Zero-Knowledge Proofs (ZKPs) are an emergent paradigm in verifiable computing. In the context of applications like cloud computing, ZKPs can be used by a client (called the verifier) to verify the service provider (called the prover) is in fact performing the correct computation based on a public input. A recently prominent variant of ZKPs is zkSNARKs, generating succinct proofs that can be rapidly verified by the end user. However, proof generation itself is very time consuming per transaction. Two key primitives in proof generation are the Number Theoretic Transform (NTT) and Multi-scalar Multiplication (MSM). These primitives are prime candidates for hardware acceleration, and prior works have looked at GPU implementations and custom RTL. However, both algorithms involve complex dataflow patterns -- standard NTTs have irregular memory accesses for butterfly computations from stage to stage, and MSMs using Pippenger's algorithm have data-dependent memory accesses for partial sum calculations. We present SZKP, a scalable accelerator framework that is the first ASIC to accelerate an entire proof on-chip by leveraging structured dataflows for both NTTs and MSMs. SZKP achieves conservative full-proof speedups of over 400$\times$, 3$\times$, and 12$\times$ over CPU, ASIC, and GPU implementations.
Modern cryptography relies heavily on the principles of algebraic structures to ensure the security and integrity of data. This paper explores the fundamental algebraic structures that underpin contemporary cryptographic systems, including groups, rings, fields, and lattices. We provide a detailed examination of how these structures are employed in various cryptographic algorithms and protocols, such as public-key cryptography, digital signatures, and hash functions. an overview of basic algebraic concepts and their properties, followed by an in-depth analysis of their applications in cryptographic schemes. For instance, the use of elliptic curve groups in Elliptic Curve Cryptography (ECC) offers enhanced security with smaller key sizes compared to traditional systems like RSA. Similarly, lattice-based cryptography presents promising solutions for post-quantum security, leveraging the hardness of lattice problems to resist attacks by quantum computers. the role of algebraic structures in the development of advanced cryptographic techniques, such as homomorphic encryption, which allows computations on encrypted data without decryption, and zero-knowledge proofs, which enable the verification of information without revealing the information itself. Through these examples, we illustrate the critical importance of algebraic structures in achieving robust and efficient cryptographic systems.
As cryptographic technologies evolve, the need for specialized hash functions to operate efficiently over different computational environments becomes necessary. Traditional symmetric algorithms like AES and SHA-3 have been optimized for traditional hardware and software implementations, which are designed over binary fields. However, protocols like zero-knowledge proofs require hash functions that are optimised over large prime fields. This thesis addresses the growing demand for Arithmetization-Oriented (AO) cryptographic hash functions for zero-knowledge applications. The performance and efficiency of many zero-knowledge applications often depends on the efficiency of the hash function used. In response to this need, this work explores a selection of these hash functions and implements them within two zero-knowledge proving systems: Dusk Network?s Plonk and Polygon?s Plonky2, with a focus on assessing the different performance tradeoffs that these hash functions offer within these sytems.
Multi-scalar multiplication (MSM) is the most computation-intensive part in proof generation of Zero-knowledge proof (ZKP). In this paper, we propose MSMAC, an FPGA accelerator for large-scale MSM. MSMAC adopts a specially designed Instruction Set Architecture (ISA) for MSM and optimizes pipelined Point Addition Unit (PAU) with hybrid Karatsuba multiplier. Moreover, a runtime system is proposed to split MSM tasks with the optimal sub-task size and orchestrate execution of Processing Elements (PEs). Experimental results show that MSMAC achieves up to 328X and 1.96X speedups compared to the state-of-the-art implementation on CPU (one core) and GPU, respectively, outperforming the state-of-the-art ASIC accelerator by 1.79X. On 4 FPGAs, MSMAC performs 1,261X faster than a single CPU core.
Biscuit is a recent multivariate signature scheme based on the MPC-in-the-Head paradigm. It has been submitted to the NIST competition for additional signature schemes. Signatures are derived from a zero-knowledge proof of knowledge of the solution of a structured polynomial system. This extra structure enables efficient proofs and compact signatures. This short note demonstrates that it also makes these polynomial systems easier to solve than random ones. As a consequence, the original parameters of Biscuit failed to meet the required security levels and had to be upgraded.
