Revisiting Atomic Patterns for Scalar Multiplications on Elliptic Curves

This paper deals with the protection of elliptic curve scalar multiplications against side-channel analysis by using the atomicity principle. Unlike other atomic patterns, we investigate new formulae with same cost for both doubling and addition. This choice is particularly well suited to evaluate double scalar multiplications with the Straus-Shamir trick. Thus, in situations where this trick is used to evaluate single scalar multiplications our pattern allows an average improvement of \(40\,\%\) when compared with the most efficient atomic scalar multiplication published so far. Surprisingly, in other cases our choice remains very efficient. Besides, we also point out a security threat when the curve parameter \(a\) is null and propose an even more efficient pattern in this case.

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