Practical product proofs for lattice commitments

We construct a practical lattice-based zero-knowledge argument for proving multiplicative relations between committed values. The underlying commitment scheme that we use is the currently most efficient one of Baum et al. (SCN 2018), and the size of our multiplicative proof (9 KB) is only slightly larger than the 7 KB required for just proving knowledge of the committed values. We additionally expand on the work of Lyubashevsky and Seiler (Eurocrypt 2018) by showing that the above-mentioned result can also apply when working over rings Zq[X]/(Xd+1) where Xd+1 splits into low-degree factors, which is a desirable property for many applications (e.g. range proofs, multiplications over

Practical Exact Proofs from Lattices: New Techniques to Exploit Fully-Splitting Rings

We propose a very fast lattice-based zero-knowledge proof system for exactly proving knowledge of a ternary solution $\vec{s} \in \{-1,0,1\}^n$ to a linear equation $A\vec{s}=\vec{u}$ over $\mathbb{Z}_q$, which improves upon the protocol by Bootle, Lyubashevsky and Seiler (CRYPTO 2019) by producing proofs that are shorter by a factor of $8$. At the core lies a technique that utilizes the module-homomorphic BDLOP commitment scheme (SCN 2018) over the fully splitting cyclotomic ring $\mathbb{Z}_q[X]/(X^d + 1)$ to prove scalar products with the NTT vector of a secret polynomial

Adding Linkability to Ring Signatures with One-Time Signatures

We propose a generic construction that adds linkability to any ring signature scheme with one-time signature scheme. Our construction has both theoretical and practical interest. In theory, the construction gives a formal and cleaner description for constructing linkable ring signature from ring signature directly. In practice, the transformation incurs a tiny overhead in size and running time. By instantiating our construction using the ring signature scheme (ACNS 2019) and the one-time signature scheme (TCHES 2018), we obtain a lattice-based linkable ring signature scheme whose signature size is logarithmic in the number of ring members. This scheme is practical, especially the signature size is very short: for $2^{30}$ ring members and 100 bit security, our signature size is only 4 MB. In addition, when proving the linkability we develop a new proof technique in the random oracle model, which might be of independent interes

A non-PCP Approach to Succinct Quantum-Safe Zero-Knowledge

Today\u27s most compact zero-knowledge arguments are based on the hardness of the discrete logarithm problem and related classical assumptions. If one is interested in quantum-safe solutions, then all of the known techniques stem from the PCP-based framework of Kilian (STOC 92) which can be instantiated based on the hardness of any collision-resistant hash function. Both approaches produce asymptotically logarithmic sized arguments but, by exploiting extra algebraic structure, the discrete logarithm arguments are a few orders of magnitude more compact in practice than the generic constructions. In this work, we present the first (poly)-logarithmic, potentially post-quantum zero-knowledge arguments that deviate from the PCP approach. At the core of succinct zero-knowledge proofs are succinct commitment schemes (in which the commitment and the opening proof are sub-linear in the message size), and we propose two such constructions based on the hardness of the (Ring)-Short Integer Solution (Ring-SIS) problem, each having certain trade-offs. For commitments to $N$ secret values, the communication complexity of our first scheme is $\tilde{O}(N^{1/c})$ for any positive integer $c$, and $O(\log^2 N)$ for the second. Both of these are a significant theoretical improvement over the previously best lattice construction by Bootle et al. (CRYPTO 2018) which gave $O(\sqrt{N})$-sized proofs

