In this short note, we give an algorithm that returns an explicit expression of the Lüroth invariant in terms of the Dixmier-Ohno invariants of plane quartic curves. We also obtain an explicit factorized expression on the locus of Ciani quartics in terms of the coefficients. After this calculation, we extend our methods to answer two open theoretical questions concerning the sub-locus of singular Lüroth quartics.

In this paper, we present a new fault attack on elliptic curve scalar product algorithms. This attack is tailored to work on the classical Montgomery ladder method when the $y$-coordinate is not used. No weakness has been reported so far on such implementations, which are very efficient and were promoted by several authors. But taking into account the twist of the elliptic curves, we show how, with few faults (around one or two faults), we can retrieve the full secret exponent even if classical countermeasures are employed to prevent fault attacks. It turns out that this attack has not been anticipated as the security of the elliptic curve parameters in most standards can be strongly reduced. Especially, the attack is meaningful on some NIST or SECG parameters.