed user groups. Recent advancements have focused on lattice-based constructions to improve efficiency, particularly for large signing rings. However, current state-of-the-art solutions suffer from significant overhead, especially for smaller rings. In this work, we present a novel NTRU-based ring signature scheme, Gandalf, tailored towards...

We revisit the question of the overhead to achieve full security (i.e., guaranteed output delivery) in secure multiparty computation (MPC). Recent works have closed the gap between full security and semi-honest security, by introducing protocols where the parties first compute the circuit using a semi-honest protocol and then run a verification step with sublinear communication in the circuit size. However, in these works the number of interaction rounds in the verification step is also...

NTRU-like cryptosystems are among the most studied lattice-based post-quantum candidates. While most NTRU proposals have been introduced over a commutative ring of quotient polynomials, other rings can be used. Noncommutative algebra has been endorsed as a direction to build new variants of NTRU a long time ago. The first attempt to construct a noncommutative variant was due to Hoffstein and Silverman motivated by more resistance to lattice attack. The scheme has been built over the group...

Fully Homomorphic Encryption (FHE) is a transformative technology that enables computations on encrypted data without requiring decryption, promising enhanced data privacy. However, its adoption has been limited due to significant performance overheads. Recent advances include the proposal of domain-specific, highly-parallel hardware accelerators designed to overcome these limitations. This paper introduces PICA, a comprehensive compiler framework designed to simplify the programming of...

Multiple works have designed or used maliciously secure honest majority MPC protocols over $\mathbb{Z}_{2^k}$ using replicated secret sharing (e.g. Koti et al. USENIX'21). A recent trend in the design of such MPC protocols is to first execute a semi-honest protocol, and then use a check that verifies the correctness of the computation requiring only sublinear amount of communication in terms of the circuit size. The so-called Galois ring extensions are needed in order to execute such checks...

This work introduces DEFI - an efficient hash-and-sign digital signature scheme based on isotropic quadratic forms over a commutative ring of characteristic 0. The form is public, but the construction is a trapdoor that depends on the scheme's private key. For polynomial rings over integers and rings of integers of algebraic number fields, the cryptanalysis is reducible to solving a quadratic Diophantine equation over the ring or, equivalently, to solving a system of quadratic Diophantine...

We revisit the alternating-moduli paradigm for constructing symmetric-key primitives with a focus on constructing efficient protocols to evaluate them using secure multi-party computation (MPC). The alternating-moduli paradigm of Boneh, Ishai, Passelègue, Sahai, and Wu (TCC 2018) enables the construction of various symmetric-key primitives with the common characteristic that the inputs are multiplied by two linear maps over different moduli. The first contribution focuses on...

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...

In this work, we present novel protocols over rings for semi-honest secure three-party computation (3PC) and malicious four-party computation (4PC) with one corruption. While most existing works focus on improving total communication complexity, challenges such as network heterogeneity and computational complexity, which impact MPC performance in practice, remain underexplored. Our protocols address these issues by tolerating multiple arbitrarily weak network links between parties...

Consider the task of secure multiparty computation (MPC) among $n$ parties with perfect security and guaranteed output delivery, supporting $t<n/3$ active corruptions. Suppose the arithmetic circuit $C$ to be computed is defined over a finite ring $\mathbb{Z}/q\mathbb{Z}$, for an arbitrary $q\in\mathbb{Z}$. It is known that this type of MPC over such ring is possible, with communication that scales as $O(n|C|)$, assuming that $q$ scales as $\Omega(n)$. However, for constant-size rings...

We suggest the family of ciphers s^E^n, n=2,3,.... with the space of plaintexts (Z*_{2^s})^n, s >1 such that the encryption map is the composition of kind G=G_1A_1G_2A_2 where A_i are the affine transformations from AGL_n(Z_{2^s}) preserving the variety (Z*_{2^s)}^n , Eulerian endomorphism G_i , i=1,2 of K[x_1, x_2,...., x_n] moves x_i to monomial term ϻ(x_1)^{d(1)}(x_2)^{d(2)}...(x_n)^{d(n)} , ϻϵ Z*_{2^s} and act on (Z*_{2^s})^n as bijective transformations. The cipher is...

In response to the evolving landscape of quantum computing and the heightened vulnerabilities in classical cryptographic systems, our paper introduces a comprehensive cryptographic framework. Building upon the pioneering work of Kuang et al., we present a unification of two innovative primitives: the Quantum Permutation Pad (QPP) for symmetric key encryption and the Homomorphic Polynomial Public Key (HPPK) for Key Encapsulation Mechanism (KEM) and Digital Signatures (DS). By harnessing...

