Lattice-Based proof of a shuffle

In this paper we present the first fully post-quantum proof of a shuffle for RLWE encryption schemes. Shuffles are commonly used to construct mixing networks (mix-nets), a key element to ensure anonymity in many applications such as electronic voting systems. They should preserve anonymity even against an attack using quantum computers in order to guarantee long-term privacy. The proof presented in this paper is built over RLWE commitments which are perfectly binding and computationally hiding under the RLWE assumption, thus achieving security in a post-quantum scenario. Furthermore we provide a new definition for a secure mixing node (mix-node) and prove that our construction satisfies this definition.Peer ReviewedPostprint (author's final draft

Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory

The present survey reports on the state of the art of the different cryptographic functionalities built upon the ring learning with errors problem and its interplay with several classical problems in algebraic number theory. The survey is based to a certain extent on an invited course given by the author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other authors/ comment of the author: quotation has been added to Theorem 5.

LNCS

We extend a commitment scheme based on the learning with errors over rings (RLWE) problem, and present efficient companion zeroknowledge proofs of knowledge. Our scheme maps elements from the ring (or equivalently, n elements fro

How to prove knowledge of small secrets

We propose a new zero-knowledge protocol applicable to additively homomorphic functions that map integer vectors to an Abelian group. The protocol demonstrates knowledge of a short preimage and achieves amortised efficiency comparable to the approach of Cramer and Damgård from Crypto 2010, but gives a much tighter bound on what we can extract from a dishonest prover. Towards achieving this result, we develop an analysis for bins-and-balls games that might be of independent interest. We also provide a general analysis of rewinding of a cut-and-choose protocol as well as a method to use Lyubachevsky\u27s rejection sampling technique efficiently in an interactive protocol when many proofs are given simultaneously. Our new protocol yields improved proofs of plaintext knowledge for (Ring-)LWE-based cryptosystems, where such general techniques were not known before. Moreover, they can be extended to prove preimages of homomorphic hash functions as well

Zero-Knowledge Arguments for Matrix-Vector Relations and Lattice-Based Group Encryption

International audienceGroup encryption (GE) is the natural encryption analogue of group signatures in that it allows verifiably encrypting messages for some anonymous member of a group while providing evidence that the receiver is a properly certified group member. Should the need arise, an opening authority is capable of identifying the receiver of any ciphertext. As introduced by Kiayias, Tsiounis and Yung (Asiacrypt'07), GE is motivated by applications in the context of oblivious retriever storage systems, anonymous third parties and hierarchical group signatures. This paper provides the first realization of group encryption under lattice assumptions. Our construction is proved secure in the standard model (assuming interaction in the proving phase) under the Learning-With-Errors (LWE) and Short-Integer-Solution (SIS) assumptions. As a crucial component of our system, we describe a new zero-knowledge argument system allowing to demonstrate that a given ciphertext is a valid encryption under some hidden but certified public key, which incurs to prove quadratic statements about LWE relations. Specifically, our protocol allows arguing knowledge of witnesses consisting of X ∈ Z m×n q , s ∈ Z n q and a small-norm e ∈ Z m which underlie a public vector b = X · s + e ∈ Z m q while simultaneously proving that the matrix X ∈ Z m×n q has been correctly certified. We believe our proof system to be useful in other applications involving zero-knowledge proofs in the lattice setting

Lattice-Based Group Signatures: Achieving Full Dynamicity (and Deniability) with Ease

In this work, we provide the first lattice-based group signature that offers full dynamicity (i.e., users have the flexibility in joining and leaving the group), and thus, resolve a prominent open problem posed by previous works. Moreover, we achieve this non-trivial feat in a relatively simple manner. Starting with Libert et al.'s fully static construction (Eurocrypt 2016) - which is arguably the most efficient lattice-based group signature to date, we introduce simple-but-insightful tweaks that allow to upgrade it directly into the fully dynamic setting. More startlingly, our scheme even produces slightly shorter signatures than the former, thanks to an adaptation of a technique proposed by Ling et al. (PKC 2013), allowing to prove inequalities in zero-knowledge. Our design approach consists of upgrading Libert et al.'s static construction (EUROCRYPT 2016) - which is arguably the most efficient lattice-based group signature to date - into the fully dynamic setting. Somewhat surprisingly, our scheme produces slightly shorter signatures than the former, thanks to a new technique for proving inequality in zero-knowledge without relying on any inequality check. The scheme satisfies the strong security requirements of Bootle et al.'s model (ACNS 2016), under the Short Integer Solution (SIS) and the Learning With Errors (LWE) assumptions. Furthermore, we demonstrate how to equip the obtained group signature scheme with the deniability functionality in a simple way. This attractive functionality, put forward by Ishida et al. (CANS 2016), enables the tracing authority to provide an evidence that a given user is not the owner of a signature in question. In the process, we design a zero-knowledge protocol for proving that a given LWE ciphertext does not decrypt to a particular message

Ring-LWE Cryptography for the Number Theorist

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.Comment: 20 Page

Reinforcing Security and Usability of Crypto-Wallet with Post-Quantum Cryptography and Zero-Knowledge Proof

Crypto-wallets or digital asset wallets are a crucial aspect of managing cryptocurrencies and other digital assets such as NFTs. However, these wallets are not immune to security threats, particularly from the growing risk of quantum computing. The use of traditional public-key cryptography systems in digital asset wallets makes them vulnerable to attacks from quantum computers, which may increase in the future. Moreover, current digital wallets require users to keep track of seed-phrases, which can be challenging and lead to additional security risks. To overcome these challenges, a new algorithm is proposed that uses post-quantum cryptography (PQC) and zero-knowledge proof (ZKP) to enhance the security of digital asset wallets. The research focuses on the use of the Lattice-based Threshold Secret Sharing Scheme (LTSSS), Kyber Algorithm for key generation and ZKP for wallet unlocking, providing a more secure and user-friendly alternative to seed-phrase, brain and multi-sig protocol wallets. This algorithm also includes several innovative security features such as recovery of wallets in case of downtime of the server, and the ability to rekey the private key associated with a specific username-password combination, offering improved security and usability. The incorporation of PQC and ZKP provides a robust and comprehensive framework for securing digital assets in the present and future. This research aims to address the security challenges faced by digital asset wallets and proposes practical solutions to ensure their safety in the era of quantum computing