Improved Lattice-Based Mix-Nets for Electronic Voting

Mix-networks were first proposed by Chaum in the late 1970s -- early 1980s as a general tool for building anonymous communication systems. Classical mix-net implementations rely on standard public key primitives (e.g. ElGamal encryption) that will become vulnerable when a sufficiently powerful quantum computer will be built. Thus, there is a need to develop quantum-resistant mix-nets. This paper focuses on the application case of electronic voting where the number of votes to be mixed may reach hundreds of thousands or even millions. We propose an improved architecture for lattice-based post-quantum mix-nets featuring more efficient zero-knowledge proofs while maintaining established security assumptions. Our current implementation scales up to 100000 votes, still leaving a lot of room for future optimisation

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.

Lattice-based zero-knowledge proofs of knowledge

(English) The main goal of this dissertation is to develop new lattice-based cryptographic schemes. Most of the cryptographic protocols that each and every one of us use on a daily basis are only secure under the assumption that two mathematical problems, namely the discrete logarithm on elliptic curves and the factorization of products of two primes, are computationally hard. That is believed to be true for classical computers, but quantum computers would be able to solve these problems much more efficiently, demolishing the foundations of plenty of cryptographic constructions. This reveals the importance of post-quantum alternatives, cryptographic schemes whose security relies on different problems intractable for both classical and quantum computers. The most promising family of problems widely believed to be hard for quantum computers are lattice-based problems. We increase the supply of lattice-based tools providing new Zero-Knowledge Proofs of Knowledge for the Ring Learning With Errors (RLWE) problem, perhaps the most popular lattice-based problem. Zero-knowledge proofs are protocols between a prover and a verifier where the prover convinces the verifier of the validity of certain statements without revealing any additional relevant information. Our proofs extend the literature of Stern-based proofs, following the techniques presented by Jacques Stern in 1994. His original idea involved a code-based problem, but it has been reiteratedly improved and generalized to be used with lattices. We illustrate our proposal defining a variant of the commitment scheme, a cryptographic primitive that allows us to ensure some message was already determined at some point without revealing it until a future time, defined by Benhamouda et al. in ESORICS 2015, and proving in zero-knowledge the knowledge of a valid opening. Most importantly we also show how to prove that the message committed in one commitment is a linear combination, with some public coefficients, of the committed messages from two other commitments, again without revealing any further information about the messages. Finally, we also present a zero-knowledge proof analogous to the previous one but for multiplicative relations, something much more involved that allows us to prove any arithmetic circuit. We give first an interactive version of these proofs and then show how to construct a non-interactive one. We diligently prove that both the commitment and the companion Zero-Knowledge Proofs of Knowledge are secure under the assumption of the hardness of the underlying lattice problems. Furthermore, we specifically develop such proofs so that the arising conditions can be directly used to compute parameters that satisfy them. This way we provide a general method to instantiate our commitment and proofs with any desired security level. Thanks to this practical approach we have been able to implement all the proposed schemes and benchmark the prototype im-plementation with actually secure parameters, which allows us to obtain meaningful results and compare its performance with the existing alternatives. Moreover, provided that multiplication of polynomials in the quotient ring ℤₚ[]/⟨ⁿ + 1⟩, with prime and a power of two, is the most basic operation when working with ideal lattices we comprehensively study what are the necessary and sufficient conditions needed for applying (a generalized version of) the Fast Fourier Transform (FFT) to obtain an efficient multiplication algorithm in quotient rings as ℤₘ[]/⟨ⁿ − ⟩ (where we consider any positive integer and generalize the quotient), as we think it is of independent interest. We believe such a theoretical analysis is fundamental to be able to determine when a given generalization can also be applied to design an efficient multiplication algorithm when the FFT is not defined for the ring we are considering. That is the case of the rings used for the commitment and proofs described before, where only a partial FFT is available.(Español) El objetivo principal de esta tesis es obtener nuevos esquemas criptográficos basados en retículos. La mayoría de los protocolos criptográficos que usamos a diario son únicamente seguros bajo la hipótesis de que el problema del logaritmo discreto en curvas elípticas y la factorización de productos de dos primos son computacionalmente difíciles. Se cree que esto es cierto para los ordenadores clásicos, pero los ordenadores cuánticos podrían resolver estos problemas de forma mucho más eficiente, acabando con las bases sobre las que se fundamenta una multitud de construcciones criptográficas. Esto evidencia la importancia de las alternativas poscuánticas, cuya seguridad se basa en problemas diferentes que sean inasumibles tanto para los ordenadores clásicos como los cuánticos. Los problemas de retículos son los candidatos más prometedores, puesto que se considera que son problemas difíciles para los ordenadores cuánticos. Presentamos nuevas herramientas basadas en retículos con unas Pruebas de Conocimiento Nulo para el problema Ring Learning With Errors (RLWE), seguramente el problema de retículos más popular. Las pruebas de Conocimiento Nulo son protocolos entre un probador y un verificador en los que el primero convence al segundo de la validez de una proposición, sin revelar ninguna información adicional relevante. Nuestras pruebas se basan en el protocolo de Stern, siguiendo sus técnicas presentadas en 1994. Su idea original involucraba un problema de códigos, pero se ha mejorado y generalizado reiteradamente para poder aplicarse a retículos. Ilustramos nuestra propuesta definiendo una variante del esquema de compromiso, una primitiva criptográfica que nos permite asegurar que un mensaje fue determinado en cierto momento sin revelarlo hasta pasado un tiempo, definido por Benhamouda et al. en ESORICS 2015, y probando que conocemos una apertura válida. Además mostramos cómo probar que el mensaje comprometido es una combinación lineal, con coeficientes públicos, de los mensajes comprometidos en otros dos compromisos. Finalmente también presentamos una prueba de Conocimiento Nulo análoga a la anterior pero para relaciones multiplicativas, algo mucho más laborioso que nos permite realizar circuitos aritméticos. Todo esto sin revelar ninguna información adicional sobre los mensajes. Mostramos tanto una versión interactiva como una no interactiva. Probamos que tanto el compromiso como las pruebas de Conocimiento Nulo que le acompañan son seguras bajo la hipótesis de que el problema de retículos subyacente sea difícil. Además planteamos estas pruebas específicamente con el objetivo de que las condiciones que surjan puedan ser utilizadas directamente para calcular los parámetros que las satisfagan. De esta forma proporcionamos un método genérico para instanciar nuestro compromiso y pruebas con cualquier nivel de seguridad. Gracias a este enfoque práctico hemos podido implementar todos los esquemas propuestos y evaluar el rendimiento con parámetros seguros, lo que nos permite obtener resultados relevantes que poder comparar con las alternativas existentes. Por otra parte, dado que la multiplicación de polinomios en el anillo cociente ℤₚ[]/⟨ⁿ + 1⟩, con primo y una potencia de 2, es la operación más utilizada al trabajar con retículos ideales, estudiamos de forma exhaustiva cuáles son las condiciones suficientes y necesarias para aplicar (una versión generalizada de) la Transformada Rápida de Fourier (FFT, por sus siglas en inglés) para obtener algoritmos de multiplicación eficientes en anillos cociente ℤₘ[]/⟨ⁿ − ⟩, (considerando cualquier positiva y generalizando el cociente), de interés por sí mismo. Creemos que este análisis teórico es fundamental para determinar cuándo puede diseñarse un algoritmo eficiente de multiplicación si la FFT no está definida para el anillo considerado. Es el caso de los anillos que utilizamos en el compromiso y las pruebas descritas anteriormente, donde solo es posible calcular una FFT parcial.DOCTORAT EN MATEMÀTICA APLICADA (Pla 2012