New Code-Based Privacy-Preserving Cryptographic Constructions

Code-based cryptography has a long history but did suffer from periods of slow development. The field has recently attracted a lot of attention as one of the major branches of post-quantum cryptography. However, its subfield of privacy-preserving cryptographic constructions is still rather underdeveloped, e.g., important building blocks such as zero-knowledge range proofs and set membership proofs, and even proofs of knowledge of a hash preimage, have not been known under code-based assumptions. Moreover, almost no substantial technical development has been introduced in the last several years. This work introduces several new code-based privacy-preserving cryptographic constructions that considerably advance the state-of-the-art in code-based cryptography. Specifically, we present $3$ major contributions, each of which potentially yields various other applications. Our first contribution is a code-based statistically hiding and computationally binding commitment scheme with companion zero-knowledge (ZK) argument of knowledge of a valid opening that can be easily extended to prove that the committed bits satisfy other relations. Our second contribution is the first code-based zero-knowledge range argument for committed values, with communication cost logarithmic in the size of the range. A special feature of our range argument is that, while previous works on range proofs/arguments (in all branches of cryptography) only address ranges of non-negative integers, our protocol can handle signed fractional numbers, and hence, can potentially find a larger scope of applications. Our third contribution is the first code-based Merkle-tree accumulator supported by ZK argument of membership, which has been known to enable various interesting applications. In particular, it allows us to obtain the first code-based ring signatures and group signatures with logarithmic signature sizes

Post-quantum adaptor signatures and payment channel networks

Adaptor signatures, also known as scriptless scripts, have recently become an important tool in addressing the scalability and interoperability issues of blockchain applications such as cryptocurrencies. An adaptor signature extends a digital signature in a way that a complete signature reveals a secret based on a cryptographic condition. It brings about various advantages such as (i) low on-chain cost, (ii) improved fungibility of transactions, and (iii) advanced functionality beyond the limitation of the blockchain’s scripting language. In this work, we introduce the first post-quantum adaptor signature, named $${\mathsf {LAS}}$$. Our construction relies on the standard lattice assumptions, namely Module-SIS and Module-LWE. There are certain challenges specific to the lattice setting, arising mainly from the so-called knowledge gap in lattice-based proof systems, that makes the realization of an adaptor signature and its applications difficult. We show how to overcome these technical difficulties without introducing additional on-chain costs. Our evaluation demonstrates that $${\mathsf {LAS}}$$ is essentially as efficient as an ordinary lattice-based signature in terms of both communication and computation. We further show how to achieve post-quantum atomic swaps and payment channel networks using $${\mathsf {LAS}}$$.Accepted author manuscriptCyber Securit

A lattice-based key-insulated and privacy-preserving signature scheme with publicly derived public key

© Springer Nature Switzerland AG 2020. As a widely used privacy-preserving technique for cryptocurrencies, Stealth Address constitutes a key component of Ring Confidential Transaction (RingCT) protocol and it was adopted by Monero, one of the most popular privacy-centric cryptocurrencies. Recently, Liu et al. [EuroS&P 2019] pointed out a flaw in the current widely used stealth address algorithm that once a derived secret key is compromised, the damage will spread to the corresponding master secret key, and all the derived secret keys thereof. To address this issue, Liu et al. introduced Key-Insulated and Privacy-Preserving Signature Scheme with Publicly Derived Public Key (PDPKS scheme), which captures the functionality, security, and privacy requirements of stealth address in cryptocurrencies. They further proposed a paring-based PDPKS construction and thus provided a provably secure stealth address algorithm. However, while other privacy-preserving cryptographic tools for RingCT, such as ring signature, commitment, and range proof, have successfully found counterparts on lattices, the development of lattice-based stealth address scheme lags behind and hinders the development of quantum-resistant privacy-centric cryptocurrencies following the RingCT approach. In this paper, we propose the first lattice-based PDPKS scheme and prove its security in the random oracle model. The scheme provides (potentially) quantum security not only for the stealth address algorithm but also for the deterministic wallet. Prior to this, the existing deterministic wallet algorithms, which have been widely adopted by most Bitcoin-like cryptocurrencies due to its easy backup/recovery and trustless audits, are not quantum resistant

Calamari and falafl: Logarithmic (linkable) ring signatures from isogenies and lattices

