This work is focused on the description of one verifiable encryption scheme, specifically a zero-knowledge proof of knowledge protocol. Verifiable encryption allows us to prove properties of data without revealing its content. The main goal of the presented method is verification of knowledge of a secret key. This method can be used for group signa- tures, multiple steps secret sharing, key escrow protocols, and many others cryptographic protocols. It is based on the hardness of the Ring-LWE problem and problems of finding solutions to linear relations over some ring. It uses the principle of rejection sampling. The method is build on two closely described cryptographic protocols, Ring-LWE and Fiat-Shamir with aborts. It uses the construction of polynomial rings R = Z[x]/(xn + 1) a Rq = Zq[x]/(xn + 1). 1
This work is focused on the description of one verifiable encryption scheme, specifically a zero-knowledge proof of knowledge protocol. Verifiable encryption allows us to prove properties of data without revealing its content. The main goal of the presented method is verification of knowledge of a secret key. This method can be used for group signa- tures, multiple steps secret sharing, key escrow protocols, and many others cryptographic protocols. It is based on the hardness of the Ring-LWE problem and problems of finding solutions to linear relations over some ring. It uses the principle of rejection sampling. The method is build on two closely described cryptographic protocols, Ring-LWE and Fiat-Shamir with aborts. It uses the construction of polynomial rings R = Z[x]/(xn + 1) a Rq = Zq[x]/(xn + 1). 1
Abstract. We design an efficient commitment scheme, and companion zero-knowledge proofs of knowledge, based on the learning with errors over rings (RLWE) problem. In particular, for rings in which almost all elements have inverses, we construct a perfectly binding commitment scheme whose hiding property relies on the RLWE assumption. Our scheme maps elements from the ring (or equivalently, n elements from Fq) to a small constant number of ring elements. We then construct Σ-protocols for proving, in a zero-knowledge manner, knowledge of the message contained in a commitment. We are able to further extend our basic protocol to allow us to prove additive and multiplicative relations among committed values. Our protocols have a communication complexity of O(Mn log q) and achieve a negligible knowledge error in one run. Here M is the constant from a rejection sampling technique that we employ, and can be set close to 1 by adjusting other parameters. Previously known Σ-protocols for LWE-related languages either relied on “smudging ” out the error (which necessitates working over large fields, resulting in poor efficiency) or only achieved a noticeable or even constant knowledge error (thus requiring many repetitions of the protocol).
We extend a commitment scheme based on the learning with errors over rings ( RLWE ) problem, and present efficient companion zero-knowledge proofs of knowledge. Our scheme maps elements from the ring (or equivalently, n elements from Fq ) to a small constant number of ring elements. We then construct Σ -protocols for proving, in a zero-knowledge manner, knowledge of the message contained in a commitment. We are able to further extend our basic protocol to allow us to prove additive and multiplicative relations among committed values. Our protocols have a communication complexity of O(Mnlogq) and achieve a negligible knowledge error in one run. Here M is the constant from a rejection sampling technique that we employ, and can be set close to 1 by adjusting other parameters. Previously known Σ -protocols for LWE-related languages only achieved a noticeable or even constant knowledge error (thus requiring many repetitions of the protocol), or relied on “smudging” out the error (which necessitates working over large fields, resulting in poor efficiency).
Key Exchange is an essential primitive in cryptography that originated from the breakthrough idea of Diffie Hellman (DH) protocol presented in 1976.Following this work, there are many successful variants of DH using Elliptic curves, Isogeny and RLWE (Ring Learning with Errors).The Ring-Leaning with Errors (RLWE), Search and Decision version are hard problems in mathematics and it has been shown that solving worst case hard problems in ideal lattices can be reduced to solving RLWE in the average case.The reduction forms the basis of provable security for RLWE based key exchange against passive adversaries.The hardness of the decision version of RLWE, which means indistinguishability of RLWE samples from Uniform, makes it challenging to validate a RLWE form public key.This proves to be an advantage for malicious users to create public keys that deviate from a protocol to recover the secret of reused keys in Key Exchange (KE) protocols.In this thesis, we provide a cryptanalysis of RLWE key exchange, presenting two polynomial time strategies to exploit key reuse.Then we propose a defense against such exploits by presenting a Zero Knowledge authentication protocol to verify the prover's knowledge of a secret corresponding to his public key.The protocol is Statistical Zero Knowledge with negligible soundness and completeness errors so that the prover does not reveal any information about the secret in the proof.This is to ensure that a malicious verifier cannot impersonate the prover after the proof.The authentication protocol uses the signal function defined in RLWE key exchange to perform the verification.Firstly, I would like to express my gratitude to my advisor Prof.Ding for his expert advice, strong encouragement and support throughout my PhD study
A Zero-Knowledge Proof basically is a protocol between two parties, the Prover and the Verifier, that allows the Prover to convince the Verifier about the truthness of a non trivial statement without revealing any additional information. Zero Knowledge Proofs have found a lot of practical applications covering most of the protocols concerning about data privacy and protocol verification. Examples of that are anonymous cash or electronic voting. The possibility to have real quantum computers with a reasonable size in a near future is forcing the cryptographic community to devise new methods to provide security that resist quantum attacks. Most of the zero-knowledge protocols used nowadays are based on computational problems like the discrete logarithm problem that can no longer be considered hard, since there are known efficient ways to solve them with quantum algorithms. Cryptographic research about the quantum security of zero knowledge proofs started nearly 20 years ago in a very theoretical approach, but not many papers on that topic appeared since then. The goal of this thesis is writing a survey including the main concepts about quantum secure zero-knowledge protocols, the state-of-the-art both from the theoretical and practical approaches, and an exploration of their potential application areas. The survey will be a good starting document for further students willing to do research in this topic.
