Breaking pairing-based cryptosystems using ηT\eta_T pairing over GF(397)GF(3^{97})

There are many useful cryptographic schemes, such as ID-based encryption, short signature, keyword searchable encryption, attribute-based encryption, functional encryption, that use a bilinear pairing. It is important to estimate the security of such pairing-based cryptosystems in cryptography. The most essential number-theoretic problem in pairing-based cryptosystems is the discrete logarithm problem (DLP) because pairing-based cryptosystems are no longer secure once the underlining DLP is broken. One efficient bilinear pairing is the $\eta_T$ pairing defined over a supersingular elliptic curve $E$ on the finite field $GF(3^n)$ for a positive integer $n$. The embedding degree of the $\eta_T$ pairing is $6$; thus, we can reduce the DLP over $E$ on $GF(3^n)$ to that over the finite field $GF(3^{6n})$. In this paper, for breaking the $\eta_T$ pairing over $GF(3^n)$, we discuss solving the DLP over $GF(3^{6n})$ by using the function field sieve (FFS), which is the asymptotically fastest algorithm for solving a DLP over finite fields of small characteristics. We chose the extension degree $n=97$ because it has been intensively used in benchmarking tests for the implementation of the $\eta_T$ pairing, and the order (923-bit) of $GF(3^{6\cdot 97})$ is substantially larger than the previous world record (676-bit) of solving the DLP by using the FFS. We implemented the FFS for the medium prime case (JL06-FFS), and propose several improvements of the FFS, for example, the lattice sieve for JL06-FFS and the filtering adjusted to the Galois action. Finally, we succeeded in solving the DLP over $GF(3^{6\cdot 97})$. The entire computational time of our improved FFS requires about 148.2 days using 252 CPU cores. Our computational results contribute to the secure use of pairing-based cryptosystems with the $\eta_T$ pairing

Key Length Estimation of Pairing-based Cryptosystems using ηT\eta_T Pairing

The security of pairing-based cryptosystems depends on the difficulty of the discrete logarithm problem (DLP) over certain types of finite fields. One of the most efficient algorithms for computing a pairing is the $\eta_T$ pairing over supersingular curves on finite fields whose characteristic is $3$. Indeed many high-speed implementations of this pairing have been reported, and it is an attractive candidate for practical deployment of pairing-based cryptosystems. The embedding degree of the $\eta_T$ pairing is 6, so we deal with the difficulty of a DLP over the finite field $GF(3^{6n})$, where the function field sieve (FFS) is known as the asymptotically fastest algorithm of solving it. Moreover, several efficient algorithms are employed for implementation of the FFS, such as the large prime variation. In this paper, we estimate the time complexity of solving the DLP for the extension degrees $n=97,163, 193,239,313,353,509$, when we use the improved FFS. To accomplish our aim, we present several new computable estimation formulas to compute the explicit number of special polynomials used in the improved FFS. Our estimation contributes to the evaluation for the key length of pairing-based cryptosystems using the $\eta_T$ pairing

Cryptographic key distribution in wireless sensor networks: a hardware perspective

In this work the suitability of different methods of symmetric key distribution for application in wireless sensor networks are discussed. Each method is considered in terms of its security implications for the network. It is concluded that an asymmetric scheme is the optimum choice for key distribution. In particular, Identity-Based Cryptography (IBC) is proposed as the most suitable of the various asymmetric approaches. A protocol for key distribution using identity based Non-Interactive Key Distribution Scheme (NIKDS) and Identity-Based Signature (IBS) scheme is presented. The protocol is analysed on the ARM920T processor and measurements were taken for the run time and energy of its components parts. It was found that the Tate pairing component of the NIKDS consumes significants amounts of energy, and so it should be ported to hardware. An accelerator was implemented in 65nm Complementary Metal Oxide Silicon (CMOS) technology and area, timing and energy figures have been obtained for the design. Initial results indicate that a hardware implementation of IBC would meet the strict energy constraint of a wireless sensor network node

