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Elliptic curves in cryptography
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Protections against Differential Analysis for Elliptic Curve Cryptography
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The Montgomery Powering Ladder
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
On Rational Number Reconstruction and Approximation
SIAM Journal on Computing
Elliptic Curve Cryptosystems in the Presence of Permanent and Transient Faults
Designs, Codes and Cryptography
New Point Addition Formulae for ECC Applications
WAIFI '07 Proceedings of the 1st international workshop on Arithmetic of Finite Fields
Fault Attack on Elliptic Curve Montgomery Ladder Implementation
FDTC '08 Proceedings of the 2008 5th Workshop on Fault Diagnosis and Tolerance in Cryptography
An Introduction to Mathematical Cryptography
An Introduction to Mathematical Cryptography
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FC'02 Proceedings of the 6th international conference on Financial cryptography
Co-Z addition formulæ and binary ladders on elliptic curves
CHES'10 Proceedings of the 12th international conference on Cryptographic hardware and embedded systems
To infinity and beyond: combined attack on ECC using points of low order
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
Improved fault analysis of signature schemes
CARDIS'10 Proceedings of the 9th IFIP WG 8.8/11.2 international conference on Smart Card Research and Advanced Application
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This paper presents a new type of fault attacks on elliptic curves cryptosystems. At EUROCRYPT 2004, Naccache et alii showed that when the result of an elliptic curve scalar multiplication [k] P (computed using a fixed scalar multiplication algorithm, such as double-and-add) is given in projective coordinates, an attacker can recover information on k. The attack is somewhat theoretical, because elliptic curve cryptosystems implementations usually convert scalar multiplication's result back to affine coordinates before outputting [k]P. This paper explains how injecting faults in the final projective-to-affine coordinate conversion enables an attacker to retrieve the projective coordinates of [k]P, making Naccache et alii's attack also applicable to implementations that output points in affine coordinates. As a result, such faults allow the recovery of information about k.