In the context of fully-homomorphic-encryption, we consider the representation of large integers by their decomposition over a product of rings (through the Chinese Remainder Theorem) and introduce a new algorithm for the determination of the sign solely through the knowledge of ring-components. Our implementation with 128 bits of security delivers a correct result and a probability higher than 1 E-9 in less than 100 milliseconds for 32-bit integers on a laptop.

Numerous applications in homomorphic encryption require an operation that moves the slots of a ciphertext to the coefficients of a different ciphertext. For the BGV and BFV schemes, the only efficient algorithms to implement this slot-to-coefficient transformation were proposed in the setting of non-power-of-two cyclotomic rings. In this paper, we devise an FFT-like method to decompose the slot-to-coefficient transformation (and its inverse) for power-of-two cyclotomic rings. The proposed...

This work introduces several algorithms related to the computation of orientations in endomorphism rings of supersingular elliptic curves. This problem boils down to representing integers by ternary quadratic forms, and it is at the heart of several results regarding the security of oriented-curves in isogeny-based cryptography. Our main contribution is to show that there exists efficient algorithms that can solve this problem for quadratic orders of discriminant $n$ up to $O(p^{4/3})$....

All anonymous identity-based encryption (IBE) schemes that are group homomorphic (to the best of our knowledge) require knowledge of the identity to compute the homomorphic operation. This paper is motivated by this open problem, namely to construct an anonymous group-homomorphic IBE scheme that does not sacrifice anonymity to perform homomorphic operations. Note that even when strong assumptions such as indistinguishability obfuscation (iO) are permitted, no schemes are known. We succeed in...

Secure Multi-Party Computation (MPC) constructions typically allow computation over a finite field or ring. While useful for many applications, certain real-world applications require the usage of decimal numbers. While it is possible to emulate floating-point operations in MPC, fixed-point computation has gained more traction in the practical space due to its simplicity and efficient realizations. Even so, current protocols for fixed-point MPC still require computing a secure truncation...

Fully Homomorphic Encryption (FHE) is a prevalent cryptographic primitive that allows for computation on encrypted data. In various cryptographic protocols, this enables outsourcing computation to a third party while retaining the privacy of the inputs to the computation. However, these schemes make an honest-but-curious assumption about the adversary. Previous work has tried to remove this assumption by combining FHE with Verifiable Computation (VC). Recent work has increased the...

This paper conducts a comprehensive benchmarking analysis of the performance of two innovative cryptographic schemes: Homomorphic Polynomial Public Key (HPPK)-Key Encapsulation Mechanism (KEM) and Digital Signature (DS), recently proposed by Kuang et al. These schemes represent a departure from traditional cryptographic paradigms, with HPPK leveraging the security of homomorphic symmetric encryption across two hidden rings without reliance on NP-hard problems. HPPK can be viewed as a...

We survey various mathematical tools used in software works multiplying polynomials in \[ \frac{\mathbb{Z}_q[x]}{\left\langle {x^n - \alpha x - \beta} \right\rangle}. \] In particular, we survey implementation works targeting polynomial multiplications in lattice-based cryptosystems Dilithium, Kyber, NTRU, NTRU Prime, and Saber with instruction set architectures/extensions Armv7-M, Armv7E-M, Armv8-A, and AVX2. There are three emphases in this paper: (i) modular arithmetic, (ii)...

In recent years, actively secure SPDZ-like protocols for dishonest majority, like SPD$\mathbb Z_{2^k}$, Overdrive2k, and MHz2k, over base rings $\mathbb Z_{2^k}$ have become more and more efficient. In this paper, we present a new actively secure MPC protocol Multipars that outperforms these state-of-the-art protocols over $\mathbb Z_{2^k}$ by more than a factor of 2 in the two-party setup in terms of communication. Multipars is the first actively secure N-party protocol over $\mathbb...

The privacy-preserving machine learning (PPML) has gained growing importance over the last few years. One of the biggest challenges is to improve the efficiency of PPML so that the communication and computation costs of PPML are affordable for large machine learning models such as deep learning. As we know, linear algebra such as matrix multiplication occupies a significant part of the computation in deep learning such as deep convolutional neural networks (CNN). Thus, it is desirable to...

The Learning with Errors (LWE) problem has been widely utilized as a foundation for numerous cryptographic tools over the years. In this study, we focus on an algebraic variant of the LWE problem called Group ring LWE (GR-LWE). We select group rings (or their direct summands) that underlie specific families of finite groups constructed by taking the semi-direct product of two cyclic groups. Unlike the Ring-LWE problem described in \cite{lyubashevsky2010ideal}, the multiplication operation in...