Efficient software implementation of elliptic curves and bilinear pairings

Orientador: Júlio César Lopez HernándezTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O advento da criptografia assimétrica ou de chave pública possibilitou a aplicação de criptografia em novos cenários, como assinaturas digitais e comércio eletrônico, tornando-a componente vital para o fornecimento de confidencialidade e autenticação em meios de comunicação. Dentre os métodos mais eficientes de criptografia assimétrica, a criptografia de curvas elípticas destaca-se pelos baixos requisitos de armazenamento para chaves e custo computacional para execução. A descoberta relativamente recente da criptografia baseada em emparelhamentos bilineares sobre curvas elípticas permitiu ainda sua flexibilização e a construção de sistemas criptográficos com propriedades inovadoras, como sistemas baseados em identidades e suas variantes. Porém, o custo computacional de criptossistemas baseados em emparelhamentos ainda permanece significativamente maior do que os assimétricos tradicionais, representando um obstáculo para sua adoção, especialmente em dispositivos com recursos limitados. As contribuições deste trabalho objetivam aprimorar o desempenho de criptossistemas baseados em curvas elípticas e emparelhamentos bilineares e consistem em: (i) implementação eficiente de corpos binários em arquiteturas embutidas de 8 bits (microcontroladores presentes em sensores sem fio); (ii) formulação eficiente de aritmética em corpos binários para conjuntos vetoriais de arquiteturas de 64 bits e famílias mais recentes de processadores desktop dotadas de suporte nativo à multiplicação em corpos binários; (iii) técnicas para implementação serial e paralela de curvas elípticas binárias e emparelhamentos bilineares simétricos e assimétricos definidos sobre corpos primos ou binários. Estas contribuições permitiram obter significativos ganhos de desempenho e, conseqüentemente, uma série de recordes de velocidade para o cálculo de diversos algoritmos criptográficos relevantes em arquiteturas modernas que vão de sistemas embarcados de 8 bits a processadores com 8 coresAbstract: The development of asymmetric or public key cryptography made possible new applications of cryptography such as digital signatures and electronic commerce. Cryptography is now a vital component for providing confidentiality and authentication in communication infra-structures. Elliptic Curve Cryptography is among the most efficient public-key methods because of its low storage and computational requirements. The relatively recent advent of Pairing-Based Cryptography allowed the further construction of flexible and innovative cryptographic solutions like Identity-Based Cryptography and variants. However, the computational cost of pairing-based cryptosystems remains significantly higher than traditional public key cryptosystems and thus an important obstacle for adoption, specially in resource-constrained devices. The main contributions of this work aim to improve the performance of curve-based cryptosystems, consisting of: (i) efficient implementation of binary fields in 8-bit microcontrollers embedded in sensor network nodes; (ii) efficient formulation of binary field arithmetic in terms of vector instructions present in 64-bit architectures, and on the recently-introduced native support for binary field multiplication in the latest Intel microarchitecture families; (iii) techniques for serial and parallel implementation of binary elliptic curves and symmetric and asymmetric pairings defined over prime and binary fields. These contributions produced important performance improvements and, consequently, several speed records for computing relevant cryptographic algorithms in modern computer architectures ranging from embedded 8-bit microcontrollers to 8-core processorsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã

The Discrete Logarithm Problem in Finite Fields of Small Characteristic

Computing discrete logarithms is a long-standing algorithmic problem, whose hardness forms the basis for numerous current public-key cryptosystems. In the case of finite fields of small characteristic, however, there has been tremendous progress recently, by which the complexity of the discrete logarithm problem (DLP) is considerably reduced. This habilitation thesis on the DLP in such fields deals with two principal aspects. On one hand, we develop and investigate novel efficient algorithms for computing discrete logarithms, where the complexity analysis relies on heuristic assumptions. In particular, we show that logarithms of factor base elements can be computed in polynomial time, and we discuss practical impacts of the new methods on the security of pairing-based cryptosystems. While a heuristic running time analysis of algorithms is common practice for concrete security estimations, this approach is insufficient from a mathematical perspective. Therefore, on the other hand, we focus on provable complexity results, for which we modify the algorithms so that any heuristics are avoided and a rigorous analysis becomes possible. We prove that for any prime field there exist infinitely many extension fields in which the DLP can be solved in quasi-polynomial time. Despite the two aspects looking rather independent from each other, it turns out, as illustrated in this thesis, that progress regarding practical algorithms and record computations can lead to advances on the theoretical running time analysis -- and the other way around.Die Berechnung von diskreten Logarithmen ist ein eingehend untersuchtes algorithmisches Problem, dessen Schwierigkeit zahlreiche Anwendungen in der heutigen Public-Key-Kryptographie besitzt. Für endliche Körper kleiner Charakteristik sind jedoch kürzlich erhebliche Fortschritte erzielt worden, welche die Komplexität des diskreten Logarithmusproblems (DLP) in diesem Szenario drastisch reduzieren. Diese Habilitationsschrift erörtert zwei grundsätzliche Aspekte beim DLP in Körpern kleiner Charakteristik. Es werden einerseits neuartige, erheblich effizientere Algorithmen zur Berechnung von diskreten Logarithmen entwickelt und untersucht, wobei die Laufzeitanalyse auf heuristischen Annahmen beruht. Unter anderem wird gezeigt, dass Logarithmen von Elementen der Faktorbasis in polynomieller Zeit berechnet werden können, und welche praktischen Auswirkungen die neuen Verfahren auf die Sicherheit paarungsbasierter Kryptosysteme haben. Während heuristische Laufzeitabschätzungen von Algorithmen für die konkrete Sicherheitsanalyse üblich sind, so erscheint diese Vorgehensweise aus mathematischer Sicht unzulänglich. Der Aspekt der beweisbaren Komplexität für DLP-Algorithmen konzentriert sich deshalb darauf, modifizierte Algorithmen zu entwickeln, die jegliche heuristische Annahme vermeiden und dessen Laufzeit rigoros gezeigt werden kann. Es wird bewiesen, dass für jeden Primkörper unendlich viele Erweiterungskörper existieren, für die das DLP in quasi-polynomieller Zeit gelöst werden kann. Obwohl die beiden Aspekte weitgehend unabhängig voneinander erscheinen mögen, so zeigt sich, wie in dieser Schrift illustriert wird, dass Fortschritte bei praktischen Algorithmen und Rekordberechnungen auch zu Fortentwicklungen bei theoretischen Laufzeitabschätzungen führen -- und umgekehrt

GPU-based Parallel Computing Models and Implementations for Two-party Privacy-preserving Protocols

In (two-party) privacy-preserving-based applications, two users use encrypted inputs to compute a function without giving out plaintext of their input values. Privacy-preserving computing algorithms have to utilize a large amount of computing resources to handle the encryption-decryption operations. In this dissertation, we study optimal utilization of computing resources on the graphic processor unit (GPU) architecture for privacy-preserving protocols based on secure function evaluation (SFE) and the Elliptic Curve Cryptographic (ECC) and related algorithms. A number of privacy-preserving protocols are implemented, including private set intersection (PSI), secret handshaking (SH), secure Edit distance (ED) and Smith-Waterman (SW) problems. PSI is chosen to represent ECC point multiplication related computations, SH for bilinear pairing, and the last two for SFE-based dynamic programming (DP) problems. They represent different types of computations, so that in-depth understanding of the benefits and limitations of the GPU architecture for privacy preserving protocols is gained. For SFE-based ED and SW problems, a wavefront parallel computing model on the CPU-GPU architecture under the semi-honest security model is proposed. Low level parallelization techniques for GPU-based gate (de-)garbler, synchronized parallel memory access, pipelining, and general GPU resource mapping policies are developed. This dissertation shows that the GPU architecture can be fully utilized to speed up SFE-based ED and SW algorithms, which are constructed with billions of garbled gates, on a contemporary GPU card GTX-680, with very little waste of processing cycles or memory space. For PSI and SH protocols and underlying ECC algorithms, the analysis in this research shows that the conventional Montgomery-based number system is more friendly to the GPU architecture than the Residue Number System (RNS) is. Analysis on experiment results further shows that the lazy reduction in higher extension fields can have performance benefits only when the GPU architecture has enough fast memory. The resulting Elliptic curve Arithmetic GPU Library (EAGL) can run 3350.9 R-ate (bilinear) pairing/sec, and 47000 point multiplication/sec at the 128-bit security level, on one GTX-680 card. The primary performance bottleneck is found to be lacking of advanced memory management functions in the contemporary GPU architecture for bilinear pairing operations. Substantial performance gain can be expected when the on-chip memory size and/or more advanced memory prefetching mechanisms are supported in future generations of GPUs

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum