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...

We have proposed a novel FHE scheme that uniquely encodes the plaintext with noise in a way that prevents the increasing noise from overflowing and corrupting the plaintext. This allows users to perform computations on encrypted data smoothly. The scheme is constructed using the Chinese Remainder Theorem (CRT), supporting a predefined number of modular operations on encrypted plaintext without the need for bootstrapping. Although FHE recently became popular after Gentry's work and various...

Threshold secret sharing enables distributing a message to $n$ parties such that no subset of fewer than $t$ parties can learn the message, whereas any subset of at least $t$ parties can recover the message. Despite being a fundamental primitive, secret sharing still suffers from one significant drawback, where its message reconstruction algorithm is computationally expensive for large privacy thresholds $t$. In this paper, we aim to address this significant drawback. We study general...

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...

Accumulation schemes are a simple yet powerful primitive that enable highly efficient constructions of incrementally verifiable computation (IVC). Unfortunately, all prior accumulation schemes rely on homomorphic vector commitments whose security is based on public-key assumptions. It is an interesting open question to construct efficient accumulation schemes that avoid the need for such assumptions. In this paper, we answer this question affirmatively by constructing an accumulation...

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 this paper, we study two problems: oblivious compression and decompression of ciphertexts. In oblivious compression, a server holds a set of ciphertexts with a subset of encryptions of zeroes whose positions are only known to the client. The goal is for the server to effectively compress the ciphertexts obliviously, while preserving the non-zero plaintexts and without learning the plaintext values. For oblivious decompression, the client, instead, succinctly encodes a sequence of...

In this work, we introduce a family of asymmetric cryptographic functions based on dynamic number theoretic transformations with multiple rounds of modular arithmetic to enhance diffusion and difficulty of inversion. This function acts as a basic cryptographic building block for a novel communication-efficient zero-knowledge crypto-system. The system as defined exhibits partial homomorphism and behaves as an additive positive accumulator. By using a novel technique to constructively embed...

Function secret sharing (FSS) for a class $\cal{F}$ allows to split a secret function $f \in \cal{F}$ into (succinct) secret shares $f_0,f_1$, such that for all $x\in \{0,1\}^n$ it holds $f_0(x)-f_1(x)=f(x)$. FSS has numerous applications, including private database queries, nearest neighbour search, private heavy hitters and secure computation in the preprocessing model, where the supported class $\cal{F}$ translates to richness in the application. Unfortunately, concretely efficient FSS...

We propose protocols in the Compressed Sigma Protocol framework that achieve a succinct verifier. Towards this, we construct a new inner product argument and cast it in the Compressed Sigma Protocol (CSP) framework as a protocol for opening a committed linear form, achieving logarithmic verification. We then use our succinct-verifier CSP to construct a zero-knowledge argument for circuit satisfiability (under the discrete logarithm assumption in bilinear groups) in the updatable...

Homomorphic encryption enables public computation over encrypted data. In the past few decades, homomorphic encryption has become a staple of both the theory and practice of cryptography. Nevertheless, while there is a general loose understanding of what it means for a scheme to be homomorphic, to date there is no single unifying minimal definition that captures all schemes. In this work, we propose a new definition, which we refer to as combinatorially homomorphic encryption, which...

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)...

We present a secret-key encryption scheme based on random rank metric ideal linear codes with a simple decryption circuit. It supports unlimited homomorphic additions and plaintext absorptions as well as a fixed arbitrary number of homomorphic multiplications. We study a candidate bootstrapping algorithm that requires no multiplication but additions and plaintext absorptions only. This latter operation is therefore very efficient in our scheme, whereas bootstrapping is usually the main...

$\Sigma$-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,...

We propose a new scheme based on ephemeral elliptic curves over the ring $\mathbb{Z}/n\mathbb{Z}$ where $n=pq$ is an RSA modulus with $p=u_p^2 v_p^2$, $q=u_q^2 v_q^2$, $u_p\equiv u_q\equiv 3\pmod 4$. The new scheme is a variant of both the RSA and the KMOV cryptosystems. The scheme can be used for both signature and encryption. We study the security of the new scheme and show that is immune against factorization attacks, discrete logarithm problem attacks, sum of two squares attacks, sum of...

This work investigates zero-knowledge protocols in subverted RSA groups where the prover can choose the modulus and where the verifier does not know the group order. We introduce a novel technique for extracting the witness from a general homomorphism over a group of unknown order that does not require parallel repetitions. We present a NIZK range proof for general homomorphisms such as Paillier encryptions in the designated verifier model that works under a subverted setup. The key...

