Zero-knowledge proof is a critical cryptographic primitive. Its most practical type, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK), has been deployed in various privacy-preserving applications such as cryptocurrencies and verifiable machine learning. Unfortunately, zkSNARK like Groth16 has a high overhead on its proof generation step, which consists of several time-consuming operations, including large-scale matrix-vector multiplication (MUL), number-theoretic transform (NTT), and multi-scalar multiplication (MSM). Therefore, this paper presents cuZK, an efficient GPU implementation of zkSNARK with the following three techniques to achieve high performance. First, we propose a new parallel MSM algorithm. This MSM algorithm achieves nearly perfect linear speedup over the Pippenger algorithm, a well-known serial MSM algorithm. Second, we parallelize the MUL operation. Along with our self-designed MSM scheme and well-studied NTT scheme, cuZK achieves the parallelization of all operations in the proof generation step. Third, cuZK reduces the latency overhead caused by CPU-GPU data transfer by 1) reducing redundant data transfer and 2) overlapping data transfer and device computation. The evaluation results show that our MSM module provides over 2.08x (up to 2.94x) speedup versus the state-of-the-art GPU implementation. cuZK achieves over 2.65x (up to 4.86x) speedup on standard benchmarks and 2.18× speedup on a GPU-accelerated cryptocurrency application, Filecoin.
Handong Cui, Kwan Yin Chan, Tsz Hon Yuen, Xin Kang · 5 authors
Abstract In most threshold Elliptic Curve Digital Signature Algorithm (ECDSA) signatures using additively homomorphic encryption, the zero-knowledge (ZK) proofs related to the ciphertext or the message space are the bottleneck in terms of bandwidth as well as computation time. In this paper, we propose a compact ZK proof for relations related to the Castagnos–Laguillaumie (CL) encryption, which is 33% shorter and 29% faster than the existing work in PKC 2021. We also give new ZK proofs for relations related to homomorphic operations over the CL ciphertext. These new ZK proofs are useful to construct a bandwidth-efficient universal composable-secure threshold ECDSA without compromising the proactive security and the non-interactivity. In particular, we lowered the communication and computation cost of the key refresh algorithm in the Paillier-based counterpart from $O(n^3)$ to $O(n^2)$. Considering a 5-signer setting, the bandwidth is better than the Paillier-based counterpart for up to 99, 95 and 35% for key generation, key refreshment and pre-signing, respectively.
In this paper, we propose a physical protocol to verify the first nonzero term of a sequence using a deck of cards. The protocol lets a prover show the value of the first nonzero term of a given sequence to a verifier without revealing which term it is. Our protocol uses $Θ(1)$ shuffles, which is asymptotically lower than that of an existing protocol of Fukusawa and Manabe which uses $Θ(n)$ shuffles, where $n$ is the length of the sequence. We also apply our protocol to construct zero-knowledge proof protocols for three well-known logic puzzles: ABC End View, Goishi Hiroi, and Toichika. These protocols enables a prover to physically show that he/she know solutions of the puzzles without revealing them.
Very recent attacks like ladder leak demonstrated feasibility to recover private key with side channel attacks using just one bit of secret nonce. ECDSA nonce bias can be exploited in many ways. Some attacks on ECDSA involve complicated Fourier analysis and lattice mathematics. In this paper will enable cryptographers to identify efficient ways in which ECDSA can be cracked on curves NIST256p, SECP256k1, NIST521p and weak nonce, kind of attacks that can crack ECDSA and how to protect yourself. Initially we begin with ECDSA signature to sign a message using private key and validate the generated signature using the shared public key. Then we use a nonce or a random value to randomize the generated signature. Every time we sign, a new verifiable random nonce value is created and way in which the intruder can discover the private key if the signer leaks any one of the nonce value. Then we use Lenstra–Lenstra–Lovasz (LLL) method as a black box, we will try to attack signatures generated from bad nonce or bad random number generator (RAG) on NIST256p, SECP256k1 curves. The analysis is performed by considering all the three curves for implementation of Elliptic Curve Digital Signature Algorithm (ECDSA).The comparative analysis for each of the selected curves in terms of computational time is done with leak of nonce and with Lenstra–Lenstra–Lovasz method to crack ECDSA. The average computational costs to break ECDSA with curves NIST256p, NIST521p and SECP256k1 are 0.016, 0.34, 0.46 respectively which is almost to zero depicts the strength of algorithm. The average computational costs to break ECDSA with curves SECP256K1 and NIST256p using LLL are 2.9 and 3.4 respectively.
