## Cryptology ePrint Archive: Report 2003/191

Projective Coordinates Leak

David Naccache and Nigel Smart and Jacques Stern

Abstract: Denoting by $P=[k]G$ the elliptic-curve double-and-add multiplication of a public base point $G$ by a secret $k$, we show that allowing an adversary access to the projective representation of $P$ results in information being revealed about $k$.

Such access might be granted to an adversary by a poor software implementation that does not erase the $Z$ coordinate of $P$ from the computer's memory or by a computationally-constrained secure token that sub-contracts the affine conversion of $P$ to the external world.

From a wider perspective, our result proves that the choice of representation of elliptic curve points {\sl can reveal} information about their underlying discrete logarithms, hence casting potential doubt on the appropriateness of blindly modelling elliptic-curves as generic groups.

As a conclusion, our result underlines the necessity to sanitize $Z$ after the affine conversion or, alternatively, randomize $P$ before releasing it out.

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