In [15], Leonardi and Ruiz-Lopez propose an additively homomorphic public key encryption scheme whose security is expected to depend on the hardness of the $\textit{learning homomorphism with noise problem}$ (LHN). Choosing parameters for their primitive requires choosing three groups $G$, $H$, and $K$. In their paper, Leonardi and Ruiz-Lopez claim that, when $G$, $H$, and $K$ are abelian, then their public-key cryptosystem is not quantum secure. In this paper, we study security for finite...

Timed commitment schemes, introduced by Boneh and Naor (CRYPTO 2000), can be used to achieve fairness in secure computation protocols in a simple and elegant way. The only known non-malleable construction in the standard model is due to Katz, Loss, and Xu (TCC 2020). This construction requires general-purpose zero knowledge proofs with specific properties, and it suffers from an inefficient commitment protocol, which requires the committing party to solve a computationally expensive...

A new CRT-based positive (non-zero) secret-sharing scheme with perfect information-theoretic (PIT) security and multiplicative homomorphism is presented. The scheme is designed to support the evaluation of multiplications of non-zero secrets of multiplicative groups. Our CRT-based scheme is partially homomorphic, supporting homomorphic multiplications. Nevertheless, our scheme has the potential to be regarded as fully homomorphic for practical scenarios, such as bounded-sized multi-cloud...

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]...

In order to analyze real-time power data without revealing user's privacy, privacy-preserving data aggregation has been extensively researched in smart grid. However, most of the existing schemes either have too much computation overhead and cannot achieve dynamic users, or require a trusted center. In this paper, we propose an efficient and robust multidimensional data aggregation scheme based on blockchain. In our scheme, a leader election algorithm in Raft protocol is used to select a...

We show that it is possible to perform $n$ independent copies of $1$-out-of-$2$ oblivious transfer in two messages, where the communication complexity of the receiver and sender (each) is $n(1 o(1))$ for sufficiently large $n$. Note that this matches the information-theoretic lower bound. Prior to this work, this was only achievable by using the heavy machinery of rate-$1$ fully homomorphic encryption (Rate-$1$ FHE, Brakerski et al., TCC 2019). To achieve rate-$1$ both on the receiver's and...

We introduce reductions of knowledge, a generalization of arguments of knowledge, which reduce checking knowledge of a witness in one relation to checking knowledge of a witness in another (simpler) relation. Reductions of knowledge unify a growing class of modern techniques as well as provide a compositional framework to modularly reason about individual steps in complex arguments of knowledge. As a demonstration, we simplify and unify recursive arguments over linear algebraic statements by...

Commitments to key-value maps (or, authenticated dictionaries) are an important building block in cryptographic applications, including cryptocurrencies and distributed file systems. In this work we study short commitments to key-value maps with two additional properties: double-hiding (both keys and values should be hidden) and homomorphism (we should be able to combine two commitments to obtain one that is the ``sum'' of their key-value openings). Furthermore, we require these commitments...

Fully homomorphic encryption (FHE) enables a simple, attractive framework for secure search. Compared to other secure search systems, no costly setup procedure is necessary; it is sufficient for the client merely to upload the encrypted database to the server. Confidentiality is provided because the server works only on the encrypted query and records. While the search functionality is enabled by the full homomorphism of the encryption scheme. For this reason, researchers have been paying...

Computation on ciphertexts of all known fully homomorphic encryption (FHE) schemes induces some noise, which, if too large, will destroy the plaintext. Therefore, the bootstrapping technique that re-encrypts a ciphertext and reduces the noise level remains the only known way of building FHE schemes for arbitrary unbounded computations. The bootstrapping step is also the major efficiency bottleneck in current FHE schemes. A promising direction towards improving concrete efficiency is to...

State-of-the-art re-keying schemes can be viewed as a tradeoff between efficient but heuristic solutions based on binary field multiplications, that are only secure if implemented with a sufficient amount of noise, and formal but more expensive solutions based on weak pseudorandom functions, that remain secure if the adversary accesses their output in full. Recent results on “crypto dark matter” (TCC 2018) suggest that low-complexity pseudorandom functions can be obtained by mixing linear...

We present probabilistic dynamic I/O automata, a framework to model dynamic probabilistic systems. Our work extends dynamic I/O Automata formalism of Attie & Lynch to probabilistic setting. The original dynamic I/O Automata formalism included operators for parallel composition, action hiding, action renaming, automaton creation, and behavioral sub-typing by means of trace inclusion. They can model mobility by using signature modification. They are also hierarchical: a dynamically changing...