Maryam Sheikhi Garjan, N. Gamze ORHON KILIÇ, Murat Cenk
The increasing demand for secure and anonymous transactions raises the popularity of ring signatures, which is a digital signature scheme that allows identifying a group of possible signers without revealing the identity of the actual signer. This paper presents efficient supersingular isogeny-based ring signature and linkable ring signature schemes that will find potential applications in post-quantum technologies. We develop the ring signature scheme by applying the Fiat-Shamir transform on the sigma protocol for a ring which we obtain from the supersingular isogeny-based interactive zero-knowledge identification scheme by adopting the scheme for a ring. We also extend our ring signature protocol with an additional parameter, i.e., a tag that provides to detect if a signer issues two signatures concerning the same ring by preserving anonymity and linkable anonymity. The signature size of our ring signature protocols increases logarithmically in the size of the ring thanks to the Merkle trees. We show the security proofs and efficiency analyses of the protocols offered. Moreover, we provide the implementation results of the supersingular isogeny-based ring signature, which offers small signature sizes for NIST post-quantum security levels.
Abstract Decomposition puzzles are pencil-and-paper logic puzzles that involve partitioning a rectangular grid into several regions to satisfy certain rules. In this paper, we construct a generic card-based protocol called printing protocol , which can be used to physically verify solutions of decompositon puzzles. We apply the printing protocol to develop card-based zero-knowledge proof protocols for two such puzzles: Five Cells and Meadows. These protocols allow a prover to physically show that he/she knows solutions of the puzzles without revealing them.
Метою роботи є оптимізувати множення точок еліптичної кривої на скаляр за допомогою модернізованого метода “Подвійне скалярне множення з використанням трюку Штрауса-Шаміра з урахуванням Skew representation” та багато інших підходів. Ще однією метою роботи - зробити використання множення точок еліптичної кривої на скаляр безпечним в межах протоколу Zero-knowledge proof. З'ясувати, який метод множення буде найдешевшим в контексті визначеної метрики. Об’єктом дослідження є множення точок еліптичної кривої на скаляр в системі гаджет бібліотеках рекурсивних zk-Snark’s. Множення повинно виконуватися до стандартів протоколу Zero-knowledge proof. Також розглянуто варіанти практичного застосування описаних методів. У роботі виконане теоретичне та практичне дослідження, огляд алгоритмів та методів розв’язання задачі оптимізації з використанням різноманітних хитростей та підходів. Кодова база була написана мовою програмування Rust в бібліотеці franklin-crypto. Арифметизація, яка застосовується в бібліотеці – Plonkish та lookup table. Крива, яка була використана для тестування множення – Bn256. Ключові слова : точки еліптичної кривої, Zero-knowledge proof, мовою програмування Rust.
The paper discusses the basic principles of construction and the main types of zeroknowledge succinct non-interactive argument of knowledge (zk-SNARK) which is used in the model of a three-way insecure computing environment and based on sets of polynomials. A number of zk-SNARK cryptographic protocols with different algorithms for generating public parameters (Trusted Setup) are given, constructing succinct proofs of reliability calculations (Prover) and public/designated verification of proofs (Verifier). The cases of satisfying the feasibility of discrete functions (arithmetic/ Boolean circuits) using different polynomial sets are presented in quadratic arithmetic programs (QAP), square arithmetic programs (SAP), quadratic span programs (QSP), square span programs (SSP), quadratic polynomial programs (QPP), etc., also the use of authenticated data are described. The cryptographic transformations needed to build zk-SNARKs based on symmetric and asymmetric hash functions, exponential knowledge problems, digital signatures, homomorphic encryption, bilinear pairings based on elliptic curves, etc. are presented. Examples of multilateral verifiable calculations based on zk-SNARK are given.
