Fast and Flexible Elliptic Curve Point Arithmetic over Prime Fields

We present an innovative methodology for accelerating the elliptic curve point formulas over prime fields. This flexible technique uses the substitution of multiplication with squaring and other cheaper operations by exploiting the fact that field squaring is generally less costly than multiplication. Applying this substitution to the traditional formulas, we obtain faster point operations in unprotected sequential implementations. We also show the significant impact our methodology has in protecting against simple side- channel (SSCA) attacks. We modify the elliptic curve cryptography (ECC) point formulas to achieve a faster atomic structure when applying side-channel atomicity protection. In contrast to previous atomic operations that assume that squarings are indistinguishable from multiplications, our new atomic structure offers true SSCA-protection because it includes squaring in its formulation. Moreover, we extend our implementation to parallel architectures such as Single-Instruction Multiple-Data (SIMD). With the introduction of a new coordinate system and the flexibility of our methodology, we present, to our knowledge, the fastest formulas for SIMD-based schemes that are capable of executing three and four operations simultaneously. Finally, a new parallel SSCA-protected scheme is proposed for multiprocessor/parallel architectures by applying the atomic structure presented in this work. Our parallel and atomic operations are shown to be significantly faster than previous implementations.