Correlated random variables are a key tool in cryptographic applications like secure multi-party computation. We investigate the power of a class of correlations that we term group correlations: A group correlation is a uniform distribution over pairs $(x,y) \in G^2$ such that $x y\in S$, where $G$ is a (possibly non-abelian) group and $S$ is a subset of $G$. We also introduce bi-affine correlations and show how they relate to group correlations. We present several structural results, new...

We present Hyperproofs, the first vector commitment (VC) scheme that is efficiently maintainable and aggregatable. Similar to Merkle proofs, our proofs form a tree that can be efficiently maintained: updating all $n$ proofs in the tree after a single leaf change only requires $O(\log{n})$ time. Importantly, unlike Merkle proofs, Hyperproofs are efficiently aggregatable, anywhere from $10\times$ to $41\times$ faster than SNARK-based aggregation of Merkle proofs. At the same time, an...

Large semigroups and groups of transformations of finite affine space of dimension n with the option of computability of the composition of n arbitrarily chosen elements in polynomial time are described in the paper. Constructions of such families are given together with effectively computed homomorphisms between members of the family. These algebraic platforms allow us to define protocols for several generators of subsemigroup of affine Cremona semigroups with several outputs. Security of...

The high demand for customer-centric applications such as secure cloud storage laid the foundation for the development of user-centric security protocols with multiple security features in recent years. But, the current state-of-art techniques primarily emphasized only one type of security feature i.e., either homomorphism or non-malleability. In order to fill this gap and provide a common platform for both homomorphic and non-malleable cloud applications, we have introduced a new public key...

Timed-release encryption (TRE) makes it possible to send information ``into the future'' such that a pre-determined amount of time needs to pass before the information can be decrypted, which has found numerous applications. The most prominent construction is based on sequential squaring in RSA groups, proposed by Rivest et al. in 1996. Malavolta and Thyagarajan (CRYPTO'19) recently proposed an interesting variant of TRE called homomorphic time-lock puzzles (HTLPs). Here one considers...

A weak pseudorandom function $F: \mathcal{K} \times \mathcal{X} \rightarrow \mathcal{Y}$ is said to be ring key-homomorphic if, given $F \left(k_{1}, x \right)$ and $F \left(k_{2}, x \right)$, there are efficient algorithms to compute $F \left(k_{1} \oplus k_{2}, x \right)$ and $F \left(k_{1} \otimes k_{2}, x \right)$ where $\oplus$ and $\otimes$ are the addition and multiplication operations in the ring $\mathcal{K}$, respectively. In this work, we initiate the study of ring...

Pseudorandom functions (PRFs) are fundamental objects in cryptography that play a central role in symmetric-key cryptography. Although PRFs can be constructed from one-way functions generically, these black-box constructions are usually inefficient and require deep circuits to evaluate compared to direct PRF constructions that rely on specific algebraic assumptions. From lattices, one can directly construct PRFs from the Learning with Errors (LWE) assumption (or its ring variant) using the...

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...

The Fiat-Shamir (FS) transform is a well known and widely used technique to convert any constant-round public-coin honest-verifier zero-knowledge (HVZK) proof or argument system $CIPC=(Prov,Ver)$ in a non-interactive zero-knowledge (NIZK) argument system $NIZK=(NIZK.Prove, NIZK.Verify)$. The FS transform is secure in the random oracle (RO) model and is extremely efficient: it adds an evaluation of the RO for every message played by $Ver$. While a major effort has been done to attack the...

In this work, we define and construct fully homomorphic non-interactive zero knowledge (FH-NIZK) and non-interactive witness-indistinguishable (FH-NIWI) proof systems. We focus on the NP complete language $L$, where, for a boolean circuit $C$ and a bit $b$, the pair $(C,b) \in L$ if there exists an input $w$ such that $C(w)=b$. For this language, we call a non-interactive proof system 'fully homomorphic' if, given instances $(C_i,b_i) \in L$ along with their proofs $\Pi_i$, for $i \in...

We present a framework for the study of a learning problem over abstract groups, and introduce a new technique which allows for public-key encryption using generic groups. We proved, however, that in order to obtain a quantum resistant encryption scheme, commutative groups cannot be used to instantiate this protocol.

Securely managing encrypted data on an untrusted party is a challenging problem that has motivated the study of a variety of cryptographic primitives. A special class of such primitives allows an untrusted party to transform a ciphertext encrypted under one key to a ciphertext under another key, using some auxiliary information that does not leak the underlying data. Prominent examples of such primitives in the symmetric-key setting are key-homomorphic PRFs, updatable encryption, and proxy...