У кваліфікаційній роботі бакалавра було проведено аналіз архітектури криптовалют та особливостей використання цифрового підпису в криптовалютах. Було розглянуто основні принципи криптовалют, зокрема децентралізацію, роль журналу транзакцій та принципи консенсусу, такі як proof of work та proof of stake. Також було досліджено застосування криптографії в криптовалютах, зокрема генерацію приватних та публічних ключів, еліптичну криптографію та хеш-функції. Було проведено аналіз основних алгоритмів цифрового підпису. Були розглянуті алгоритми RSA, DSA, ECDSA та EdDSA. Досліджено структуру та безпеку цих алгоритмів, а також механізми, на яких вони ґрунтуються. Виявлено, що EdDSA є простішим у реалізації та розумінні порівняно з ECDSA, при цьому забезпечуючи подібний рівень безпеки. Було реалізовано програму для використання цифрових підписів з використанням обговорених алгоритмів. Було описано процес реалізації програми, вибір та підключення необхідних бібліотек, а також розроблено інтерфейси та функціонал для роботи з цифровим підписом. Ключові слова : цифрові підписи, еліптичні криві, еліптичні криві Едвардса, криптовалюта, JAVA.
Open access
Cryptography and Residue Arithmetic
Cybersecurity and Information Systems
Advanced Steganography and Watermarking Techniques
Diego F. Aranha, Youssef El Housni, Aurore Guillevic
Elliptic curves have become key ingredients for instantiating zero-knowledge proofs and more generally proof systems. Recently, there have been many tailored constructions of these curves that aim at efficiently implementing different kinds of proof systems. In this survey we provide the reader with a comprehensive overview on existing work and revisit the contributions in terms of efficiency and security. We present an overview at three stages of the process: curves to instantiate a SNARK, curves to instantiate a recursive SNARK, and also curves to express an elliptic-curve related statement. We provide new constructions of curves for SNARKs and generalize the state-of-the-art constructions for recursive SNARKs. We also exhaustively document the existing work and open-source implementations.
Zk-SNARK unleashes the great potential of ZKP (zero-knowledge proof) in the blockchain, distributed storage, etc. However, the proof-generation of zk-SNARK is excessively time intensive, making it a challenge to deploy a high-performance zk-SNARK in most real applications. As a result, NTT (Number Theoretic Transform), one of the most time-consuming parts in proof-generation, needs to be accelerated significantly. To address this issue, we propose a novel and efficient “data reordering” technique to enable a highly pipelined architecture, on which an FPGA-based hardware accelerator is designed to support the large-bitwidth and large-scale NTT tasks in zk-SNARK. Our architecture achieves a two-level pipeline: 1) the top-level pipeline is achieved among smaller NTT sub-tasks, which are decomposed from a large-scale NTT task; 2) the bottom-level pipeline is achieved in each sub-task, among butterfly operations with different step sizes. This architecture can effectively reduce the data dependency and memory access requirements, meanwhile, can be flexibly scaled to different scales of FPGAs. To balance computing efficiency and flexibility, the OpenCL equipped with HLS is used to implement the heterogeneous acceleration system. We prototype the accelerator on the AMD-Xilinx Alveo U50 card (UltraScale+ XCU50 FPGA). The evaluation results show that 1) our accelerator shows high scalability for different scales of FPGAs with a stable performance improvement; 2) it performs 1.95× faster than the one in PipeZK; 3) and it achieves 27.98×, 1.74× speedup and 6.9×, 6× energy efficiency improvement than AMD Ryzen 9 5900X single core and 12 cores respectively when integrated into the well-known ZKP open-source project, Bellman.
We study the problem of constructing succinct zero knowledge proof systems for floating point computations. The standard approach to handle floating point computations requires conversion to binary circuits, following the IEEE-754 floating point standard. This approach incurs a poly(w) overhead in prover efficiency for computations with w-bit precision, resulting in very high prover runtimes -- already the key bottleneck in the design of succinct arguments. We make the following contributions: -We propose a new model for verifying floating point computations that guarantees approximate correctness w.r.t. a relative error bound. This model is inspired by numerical analysis, and is very meaningful for applications such as machine learning and scientific computing. -Using this model, we present a general method for constructing succinct zero-knowledge proofs for floating point computations starting from existing public-coin "commit-and-prove'' systems. For computations with w-bit precision, our approach incurs only a log(w) overhead in prover running time. Our compiler nearly preserves (up to a factor of 2) the communication complexity of the underlying protocol, and requires sub-linear verification time. The resulting proof can be made non-interactive in the random oracle model. Concretely, our scheme is ~57x faster than the method following IEEE standard exactly [35] for 32-bit floating point computations. Central to our main result, and of independent interest, is a new batch range proof system in standard prime order groups that does not rely on bit decomposition.