Abstract Verifiable encryption enables the decryption to be taken on properly generated ciphertexts, by making the encryptor provide a zero-knowledge proof. To meet the quantum-safe application requirements, such as key escrow, Lyubashevsky et al. proposed a one-shot verifiable encryption (LN17 scheme) based on the hardness of lattice problems. In their scheme, the FSwA-type zero-knowledge proof was obtained using rejection sampling on a discrete Gaussian distribution. In this paper, we present a construction of verifiable encryption that utilizes rejection sampling on bimodal Gaussian to get the associated zero-knowledge proof. Our new construction, while exhibiting a weaker soundness property than LN17 scheme, benefits from a smaller proof size, leading to a reduced size of the verifiable ciphertext. As for the weaker soundness property, it supports some applications such as key escrow where honestly generated verifiable ciphertexts are more useful to be decrypted out in the hope of doing some further computation tasks. We provide the efficiency comparison of the new construction by instantiating it with several sets of concrete parameters.
Lattice-based cryptography is one of the most active research topics in cryptography in recent years. In many cryptographic constructions based on lattice assumptions, the building block is a proof of knowledge of a solution to an instance of the Inhomogeneous Small Integer Solution (ISIS) problem. However, all known such proof \n \nsystems have relatively weak security guarantees: Breaking each of these protocols is potentially easier than solving the underlying instance of the ISIS problem. As a consequence, cryptographic constructions relying on these proof systems typically inherit the sub-optimal security guarantees. Therefore, proof systems with stronger security guarantees are highly desirable. Such protocols are not only interesting from a theoretical point of view, but they also lead to lattice-based cryptographic constructions relying on weaker hardness assumptions than the contemporary schemes. \n \n \n \nIn this thesis, we construct a series of zero-knowledge proofs of knowledge with strong security guarantees and reasonable communication costs, that can find various applications in lattice-based cryptography. Our constructions rely on a simple, yet versatile and effective technique, called Decomposition-Extension. When adapting this technique to the Stern-KTX proof system (Stern ‘96 - Kawachi, Tanaka, Xagawa ‘08), we obtain a zero-knowledge proof of knowledge for the ISIS problem (in the infinity norm) with a very strong security guarantee: Breaking the protocol is as least as hard as solving the underlying ISIS instance. We then develop our technique to design the following lattice-based cryptographic constructions: \n \n \n \n- Efficient zero-knowledge proofs of plaintext knowledge with strong security guarantees \n \nfor 4 encryption schemes based on the Learning with Errors problem: Regev’s scheme (Regev ‘05); the Dual-Regev scheme (Gentry, Peikert, Vaikuntanathan ‘08); the PVW scheme (Peikert, Vaikuntanathan, Waters ‘08); and the GHV scheme (Gentry, Halevi, Vaikuntanathan ‘10). Our results immediately yield 4 lattice-based interactive encryption protocols that are secure under chosen ciphertext attacks. Previously, only zero-knowledge proofs of plaintext knowledge for Regev’s scheme were known, and they are relatively inefficient with rather weak security guarantees. \n \n \n \n- A lattice-based identity-based identification scheme relying on a weaker hardness assumption than in the previous works. Furthermore, we introduce an identity-based ring identification scheme based on the worst-case hardness of lattice problems. To the best of our knowledge, this is the first such scheme. \n \n \n \n- An improved lattice-based group signature scheme relying on relatively weak hardness assumptions, in which the signature size is logarithmic in the number of group users. Earlier lattice-based group signature schemes, which were published before 2013, rely on relatively weak hardness assumptions but have the undesirable property that the size of the signature is linear in the number of group users. A recent scheme (Laguillaumie, Langlois, Libert, Stehlé ‘13) achieves logarithmic signature size but it has to rely on relatively strong hardness assumptions. Our construction, thus, simultaneously achieves the good features of the existing schemes.