Noncommutative cryptography is based on applications of algebraic structures like noncommutative groups, semigroups and non-commutative rings. Its inter-section with Multivariate cryptography contains studies of cryptographic applications of subsemigroups and subgroups of affine Cremona semigroups defined overfinite commutative rings. Efficiently computed homomorphisms between stable subsemigroups of affine Cremona semigroups can be used in tame homomorphisms protocols schemes and...

Based on the identity-based encryption (IBE) from lattices by Agrawal et al. (Eurocrypt'10), Micciancio and Peikert (Eurocrypt'12) presented a CCA1-secure public-key encryption (PKE), which has the best known efficiency in the standard model and can be used to obtain a CCA2-secure PKE from lattices by using the generic BCHK transform (SIAM J. Comput., 2006) with a cost of introducing extra overheads to both computation and storage for the use of other primitives such as signatures and...

Electronic cash (e-cash) is the digital analogue of regular cash which aims at preserving users' privacy. Following Chaum's seminal work, several new features were proposed for e-cash to address the practical issues of the original primitive. Among them, divisibility has proved very useful to enable efficient storage and spendings. Unfortunately, it is also very difficult to achieve and, to date, quite a few constructions exist, all of them relying on complex mechanisms that can only be...

Algebraic structure lies at the heart of much of Cryptomania as we know it. An interesting question is the following: instead of building (Cryptomania) primitives from concrete assumptions, can we build them from simple Minicrypt primitives endowed with additional algebraic structure? In this work, we affirmatively answer this question by adding algebraic structure to the following Minicrypt primitives: • One-Way Function (OWF) • Weak Unpredictable Function (wUF) • Weak Pseudorandom...

We present an identity-Based encryption (IBE) scheme that is group homomorphic for addition modulo a ``large'' (i.e. superpolynomial) integer, the first such group homomorphic IBE. Our first result is the construction of an IBE scheme supporting homomorphic addition modulo a poly-sized prime $e$. Our construction builds upon the IBE scheme of Boneh, LaVigne and Sabin (BLS). BLS relies on a hash function that maps identities to $e$-th residues. However there is no known way to securely...

Quantum homomorphic encryption (QHE) is an important cryptographic technology for delegated quantum computation. It enables remote Server performing quantum computation on encrypted quantum data, and the specific algorithm performed by Server is unnecessarily known by Client. Quantum fully homomorphic encryption (QFHE) is a QHE that satisfies both compactness and $\mathcal{F}$-homomorphism, which is homomorphic for any quantum circuits. However, Yu et al.[Phys. Rev. A 90, 050303(2014)]...

Homomorphic universally composable (UC) commitments allow for the sender to reveal the result of additions and multiplications of values contained in commitments without revealing the values themselves while assuring the receiver of the correctness of such computation on committed values. In this work, we construct essentially optimal additively homomorphic UC commitments from any (not necessarily UC or homomorphic) extractable commitment. We obtain amortized linear computational complexity...

In this paper, we focus on the research of non-interactive secure multi-party computation (MPC). At first, we propose a fully homomorphic non-interactive verifiable secret sharing (FHNVSS) scheme. In this scheme, shareholders can generate shares of any-degree polynomials of shared numbers without interaction, and the dealer can verify the correctness of shares sent by shareholders without interaction. We implemented the FHNVSS scheme in Python with a detailed performance evaluation....

Homomorphic encryption schemes allow to perform computations over encrypted data. In schemes based on RLWE assumption the plaintext data is a ring polynomial. In many use cases of homomorphic encryption only the degree-0 coefficient of this polynomial is used to encrypt data. In this context any computation on encrypted data can be performed. It is trickier to perform generic computations when more than one coefficient per ciphertext is used. In this paper we introduce a method to...

We propose a lattice-based electronic voting scheme, EVOLVE (Electronic Voting from Lattices with Verification), which is conjectured to resist attacks by quantum computers. Our protocol involves a number of voting authorities so that vote privacy is maintained as long as at least one of the authorities is honest, while the integrity of the result is guaranteed even when all authorities collude. Furthermore, the result of the vote can be independently computed by any observer. At the core of...

We define and study zero-testable homomorphic encryption (ZTHE) -- a semantically secure, somewhat homomorphic encryption scheme equipped with a weak zero test that can identify trivial zeros. These are ciphertexts that result from homomorphically evaluating an arithmetic circuit computing the zero polynomial over the integers. This is a relaxation of the (strong) zero test provided by the notion of graded encodings, which identifies all encodings of zero. We show that ZTHE can suffice for...

