Welcome to the resource topic for
**2002/007**

**Title:**

Parallel scalar multiplication on general elliptic curves over \mathbb{F}_p hedged against Non-Differential Side-Channel Attacks

**Authors:**
Wieland Fischer, Christophe Giraud, Erik Woodward Knudsen, Jean-Pierre Seifert

**Abstract:**

For speeding up elliptic curve scalar multiplication and making it secure against side-channel attacks such as timing or power analysis, various

methods have been proposed using specifically chosen elliptic curves. We show that both goals can be achieved simultaneously even for conventional

elliptic curves over \mathbb{F}_p. This result is shown via two facts.

First, we recall the known fact that every elliptic curve over \mathbb{F}_p admits a scalar

multiplication via a (Montgomery ladder) Lucas chain.

As such chains are known to be resistant against timing- and simple power/electromagnetic

radiation analysis attacks, the security of our scalar multiplication against timing and

simple power/electromagnetic radiation analysis follows.

Second, we show how to parallelize the 19 multiplications within the resulting

\lq\lq double" and \lq\lq add" formulas of the Lucas chain for the

scalar multiplication.

This parallelism together with the Lucas chain results in 10 parallel field multiplications per bit of the scalar.

Finally, we also report on a concrete successful implementation of the above mentioned scalar multiplication algorithm

on a very recently developed and commercially available coprocessor for smart cards.

**ePrint:**
https://eprint.iacr.org/2002/007

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