A New Method for Geometric Interpretation of Elliptic Curve Discrete Logarithm Problem

In this paper, we intend to study the geometric meaning of the discrete logarithm problem defined over an Elliptic Curve. The key idea is to reduce the Elliptic Curve Discrete Logarithm Problem (EC-DLP) into a system of equations. These equations arise from the interesection of quadric hypersurfaces in an affine space of lower dimension. In cryptography, this interpretation can be used to design attacks on EC-DLP. Presently, the best known attack algorithm having a sub-exponential time complexity is through the implementation of Summation Polynomials and Weil Descent. It is expected that the proposed geometric interpretation can result in faster reduction of the problem into a system of equations. These overdetermined system of equations are hard to solve. We have used F4 (Faugere) algorithms and got results for primes less than 500,000. Quantum Algorithms can expedite the process of solving these over-determined system of equations. In the absence of fast algorithms for computing summation polynomials, we expect that this could be an alternative. We do not claim that the proposed algorithm would be faster than Shor's algorithm for breaking EC-DLP but this interpretation could be a candidate as an alternative to the 'summation polynomial attack' in the post-quantum era

Harder, Better, Faster, Stronger - Elliptic Curve Discrete Logarithm Computations on FPGAs

Computing discrete logarithms takes time. It takes time to develop new algorithms, choose the best algorithms, implement these algorithms correctly and efficiently, keep the system running for several months, and, finally, publish the results. In this paper, we present a highly performant architecture that can be used to compute discrete logarithms of Weierstrass curves defined over binary fields and Koblitz curves using FPGAs. We used the architecture to compute for the first time a discrete logarithm of the elliptic curve \texttt{sect113r1}, a previously standardized binary curve, using 10 Kintex-7 FPGAs. To achieve this result, we investigated different iteration functions, used a negation map, dealt with the fruitless cycle problem, built an efficient FPGA design that processes 900 million iterations per second, and we tended for several months the optimized implementations running on the FPGAs

Atac quàntic a la criptografia amb corbes el·líptiques

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Elois Sans Gispert[en] Generally, the majority of classical cryptography methods that use elliptical curves are based on the difficulty of classical computers to solve the discrete logarithm problem in polynomial time. Here we will study Peter Shor’s quantum attack derived from the algorithm for the discrete logarithm problem, published in 1994 [14]. We start with an introduction to elliptical curves and their use in criptography. Then we understand the mathematical foundations of the corresponding quantum computation to understand Shor’s algorithm, from which we will study how to reach it and why it works

Elliptic Curve Arithmetic for Cryptography

The advantages of using public key cryptography over secret key cryptography include the convenience of better key management and increased security. However, due to the complexity of the underlying number theoretic algorithms, public key cryptography is slower than conventional secret key cryptography, thus motivating the need to speed up public key cryptosystems. A mathematical object called an elliptic curve can be used in the construction of public key cryptosystems. This thesis focuses on speeding up elliptic curve cryptography which is an attractive alternative to traditional public key cryptosystems such as RSA. Speeding up elliptic curve cryptography can be done by speeding up point arithmetic algorithms and by improving scalar multiplication algorithms. This thesis provides a speed up of some point arithmetic algorithms. The study of addition chains has been shown to be useful in improving scalar multiplication algorithms, when the scalar is fixed. A special form of an addition chain called a Lucas chain or a differential addition chain is useful to compute scalar multiplication on some elliptic curves, such as Montgomery curves for which differential addition formulae are available. While single scalar multiplication may suffice in some systems, there are others where a double or a triple scalar multiplication algorithm may be desired. This thesis provides triple scalar multiplication algorithms in the context of differential addition chains. Precomputations are useful in speeding up scalar multiplication algorithms, when the elliptic curve point is fixed. This thesis focuses on both speeding up point arithmetic and improving scalar multiplication in the context of precomputations toward double scalar multiplication. Further, this thesis revisits pairing computations which use elliptic curve groups to compute pairings such as the Tate pairing. More specifically, the thesis looks at Stange's algorithm to compute pairings and also pairings on Selmer curves. The thesis also looks at some aspects of the underlying finite field arithmetic

High Speed and Low-Complexity Hardware Architectures for Elliptic Curve-Based Crypto-Processors