We try to propose two fully homomorphic encryption (FHE) schemes, one for symmetric (aka. secret-key) settings and another under asymmetric (aka. public-key) scenario. The presented schemes are noiseless in the sense that there is no noise" factor contained in the ciphertexts. Or equivalently, before performing fully homomorphic computations, our schemes do not incorporate any noise-control process (such as bootstrapping, modulus switching, etc) to refresh the ciphertexts, since our fully...

In a constrained PRF, the owner of the PRF key K can generate constrained keys K_f that allow anyone to evaluate the PRF on inputs x that satisfy the predicate f (namely, where f(x) is “true”) but reveal no information about the PRF evaluation on the other inputs. A private constrained PRF goes further by requiring that the constrained key Kf hides the predicate f. Boneh, Kim and Montgomery (EUROCRYPT 2017) presented a construction of private constrained PRF for point function constraints,...

Group Homomorphic Encryption (GHE), formally defined by Armknecht, Katzenbeisser and Peter, is a public-key encryption primitive where the decryption algorithm is a group homomorphism. Hence it supports homomorphic evaluation of a single algebraic operation such as modular addition or modular multiplication. Most classical homomorphic encryption schemes such as as Goldwasser-Micali and Paillier are instances of GHE. In this work, we extend GHE to the attribute-based setting. We introduce and...

In this paper, we report on our implementation of a lattice-based Key-Policy Attribute-Based Encryption (KP-ABE) scheme, which uses short secret keys. The particular KP-ABE scheme can be used directly for Attribute-Based Access Control (ABAC) applications, as well as a building block in more involved applications and cryptographic schemes such as audit log encryption, targeted broadcast encryption, functional encryption, and program obfuscation. We adapt a recently proposed KP-ABE scheme [1]...

Fully homomorphic encryption over the integers (FHE-OI) is currently the only alternative to lattice-based FHE. FHE-OI includes a family of schemes whose security is based on the hardness of different variants of the approximate greatest common divisor (AGCD) problem. The majority of these works is based on the noise-free variant of AGCD which is potentially weaker than the general one. In particular, the noise-free variant relies on the hardness of factoring and is thus vulnerable to...

Key-homomorphic properties of cryptographic objects, i.e., homomorphisms on their key space, have proven to be useful, both from a theoretical as well as a practical perspective. Important cryptographic objects such as pseudorandom functions or (public key) encryption have been studied previously with respect to key-homomorphisms. Interestingly, however, signature schemes have not been explicitly investigated in this context so far. We close this gap and initiate the study of...

In (key-policy) attribute based encryption (ABE), messages are encrypted respective to attributes $x$, and keys are generated respective to policy functions $f$. The ciphertext is decryptable by a key only if $f(x)=0$. Adding homomorphic capabilities to ABE is a long standing open problem, with current techniques only allowing compact homomorphic evaluation on ciphertext respective to the same $x$. Recent advances in the study of multi-key FHE also allow cross-attribute homomorphism with...

This work considers a theoretic problem of merging the digests of two ordered lists “homomorphically.” This problem has potential applications to efficient and verifiable data outsourcing, which is especially desirable in the public cloud computing where the cloud is not trustworthy. We consider the case of merge-sort as it is fun- damental to many cloud-side operations, such as database join [32], data maintenance [10], among others. Informally, a merge-homomorphic digest enables that the...

We suggest a method to construct a homomorphic encryption scheme for approximate arithmetic. It supports an approximate addition and multiplication of encrypted messages, together with a new rescaling procedure for managing the magnitude of plaintext. This procedure truncates a ciphertext into a smaller modulus, which leads to rounding of plaintext. The main idea is to add a noise following significant figures which contain a main message. This noise is originally added to the plaintext for...

The rapid expansion and increased popularity of cloud computing comes with no shortage of privacy concerns about outsourcing computation to semi-trusted parties. Leveraging the power of encryption, in this paper we introduce Cryptoleq: an abstract machine based on the concept of One Instruction Set Computer, capable of performing general-purpose computation on encrypted programs. The program operands are protected using the Paillier partially homomorphic cryptosystem, which supports addition...

Abstract. We propose generic constructions of public-key encryption schemes, satisfying key- dependent message (KDM) security for projections and different forms of key-leakage resilience, from CPA-secure private key encryption schemes with two main abstract properties: (1) additive homomorphism with respect to both messages and randomness, and (2) reproducibility, providing a means for reusing encryption randomness across independent secret keys. More precisely, our construction transforms...

