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

Abstract

Although it is one of the most popular signature schemes today, ECDSA presents a number of implementation pitfalls, in particular due to the very sensitive nature of the random value (known as the nonce) generated as part of the signing algorithm. It is known that any small amount of nonce exposure or nonce bias can in principle lead to a full key recovery: the key recovery is then a particular instance of Boneh and Venkatesan's hidden number problem (HNP). That observation has been practically exploited in many attacks in the literature, taking advantage of implementation defects or side-channel vulnerabilities in various concrete ECDSA implementations. However, most of the attacks so far have relied on at least 2 bits of nonce bias (except for the special case of curves at the 80-bit security level, for which attacks against 1-bit biases are known, albeit with a very high number of required signatures). In this paper, we uncover LadderLeak, a novel class of side-channel vulnerabilities in implementations of the Montgomery ladder used in ECDSA scalar multiplication. The vulnerability is in particular present in several recent versions of OpenSSL. However, it leaks less than 1 bit of information about the nonce, in the sense that it reveals the most significant bit of the nonce, but with probability <1. Exploiting such a mild leakage would be intractable using techniques present in the literature so far. However, we present a number of theoretical improvements of the Fourier analysis approach to solving the HNP (an approach originally due to Bleichenbacher), and this lets us practically break LadderLeak-vulnerable ECDSA implementations instantiated over the sect163r1 and NIST P-192 elliptic curves. In so doing, we achieve several significant computational records in practical attacks against the HNP.