# Elliptic Curve Cryptosystems in the Presence of Permanent and Transient Faults

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## Abstract

Elliptic curve cryptosystems in the presence of faults were studied by Biehl et al., *Advances in Cryptology CRYPTO 2000*, Springer Verlag (2000) pp. 131–146. The first fault model they consider requires that the input point **P** in the computation of d**P** is chosen by the adversary. Their second and third fault models only require the knowledge of **P**. But these two latter models are less ‘practical’ in the sense that they assume that only a few bits of error are inserted (typically exactly one bit is supposed to be disturbed) either into **P** just prior to the point multiplication or during the course of the computation in a chosen location.

This paper relaxes these assumptions and shows how random (and thus unknown) errors in either coordinates of point **P**, in the elliptic curve parameters or in the field representation enable the (partial) recovery of multiplier *d*. Then, from multiple point multiplications, we explain how this can be turned into a total key recovery. Simple precautions to prevent the leakage of secrets are also discussed.

## Keywords

elliptic curve cryptography fault analysis fault attacks information leakage## Preview

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