Jonathan Ku, Junyao Zhang, Haoxuan Shan, Saichand Samudrala · 9 authors
Elliptic curve cryptography (ECC) is widely used in security applications such as public key cryptography (PKC) and zero-knowledge proofs (ZKP). ECC is composed of modular arithmetic, where modular multiplication takes most of the processing time. Computational complexity and memory constraints of ECC limit the performance. Therefore, hardware acceleration on ECC is an active field of research. Processing-in-memory (PIM) is a promising approach to tackle this problem. In this work, we design ModSRAM, the first 8T SRAM PIM architecture to compute large-number modular multiplication efficiently. In addition, we propose R4CSA-LUT, a new algorithm that reduces the cycles for an interleaved algorithm and eliminates carry propagation for addition based on look-up tables (LUT). ModSRAM is co-designed with R4CSA-LUT to support modular multiplication and data reuse in memory with 52% cycle reduction compared to prior works with only 32% area overhead.
Ruben De Smet, Robrecht Blancquaert, Tom Godden, Kris Steenhaut · 5 authors
Elliptic curve cryptography is a widely deployed technology for securing digital communication. It is the basis of many cryptographic primitives such as key agreement protocols, digital signatures, and zero-knowledge proofs. Fast elliptic curve cryptography relies on heavily optimised modular arithmetic operations, which are often tailored to specific micro-architectures. In this article, we study and evaluate optimisations of the popular elliptic curve Curve25519 for ARM processors. We specifically target the ARM NEON single instruction, multiple data (SIMD) architecture, which is a popular architecture for modern smartphones. We introduce a novel representation for 128-bit NEON SIMD vectors, optimised for SIMD parallelisation, to accelerate elliptic curve operations significantly. Leveraging this representation, we implement an extended twisted Edwards curve Curve25519 back-end within the popular Rust library "curve25519-dalek". We extensively evaluate our implementation across multiple ARM devices using both cryptographic benchmarks and the benchmark suite available for the Signal protocol. Our findings demonstrate a substantial back-end speed-up of at least 20% for ARM NEON, along with a noteworthy speed improvement of at least 15% for benchmarked Signal functions.
Nicholas Brandt, Dennis Hofheinz, Michael Klooß, Michael Reichle
We construct the first blind signature scheme that achieves all of the following properties simultaneously: – it is tightly secure under a standard (i.e., non-interactive, non-q-type) computational assumption, – it does not require pairings, – it does not rely on generic, non-black-box techniques (like generic NIZK proofs). The third property enables a reasonably efficient solution, and in fact signatures in our scheme comprise 10 group elements and 29 Zp-elements. Our scheme starts from a pairing-based non-blind signature scheme (Abe et al., JoC 2023), and uses recent techniques of Chairattana-Apirom, Tessaro, and Zhu (CRYPTO 2024) to replace the pairings used in this scheme with non-interactive zero-knowledge proofs in the random oracle model. This conversion is not generic or straightforward (also because the mentioned previous works have converted only significantly simpler signature schemes), and we are required to improve upon and innovate existing techniques in several places. As an interesting side note, and unlike previous works, our techniques only require a non-programmable random oracle, and our signature scheme achieves predicate blindness (which means that the user can prove state ments about the signed message during the signing process).
Threshold signatures, notably ECDSA, are fundamental for securing decentralized applications.Their non-linear structure poses challenges in distributed signing, often tackled by pairwise multiplicative-to-additive share conversion, leading to O(n) communication and O(n 2 ) verification costs for each of n signers.Moreover, most schemes lack robustness, necessitating a complete restart upon fault.A pioneering work by Wong et al. (NDSS '23) still requires rolling back to the preceding round to resume signing after another round to convince all other signers.We revisit secure multiparty computation from threshold linearly homomorphic encryption (LHE).Realizing its public verifiability and fault recovery, we encompass two technical contributions to Castagnos-Laguillaumie LHE (CT-RSA '15): a 2-round robust distributed key generation (DKG) protocol in the dishonest majority setting and an accompanying zero-knowledge proof allowing extraction in an unknown-order group.We extend the DKG with dual-code-based verification (ACNS '17), upgrading its O(tn 2 )-cost private verifiability to an O(n 2 ) public one.Built on our DKG, we present the first threshold ECDSA protocol with O(1) communication and O(n) verification perparty costs while matching the lowest round complexity of nonrobust schemes (CCS '20).Empirically, we halve the computation and communication costs of the signing phase compared to stateof-the-art robust threshold ECDSA (NDSS '23).We also illustrate the versatility of our techniques with an improved threshold extension (IEEE S&P '23) of BBS+ signatures (IEEE Syst.J. '13).