In their 2022 study, Kuang et al. introduced the Multivariable Polynomial Public Key (MPPK) cryptography, a quantum-safe public key cryptosystem leveraging the mutual inversion relationship between multiplication and division. MPPK employs multiplication for key pair construction and division for decryption, generating public multivariate polynomials. Kuang and Perepechaenko expanded the cryptosystem into the Homomorphic Polynomial Public Key (HPPK), transforming product polynomials over...

Private Simultaneous Messages (PSM) is a minimal model of secure computation, where the input players with shared randomness send messages to the output player simultaneously and only once. In this field, finding upper and lower bounds on communication complexity of PSM protocols is important, and in particular, identifying the optimal one where the upper and lower bounds coincide is the ultimate goal. However, up until now, functions for which the optimal communication complexity has been...

We present a general framework for polynomial-time lattice Gaussian sampling. It revolves around a systematic study of the discrete Gaussian measure and its samplers under extensions of lattices; we first show that given lattices $\Lambda'\subset \Lambda$ we can sample efficiently in $\Lambda$ if we know how to do so in $\Lambda'$ and the quotient $\Lambda/\Lambda'$, \emph{regardless} of the primitivity of $\Lambda'$. As a direct application, we...

$\Sigma$-protocols are a widely utilized, relatively simple and well understood type of zero-knowledge proofs. However, the well known Schnorr $\Sigma$-protocol for proving knowledge of discrete logarithm in a cyclic group of known prime order, and similar protocols working over this type of groups, are hard to generalize to dealing with other groups. In particular with hidden order groups, due to the inability of the knowledge extractor to invert elements modulo the order. In this paper,...

Finding isogenies between supersingular elliptic curves is a natural algorithmic problem which is known to be equivalent to computing the curves' endomorphism rings. When the isogeny is additionally required to have a specific known degree $d$, the problem appears to be somewhat different in nature, yet its hardness is also required in isogeny-based cryptography. Let $E_1,E_2$ be supersingular elliptic curves over $\mathbb{F}_{p^2}$. We present improved classical and quantum...

The Number Theoretic Transform (NTT) is a powerful mathematical tool with a wide range of applications in various fields, including signal processing, cryptography, and error correction codes. In recent years, there has been a growing interest in efficiently implementing the NTT on hardware platforms for lattice-based cryptography within the context of NIST's Post-Quantum Cryptography (PQC) competition. The implementation of NTT in cryptography stands as a pivotal advancement,...

We present a verifiable delay function based on isogenies of supersingular elliptic curves, using Deuring correspondence and computation of endomorphism rings for the delay. For each input x a verifiable delay function has a unique output y and takes a predefined time to evaluate, even with parallel computing. Additionally, it generates a proof by which the output can efficiently be verified. In our approach the input is a path in the 2-isogeny graph and the output is the maximal order...

Recently a completely new post-quantum digital signature scheme was proposed using the so called ``scrap automorphisms''. The structure is inherently multivariate, but differs significantly from most of the multivariate literature in that it relies on sparsity and rings containing zero divisors. In this article, we derive a complete and total break of Scrap, performing a key recovery in not much more time than verifying a signature. We also generalize the result, breaking unrealistic...

Gentry's groundbreaking work showed that a fully homomorphic, provably secure scheme is possible via bootstrapping a somewhat homomorphic scheme. However, a major drawback of bootstrapping is its high computational cost. One alternative is to use a different metric for noise so that homomorphic operations do not accumulate noise, eliminating the need for boostrapping altogether. Leonardi and Ruiz-Lopez present a group-theoretic framework for such a ``noise non-accumulating'' multiplicative...

The recent devastating attacks on SIDH rely on the fact that the protocol reveals the images $\varphi(P)$ and $\varphi(Q)$ of the secret isogeny $\varphi : E_0 \rightarrow E$ on a basis $\{P, Q\}$ of the $N$-torsion subgroup $E_0[N]$ where $N^2 > \deg(\varphi)$. To thwart this attack, two recent proposals, M-SIDH and FESTA, proceed by only revealing the images upto unknown scalars $\lambda_1, \lambda_2 \in \mathbb{Z}_N^\times$, i.e., only $\lambda_1 \varphi(P)$ and $\lambda_2 \varphi(Q)$...

Reversed multiplication friendly embedding (RMFE) amortization has been playing an active role in the state-of-the-art constructions of MPC protocols over rings (in particular, the ring $\mathbb{Z}_{2^k}$). As far as we know, this powerful technique has NOT been able to find applications in the crown jewel of two-party computation, the non-interactive secure computation (NISC), where the requirement of the protocol being non-interactive constitutes a formidable technical bottle-neck. We...

