Accelerating the Scalar Multiplication on Elliptic Curve Cryptosystems over Prime Fields

Elliptic curve cryptography (ECC), independently introduced by Koblitz and Miller in the 80\u27s, has attracted increasing attention in recent years due to its shorter key length requirement in comparison with other public-key cryptosystems such as RSA. Shorter key length means reduced power consumption and computing effort, and less storage requirement, factors that are fundamental in ubiquitous portable devices such as PDAs, cellphones, smartcards, and many others. To that end, a lot of research has been carried out to speed-up and improve ECC implementations, mainly focusing on the most important and time-consuming ECC operation: scalar multiplication. In this thesis, we focus in optimizing such ECC operation at the point and scalar arithmetic levels, specifically targeting standard curves over prime fields. At the point arithmetic level, we introduce two innovative methodologies to accelerate ECC formulae: the use of new composite operations, which are built on top of basic point doubling and addition operations; and the substitution of field multiplications by squarings and other cheaper operations. These techniques are efficiently exploited, individually or jointly, in several contexts: to accelerate computation of scalar multiplications, and the computation of pre-computed points for window-based scalar multiplications (up to 30% improvement in comparison with previous best method); to speed-up computations of simple side-channel attack (SSCA)-protected implementations using innovative atomic structures (up to 22% improvement in comparison with scalar multiplication using original atomic structures); and to develop parallel formulae for SIMD-based applications, which are able to execute three and four operations simultaneously (up to 72% of improvement in comparison with a sequential scalar multiplication). At the scalar arithmetic level, we develop new sublinear (in terms of Hamming weight) multibase scalar multiplications based on NAF-like conversion algorithms that are shown to be faster than any previous scalar multiplication method. For instance, proposed multibase scalar multiplications reduce computing times in 10.9% and 25.3% in comparison with traditional NAF for unprotected and SSCA-protected scenarios, respectively. Moreover, our conversion algorithms overcome the problem of converting any integer to multibase representation, solving an open problem that was defined as hard. Thus, our algorithms make the use of multiple bases practical for applications as ECC scalar multiplication for first time

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