Lennart Braun, Guilhem Castagnos, Ivan Damgård, Fabien Laguillaumie · 7 authors
We present distributed key generation and decryption protocols for an additively homomorphic cryptosystem based on class groups, improving on a similar system proposed by Braun, Damgård, and Orlandi at CRYPTO ‘23. Our key generation is similarly constant round but achieves lower communication complexity than the previous work. This improvement is in part the result of relaxing the reconstruction property required of the underlying integer verifiable secret sharing scheme. This eliminates the reliance on potentially costly proofs of knowledge in unknown order groups. We present a new method to batch zero-knowledge proofs in unknown order groups which strengthens these improvements. We also present a protocol which is proven secure against adaptive adversaries in the single inconsistent player (SIP) model. Our protocols are secure in the universal composability (UC) framework and provide guaranteed output delivery. We demonstrate the relative efficiency of our techniques by presenting the running times and communication costs associated with our implementation of the statically secure protocol and provide a direct comparison with alternate state of the art constructions.
Zero-knowledge proof (ZKP) systems have surged attention and held a fundamental role in contemporary cryptography. Zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK) protocols dominate the ZKP usage, implemented through arithmetic circuit programming paradigm. However, underconstrained or overconstrained circuits may lead to bugs. The former refers to circuits that lack the necessary constraints, resulting in unexpected solutions and causing the verifier to accept a bogus witness, and the latter refers to circuits that are constrained excessively, resulting in lacking necessary solutions and causing the verifier to accept no witness. This article introduces a novel approach for pinpointing two distinct types of bugs in ZKP circuits. The method involves encoding the arithmetic circuit constraints to polynomial equation systems and solving them over finite fields by the computer algebra system . The classification of verification results is refined, greatly enhancing the expressive power of the system. A tool, AC 4 , is proposed to represent the implementation of the method. Experiments show that AC 4 demonstrates an increase in the solved rate, showing a 36.7% improvement over Picus and CIVER, and a slight improvement over halo2-analyzer, a checker for halo2 circuits. Within a solvable range, the checking time has also exhibited noticeable improvement, demonstrating a magnitude increase compared to previous efforts.
Background. To ensure the protection of the biometric access control system used in unsecured communication channels, it is necessary to exclude the storage and transfer, transfer of biometric data as well as sequences generated on their basis. The paper proposes a cryptographic protocol of two-factor authentication with the zero-knowledge over the extended field GF(2m) on elliptic curves using biometric data and the private key of the user. Objective. The aim of the article is to develop a cryptographic protocol for zero-knowledge two-factor authentication based on elliptic curves using biometric data and the user’s private key, which allows increasing cryptographic strength and reducing the duration of the authentication process. Methods. The process of implementing zero-knowledge proof protocols is as follows: one user (proofer) can convince another user (verifier) that he has some secret without disclosing the secret itself. Results. A cryptographic protocol for two-factor authentication with zero-knowledge over the extended field GF(2m) of elliptic curves using user biometric data is proposed, which significantly reduces the size of the protocol parameters and increases cryptographic strength (computational complexity of the breaking). There is no leakage of private key information and biometric data of the user during the execution of the zero-knowledge proof protocol. Conclusions. The implementation of a cryptographic protocol with zero-knowledge proof two-factor authentication based on elliptic curves allows significantly reducing the size of protocol parameters and increasing the cryptographic strength (computational complexity of the breaking).
Ірина Стрелковська, Олексій Онацький, Лариса Григорівна Йона
Background. To ensure the protection of the biometric access control system used in unsecured communication channels, it is necessary to exclude the storage and transfer, transfer of biometric data as well as sequences generated on their basis. The paper proposes a cryptographic protocol of two-factor authentication with the zero-knowledge over the extended field GF(2m) on elliptic curves using biometric data and the private key of the user. Objective. The aim of the article is to develop a cryptographic protocol for zero-knowledge two-factor authentication based on elliptic curves using biometric data and the user’s private key, which allows increasing cryptographic strength and reducing the duration of the authentication process. Methods. The process of implementing zero-knowledge proof protocols is as follows: one user (proofer) can convince another user (verifier) that he has some secret without disclosing the secret itself. Results. A cryptographic protocol for two-factor authentication with zero-knowledge over the extended field GF(2m) of elliptic curves using user biometric data is proposed, which significantly reduces the size of the protocol parameters and increases cryptographic strength (computational complexity of the breaking). There is no leakage of private key information and biometric data of the user during the execution of the zero-knowledge proof protocol. Conclusions. The implementation of a cryptographic protocol with zero-knowledge proof two-factor authentication based on elliptic curves allows significantly reducing the size of protocol parameters and increasing the cryptographic strength (computational complexity of the breaking).
Protecting the privacy of blockchain transactions is extremely important for users. Stealth address protocols (SAP) allow users to receive assets via stealth addresses that they do not associate with their stealth meta-addresses. SAP can be generated using different cryptographic approaches. DKSAP uses an elliptic curve multiplication and hashing of the resulting shared secret. Another approach is to use a elliptic curve pairing. This paper presents four SA protocols that use elliptic curve pairing as a cryptographic solution. ECPDKSAPs are pairing-based protocols that include viewing key and spending key, while ECPSKSAP is a pairing-based protocol that uses a single key with which spending and the viewing key are derived. We find that ECPDKSAPs give significantly better results than DKSAP with the view tag. The best results are achieved with Protocol 3 (Elliptic Curve Pairing Dual Key Stealth Address Protocol), which is Ethereum-friendly. ECPSKSAP is significantly slower, but it provides an interesting theoretical result as it uses only one private key.