In this paper, we introduce a novel trapdoor generation technique for Prest's hybrid sampler over NTRU lattices. Prest's sampler is used in particular in the recently proposed Mitaka signature scheme (Eurocrypt 2022), a variant of the Falcon signature scheme, one of the candidates selected by NIST for standardization. Mitaka was introduced to address Falcon's main drawback, namely the fact that the lattice Gaussian sampler used in its signature generation is highly...

Column Parity Mixers, or CPMs in short, are a particular type of linear maps, used as the mixing layer in permutation-based cryptographic primitives like Keccak-f (SHA3) and Xoodoo. Although being successfully applied, not much is known regarding their algebraic properties. They are limited to invertibility of CCPMs, and that the set of invertible CCPMs forms a group. A possible explanation is due to the complexity of describing CPMs in terms of linear algebra. In this paper, we introduce a...

Masking is a well-studied method for achieving provable security against side-channel attacks. In masking, each sensitive variable is split into $d$ randomized shares, and computations are performed with those shares. In addition to the computational overhead of masked arithmetic, masking also has a storage cost, increasing the requirements for working memory and secret key storage proportionally with $d$. In this work, we introduce mask compression. This conceptually simple technique is...

In this work, we extend the MPC-in-the-head framework, used in recent efficient zero-knowledge protocols, to work over the ring $\mathbb{Z}_{2^k}$, which is the primary operating domain for modern CPUs. The proposed schemes are compatible with any threshold linear secret sharing scheme and draw inspiration from MPC protocols adapted for ring operations. Additionally, we explore various batching methodologies, leveraging Shamir's secret sharing schemes and Galois ring extensions, and...

Lattice-based cryptographic schemes such as Crystals-Kyber and Dilithium are post-quantum algorithms selected to be standardized by NIST as they are considered to be secure against quantum computing attacks. The multiplication in polynomial rings is the most time-consuming operation in many lattice-based cryptographic schemes, which is also subject to side-channel attacks. While NTT-based polynomial multiplication is almost a norm in a wide range of implementations, a relatively new method,...

Most protocols for secure multi-party computation (MPC) work over fields or rings, which means that encoding techniques are needed to map rational-valued data into the algebraic structure being used. Leveraging an encoding technique introduced in recent work of Harmon et al. that is compatible with any MPC protocol over a prime-order field, we present Mercury - a family of protocols for addition, multiplication, subtraction, and division of rational numbers. Notably, the output of our...

Symmetric primitives are a cornerstone of cryptography, and have traditionally been defined over fields, where cryptanalysis is now well understood. However, a few symmetric primitives defined over rings Z_q for a composite number q have recently been proposed, a setting where security is much less studied. In this paper we focus on studying established algebraic attacks typically defined over fields and the extent of their applicability to symmetric primitives defined over the ring of...

This article is dedicated to a new generation method of two ``independent'' $\mathbb{F}_{\!q}$-points $P_0$, $P_1$ on almost any ordinary elliptic curve $E$ over a finite field $\mathbb{F}_{\!q}$ of large characteristic. In particular, the method is relevant for all standardized and real-world elliptic curves of $j$-invariants different from $0$, $1728$. The points $P_0$, $P_1$ are characterized by the fact that nobody (even a generator) knows the discrete logarithm $\log_{P_0}(P_1)$ in the...

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...

We present new protocols for *Asynchronous Verifiable Secret Sharing* for Shamir (i.e., threshold $t<n$) sharing of secrets. Our protocols: * Use only "lightweight" cryptographic primitives, such as hash functions; * Can share secrets over rings such as $\mathbb{Z}_{p^k}$ as well as finite fields $\mathbb{F}_q$; * Provide *optimal resilience*, in the sense that they tolerate up to $t < n/3$ corruptions, where $n$ is the total number of parties; * Are *complete*, in the sense that they...

In the recent work of (Cheon & Lee, Eurocrypt'22), the concept of a degree-$D$ packing method was formally introduced, which captures the idea of embedding multiple elements of a smaller ring into a larger ring, so that element-wise multiplication in the former is somewhat "compatible" with the product in the latter. Then, several optimal bounds and results are presented, and furthermore, the concept is generalized from one multiplication to degrees larger than two. These packing...

A recent line of works on zero-knowledge (ZK) protocols with a vector oblivious linear function evaluation (VOLE)-based offline phase provides a new paradigm for scalable ZK protocols featuring fast proving and small prover memory. Very recently, Baum et al. (Crypto'23) proposed the VOLE-in-the-head technique, allowing such protocols to become publicly verifiable. Many practically efficient protocols for proving circuit satisfiability over any Galois field are implemented, while protocols...

Proposed by Thang and Binh (NICS, 2015 ), DBTRU is a variant of NTRU, where the integer polynomial ring is replaced by two binary truncated polynomial rings GF(2)[x]/(x^n 1). DBTRU has significant advantages over NTRU in terms of security and performance. NTRU is a probabilistic public key cryptosystem having security related to some hard problems in lattices. In this paper we will present a polynomial-time linear algebra attack on the DBTRU cryptosystem which can break DBTRU for all...

Constructing a supersingular elliptic curve whose endomorphism ring is isomorphic to a given quaternion maximal order (one direction of the Deuring correspondence) is known to be polynomial-time assuming the generalized Riemann hypothesis [KLPT14; Wes21], but notoriously daunting in practice when not working over carefully selected base fields. In this work, we speed up the computation of the Deuring correspondence in general characteristic, i.e., without assuming any special form of the...

We introduce the first candidate Lattice-based designated verifier (DV) zero knowledge sUccinct Non-interactive Argument (ZK-SNARG) protocol, LUNA, with quasi-optimal proof length (quasi-linear in the security/privacy parameter). By simply relying on mildly stronger security assumptions, LUNA is also a candidate ZK-SNARK (i.e. argument of knowledge). LUNA achieves significant improvements in concrete proof sizes, reaching below 6 KB (compared to >32 KB in prior work) for 128-bit...

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...

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...

Homogeneous algebraic graphs defined over arbitrary field are classical objects of Algebraic Geometry. This class includes geometries of Chevalley groups $A_2(F)$, $B_2(F)$ and $G_2(F)$ defined over arbitrary field $F$. Assume that codimension of homogeneous graph is the ratio of dimension of variety of its vertices and the dimension of neighbourhood of some vertex. We evaluate minimal codimension $v(g)$ and $u(h)$ of algebraic graph of prescribed girth $g$ and cycle indicator....

The history of equations dates back to thousands of years ago, though the equals sign "=" was only invented in 1557. We formalize the processes of "decomposition" and "restoration" in mathematics and physics by defining "discrete exponential equations" and "noisy equation systems" over an abstract structure called a "land", which is more general than fields, rings, groups, and monoids. Our abstract equations and systems provide general languages for many famous computational problems such as...

All known instantiations for fully homomorphic encryption (FHE) produce noisy ciphertexts and rely on a technique called bootstrapping to reduce the noise so as to enable an arbitrary number of homomorphic operations. Bootstrapping is the main performance bottleneck and arguably the biggest obstacle to widespread adoption of FHE. Among the FHE schemes, TFHE and its variations present the appealing property of having a bootstrapping procedure---as well as its extension to programmable ...

Previously [4], Orbis Labs presented a method for compiling (“arithmetizing”) relations, expressed as Σ¹₁ formulas in the language of rings, into Halo 2 [1, 2, 3] arithmetic circuits. In this research, we extend this method to support polynomial quantifier bounds, in addition to constant quantifier bounds. This allows for more efficient usage of rows in the resulting circuit.

The offline-online model is a leading paradigm for practical secure multi-party computation (MPC) protocol design that has successfully reduced the overhead for several prevalent privacy-preserving computation functionalities common to diverse application domains. However, the prohibitive overheads associated with secure comparison -- one of these vital functionalities -- often bottlenecks current and envisioned MPC solutions. Indeed, an efficient secure comparison solution has the potential...

We propose NTT implementations with each supporting at least one parameter of NTRU and one parameter of NTRU Prime. Our implementations are based on size-1440, size-1536, and size-1728 convolutions without algebraic assumptions on the target polynomial rings. We also propose several improvements for the NTT computation. Firstly, we introduce dedicated radix-(2,3) butterflies combining Good–Thomas FFT and vector-radix FFT. In general, there are six dedicated radix-(2, 3) butterflies and they...

Public-key cryptography, including conventional cryptosystems and post-quantum cryptography, involves computation-intensive workloads. With noticing the extraordinary computing power of AI accelerators, in this paper, we further explore the feasibility to introduce AI accelerators into high-performance cryptographic computing. Since AI accelerators are dedicated to machine learning or neural networks, the biggest challenge is how to transform cryptographic workloads into their operations,...

Machine learning algorithms crucially depend on non-linear mathematical functions such as division (for normalization), exponentiation (for softmax and sigmoid), tanh (as an activation function), logarithm (for cross-entropy loss), and square root (for back-propagation of normalization layers). However, when machine learning is performed over secure computation, these protocols incur a large communication overhead and high round complexity. In this work, we propose new multi-party...