Recent advanced Zero-Knowledge protocols, along with other high-level constructions such as Multi-Party Computations (MPC), have highlighted the need for a new type of symmetric primitives that are not optimized for speed on the usual platforms (desktop computers, servers, microcontrollers, RFID tags...), but for their ability to be implemented using arithmetic circuits.Several primitives have already been proposed to satisfy this need. In order to enable an efficient arithmetization, they operate over large finite fields, and use round functions that can be modelled using low degree equations. The impact of these properties on their security remains to be completely assessed. In particular, algebraic attacks relying on polynomial root-finding become extremely relevant. Such attacks work by writing the cryptanalysis as systems of polynomial equations over the large field, and solving them with off-the-shelf tools (SageMath, NTL, Magma, . . . ).The need for further analysis of these new designs has been recently highlighted by the Ethereum Foundation, as it issued bounties for successful attacks against round-reduced versions of several of them.In this paper, we show that the security analysis performed by the designers (or challenge authors) of four such primitives is too optimistic, and that it is possible to improve algebraic attacks using insights gathered from a careful study of the round function.First, we show that univariate polynomial root-finding can be of great relevance n practice, as it allows us to solve many of the Ethereum Foundation’s challenges on Feistel–MiMC. Second, we introduce a trick to essentially shave off two full rounds at little to no cost for Substitution-Permutation Networks (SPN). This can be combined with univariate (resp. multivariate) root-finding, which allowed to solve some challenges for Poseidon (resp. Rescue–Prime). Finally, we also find an alternative way to set up a system of equations to attack Ciminion, leading to much faster attacks than expected by the designers.
Antonio J. Di Scala, Andrea Gangemi, Giuliano Romeo, Gabriele Vernetti
In 2020, Sala, Sogiorno and Taufer were able to find the private keys of some Bitcoin addresses, thus being able to spend the cryptocurrency linked to them. This was unexpected since the recovery of non-trivial private keys for blockchain addresses is deemed to be an infeasible problem. In this paper, we widen this analysis by mounting a similar attack on other small subsets of the set of private keys. We then apply it to other blockchains as well, examining Ethereum, Dogecoin, Litecoin, Dash, Zcash and Bitcoin Cash. In addition to the results, we also explain the techniques we have used to perform this exhaustive search for all the addresses that have ever appeared in these blockchains, and we give an estimate of the time needed to perform all the computations. Finally, we also examine the possibility of mounting a similar attack on other elliptic curves used in blockchains, i.e., Curve25519 and NIST P-256.
In recent years a new class of symmetric-key primitives over $\mathbb{F}_p$ that are essential to Multi-Party Computation and Zero-Knowledge Proofs based protocols have emerged. Towards improving the efficiency of such primitives, a number of new block ciphers and hash functions over $\mathbb{F}_p$ were proposed. These new primitives also showed that following alternative design strategies to the classical Substitution-Permutation Network (SPN) and Feistel Networks leads to more efficient cipher and hash function designs over $\mathbb{F}_p$ specifically for large odd primes $p$. In view of these efforts, in this work we build an \emph{algebraic framework} that allows the systematic exploration of viable and efficient design strategies for constructing symmetric-key (iterative) permutations over $\mathbb{F}_p$. We first identify iterative polynomial dynamical systems over finite fields as the central building block of almost all block cipher design strategies. We propose a generalized triangular polynomial dynamical system (GTDS), and based on the GTDS we provide a generic definition of an iterative (keyed) permutation over $\mathbb{F}_p^n$. Our GTDS-based generic definition is able to describe the three most well-known design strategies, namely SPNs, Feistel networks and Lai--Massey. Consequently, the block ciphers that are constructed following these design strategies can also be instantiated from our generic definition. Moreover, we find that the recently proposed \texttt{Griffin} design, which neither follows the Feistel nor the SPN design, can be described using the generic GTDS-based definition. We also show that a new generalized Lai--Massey construction can be instantiated from the GTDS-based definition. We further provide generic analysis of the GTDS including an upper bound on the differential uniformity and the correlation.
Elliptic curve pairings are a powerful tool and a popular way to construct zero-knowledge proofs, which are beginning to be used in blockchains as a way to provide privacy in the transaction ledger.