Aggregating Falcon Signatures with LaBRADOR

Several prior works have suggested to use non-interactive arguments of knowledge with short proofs to aggregate signatures of Falcon, which is part of the first post-quantum signatures selected for standardization by NIST. Especially LaBRADOR, based on standard structured lattice assumptions and published at CRYPTO’23, seems promising to realize this task. However, no prior work has tackled this idea in a rigorous way. In this paper, we thoroughly prove how to aggregate Falcon signatures using LaBRADOR. First, we improve LaBRADOR by moving from a low-splitting to a high-splitting ring, allowing for faster computations. This modification leads to some additional technical challenges for proving the knowledge soundness of LaBRADOR. Moreover, we provide the first complete knowledge soundness analysis for the non-interactive version of LaBRADOR. Here, the multi-round and recursive nature of LaBRADOR requires a complex and thorough analysis. For this purpose, we introduce the notion of predicate special soundness (PSS). This is a general framework for evaluating the knowledge error of complex Fiat-Shamir arguments of knowledge protocols in a modular fashion, which we believe to be of independent interest. Lastly, we explain the exact steps to take in order to adapt the LaBRADOR proof system for aggregating Falcon signatures and provide concrete estimates for proof sizes. Additionally, we formalize the folklore approach of obtaining aggregate signatures from the class of hash-then-sign signatures through arguments of knowledge