The elliptic curve cryptography (ECC) has been identified as an efficient scheme for public-key cryptography. This thesis studies efficient implementation of ECC crypto-processors on hardware platforms in a bottom-up approach. We first study efficient and low-complexity architectures for finite field multiplications over Gaussian normal basis (GNB). We propose three new low-complexity digit-level architectures for finite field multiplication. Architectures are modified in order to make them more suitable for hardware implementations specially focusing on reducing the area usage. Then, for the first time, we propose a hybrid digit-level multiplier architecture which performs two multiplications together (double-multiplication) with the same number of clock cycles required as the one for one multiplication. We propose a new hardware architecture for point multiplication on newly introduced binary Edwards and generalized Hessian curves. We investigate higher level parallelization and lower level scheduling for point multiplication on these curves. Also, we propose a highly parallel architecture for point multiplication on Koblitz curves by modifying the addition formulation. Several FPGA implementations exploiting these modifications are presented in this thesis. We employed the proposed hybrid multiplier architecture to reduce the latency of point multiplication in ECC crypto-processors as well as the double-exponentiation. This scheme is the first known method to increase the speed of point multiplication whenever parallelization fails due to the data dependencies amongst lower level arithmetic computations. Our comparison results show that our proposed multiplier architectures outperform the counterparts available in the literature. Furthermore, fast computation of point multiplication on different binary elliptic curves is achieved

On the Cryptanalysis of Public-Key Cryptography

Nowadays, the most popular public-key cryptosystems are based on either the integer factorization or the discrete logarithm problem. The feasibility of solving these mathematical problems in practice is studied and techniques are presented to speed-up the underlying arithmetic on parallel architectures. The fastest known approach to solve the discrete logarithm problem in groups of elliptic curves over finite fields is the Pollard rho method. The negation map can be used to speed up this calculation by a factor √2. It is well known that the random walks used by Pollard rho when combined with the negation map get trapped in fruitless cycles. We show that previously published approaches to deal with this problem are plagued by recurring cycles, and we propose effective alternative countermeasures. Furthermore, fast modular arithmetic is introduced which can take advantage of prime moduli of a special form using efficient "sloppy reduction." The effectiveness of these techniques is demonstrated by solving a 112-bit elliptic curve discrete logarithm problem using a cluster of PlayStation 3 game consoles: breaking a public-key standard and setting a new world record. The elliptic curve method (ECM) for integer factorization is the asymptotically fastest method to find relatively small factors of large integers. From a cryptanalytic point of view the performance of ECM gives information about secure parameter choices of some cryptographic protocols. We optimize ECM by proposing carry-free arithmetic modulo Mersenne numbers (numbers of the form 2M – 1) especially suitable for parallel architectures. Our implementation of these techniques on a cluster of PlayStation 3 game consoles set a new record by finding a 241-bit prime factor of 21181 – 1. A normal form for elliptic curves introduced by Edwards results in the fastest elliptic curve arithmetic in practice. Techniques to reduce the temporary storage and enhance the performance even further in the setting of ECM are presented. Our results enable one to run ECM efficiently on resource-constrained platforms such as graphics processing units

Hardware Implementations of Scalable and Unified Elliptic Curve Cryptosystem Processors

As the amount of information exchanged through the network grows, so does the demand for increased security over the transmission of this information. As the growth of computers increased in the past few decades, more sophisticated methods of cryptography have been developed. One method of transmitting data securely over the network is by using symmetric-key cryptography. However, a drawback of symmetric-key cryptography is the need to exchange the shared key securely. One of the solutions is to use public-key cryptography. One of the modern public-key cryptography algorithms is called Elliptic Curve Cryptography (ECC). The advantage of ECC over some older algorithms is the smaller number of key sizes to provide a similar level of security. As a result, implementations of ECC are much faster and consume fewer resources. In order to achieve better performance, ECC operations are often offloaded onto hardware to alleviate the workload from the servers' processors. The most important and complex operation in ECC schemes is the elliptic curve point multiplication (ECPM). This thesis explores the implementation of hardware accelerators that offload the ECPM operation to hardware. These processors are referred to as ECC processors, or simply ECPs. This thesis targets the efficient hardware implementation of ECPs specifically for the 15 elliptic curves recommended by the National Institute of Standards and Technology (NIST). The main contribution of this thesis is the implementation of highly efficient hardware for scalable and unified finite field arithmetic units that are used in the design of ECPs. In this thesis, scalability refers to the processor's ability to support multiple key sizes without the need to reconfigure the hardware. By doing so, the hardware does not need to be redesigned for the server to handle different levels of security. Unified refers to the ability of the ECP to handle both prime and binary fields. The resultant designs are valuable to the research community and industry, as a single hardware device is able to handle a wide range of ECC operations efficiently and at high speeds. Thus, improving the ability of network servers to handle secure transaction more quickly and improve productivity at lower costs