Zero-knowledge proof systems are usually designed to support computations for circuits over $\mathbb{F}_2$ or $\mathbb{F}_p$ for large $p$, but not for computations over $\mathbb{Z}_{2^k}$, which all modern CPUs operate on. Although $\mathbb{Z}_{2^k}$-arithmetic can be emulated using prime moduli, this comes with an unavoidable overhead. Recently, Baum et al. (CCS 2021) suggested a candidate construction for a designated-verifier zero-knowledge proof system that natively runs over...

In this work we present a novel actively secure multiparty computation protocol in the dishonest majority setting, where the computation domain is a ring of the type $\mathbb{Z}_{2^k}$. Instead of considering an "extension ring" of the form $\mathbb{Z}_{2^{k \kappa}}$ as in SPD$\mathbb{Z}_{2^k}$ (Cramer et al, CRYPTO 2018) and its derivatives, we make use of an actual ring extension, or more precisely, a Galois ring extension $\mathbb{Z}_{p^k}[\mathtt{X}]/(h(\mathtt{X}))$ of large enough...

Katz et al., CCS 2018 (KKW) is a popular and efficient MPC-in-the-head non-interactive ZKP (NIZK) scheme, which is the technical core of the post-quantum signature scheme Picnic, currently considered for standardization by NIST. The KKW approach simultaneously is concretely efficient, even on commodity hardware, and does not rely on trusted setup. Importantly, the approach scales linearly in the circuit size with low constants with respect to proof generation time, proof verification time,...

Orbis Labs presents a method for compiling (“arithmetizing”) relations, expressed as Σ¹₁ formulas in the language of rings, into Halo 2 arithmetic circuits. This method offers the possibility of creating arithmetic circuits without laborious and error-prone manual circuit design and implementation, by instead expressing the relation to be arithmetized in a concise mathematical notation and generating the circuit based on that expression.

Learning parity with noise (LPN) has been widely studied and used in cryptography. It was recently brought to new prosperity since Boyle et al. (CCS'18), putting LPN to a central role in designing secure multi-party computation, zero-knowledge proofs, private set intersection, and many other protocols. In this paper, we thoroughly studied the security of LPN problems in this particular context. We found that some important aspects have long been ignored and many conclusions from classical...

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...

In this paper, we propose a noncommutative-ring based unbalanced oil and vinegar signature scheme with key-randomness alignment: NOVA (Noncommutative Oil and Vinegar with Alignment). Instead of fields or even commutative rings, we show that noncommutative rings can be used for algebraic cryptosystems. At the same or better level of security requirement, NOVA has a much smaller public key than UOV (Unbalanced Oil and Vinegar), which makes NOVA practical in most situations. We use Magma to...

Threshold signature schemes enable distribution of the signature issuing capability to multiple users, to mitigate the threat of signing key compromise. Though a classic primitive, these signatures have witnessed a surge of interest in recent times due to relevance to modern applications like blockchains and cryptocurrencies. In this work, we study round-optimal threshold signatures in the post- quantum regime and improve the only known lattice-based construction by Boneh et al [CRYPTO’18]...

We introduce the first proof system for layered arithmetic circuits over an arbitrary ring $R$ that is (possibly) non-commutative and (possibly) infinite, while only requiring black-box access to its arithmetic and a subset $A \subseteq R$. Our construction only requires limited commutativity and regularity properties from $A$, similar to recent work on efficient information theoretic multi-party computation over non-commutative rings by Escudero and Soria-Vazquez (CRYPTO 2021), but...

We present a much-improved practical protocol, based on the hardness of Module-SIS and Module-LWE problems, for proving knowledge of a short vector $s$ satisfying $As=t\bmod q$. The currently most-efficient technique for constructing such a proof works by showing that the $\ell_\infty$ norm of $s$ is small. It creates a commitment to a polynomial vector $m$ whose CRT coefficients are the coefficients of $s$ and then shows that (1) $A\cdot \mathsf{CRT}(m)=t\bmod\,q$ and (2) in the case that...

We consider the efficiency of protocols for secure multiparty computation (MPC) with a dishonest majority. A popular approach for the design of such protocols is to employ preprocessing. Before the inputs are known, the parties generate correlated secret randomness, which is consumed by a fast and possibly ``information-theoretic'' online protocol. A powerful technique for securing such protocols against malicious parties uses homomorphic MACs to authenticate the values produced by the...

Compressed $\Sigma$-Protocol Theory (CRYPTO 2020) presents an ``alternative'' to Bulletproofs that achieves the same communication complexity while adhering more elegantly to existing $\Sigma$-protocol theory, which enables their techniques to be directly applicable to other widely used settings in the context of ``plug \& play'' algorithmics. Unfortunately, their techniques are restricted to arithmetic circuits over \emph{prime} fields, which rules out the possibility of using more...

