Using Bleichenbacher”s Solution to the Hidden Number Problem to Attack Nonce Leaks in 384-Bit ECDSA

Abstract

Abstract. In this paper we describe an attack against nonce leaks in 384-bit ECDSA using an FFT-based attack due to Bleichenbacher. The signatures were computed by a modern smart card. We extracted the low-order bits of each nonce using a template-based power analysis attack against the modular inversion of the nonce. We also developed a BKZ-based method for the range reduction phase of the attack, as it was impractical to collect enough signatures for the collision searches originally used by Bleichenbacher. We confirmed our attack by extracting the entire signing key using a 5-bit nonce leak from 4 000 signatures.

LadderLeak: Breaking ECDSA with Less than One Bit of Nonce Leakage

“…Moreover, all these works only considered the setting where HNP samples come without any errors in the MSB information. In such an ideal case, our tradeoff formula actually allows to mount the attack given only 2 23 samples with almost the same time and space complexity. Appendix B describes how it can be achieved in detail.…”

“…The concrete attack parameters for the former are described in Appendix B and the ones for the latter were already described in Section 4.3. We first generated 2 23 and 2 24 ECDSA signatures like in the case of P-192, which took 1.8 and 3.6 CPU hours respectively. The measured experimental results are in Table 3.…”

“…For ℓ = 162 we present a solution to the linear programming problem in Table 2 for the optimized data complexity. Suppose the attacker is given 2 m in = 2 23…”

“…Despite those features, it has garnered far less attention from the community than latticebased approaches, perhaps in part due to the lack of formal publications describing the attack until recently (even though some attack records were announced publicly [14]). It was only in 2013 that De Mulder et al [23] revisited the theory of Bleichenbacher's approach in a formally published scholarly publication, followed soon after by Aranha et al [6], who overcame the 1-bit "lattice barrier" by breaking 160-bit ECDSA with a single bit of nonce bias. Takahashi, Table 1: Comparison with the previous records of solutions to the hidden number problem with small nonce leakages.…”

Randomizing the Montgomery Powering Ladder

“…Howgrave-Graham and Smart [19] noted that if a few bits of the ephemeral exponent are known for sufficiently many signatures, then the scheme can be broken, based on the so-called hidden number problem introduced by Boneh and Venkatesan [27]. Recently, De Mulder et al [28] demonstrated that, in practice, the lattice attack required as few as four bits of the ephemeral exponent assuming that some hundreds of such signatures could be obtained. However, this information would not be available to an adversary.…”