A Refined Power-Analysis Attack on Elliptic Curve Cryptosystems

Abstract

As Elliptic Curve Cryptosystems are becoming more and more popular and are included in many standards, an increasing demand has appeared for secure implementations that are not vulnerable to sidechannel attacks. To achieve this goal, several generic countermeasures against Power Analysis have been proposed in recent years. In particular, to protect the basic scalar multiplication-on an elliptic curve-against Differential Power Analysis (DPA), it has often been recommended using "random projective coordinates", "random elliptic curve isomorphisms" or "random field isomorphisms". So far, these countermeasures have been considered by many authors as a cheap and secure way of avoiding the DPA attacks on the "scalar multiplication" primitive. However we show in the present paper that, for many elliptic curves, such a DPA-protection of the "scalar" multiplication is not sufficient. In a chosen message scenario, a Power Analysis attack is still possible even if one of the three aforementioned countermeasures is used. We expose a new Power Analysis strategy that can be successful for a large class of elliptic curves, including most of the sample curves recommended by standard bodies such as ANSI, IEEE, ISO, NIST, SECG or WTLS. This result means that the problem of randomizing the basepoint may be more difficult than expected and that "standard" techniques have still to be improved, which may also have an impact on the performances of the implementations.

Defeating Countermeasures Based on Randomized BSD Representations

“…They include randomizing the secret scalar, blinding the point and using randomized projective coordinates. New directions for attacking these countermeasures have recently been proposed [5,6] but none of them works when all protections proposed by Coron are applied An alternative to Coron's countermeasures is to randomize the multiplication algorithm itself by introducing some random decisions. Two recent propositions [7,20] use a randomized addition-subtraction chain, which is equivalent to represent the scalar with the alternative set of digits {0, 1, −1}.…”

Blinded Fault Resistant Exponentiation

“…In particular, this is the case for RSA as the factorization of the modulus and the public exponent are rarely available to the device. Note that our countermeasure also fully applies to the ECC setting since the randomization of projective coordinates, introduced by Coron in [5], was later proven insufficiant by Goubin in [6]. As pointed out recently by Dupuy and Kuntz-Jacques [7], when the attacker can tamper with the base element, scalar point multiplications also require randomization of the computation flow to provide DPA resistance.…”

Enhanced Doubling Attacks on Signed-All-Bits Set Recoding

“…Our attack seems to be Goubin's Refined Power-analysis Attack [6], but ours uses the discriminative method that compares two waveforms (The method is introduced in Section 4). Also, while Goubin's attack can use only "special point" with zero coordinate, our attack has a merit that can use almost every points over elliptic curve.…”