In this note, we prove the conjecture posed by Keller and Rosemarin at Eurocrypt 2021 on the nullity of a matrix polynomial of a block matrix with Hadamard type blocks over commutative rings of characteristic 2. Therefore, it confirms the conjectural optimal bound on the dimension of invariant subspace of the Starkad cipher using the HADES design strategy. We also give characterizations of the algebraic structure formed by Hadamard matrices over commutative rings.

This work revisits the security of classical signatures and ring signatures in a quantum world. For (ordinary) signatures, we focus on the arguably preferable security notion of blind-unforgeability recently proposed by Alagic et al. (Eurocrypt'20). We present two short signature schemes achieving this notion: one is in the quantum random oracle model, assuming quantum hardness of SIS; and the other is in the plain model, assuming quantum hardness of LWE with super-polynomial modulus. Prior...

We describe a technique to backdoor a prime factor of a composite odd integer $N$, so that an attacker knowing a possibly secret factor base $\mathcal{B}$, can efficiently retrieve it from $N$. Such method builds upon Complex Multiplication theory for elliptic curves, by generating primes $p$ associated to $\mathcal{B}$-smooth order elliptic curves over $\mathbb{F}_p$. When such primes $p$ divide an integer $N$, the latter can be efficiently factored using a generalization of Lenstra's...

Solving a system of $m$ multivariate quadratic equations in $n$ variables over finite fields (the MQ problem) is one of the important problems in the theory of computer science. The XL algorithm (XL for short) is a major approach for solving the MQ problem with linearization over a coefficient field. Furthermore, the hybrid approach with XL (h-XL) is a variant of XL guessing some variables beforehand. In this paper, we present a variant of h-XL, which we call the polynomial XL (PXL). In PXL,...

This paper focuses on isogeny representations, defined as ways to evaluate isogenies and verify membership to the language of isogenous supersingular curves (the set of triples $D,E_1,E_2$ with a cyclic isogeny of degree $D$ between $E_1$ and $E_2$). The tasks of evaluating and verifying isogenies are fundamental for isogeny-based cryptography. Our main contribution is the design of the suborder representation, a new isogeny representation targeted at the case of (big) prime...

We propose a multi-party computation (MPC) protocol over $\mathbb{Z}_{2^k}$ secure against actively corrupted majority from somewhat homomorphic encryption. The main technical contributions are: (i) a new efficient packing method for $\mathbb{Z}_{2^k}$-messages in lattice-based somewhat homomorphic encryption schemes, (ii) a simpler reshare protocol for level-dependent packings, (iii) a more efficient zero-knowledge proof of plaintext knowledge on cyclotomic rings $\mathbb{Z}[X]/\Phi_M(X)$...

There are two main aims to this paper. Firstly, we survey the relevant existing attack strategies known to apply to the most commonly used lattice-based cryptographic problems as well as to a number of their variants. In particular, we consider attacks against problems in the style of LWE, SIS and NTRU defined over rings of the form $\mathbb{Z}[X]/(f(X), g(X))$, where classically $g(X) = q$ is an integer modulus. We also include attacks on variants which use only large integer arithmetic,...

Cryptography based on the hardness of lattice problems over polynomial rings currently provides the most practical solution for public key encryption in the quantum era. The first encryption scheme utilizing properties of polynomial rings was NTRU (ANTS '98), but in the recent decade, most research has focused on constructing schemes based on the hardness of the somewhat related Ring/Module-LWE problem. Indeed, 14 out of the 17 encryption schemes based on the hardness of lattice problems in...

Oblivious Polynomial Evaluation (OPE) schemes are interactive protocols between a sender with a private polynomial and a receiver with a private evaluation point where the receiver learns the evaluation of the polynomial in their point and no additional information. In this work, we introduce MyOPE, a ``short-sighted'' non-interactive polynomial evaluation scheme with a poly-logarithmic communication complexity in the presence of malicious senders. In addition to strong privacy guarantees,...

We introduce a novel generic ring signature construction, called DualRing, which can be built from several canonical identification schemes (such as Schnorr identification). DualRing differs from the classical ring signatures by its formation of two rings: a ring of commitments and a ring of challenges. It has a structural difference from the common ring signature approaches based on accumulators or zero-knowledge proofs of the signer index. Comparatively, DualRing has a number of unique...

Some key agreement schemes, such as Diffie--Hellman key agreement, reduce to Rabi--Sherman key agreement, in which Alice sends $ab$ to Charlie, Charlie sends $bc$ to Alice, they agree on key $a(bc) = (ab)c$, where multiplicative notation here indicates some specialized associative binary operation. All non-interactive key agreement schemes, where each peer independently determines a single delivery to the other, reduce to this case, because the ability to agree implies the existence of an...