A pseudorandom function F : K x X -> Y is said to be key homomorphic if given F(k1, x) and F(k2, x) there is an efficient algorithm to compute F(k1 xor k2, x), where xor denotes a group operation on k1 and k2 such as xor. Key homomorphic PRFs are natural objects to study and have a number of interesting applications: they can simplify the process of rotating encryption keys for encrypted data stored in the cloud, they give one round distributed PRFs, and they can be the basis of a...

Private query processing on encrypted databases allows users to obtain data from encrypted databases in such a way that the user's sensitive data will be protected from exposure. Given an encrypted database, the users typically submit queries similar to the following examples: - How many employees in an organization make over 100,000? - What is the average age of factory workers suffering from leukemia? Answering the above questions requires one to \textbf{search} and then \textbf{compute}...

This survey, aimed mainly at mathematicians rather than practitioners, covers recent developments in homomorphic encryption (computing on encrypted data) and program obfuscation (generating encrypted but functional programs). Current schemes for encrypted computation all use essentially the same "noisy" approach: they encrypt via a noisy encoding of the message, they decrypt using an "approximate" ring homomorphism, and in between they employ techniques to carefully control the noise as...

In this paper, we present a construction of public key encryption secure against related key attacks from hash proof systems in the standard model. We show that the schemes presented by Jia et al. (Provsec2013) are special cases of our general theory, and also give other instantiations based on the QR and DCR assumptions. To fulfill the related key security, we require the hash proof systems to satisfy the key homomorphism and computational finger-printing properties. Compared with the...

We investigate structure-preserving signatures in asymmetric bilinear groups with an efficiently computable homomorphism from one source group to the other, i.e., the Type II setting. It has been shown that in the Type I and Type III settings (with maximal symmetry and maximal asymmetry respectively), structure-preserving signatures need at least 2 verification equations and 3 group elements. It is therefore natural to conjecture that this would also be required in the intermediate Type II...

In CRYPTO 2008, one year earlier than Gentry's pioneering \lq\lq bootstrapping'' technique on constructing the first fully homomorphic encryption (FHE) scheme, Ostrovsky and Skeith III had suggested a completely different approach towards achieving FHE. Namely, they showed that the $\mathsf{NAND}$ operator can be realized in some \emph{non-commutative} groups; consequently, in combination with the $\mathsf{NAND}$ operator realized in such a group, homomorphically encrypting the elements of...

We construct a structure-preserving signature scheme that is selectively randomizable and works in all types of bilinear groups. We give matching lower bounds showing that our structure-preserving signature scheme is optimal with respect to both signature size and public verification key size. State of the art structure-preserving signatures in the asymmetric setting consist of 3 group elements, which is known to be optimal. Our construction preserves the signature size of 3 group elements...

A \emph{key-homomorphic} pseudorandom function (PRF) family $\set{F_{s} \colon D \to R}$ allows one to efficiently compute the value $F_{s t}(x)$ given $F_{s}(x)$ and $F_{t}(x)$. Such functions have many applications, such as distributing the operation of a key-distribution center and updatable symmetric encryption. The only known construction of key-homomorphic PRFs without random oracles, due to Boneh \etal (CRYPTO~2013), is based on the learning with errors (\lwe) problem and hence on...

Identity-based encryption (IBE) is a special case of public-key encryption where user identities replace public keys. Every user is given a corresponding secret key for decryp- tion, and encryptions for his or her identity must remain confidential even to attackers who learn the secret keys associated with other identities. Several IBE constructions are known to date, but their security relies on specific assumptions, such as quadratic residuosity, as well as different pairing-based and...

At Eurocrypt 2013, Joye and Libert proposed a method for constructing public key cryptosystems (PKCs) and lossy trapdoor functions (LTDFs) from $(2^\alpha)^{th}$-power residue symbols. Their work can be viewed as non-trivial extensions of the well-known PKC scheme due to Goldwasser and Micali, and the LTDF scheme due to Freeman et al., respectively. In this paper, we will demonstrate that this kind of work can be extended \emph{more generally}: all related constructions can work for any...

At Eurocrypt 2013, Joye and Libert proposed a method for constructing public key cryptosystems (PKCs) and lossy trapdoor functions (LTDFs) from $(2^\alpha)^{th}$-power residue symbols. Their work can be viewed as non-trivial extensions of the well-known PKC scheme due to Goldwasser and Micali, and the LTDF scheme due to Freeman et al., respectively. In this paper, we will demonstrate that this kind of work can be extended \emph{more generally}: all related constructions can work for any...

