Resistance Against Differential Power Analysis For Elliptic Curve Cryptosystems

Abstract

Differential Power Analysis, first introduced by Kocher et al. in [14], is a powerful technique allowing to recover secret smart card information by monitoring power signals. In [14] a specific DPA attack against smart-cards running the DES algorithm was described. As few as 1000 encryptions were sufficient to recover the secret key. In this paper we generalize DPA attack to elliptic curve (EC) cryptosystems and describe a DPA on EC Diffie-Hellman key exchange and EC El-Gamal type encryption. Those attacks enable to recover the private key stored inside the smart-card. Moreover, we suggest countermeasures that thwart our attack.

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.…”

“…2) introduced by Coron [5] were designed to prevent simple side-channel attacks by performing dummy operations. However, such algorithms bring specific weaknesses with respect to so-called safe-error attacks [8].…”

“…Randomization of the group structure was not explored in this paper. Known techniques targetting the exponent and the base element are presented in more details in [5,9]. Blinding the exponent is not well suited for exponentiations in finite abelian groups.…”

“…Usually, blinding the base element consists in multiplying the input x ∈ G with a random element r picked at random from G; the value of x d ∈ G is then obtained as (xr) d × (r −1 ) d . This countermeasure introduced in [5] requires two balanced group exponentiations, or a subtle but unpractical pre-computation trick that may be difficult to handle by the personalization process.…”

Unified Point Addition Formulæ and Side-Channel Attacks

“…If, in a side-channel analysis, a point addition is distinguishable from a point doubling, then the bits of the secret key can be determined; this has been demonstrated experimentally using timing [3], power analysis [4], and electromagnetic emissions [5]. Techniques for counteracting this problem include: performing dummy operations, such as forcing a point addition at each iteration [6]; using alternate point multiplication algorithms, such as Montgomery point multiplication [7]; using alternate curve parameterizations, such as the Jacobi or Hessian forms; and unifying the algorithms for point addition and point doubling so that they use the same sequence of field operations and hence are indistinguishable. It is this last technique that we address in this paper.…”