Abstract. We describe e cient formulas for computing pairings on ordinary elliptic curves over prime elds. First, we generalize lazy reduction techniques, previously considered only for arithmetic in quadratic extensions, to the whole pairing computation, including towering and curve arithmetic. Second, we introduce a new compressed squaring formula for cyclotomic subgroups and a new technique to avoid performing an inversion in the nal exponentiation when the curve is parameterized by a negative integer. The techniques are illustrated in the context of pairing computation over Barreto-Naehrig curves, where they have a particularly e cient realization, and also combined with other important developments in the recent literature. The resulting formulas reduce the number of required operations and, consequently, execution time, improving on the state-of-the-art performance of cryptographic pairings by 27%-33% on several popular 64-bit computing platforms. In particular, our techniques allow to compute a pairing under 2 million cycles for the rst time on such architectures.

Abstract-Key distribution in Wireless Sensor Networks (WSNs) is challenging. Symmetric cryptosystems can perform it efficiently, but they often do not provide a perfect trade-off between resilience and storage. Further, even though conventional public key and elliptic curve cryptosystems are computationally feasible on sensor nodes, protocols based on them are not. They require exchange and storage of large keys and certificates, which is expensive. Using Pairing-based Cryptography (PBC) protocols, conversely, parties can agree on keys without any interaction. In this work, we (i) show how security in WSNs can be bootstrapped using an authenticated identitybased non-interactive protocol and (ii) present TinyPBC, to our knowledge, the most efficient implementation of PBC primitives for an 8-bit processor. TinyPBC is able to compute pairings in about 5.5s on an ATmega128L clocked at 7.3828-MHz (the MICA2 and MICAZ node microcontroller).

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The availability of a new carry-less multiplication instruction in the latest Intel desktop processors significantly accelerates multiplication in binary fields and hence presents the opportunity for reevaluating algorithms for binary field arithmetic and scalar multiplication over elliptic curves. We describe how to best employ this instruction in field multiplication and the effect on performance of doubling and halving operations. Alternate strategies for implementing inversion and half-trace are examined to restore most of their competitiveness relative to the new multiplier. These improvements in field arithmetic are complemented by a study on serial and parallel approaches for Koblitz and random curves, where parallelization strategies are implemented J. and compared. The contributions are illustrated with experimental results improving the state-of-the-art performance of halving and doubling-based scalar multiplication on NIST curves at the 112-and 192-bit security levels and a new speed record for side-channel-resistant scalar multiplication in a random curve at the 128-bit security level. The algorithms presented in this work were implemented on Westmere and Sandy Bridge processors, the latest generation Intel microarchitectures.

After a few years of intense research, Wireless Sensor Networks (WSNs) still demand new secure and cryptographic schemes. On the other hand, the advent of cryptography from pairings has enabled a wide range of novel cryptosystems. In this work we present TinyTate, the first known implementation of pairings for sensor nodes based on the 8-bit/7.3828-MHz ATmega128L microcontroller (e.g., MICA2 and MICAz motes). We then conclude that cryptography from pairings is indeed viable in resource-constrained nodes.