Lennart Braun, Cyprien Delpech de Saint Guilhem, Robin Jadoul, Emmanuela Orsini · 6 authors
In this work, we extend the MPC-in-the-Head framework, used in recent efficient zero-knowledge protocols, to work over the ring $$\mathbb {Z}_{2^k}$$ , which is the primary operating domain for modern CPUs. The proposed schemes are compatible with any threshold linear secret sharing scheme and draw inspiration from MPC protocols adapted for ring operations. Additionally, we explore various batching methodologies, leveraging Shamir’s secret sharing schemes and Galois ring extensions, and show the applicability of our approach in RAM program verification. Finally, we analyse different options for instantiating the resulting ZK scheme over rings and compare their communication costs.
Mohammed Mujeer Ulla, Preethi Preethi, Md. Sameeruddin Khan, Deepak S. Sakkari
Very recent attacks like ladder leaks demonstrated the feasibility of recovering private keys 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. 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 an ECDSA signature to sign a message using the 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 a way in which the intruder can discover the private key if the signer leaks any one of the nonce values. 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 combination of nonce generation, post-message signing, and validation in ECDSA helps achieve Uniqueness, Authentication, Integrity, and Non-Repudiation. The analysis is performed by considering all three curves for the implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA). The comparative analysis for each of the selected curves in terms of computational time is done with the leak of nonce and with the 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 zero depicting the strength of the algorithm. The average computational costs to break ECDSA with curves SECP256K1 and NIST256p using LLL are 2.9 and 3.4 respectively
Signatures post-quantiques à partir de techniques de calcul multipartite Le développement actuel des ordinateurs quantiques pousse la communauté cryptographique à mettre au point de nouveaux cryptosystèmes dont la sécurité se fonde sur la difficulté à résoudre des problèmes cryptographiques résistant au calcul quantique. Dans le cadre de cette thèse, nous nous sommes focalisés sur la conception de schémas de signatures électroniques construits à partir de preuves à divulgation nulle de connaissance (zero-knowledge proofs of knowledge). Plus précisément, nous nous sommes intéressés au paradigme “MPC-in-the-Head” (littéralement, “calcul-multipartite-dans-la-tête”) qui fournit une méthode générique de construire de telles preuves en utilisant des techniques de calcul multipartite sécurisé. Nous proposons plusieurs nouveaux schémas de signatures utilisant le paradigme “MPC-in-the-Head”. La plupart d’entre eux sont compétitifs avec les schémas existants dans l’état de l’art post-quantique. Ils produisent des signatures ayant des tailles entre 5 et 20 kylo-octets (pour un niveau de sécurité de 128 bits) et possèdent de très petites clés (de moins de 200 octets). Les problèmes difficiles sur lesquels la sécurité de ces schémas se fonde sont très variés. Certains schémas s’appuient sur des hypothèses de sécurité issues de la théorie des codes correcteurs d’erreurs, telle que celle sur la difficulté à résoudre le problème de décodage par syndrome pour des codes linéaires aléatoires. Les autres schémas s’appuient sur la difficultés à résoudre un système d’équations quadratiques, le problème de la somme de sous-ensembles ou le problème MinRank. Nous avons également mis au point deux nouvelles techniques de MPC-in-the-Head. La première vise à gérer efficacement les situations où le secret est de petite taille avec un grand modulus. La seconde consiste en une nouvelle méthode pour transformer un protocole de calcul multipartite en preuve de divulgation nulle de connaissance. Cette nouvelle transformation offre des nouveaux compromis entre coût de communication et temps de calcul. En particulier, elle permet de produire des algorithmes de vérification très rapides. Plusieurs soumissions à l’appel du NIST pour des schémas de signatures post-quantiques supplémentaires s'appuient (parfois partiellement) sur des idées développées dans le cadre de cette thèse.
(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[𝑥]/⟨𝑥ⁿ + 1⟩, with 𝑝 prime and 𝑛 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[𝑥]/⟨𝑥ⁿ − 𝑎⟩ (where we consider any positive integer 𝑚 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[𝑥]/⟨𝑥ⁿ + 1⟩, con 𝑝 primo y 𝑛 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[𝑥]/⟨𝑥ⁿ − 𝑎⟩, (considerando cualquier 𝑚 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.