We construct efficient ring signatures (RS) from isogeny and lattice assumptions. Our ring signatures are based on a logarithmic OR proof for group actions. We instantiate this group action by either the CSIDH group action or an MLWE-based group action to obtain our isogeny-based or lattice-based RS scheme, respectively. Even though the OR proof has a binary challenge space and therefore requires a number of repetitions which is linear in the security parameter, the sizes of our ring signatures are small and scale better with the ring size N than previously known post-quantum ring signatures. We also construct linkable ring signatures (LRS) that are almost as efficient as the non-linkable variants. The isogeny-based scheme produces signatures whose size is an order of magnitude smaller than all previously known logarithmic post-quantum ring signatures, but it is relatively slow (e.g. 5.5 KB signatures and 79 s signing time for rings with 8 members). In comparison, the latticebased construction is much faster, but has larger signatures (e.g. 30 KB signatures and 90 ms signing time for the same ring size). For small ring sizes our lattice-based ring signatures are slightly larger than state-of-the-art schemes, but they are smaller for ring sizes larger than N ≈ 1024

A lattice-based linkable ring signature supporting stealth addresses

First proposed in CryptoNote, a collection of popular privacy-centric cryptocurrencies have employed Linkable Ring Signature and a corresponding Key Derivation Mechanism (KeyDerM) for keeping the payer and payee of a transaction anonymous and unlinkable. The KeyDerM is used for generating a fresh signing key and the corresponding public key, referred to as a stealth address, for the transaction payee. The stealth address will then be used in the linkable ring signature next time when the payee spends the coin. However, in all existing works, including Monero, the privacy model only considers the two cryptographic primitives separately. In addition, to be applied to cryptocurrencies, the security and privacy models for Linkable Ring Signature should capture the situation that the public key ring of a signature may contain keys created by an adversary (referred to as adversarially-chosen-key attack), since in cryptocurrencies, it is normal for a user (adversary) to create self-paying transactions so that some maliciously created public keys can get into the system without being detected . In this paper, we propose a new cryptographic primitive, referred to as Linkable Ring Signature Scheme with Stealth Addresses (SALRS), which comprehensively and strictly captures the security and privacy requirements of hiding the payer and payee of a transaction in cryptocurrencies, especially the adversarially-chosen-key attacks. We also propose a lattice-based SALRS construction and prove its security and privacy in the random oracle model. In other words, our construction provides strong confidence on security and privacy in twofolds, i.e., being proved under strong models which capture the practical scenarios of cryptocurrencies, and being potentially quantum-resistant. The efficiency analysis also shows that our lattice-based SALRS scheme is practical for real implementations.NRF (Natl Research Foundation, S’pore)MOE (Min. of Education, S’pore)Accepted versio

A Lattice-Based Linkable Ring Signature Supporting Stealth Addresses

First proposed in CryptoNote, a collection of popular privacy-centric cryptocurrencies have employed Linkable Ring Signature and a corresponding Key Derivation Mechanism (KeyDerM) for keeping the payer and payee of a transaction anonymous and unlinkable. The KeyDerM is used for generating a fresh signing key and the corresponding public key, referred to as a stealth address, for the transaction payee. The stealth address will then be used in the linkable ring signature next time when the payee spends the coin. However, in all existing works, including Monero, the privacy model only considers the two cryptographic primitives separately. In addition, to be applied to cryptocurrencies, the security and privacy models for Linkable Ring Signature should capture the situation that the public key ring of a signature may contain keys created by an adversary (referred to as adversarially-chosen-key attack), since in cryptocurrencies, it is normal for a user (adversary) to create self-paying transactions so that some maliciously created public keys can get into the system without being detected. In this paper, we propose a new cryptographic primitive, referred to as Linkable Ring Signature Scheme with Stealth Addresses (SALRS), which comprehensively and strictly captures the security and privacy requirements of hiding the payer and payee of a transaction in cryptocurrencies, especially the adversarially-chosen-key attacks. We also propose a lattice-based SALRS construction and prove its security and privacy in the random oracle model. In other words, our construction provides strong confidence on security and privacy in twofolds, i.e., being proved under strong models which capture the practical scenarios of cryptocurrencies, and being potentially quantum-resistant. The efficiency analysis also shows that our lattice-based SALRS scheme is practical for real implementations