Efficient SIMD arithmetic modulo a Mersenne number

This paper describes carry-less arithmetic operations modulo an integer 2^M − 1 in the thousand-bit range, targeted at single instruction multiple data platforms and applications where overall throughput is the main performance criterion. Using an implementation on a cluster of PlayStation 3 game consoles a new record was set for the elliptic curve method for integer factorization

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

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 Factorization Based on Elliptic Curve Method

The security of the most popular asymmetric cryptographic scheme RSA depends on the hardness of factoring large numbers. The best known method for factorization large integers is the General Number Field Sieve (GNFS). Recently, architectures for special purpose hardware for the GNFS have been proposed [5, 12]. One important step within the GNFS is the factorization of mid-size numbers for smoothness testing, an efficient algorithm for which is the Elliptic Curve Method (ECM). Since the smoothness testing is also suitable for parallelization, it is promising to improve ECM via special-purpose hardware. We show that massive parallel and cost efficient ECM hardware engines can improve the cost-time product of the RSA moduli factorization via the GNFS considerably. The computation of ECM is a classical example for an algorithm that can be significantly accelerated through special-purpose hardware. In this work, we present an efficient hardware implementation of ECM to factor numbers up to 200 bits, which is also scalable to other bit lengths. For proof-of-concept purposes, ECM is realized as a software-hardware co-design on an FPGA and an embedded microcontroller. This appears to be the first publication of a realized hardware implementation of ECM, and the first description of GNFS acceleration through hardware-based ECM