Batch Verification of Elliptic Curve Digital Signatures

This thesis investigates the efficiency of batching the verification of elliptic curve signatures. The first signature scheme considered is a modification of ECDSA proposed by Antipa et al.\ along with a batch verification algorithm by Cheon and Yi. Next, Bernstein's EdDSA signature scheme and the Bos-Coster multi-exponentiation algorithm are presented and the asymptotic runtime is examined. Following background on bilinear pairings, the Camenisch-Hohenberger-Pedersen (CHP) pairing-based signature scheme is presented in the Type 3 setting, along with the derivative BN-IBV due to Zhang, Lu, Lin, Ho and Shen. We proceed to count field operations for each signature scheme and an exact analysis of the results is given. When considered in the context of batch verification, we find that the Cheon-Yi and Bos-Coster methods have similar costs in practice (assuming the same curve model). We also find that when batch verifying signatures, CHP is only 11\% slower than EdDSA with Bos-Coster, a significant improvement over the gap in single verification cost between the two schemes

On the Analysis of Public-Key Cryptologic Algorithms

The RSA cryptosystem introduced in 1977 by Ron Rivest, Adi Shamir and Len Adleman is the most commonly deployed public-key cryptosystem. Elliptic curve cryptography (ECC) introduced in the mid 80's by Neal Koblitz and Victor Miller is becoming an increasingly popular alternative to RSA offering competitive performance due the use of smaller key sizes. Most recently hyperelliptic curve cryptography (HECC) has been demonstrated to have comparable and in some cases better performance than ECC. The security of RSA relies on the integer factorization problem whereas the security of (H)ECC is based on the (hyper)elliptic curve discrete logarithm problem ((H)ECDLP). In this thesis the practical performance of the best methods to solve these problems is analyzed and a method to generate secure ephemeral ECC parameters is presented. The best publicly known algorithm to solve the integer factorization problem is the number field sieve (NFS). Its most time consuming step is the relation collection step. We investigate the use of graphics processing units (GPUs) as accelerators for this step. In this context, methods to efficiently implement modular arithmetic and several factoring algorithms on GPUs are presented and their performance is analyzed in practice. In conclusion, it is shown that integrating state-of-the-art NFS software packages with our GPU software can lead to a speed-up of 50%. In the case of elliptic and hyperelliptic curves for cryptographic use, the best published method to solve the (H)ECDLP is the Pollard rho algorithm. This method can be made faster using classes of equivalence induced by curve automorphisms like the negation map. We present a practical analysis of their use to speed up Pollard rho for elliptic curves and genus 2 hyperelliptic curves defined over prime fields. As a case study, 4 curves at the 128-bit theoretical security level are analyzed in our software framework for Pollard rho to estimate their practical security level. In addition, we present a novel many-core architecture to solve the ECDLP using the Pollard rho algorithm with the negation map on FPGAs. This architecture is used to estimate the cost of solving the Certicom ECCp-131 challenge with a cluster of FPGAs. Our design achieves a speed-up factor of about 4 compared to the state-of-the-art. Finally, we present an efficient method to generate unique, secure and unpredictable ephemeral ECC parameters to be shared by a pair of authenticated users for a single communication. It provides an alternative to the customary use of fixed ECC parameters obtained from publicly available standards designed by untrusted third parties. The effectiveness of our method is demonstrated with a portable implementation for regular PCs and Android smartphones. On a Samsung Galaxy S4 smartphone our implementation generates unique 128-bit secure ECC parameters in 50 milliseconds on average

Isogeny-based key compression without pairings

SIDH/SIKE-style protocols benefit from key compression to minimize their bandwidth requirements, but proposed key compression mechanisms rely on computing bilinear pairings. Pairing computation is a notoriously expensive operation, and, unsurprisingly, it is typically one of the main efficiency bottlenecks in SIDH key compression, incurring processing time penalties that are only mitigated at the cost of trade-offs with precomputed tables. We address this issue by describing how to compress isogeny-based keys without pairings. As a bonus, we also substantially reduce the storage requirements of other operations involved in key compression