We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the field with $q$ elements. Modulo a well supported number theoretic hypothesis, which holds in particular for concrete homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions: upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the...

Structure-preserving signatures (SPS) are signature schemes where messages, signatures and public keys all consist of elements of a group over which a bilinear map is efficiently computable. This property makes them useful in cryptographic protocols as they nicely compose with other algebraic tools (like the celebrated Groth-Sahai proof systems). In this paper, we consider SPS systems with homomorphic properties and suggest applications that have not been provided before (in...

In 1978, Rivest, Adleman and Dertouzos introduced the basic concept of privacy homomorphism that allows computation on encrypted data without decryption. It was elegant work that precedes the recent development of fully homomorphic encryption schemes although there were found some security flaws, e.g., ring homomorphic schemes are broken by the known-plaintext attacks. In this paper, we revisit one of their proposals, in particular the third scheme which is based on the Chinese Remainder...

\emph{Succinct arguments of knowledge} are computationally-sound proofs of knowledge for NP where the verifier's running time is independent of the time complexity $t$ of the nondeterministic NP machine $M$ that decides the given language. Existing succinct argument constructions are, typically, based on techniques that combine cryptographic hashing and probabilistically-checkable proofs (PCPs). Yet, even when instantiating these constructions with state-of-the-art PCPs, the prover needs...

We investigate a new class of authenticate codes (A-codes) that support verification by a group of message recipients in the network coding setting. That is, a sender generates an A-code over a message such that any intermediate node or recipient can check the authenticity of the message, typically to detect pollution attacks. We call such an A-code as multi-receiver homomorphic A-code (MRHA-code). In this paper, we first formally define an MRHA-code. We then derive some lower bounds on the...

We show that an encryption scheme cannot have a simple decryption function and be homomorphic at the same time, even with added noise. Specifically, if a scheme can homomorphically evaluate the majority function, then its decryption cannot be weakly-learnable (in particular, linear), even if large decryption error is allowed. (In contrast, without homomorphism, such schemes do exist and are presumed secure, e.g. based on LPN.) An immediate corollary is that known schemes that are based on...

The Schnorr signature scheme has been known to be provably secure in the Random Oracle Model under the Discrete Logarithm (DL) assumption since the work of Pointcheval and Stern (EUROCRYPT '96), at the price of a very loose reduction though: if there is a forger making at most $q_h$ random oracle queries, and forging signatures with probability $\varepsilon_F$, then the Forking Lemma tells that one can compute discrete logarithms with constant probability by rewinding the forger...

We present a generic method to secure various widely-used cryptosystems against \emph{arbitrary} side-channel leakage, as long as the leakage adheres three restrictions: first, it is bounded per observation but in total can be arbitrary large. Second, memory parts leak \emph{independently}, and, third, the randomness that is used for certain operations comes from a simple (non-uniform) distribution. As a fundamental building block, we construct a scheme to store a cryptographic secret such...

We aim at constructing adaptive oblivious transfer protocols, enjoying fully simulatable security, from various well-known assumptions such as DDH, $d$-Linear, QR, DCR, and LWE. To this end, we present two generic constructions of adaptive OT, one of which utilizes verifiable shuffles together with threshold decryption schemes, while the other uses permutation networks together with what we call {\em loosely-homomorphic} key encapsulation schemes. We then show that specific choices of the...

Homomorphic encryption schemes are powerful cryptographic primitives that allow for a variety of applications. Consequently, a variety of proposals have been made in the recent decades but none of them was based on coding theory. The existence of such schemes would be interesting for several reasons. First, it is well known that having multiple schemes based on different hardness assumptions is advantageous. In case that one hardness assumption turns out be wrong, one can switch over to...

Verifiable secret sharing (VSS) is an important primitive in distributed cryptography that allows a dealer to share a secret among n parties in the presence of an adversary controlling at most t of them. In the computational setting, the feasibility of VSS schemes based on commitments was established over two decades ago. Interestingly, all known computational VSS schemes rely on the homomorphic nature of these commitments or achieve weaker guarantees. As homomorphism is not inherent to...

In this work we propose a new cryptographic primitive called Delegatable Homomorphic Encryption (DHE). This allows a Trusted Authority to control/delegate the capability to evaluate circuits over encrypted data to untrusted workers/evaluators by issuing tokens. This primitive can be both seen as a public-key counterpart to Verifiable Computation, where input generation and output verification are performed by different entities, or as a generalisation of Fully Homomorphic Encryption enabling...

Method of secure evaluation of polynomial y=F(x_1, …, x_k) over some rings on untrusted computer is proposed. Two models of untrusted computer are considered: passive and active. In passive model untrusted computer correctly computes polynomial F and tries to know secret input (x_1, …, x_k) and output y. In active model untrusted computer tries to know input and output and tries to change correct output y so that this change cannot be determined. Secure computation is proposed by using...

The Pollard kangaroo method solves the discrete logarithm problem (DLP) in an interval of size $N$ with heuristic average case expected running time approximately $2 \sqrt{N}$ group operations. A recent variant of the kangaroo method, requiring one or two inversions in the group, solves the problem in approximately $1.71 \sqrt{N}$ group operations. It is well-known that the Pollard rho method can be sped-up by using equivalence classes (such as orbits of points under an efficiently computed...

This thesis explores the notion of isogenies and its applications to cryptography. Elliptic curve cryptography (ECC) is an efficient public cryptosystem with a short key size. For this reason it is suitable for implementing on memory-constraint devices such as smart cards, mobile devices, etc. However, these devices leak information about their private key through side channels (power consumption, electromagnetic radiation, timing etc) during cryptographic processing. In this thesis we have...

In a related-key attack (RKA) an adversary attempts to break a cryptographic primitive by invoking the primitive with several secret keys which satisfy some known, or even chosen, relation. We initiate a formal study of RKA security for \emph{randomized encryption} schemes. We begin by providing general definitions for semantic security under passive and active RKAs. We then focus on RKAs in which the keys satisfy known linear relations over some Abelian group. We construct simple and...

Providing security services for multicast, such as traffic integrity, authentication, and confidentiality, requires securely distributing a group key to group receivers. In the literature, this problem is called multicast key distribution (MKD). A famous MKD protocol—one-way function tree (OFT)—has been found vulnerable to collusion attacks. Solutions to prevent these attacks have been proposed, but at the cost of a higher communication overhead than the original protocol. In this paper, we...

When using pairing-friendly ordinary elliptic curves over prime fields to implement identity-based protocols, there is often a need to hash identities to points on one or both of the two elliptic curve groups of prime order $r$ involved in the pairing. Of these $G_1$ is a group of points on the base field $E(\F_p)$ and $G_2$ is instantiated as a group of points with coordinates on some extension field, over a twisted curve $E'(\F_{p^d})$, where $d$ divides the embedding degree $k$. While...

While the idea of database outsourcing is becoming increasingly popular, the associated security risks still prevent many potential users from deploying it. In particular, the need to give full access to one's data to a third party, the database service provider, remains a major obstacle. A seemingly obvious solution is to encrypt the data in such a way that the service provider retains the ability to perform relational operations on the encrypted database. In this paper we present a model...

A new general mathematical problem, suitable for public-key cryptosystems, is proposed: morphism computation in a category of Abelian groups. In connection with elliptic curves over finite fields, the problem becomes the following: compute an isogeny (an algebraic homomorphism) between the elliptic curves given. The problem seems to be hard for solving with a quantum computer. ElGamal public-key encryption and Diffie-Hellman key agreement are proposed for an isogeny cryptosystem. The paper...

Mix-nets can be used to shuffle vectors of shared secrets. This operation can be an important building block for solving combinatorial problems where constraints depend on secrets of different participants. A main contribution of this paper is to show how participants in the mix-net can provide Zero-Knowledge proofs to convince each other that they do not tamper with the shuffled secrets, and that inverse permutations are correctly applied at unshuffling. The approach is related to the...

We propose a cryptanalysis of the original Domingo-Ferrer's algebraic privacy homomorphism. We show that the scheme over $\Z_n$ can be broken by $d 1$ known plaintexts in $O(d^3\log^2 n)$ time when it has $d$ times expansion through the encryption. Furthermore even when the public modulus $n$ is kept secret, it can be broken by $d 2$ known plaintexts in time at most $O(d^5\log^2(dn))$.

We investigate relations among the discrete logarithm (DL) problem, the Diffie-Hellman (DH) problem and the bilinear Diffie-Hellman (BDH) problem when we have an efficient computable non-degenerate bilinear map $e:G\times G \rightarrow H$. Under a certain assumption on the order of $G$, we show that the DH problem on $H$ implies the DH problem on $G$, and both of them are equivalent to the BDH problem when $e$ is {\it weak-invertible}. Moreover, we show that given the bilinear map $e$ an...