Round-optimal Verifiable Oblivious Pseudorandom Functions from Ideal Lattices

timestamp: Fri, 07 May 2021 15:40:46 +0200 biburl: https://dblp.org/rec/conf/pkc/AlbrechtDDS21.bib bibsource: dblp computer science bibliography, https://dblp.orgstatus: publishe

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

Ring/Module Learning with Errors under Linear Leakage -- Hardness and Applications

This paper studies the hardness of decision Module Learning with Errors (\MLWE) under linear leakage, which has been used as a foundation to derive more efficient lattice-based zero-knowledge proofs in a recent paradigm of Lyubashevsky, Nguyen, and Seiler (PKC 21). Unlike in the plain \LWE~setting, it was unknown whether this problem remains provably hard in the module/ring setting. This work shows a reduction from the search \MLWE~to decision \MLWE~with linear leakage. Thus, the main problem remains hard asymptotically as long as the non-leakage version of \MLWE~is hard. Additionally, we also refine the paradigm of Lyubashevsky, Nguyen, and Seiler (PKC 21) by showing a more fine-grained tradeoff between efficiency and leakage. This can lead to further optimizations of lattice proofs under the paradigm

On the Non-Existence of Short Vectors in Random Module Lattices

Recently, Lyubashevsky & Seiler (Eurocrypt 2018) showed that small polynomials in the cyclotomic ring $Z_q[X]/(X^n+1)$, where $n$ is a power of two, are invertible under special congruence conditions on prime modulus $q$. This result has been used to prove certain security properties of lattice-based constructions against unbounded adversaries. Unfortunately, due to the special conditions, working over the corresponding cyclotomic ring does not allow for efficient use of the Number Theoretic Transform (NTT) algorithm for fast multiplication of polynomials and hence, the schemes become less practical. In this paper, we present how to overcome this limitation by analysing zeroes in the Chinese Remainder (or NTT) representation of small polynomials. Concretely, we follow the proof techniques from Stehlé and Steinfeld (Eprint 2013/004) and provide upper bounds on the probabilities related to the (non)-existence of a short vector in a random module lattice with no assumptions on the prime modulus. Then, we apply these results, along with the generic framework by Kiltz et al. (Eurocrypt 2018), to a number of lattice-based Fiat-Shamir signatures so they can both enjoy tight security in the quantum random oracle model and support fast multiplication algorithms (at the cost of slightly larger public keys and signatures), such as the Bai-Galbraith signature scheme (CT-RSA 2014), Dilithium-QROM (Kiltz et al., Eurocrypt 2018) and qTESLA (Alkim et al., PQCrypto 2017). These techniques can also be applied to prove that recent commitment schemes by Baum et al. (SCN 2018) are statistically binding with no additional assumptions on $q$

Lattice-based Zero-Knowledge Proofs: New Techniques for Shorter and Faster Constructions and Applications

We devise new techniques for design and analysis of efficient lattice-based zero-knowledge proofs (ZKP). First, we introduce one-shot proof techniques for non-linear polynomial relations of degree $k\ge 2$, where the protocol achieves a negligible soundness error in a single execution, and thus performs significantly better in both computation and communication compared to prior protocols requiring multiple repetitions. Such proofs with degree $k\ge 2$ have been crucial ingredients for important privacy-preserving protocols in the discrete logarithm setting, such as Bulletproofs (IEEE S&P \u2718) and arithmetic circuit arguments (EUROCRYPT \u2716). In contrast, one-shot proofs in lattice-based cryptography have previously only been shown for the linear case ($k=1$) and a very specific quadratic case ($k=2$), which are obtained as a special case of our technique. Moreover, we introduce two speedup techniques for lattice-based ZKPs: a CRT-packing technique supporting ``inter-slot\u27\u27 operations, and ``NTT-friendly\u27\u27 tools that permit the use of fully-splitting rings. The former technique comes at almost no cost to the proof length, and the latter one barely increases it, which can be compensated for by tweaking the rejection sampling parameters while still having faster computation overall. To illustrate the utility of our techniques, we show how to use them to build efficient relaxed proofs for important relations, namely proof of commitment to bits, one-out-of-many proof, range proof and set membership proof. Despite their relaxed nature, we further show how our proof systems can be used as building blocks for advanced cryptographic tools such as ring signatures. Our ring signature achieves a dramatic improvement in length over all the existing proposals from lattices at the same security level. The computational evaluation also shows that our construction is highly likely to outperform all the relevant works in running times. Being efficient in both aspects, our ring signature is particularly suitable for both small-scale and large-scale applications such as cryptocurrencies and e-voting systems. No trusted setup is required for any of our proposals