A ring signature allows a party to sign messages anonymously on behalf of a group, which is called ring. Traceable ring signatures are a variant of ring signatures that limits the anonymity guarantees, enforcing that a member can sign anonymously at most one message per tag. Namely, if a party signs two different messages for the same tag, it will be de-anomymized. This property is very useful in decentralized platforms to allow members to anonymously endorse statements in a...

The Learning with Rounding (LWR) problem is an important variant of the Learning with Errors (LWE) problem. Recently, Liu {\it{et al.}} proposed a comprehensive study of LWR problems defined over algebraic number fields in CRYPTO 2020. However, their search-to-decision reductions of LWR problems depend heavily on the existence of the so-called {\it{Normal Integral Basis}} (NIB). Meanwhile, the aesthetic deficiency is a lack of discussions of choices of secret $s$, and one may could not show...

We construct the first efficient MPC protocol that only requires black-box access to a non-commutative ring $R$. Previous results in the same setting were efficient only either for a constant number of corruptions or when computing branching programs and formulas. Our techniques are based on a generalization of Shamir's secret sharing to non-commutative rings, which we derive from the work on Reed Solomon codes by Quintin, Barbier and Chabot (IEEE Transactions on Information Theory, 2013)....

We examine Multi-Party Computation protocols in the active-security-with-abort setting for $Q_2$ access structures over small and large finite fields $F_p$ and over rings $Z_{p^k}$. We give general protocols which work for any $Q_2$ access structure which is realised by a multiplicative Extended Span Program. We generalize a number of techniques and protocols from various papers and compare the different methodologies. In particular we examine the expected communication cost per...

Mixing arithmetic and boolean circuits to perform privacy-preserving machine learning has become increasingly popular. Towards this, we propose a framework for the case of four parties with at most one active corruption called Tetrad. Tetrad works over rings and supports two levels of security, fairness and robustness. The fair multiplication protocol costs 5 ring elements, improving over the state-of-the-art Trident (Chaudhari et al. NDSS'20). A key feature of Tetrad is that robustness...

Zero-knowledge proofs are highly flexible cryptographic protocols that are an important building block for many secure systems. Typically, these are defined with respect to statements that are formulated as arithmetic operations over a fixed finite field. This inflexibility is a disadvantage when it comes to complex programs, as some fields are more amenable to express certain operations than others. At the same time, there do not seem to be many proofs with a programming model similar to...

Post-quantum cryptography (PQC) has a well-deserved NIST status. Our approach (R-Propping) replaces all numeric field arithmetic with GF(2^8) field operations. This method yields both classical and quantum secure protocols. The present work is dedicated to strengthening a chaotic Wolfram Class III cellular automata and discuss its usability as a cryptographical secure PRBG (pseudorandom bit generator), a building block for stream-ciphers, hashing, and other random numbers requiring protocols.

It was recently demonstrated that the Matrix Action Key Exchange (MAKE) algorithm, a new type of key exchange protocol using the semidirect product of matrix groups, is vulnerable to a linear algebraic attack if the matrices are over a commutative ring. In this note, we establish conditions under which protocols using matrices over a non-commutative ring are also vulnerable to this attack. We then demonstrate that group rings $R[G]$, where $R$ is a commutative ring and $G$ is a non-abelian...

Whilst lattice-based cryptosystems are believed to be resistant to quantum attack, they are often forced to pay for that security with inefficiencies in implementation. This problem is overcome by ring- and module-based schemes such as Ring-LWE or Module-LWE, whose keysize can be reduced by exploiting its algebraic structure, allowing for neater and faster computations. Many rings may be chosen to define such cryptoschemes, but cyclotomic rings, due to their cyclic nature allowing for easy...

We introduce MatRiCT , a practical private blockchain payment protocol based on ``post-quantum'' lattice assumptions. MatRiCT builds on MatRiCT due to Esgin et al. (ACM CCS'19) and, in general, follows the Ring Confidential Transactions (RingCT) approach used in Monero, the largest privacy-preserving cryptocurrency. In terms of the practical aspects, MatRiCT has 2-18x shorter proofs (depending on the number of input accounts, M) and runs 3-11x faster (for a typical transaction) in...

Efficient shuffle arguments are essential in mixnet-based e-voting solutions. Terelius and Wikström (TW) proposed a 5-round shuffle argument based on unique factorization in polynomial rings. Their argument is available as the Verificatum software solution for real-world developers, and has been used in real-world elections. It is also the fastest non-patented shuffle argument. We will use the same basic idea as TW but significantly optimize their approach. We generalize the...